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(Y)
+ F ({ O"} , D, R)
(4-7S)
In this case, the damage growth Eq.(4-7S) and the increment of accumulative hardening parameter given by Eq.( 4-76) are similar to those of Lemaitre's model, but the increment of plastic strain given by Eq.(4-74) and the damage elasto-plastic stiffness matrix [Dep] are not the same as those of Lemaitre's.
dG
Since the damage-plastic flow vector d{ O"} due to Eq. (4-7S) consists of two
dF
dd> dY
parts, one is from the classical plastic flow d{ 0" } , the other, dY d{ 0" } , is due to damage growth (kinetic) flow
dG d{O"}
dF d{O"}
dd> dY dYd{O"}
-- = -- + ---
(4-79)
It should be noted here that the second part of Eq. (4- 79) , which presents the contribution of damage growth to the plastic flow , was not included in Lemaitre's model. Therefore, the influence of damage growth on the plastic flow has been missed in Lemaitre's model due to the independence of plastic potential and damage potential. Using Eq.(3-53b) for models A and B, Eq.(4-79) becomes
dG d{ O"} Thus, we have
dF d[D*rl dd> d{ O"} dD {0" } dY
(4-S0)
4.3 Non-Associat ed Flow Rule Model
153
de ( dF ) T] [dF ( dF ) T] d[D*r l [ (dF ) T] dt} [d{a} d{a} = d{a} d{a} dD {a} d{a} dY
[D*] de ( dF) d{a} d{a}
T[D*] = [D*]
dF ( dF d{a} d{a}
(4-81 )
)T[D*]
_ [D*] d[D*rl [{ } ( dF ) T] [D*] dt} dD a d{a} dY U sing the relation Eq. (4-62),
[D*] de ( dF )T [D*] = [D*] dF ( dF )T [D*] d{a} d{a} d{a} d{a}
+ d1~*][D*rl
(4-82)
[{a}(d~~})T] [D*] ~:
[D*] de ( dF )T [D*] = [D*] dF ( dF )T [D*] d{a} d{a} d{a} d{a}
+ d 1~*] [D*rl [{a} (d~~} ) T] [D*]
(4-83)
~:
Substituting Eqs.(4-80) to (4-83) into Eqs.(4-69) to (4-72), we have
[D; ] H (F) =
[D*]
X
dF dR dF dR d'Y dR
(j~:} ((j~:}) T [D*] + 2d1~*l [D*r
1
[{a} ((j~:}) T] [D*]
*
+ ( dF )T [D*] dF + (dF + 2( dF )T d[D'l [D*]-l {a}) d
0"3 and to the Lode angle.
-'IT
/6
~
8*
~
'IT
(4-101)
/6 . The term 8* is essentially similar
4.4 Damage Plastic Criteria for Numerical Analysis
159
4.4.3 Different Modeling of Damage Yield Criteria 4.4.3.1 Modification of Tresca Yield Criterion The Tresca yield criterion of isotropic damaged material in terms of effective stresses is
F =
~ (J;) ~
[sin
(8*+ 2;) - sin (8*+ ~)] - R ("() = 0
(4-102)
or expanding, we have 1
F = 2(J;)2 cos
8* - R("() = 0
(4-103)
Substituting Eqs.(4-99) and (4-100) into Eq.(4-103), the modified Tresca yield criterion in t erms of invariants of deviatoric Cauchy stress is represented as
[1
F = 2J221 cos "3 arcsin
( 3V3h)] - (l - Sl)R("() = O -
2J2~
(4-104)
4.4.3.2 Modification of von Mises Yield Criterion The von Mises yield criterion of isotropic damaged material in terms of effective stress is 1
F = (3J;)2 - R("() = 0
(4-105)
Substituting Eq.( 4-99) into Eq.( 4-105), the modified von Mises yield criterion in t erms of the invariant of deviatoric Cauchy stress is represented as (4-106) 4.4.3.3 Modification of Mohr-Coulomb Yield Criterion The Mohr-Coulomb yield criterion of isotropic damaged material is a generalization of Coulomb friction failure law by introducing the effective shearing stress T * and the effective normal stress O"~ on the friction failure surface as T*
= c-
O"~
t an 'P
(4-107)
where c is the cohesion and 'P is the angle of internal friction. Eq.(4-107) can be rewritten in terms of effective principal stress as F
= (O"r - 0"i3 ) - 2ccos 'P + (O"~ + 0"i3 ) sin 'P = 0
Substituting Eq.(4-101) into Eq.(4-108) , we have
(4-108)
160
4 Isotropic Elasto-Plastic Damage Mechanics
. cp + (J*) F = :31 J*1 sm 2 12
COS B* -
(
1 sm . B*· J3 sm cp )
-
C cos
cp = 0
(4-109)
Substituting Eqs.(4-99) and (4-100) into Eq.(4-103), the modified Tresca yield criterion in terms of invariants of deviatoric Cauchy stress is represented as F =
~J~ sin cp+(h)~
( COSB* -
~ sinB* sincp)
-c(1-D) coscp = 0 (4-110)
where . ( - ----y3J3h) B* =:31 arcsm 2J22
(4-111)
The cohesion c can be equivalently expressed by the hardening rule R( ')') [4-30] as
Rh) c= - -
(4-112)
cos cp
and when ,),=0, it gives RI"'I=o = R o, and cl"'l=o = Co = Ro/coscp, we can obtain from Eq.(4-96) C
k 1 = Co + __ ')'m
(4-113)
cos cp
4.4.3.4 Modification of Drucker-Prager Yield Criterion The influence of a hydrostatic stress component on yielding was introduced by inclusion of an additional term in von Mises expression to give F
=
(3oJ~
1
+ (J~p
- Rh) = 0
(4-114)
This yield surface has the form of a circular cone. In order to make the Drucker-Prager criterion with the inner or outer apices of the Mohr-Coulomb hexagon at any section, it can be shown that (30 =
Rh) =
2 sincp
(4-115)
6ccos cp
(4-116)
J3(3 ± sin cp) J3(3 ± cos cp)
where "+"for inner apex, "- " for outer apex. Substituting Eqs.(4-115) and (4-116) into Eq.(4-114), it gives F =
2 sin cp
J3 (3 ± sin cp)
J~ + (J~) ~ _
6c cos cp
J3 (3 ± cos cp)
= 0
(4-117)
4.4 Damage Plastic Criteria for Numerical Analysis
161
Substituting Eqs.(4-98) and (4-99) into Eq.(4-117), the modified DruckerPrager criterion3 in terms of invariants of the Cauchy stress deviator is represented as F =
2 sin 'P h v3 (3 ± sin 'P)
+ (J2)~
-
6ccos'P (1 - D) = 0 v3 (3 ± cos 'P)
(4-118)
where the cohesion C can also be equivalently expressed by the hardening rule R(r) [4-30] as C= v3(3 ± sin 'P)R(r) 6 cos 'P
(4-119)
and whewy=O, it gives R I')'=o = Ro , and 4'1=0 = Co = v3(3±sin 'P)Ro/(6cos'P) , we can obtain C = Co
+ v3(3 ±
sin 'P) k .1. "(m 6 cos 'P
(4-120)
4.4.4 Expression for Numerical Computation 4.4.4.1 Basic Expressions for Three-Dimensional Problems For the purposes of numerical computation, it is required to express the above formulation in the following form For the purposes of numerical computation, it is required to express the above formulation in the following form
dF {b} = d{a}
(4-121)
for model A,
{d* } = [D*]
d~~}
= (1 - D) [D] {b} = (1 - D) {d}
(4-122a)
for model B,
{d* } = (1 - D)2{d}
(4-122b)
dF B = dD
(4-123)
A
A=
dFdRdF dR d"( dR
(4-124)
where (4-125a)
162
4 Isotropic Elasto-Plastic Damage Mechanics
(4-125b)
{d}T = {dl,d2 ,d3, d4,d5,d6}
(4-125c)
The vector {b} can be written as 1
{b} =
ClF Clh Clh Cl{o-}
+
ClF ClJ22 ") ~ Cl{o-} oJ2
ClF ClB*
+ ClB* Cl{o-}
(4-126)
Differentiating Eq.(4-111) with respect to {o-}, it gives
ClB* _ _ J3 Cl{o-} 2 cos 3B*
[J.... Clh
_ 3h
J2~ Cl{o-}
]',1
ClJ2~ 1
Cl{o-}
(4-127)
Substituting Eq.(4-1 27) into Eq.(4-1 26) and using Eq.(4-111) , we can then rewrite the vector in the form of (4-128) where
dI1 T {al} = Cl{o-} = {l , l,l,O,O, O}
(4-129a)
(4-129b)
Clh
{a3} = Cl{o-}
= { (SyS Z - o-;z +
~2) , (S zSx -
o-;x
+ ~2),
(S XSy _ o-;y
+ ~2),
= 2 (o- zx o-xy - Syo-y z ) , 2 (o-xy o-yz - SXO-ZX), 2 (o-y zo-zx _ Szo-xy) } T (4-129c) and ClF
C1 C _ ClF 2ClJ22
= Clh
tan3B* ClF
---r - --l-ClB* J 22
(4-130a) (4-130b)
4.4 Damage Plastic Criteria for Numerical Analysis
G3 = -
V3 2 cos 38*
1 dF 1. d8* J 22
163
(4-130c)
--
Only the constants G l , G2 and G3 are then necessary to define the yield surface. Thus, we can achieve simplicity in programming as only these three constants have to be varied from one yield surface to another. The constants G l , G2 and G3 are given in Table 4-1 for four yield criteria, and if necessary other yield functions can also be expressed in similar forms. Table 4-1 Constants defining the yield surface for numerical analysis Yield Criterion C1 C2 C3 V3sin6l* 2 cos 6l* (1 +tan 6l* tan 36l*) Tresca o h cos36l* von Mises o o V3 V3 sin 6l* + cos 6l* sin
0)
dt
(4-245)
where c and p are material constantsthe damage thermodynamic drive force Y is expressed herein as the same as the strain energy release rate
4.7 Analysis of Coupled Isotropic Damage and Fracture Mechanics
y
=
~{(Jij}T [Cijkl ~- 2]{(Jkd = ~{(J:j}T[Cijkl]{(Jkl}
187
(4-246)
where [Cijkd is the flexibility tensor of undamaged materials.
4.7.2 Basic Equation of Gradual Field near Developing Crack 4.7.2.1 Compatibility Equation of D e formations in Gradual Fie ld Let us consider the gradual field of a developing crack of type-I under monotonous loading as shown in Fig.4-10 (in which the area with distributional points represents the damage zone). Assuming Airy's stress function is
Y
...... , . . . ...
...
... #
Q,- princ ip al region
(r ,8)
..
. ,,
...
\ '
x
a
Fig. 4-10 Illustration of regions in damage process
A=
a,·\+2 A( B)
(4-247)
then the stresses can be represented as follows:
(Jrr = a,A(jrr(B) (Jee = a,A(jee(B) (Jre = a,A(jre(B)
(4-248)
188
4 Isotropic Elasto-Plastic Damage Mechanics
d2 A
_
i7rr = (.\ + 2)A + dB2
i7ee = (.\ + 2)(.\ + l)A
(4-249)
dA
i7 r e = - (.\ + 1) dB
Consequently, using Eqs. (4-243) , (4-248) and (4-249), we give the divatoric stress as (4-250)
Srr(B) = zli7rr(B)-Z2i7ee(B),
see(B) = zli7ee(B)-z2i7rr(B),
sre(B) = i7re(B)
(4-251 ) where Zl = 2/3, Z2 = 1/3 are applicable for the plane stress state; Zl = z2=1/2 is applicable for the plane strain state. Substituting Eqs.(4-250) and (4-251) into Eq.(4-243) gives (4-252) where
{
(i7;r
+ i7~e - i7rr i7ee + 3i7;e) 1/2
[3 (i7 rr - i7ee)2
+ 3i7;el 1/ 2
(plane stress) (plane strain)
(4-253)
In the damage field , since near the crack tip the material is fully damaged , the continuity of which is zero, whereas far from the crack tip the damage is smaller and the continuity is better, we may therefore assume (4-254) in which JL > O. Using Eqs.(4-244), (4-250) , (4-252) and (4-254) ,the strain field can be obtained as
{
err = h
(~) nrn().. -/L)srr (B) , eee = b1 (~) nrn()..-/L)see (B)
ere = b1 (~) nrn()..-/L)sre (B)
(4-255)
(4-256) Further substituting Eq.(4-255) into the compatibility equation (4-257) we obtain
4.7 Analysis of Coupled Isotropic Damage and Fracture Mecha nics
el = - 2 (e + 1)
e2 = -e
}
189
(4-259)
e3 = e (e + 1) e = n (A - f-L) Eq.( 4-258) becomes a 4th order ordinary differential equation of involving eigenvalues of A and f-L.
A and 1j;
4.7.2.2 Compatibility Condition of Damage Evolution Now it is possible to discuss the gradual form of the damage evolution equation. From Eqs.(4-246), (4-248) and (4-254) we obtain y
=
~ (~rr2(A-{L)Y(B)
Y(B) =
(4-260)
~E{O'ij f[Cijkl 1P- 2]{O'kl}
y
Fixed element
,,
,,
, ,,
,,
, ,,
,,
, ,,
,,
,
,'''-- dB
B+dB x
a
da
Fig. 4-11 Geometrical relation in crack developing
Regarding t he geometrical relation in Fig.4-11, it is dr } ~e = ~ c~s B
- = -smB da
r
(r > > da)
(4-261 )
190
4 Isotropic Elasto-Plastic Damage Mechanics
as well as considering Eq.(4-260), we have
(4-262)
When [d(a/ ;3)2]/(oJ;3)2 »dalr, the area near the crack but a little way from the crack tip can still be mainly considered to be in a proportional loading state, and taking the principal term in Eq.(4-262) gives
~ r 2(A-IL) Y :t (~ ) 2
y=
(4-263)
Substituting Eqs.(4-260) and (4-263) into Eq.(4-245) gives
d1j; dt
= __ c
EP+l
(~)2Pr2(p+l)(A-IL) YP+1i.(~)2 [i.(~)2 > 0] (3 dt (3 dt (3
(4-264)
Considering the geometrical relation in Fig.4-11 and Eq. (4-254) , we can obtain d1j; da = [d(3 -d1j; = - r IL 1j;- + (3r IL - 1 (d~ - sine - fJ1j;- cos e )] -da dt
da dt
da
de
dt
(4-265)
According to the fact that the closer the point to the crack tip, the greater is the damage and the lower is the continuity, this property needs to keep a hold on the second term in Eq.(4-265) as
-d1j; = (3r IL - 1 (d~ - sin e - fJ1j;- cos e) -da de
ili
ili
(4-266)
which presents the strange property of cJ; when r ---+0 and fJ < 1. Comparing Eq.(4-264) to Eq.(4-266) for independence of terms including e, rand (al(3 ) respectively, may only give a set of compatibility conditions of damage evolution from each term as
d~ - sin
de
2 (p
- +1 e - fJ1j;- cos e = - YP
+ 1) (>. -
fJ)
= fJ - 1
(4-267) (4-268)
4.7 Analysis of Coupled Isotropic Damage and Fracture Mechanics
da 2c d(aj;3) = EP+l (3
P+l (a)2 73
191
(4-269)
where Eq.( 4-268) presents independence between ,\ and f.J,. Therefore Eqs.( 4268) and (4-267) consist of the problem of solving the eigen-functions A and 1[J as well as the eigenvalue f.J,. Eq.(4-269) will be used to determine the crack developing rate. 4.7.3 Boundary Condition and Solution Method of Studied Problem 4.7.3.1 Boundary Condition of Studied Problem Solving Eqs.(4-258) and (4-267) belongs to the boundary problem at the two end points (see B= O and B=7r). Because of the symmetry in the type-I crack field we have
dA(o) d 3 A(O) = d1[J (O) = 0 (4-270) dB3 dB dB Since the angle distribution function A(B) of Ariy's stress function mostly represents the mode of angle distribution, thus we can assume A(O) = 1
(4-271)
Using Eqs.(4-258) and (4-267) we can derive the following
-
7/J (0) =
II
(d f.J"
2 Ao) d21[J" (0) dB2 ' dB2 =
12
(d f.J"
2 Ao) dB2
(4-272)
d2A d2A(0) where dB20 represents dB2 . To save space, the particular form in Eq.( 4272) is no longer presented in detail. On the crack surface, one needs (4-273) (4-274) Since 1[J (0) > 0, and 1[J is gradually decreasing in the region from B= O to B=7r , it is possible to assume that 1[J decreases to zero at a certain angle Bd E (O , 7r). If this case happens, it means that the region [0, 7r] will be divided into two parts: where [0, Bd ] is named the damage developing region or the damage active region,whereas, [Bd , 7r] is named the full damaged region or the damage stop station (see Fig.4-10), which satisfies 1[J ;? 0 and 1[J = 0 correspondingly. Since the fully damaged medium cannot bear any stresses, in the fully damaged region it therefore always has
192
4 Isotropic Elasto-Plastic Damage Mechanics
(4-275) Obviously, Eqs.(4-273) and (4-274) are satisfied together. Next , the situation of the damage active region will be mostly discussed . According to the definition of ed and Eq.(4-275), the interfacial condition on both sides of e = ed can be obtained as follows
1jj(ed- ) = 1jj (ed+) = 0
}
i5"ee(ed+) = i5"ee(ed+) = 0, i5"re(ed- ) = i5"re(ed+) = 0
(4-276)
Applying Eq.( 4-249) and taking a limit, the above equation becomes (4-277) Since the deformation in the full damage region is undetermined (arbitrary), it is not necessary to add the deformational compatibility condition to the interfacial condition. Anyway,the properties of the full damaged region are determined, such as Eq.( 4-275); the control equations of the damage active region are Eqs.(4-258) and (42-267). The functions to be determined are A d2 A(O) and 'IjJ; the parameters to be determined are j.L , ~ and ed ; the boundary conditions are Eqs.(4-270), (4-271), (4-272) and (4-277). 4.7.3.2 Solving Algorithm of Studied Problem
The steps of the solution algorithm are listed as follows: (1) According to an arbitrarily given
j.L (j.L
d2A > 0) and de 20 ' using Eqs.(4-270), (4-271) and (4-
272) to integrate Eqs. (4-258) and (4-267) (it is possible to use the 4th order Runge-Kutta's method with varied steps), it may give a region as e* E [0, 7r] satisfying 1jj (e* ) = 0 (4-278) thus e* can be considered as a function of
_
(2) Since A(e* ) and
j.L
d2 A
and de 2o .
d2 A(e*) de 2 are determined at the same time when solving d2A
e* , thus they can be considered as a function of j.L and de 20 and expressed as A (e* ) = A [e* dA(e* ) =
de
~
de
(j.L,
[A (e* (
d:~o )] j.L,
d2Ao))] de 2
}
(4-279)
4.8 Verify Isotropic Damage Mechanics Model by Numerical Examples
193
d2 A dA(B*) obviously, dB2o , A (B*) and dB usually are not zero for any f-L.
(3) Solving the following non-linear equations
(4-280)
d2 A We can obtain the necessary f-L and dB2o , moreover the corresponding B* actually is Bd, and at the same time the angle distribution functions of different fields are obtained.
4.8 Verify Isotropic Damage Mechanics Model by Numerical Examples In order to verify the formulations presented above, some results of the finite element model have been compared with the available solutions in this section.
4.8.1 Example of Bar Specimen The first example is a simulation of experimental measures for damage evolution presented in Chapter 3. The experimental results of damage evolution were obtained in terms of measures of the effective Young's modulus as shown in Fig.4-12(a) determined for the metals 99.9% Copper, Alloy INCO 718 and Steel 30CD4 in [4-9, 4-11]. The theoretical results are obtained using Eq.(428); the finite element results are obtained by the application of Eq.( 4-68) to Eq.( 4-72). As mentioned by Lemaiture [4-43], it is easier to measure damage variable rl through the variations of the elastic modulus. From the relationship between Eq.(3-25) and Eq.(3-26) , the damage can be evaluated by different definitions, say model A and model B (see Fig.4-12(a)). The finite element mesh for simulation is shown in Fig.4-12(b). In order to obtain more accurate results a higher order Gaussian point scheme has been employed in F. E. analysis. Considering the damage localization, we can assume that the sensitive coefficient of damage growth ex is not equal to zero in the central element only. The damage growth can be observed by processing the results at each stage of the load increment. It should be pointed out that the material constant, ex, is the sensitive coefficient of damage growth. ex can be approximately estimated through experimental results at points (rl=O, C d ) and (rlc, c R ) as shown in Fig.4-2 and Fig.4-12. rlc and c R are the rupture values of rl and c p . The initial (first approximate) value of ex can be taken as the average slope of the experimental
194
4 Isotropic Elasto-Plastic Damage Mechanics il ,= I - E"IE il .= I-v£'iE
,, ,:
E"
(a)
q
p
Local damage b--*] =
cos 2 (}
sin 2 (}
'l/Jl
0
2(} (1 -1 )sin'l/Jl 'l/J2 2 ( 1 1 ) sin 2(} 0 ----'l/Jl 'l/J2 2 1 0 0
'l/J2
'l/J2 0
2(} (1 -1 )sin0 'l/Jl 'l/J2 2
(5-36)
cos 2 (}
sin 2 (}
--+-'l/Jl
'l/J2
cos 2 (}
sin 2 (}
'l/Jl
'l/J2
--+--
5.4 Decomposition Model of Anisotropic Damage Tensor 5.4.1 Review of Definition of Damage Variable The principles of continuum damage mechanics are first reviewed in the case of uniaxial tension. In this case, isotropic damage is assumed throughout. Consider a cylindrical bar subjected to a uniaxial tensile force T as shown in Fig.5-8(a). The cross-sectional area of the bar is A and it is assumed that T
T
-
~yo
01 " 0/I \0/0\ 0
CT
0
o \0 0/0 / "0
(a)
Remove both voids and cracks
n A
A
-
CT
(b)
Fig. 5-8 (a) A cylindrical bar subjected to uniaxial tension; (b) Both voids and cracks are removed simultaneously
5.4 Decomposition Model of Anisotropic Damage Tensor
233
both voids and cracks appear as damage in the bar. The uniaxial stress IJ in the bar is found easily from the formula T = IJ A. In order to use the principles of continuum damage mechanics, we consider a fictitious undamaged configuration of the bar as shown in Fig.5-8(b). In this configuration all types of damage, including both voids and cracks, are removed from the bar. The effective cross sectional area of the bar in this configuration is denoted by A * and the effective uniaxial stress is IJ*. The bars in both the damaged configuration and the equivalent undamaged configuration are subjected to the same tensile force T. Therefore, considering the equivalent undamaged configuration, we have the formula T = IJ* A *. Equating the two expressions of T obtained from both configurations, one obtains the following expression for the equivalent uniaxial stress IJ*
IJ*
=
IJA/A*
(5-37)
One uses the definition of the damage variable fl, as originally proposed by Kachanov [5-21]
fl
(A - A*)/A
=
(5-38)
Thus the damage variable is defined as the ratio of the total area of voids and cracks to the total area. Its value ranges from zero (in the case of an undamaged specimen) to 1 ( in the case of complete rupture). Substituting for A/A* from Eq.(5-38) into Eq.(5-37), one obtains the expression for the equivalent uniaxial stress defined in Chapters 3 and 4 as
IJ*
=
1J/(1- fl)
(5-39)
Eq.(5-39) above was originally derived by Kachanov in 1958. It is clear from Eq.(5-39) that the case of complete rupture (fl = 1) is unattainable because the damage variable fl is not allowed to take the value 1 in the denominator. 5.4.2 Decomposition of Damage Variable in One Dimension
The principles of continuum damage mechanics are now applied to the problem of decomposition of the damage tensor in a damaged uniaxial bar subjected to a tensile force T. Isotropic damage is assumed throughout the formulation. It is also assumed that the damaged state is defined by voids and cracks only. Therefore, the cross sectional area A * of the damaged bar can be decomposed as follows: (5-40)
where AV is the total area of voids in the cross-section and AC is the total area of cracks (measured lengthwise) in the cross-section (The superscripts 'v' and 'c' denote voids and cracks, respectively). In addition to the total
234
5 Basis of Anisotropic Damage Mechanics
damage variable D, the two damage variables DV and DC are introduced to represent the damage state due to voids and cracks, respectively. Our goal is to find a representation for the total damage variable D in terms of DV and DC. In order to do this, we need to theoretically separate the damage due to voids and cracks when constructing the effective undamaged configuration. This separation can be performed by two different methods. We can start by removing the voids only, then we remove the cracks separately, or we can start by removing the cracks only, and then we can remove the voids separately. The detailed formulation based on each of these two methods is discussed below and is shown schematically in Fig.5-9 and Fig.5-1O. It is emphasized that this separation of voids and cracks is theoretical in the sense that it is an acceptable method of mathematical analysis and has no physical basis. In fact, the physics of the problem indicates a coupling between the two damage mechanisms, which is apparent in the next section in the general three-dimensional case. T O'Yo
0\0--;;
'- 0/
Remove voids
/0\0 j
D'
10/0\ I "() ,..........0
a--.
A
'rju
A
u
I
*
(Tij
=-=-= { 1
(Tij
'rju
_
2
(1 + __
I-D.
1)
1- Di
for i = j for i
(5-236a)
=f. j
(5-236b)
for i = j (5-236c) These ratios can be used as a "measure" of deviation of a symmetrization treatment against the non-symmetrization case as given in Eq.(5-226). The third group of ratios will be given below where the variation of the magnitude of the resultant net-stress tensor components under a symmetrization treatment may be assessed by defining the following
(5-237a)
for i = j for i
5u
=
"'* (Tij (Tij
.:,
-
{ (1 - D.)
- Dj
(5-237b)
for i = j
1
2
+ (1
=f. j
)
1 - - - - for i 1- D j
=f. j
(5-237c)
From above Eqs.(5-237a), (5-237b) and (5-237c), it is obvious that all symmetrization schemes do not affect the magnitude of the normal stress components (i.e. i = j case), but shear stress components are affected. The effects on shear stress components due to the symmetrization schemes may be significant and this will be discussed in subsequent sections. A simple conclusion that can be immediately drawn, in view of above equations, would be the concern that the behaviour of the damaged continuum after a symmetrization
294
5 Basis of Anisotropic Damage Mechanics
treatment may be significantly different from what the real behaviour is supposed to be. Furthermore, in granular materials (soils and rocks alike) shear stress components have no-doubt an important role in their failure behaviour, such that when such a symmetrization scheme is not carefully exercised , it may produce a spurious result and/or an irreversible effect in a materially nonlinear analysis. The variation of the magnification factors 7],n fj(n fj(n 7](J for a range of o ~ [l ~ 0.8 is comparatively shown in Figs.5-13(a), (b), (c) where the referenced ratio 1J(J defined by Eq.(5-235a) has been drawn as a dotted line. For the practical range of [l used , all magnification factors do not become larger than 5. Magnification factors fj(J ' fj(J and 7](J differ significantly from the unsymmetrized 1J(J when [li and [lj attain their opposite extreme values (i.e. [li = 0 and [lj = 0.8, or [li = 0.8 and [lj = 0). 5
5~------------'
~
~
17"
T/,,--
1Z •••
4
4 £),=0.0.0.2:0.4.0.6.0.8
3'----
3
2
2
1
0.0
0.2
0.4 ilj
(a)
0.6
0.8
I
0.0
0.2
0.4
ilj (b)
0.6
0.8
I
0.0
0.2
0.4
ilj
0.6
0.8
(c)
Fig. 5-13 Magnification factor of net-stress t ensor based on models I (a), II (b) , III (c) of symmetrization schemes respectively for different damage states
The effects of these symmetrization treatments on the shear stress component may be visualized by plotting Eqs.(5-236a), (5-236b) and (5-236c) , as presented in Figs.5-14(a), (b) , (c). All symmetrization schemes considered here show a significant influence on shear stress components. These effects again are most severe when [li and [lj attain their opposite extreme values.
t)
It should be also noted that in all symmetrization schemes for (~(J ' ),(J ' the shear stress components increase with the damage component [li increasing and decrease with the damage component [lj increasing. By comparing these figures it can be seen that the maximum values of ratios, (~(J' ),(J ' ~(J) ' defined for shear stresses, may reach the values of about 3.0, 2.0 and 1.7 respectively. Since symmetrization schemes considered here lack a physical basis, the significant deviation as demonstrated in Fig.5-14 is not at all surprising. The mean of deviation of stress magnitude may be seen from the diagrams shown in Figs.5-15(a) , (b) , (c) where deviations b(J ' 8(J ' defined in Eq.(5237) are plotted against [l. Although the deviations due to a symmetrization scheme may not be significant for a small value of [li, they can be signifi-
t
5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models
295
3.0 ,.--- - - - - - - - - , 2.2 t:"""":X:-.- ; "' - j-' . : : : . - - - - - - - , 2.0 i~j •••• ! '~'.j2.5 , ; ; ; J •••• 1. 8 1.6 2.0 1.4 1.2 1.5
t.
1.0~~"""
1.0~~~~. 0.0
0.2
0.4 [2 j (a)
0.6
0.8 0.6
0.8
Fig. 5-14 Effects of symmetrization models I (a) , II (b) , III (c) on deviations of net-stress magnitudes for different damage states cant when a combination of opposite extreme values of [2j are attained, as has been mentioned previously. It is suggested from this assessment that a symmetrization scheme should be a bandoned in a more accurate anisotropic damage model. 3 2
8.
i'#j i =j
-
---
3 2
8.
i'#j i =j
3 2
~
8.
i'#j i =j
----
0
0
0
- I
- I
- I
-2
-2
-3
_ 3 ~L-L-~~~-L-L-J
-2
[2,=0.0.2,0.4,0.6,0.8
-3
0.00. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 [2; [2; [2, (a) for mode l I
(b) for model \I
(c) for model 11\
Fig. 5-15 Effects of symmetrization models I (a) , II (b) , III (c) on deviations of net-stress magnitudes for different damage states
5.9.4 E ffects of Symmetrization on N et-Stress I nvariant Most failure criteria are usually expressed in t erms of stress invariants because these invariants do not change the magnitude in any stress transformation. Generally, principal stresses and directions of any stress state may be found by considering the following eigenvalue problem (either for an undamaged or a damaged model) as,
(5-238)
296
5 Basis of Anisotropic Damage Mechanics
where lJij is the stress tensor considered; A = {Al,A2,A3} are eigenvalues (in a 3D state of stress), lij is usually called "eigenvector" (direction cosine) corresponding to the eigenvalue Ai, and 6ij is the Kronecher delta. Hence, the corresponding principal stresses and directions of a damaged stress state (lJ:j or &:j' ij:j and j ) may readily be found by substituting an appropriate expression into Eq.(5-238) and solving the resultant cubic equation as given below
-u:
(5-239) where h, 12 , h are the first, second and third stress invariants. For the non-symmetrized model the corresponding stress invariant can be derived as in [5-34]' and only the final results are represented here (5-240a)
(5-240b)
(5-240c) By applying the considered symmetrization treatments, the resultant net stress invariants may be expressed as
j~
=
1~
(5-241a)
(5-241b)
and similarly (5-242a)
5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models
I~ = I~
297
(5-242b)
0"11
0"13
1- 0 1 1
0"12
I*3 --"2 J(l - 0 )(1 - O ) 1 2
J(l - O2 )(1 - 0 3 )
0"13
0"33
1- 0 3
(5-242c) (5-243a)
0" 110"22
0"220"33
(1 - 0 1 )(1 - O 2 )
(1 - O 2 )(1 - 0 3 )
0"11
0"12
(1 - 0 3 )(1 - 0 1 )
(5-243b)
0"13
1- 0 1
1 - [h+n2
1- nl+ n 3
0"21
0"22
0"23
O2 + 0 1 j* _ 1 3 - - 12 0"31
1-
03
2
2
1- O 2
1 _ n2+n 3
0"32
0"33
2
+01
2
1-
0 3 +02 2
2
(5-243c)
1- 0 3
where 1:::1 in Eqs.(5-241c), (5-242c) and (5-243c) indicate the determinant of the matrix. Because of the complicated nature of stress invariants, it would be easier to consider a two-dimensional case. The first stress invariant in all four cases shown above remains the same, thus this indicates that the normal stresses do not change under any symmetrizations The second stress invariant, however, may be reduced in the following manner: (5-244a) (5-244b) (5-244c)
298
5 Basis of Anisotropic Damage Mechanics (5-244d)
The corresponding deviation ratios of the stress invariant can be defined as follows (5-245a) (5-245b)
(5-245c) (5-245d) The above relations show that the second stress invariant may be significantly different after symmetrization treatment using models I and III whereas the symmetrization treatment using model II does not change the magnitude of the second stress invariant. In order to make a comparison, the deviation of the second stress invariant between the unsymmetrized model and the undamaged model is also presented in this group by Eq.(5-245). 5.9.5 Effects of Symmetrization on Net Principal Stresses and Directions By considering a 2D case again, Eq.(5-239) obtaining principal stresses and directions may be reduced to the formulae
(5-246a) (5-246b) In the damage case, irrespective of whether the symmetrization is applied or not, the governing Eqs.(5-246a) and (5-246b) are still applicable, and hence for unsymmetrization
(5-247a)
5.9 Effects of Symmetrization of Net-Stress Tensor in Anisotropic Damage Models
tan (20: *) =
(-1-_1_fl-1
+ -1-_1_fl-2 )
299
0"12
(5-247b)
--'--0""1::-=1---0""2::':::2-
IJdi,
i = 1,2,3 (5-269)
where Ai > 0, ni > 0 (i = 1,2,3) are material constants in the direction of anisotropic principal axes, and IJdi represents the threshold stress in the ith principal direction. It is assumed that the principal axes of damage maintain a prescribed orientation in a material-fixed co-ordinate system and that the damage accumulation on the principal planes depends only upon the applied tensile stress. It should be noted that the material constants Ai and ni may be obtained from experimental results, as will be explained in other section. It may be complex to adopt Eq.(5-269) in a multiaxial state of anisotropic materials for a three dimensional case, because a total of 9 material constants Ai, ni and IJdi(i = 1,2,3) need to be determined by experimental tests. The following damage kinetic equation of anisotropic damaged materials, based on the equivalent stress IJ eq may give a simplified model,
(5-270)
5.10 Simple Damage Evolution Modeling
in which only 5 material constants mined.
O'di
309
(i =1,2,3) and A, n need to be deter-
5.10.1.2 Damage Evolution Equation Based on Damaged Strain Energy Release Rate The concept of an elastic damage strain energy release rate, necessary to propagate the micro-cracks is justified by comparing the stiffness before and after the growth of micro-cracks. In fact, the elastic damage strain energy release rate is an extension of the concept adopted in linear fracture mechanics for the crack strain energy release rate. For the damage evolution, consistent with experimental results, the following kinematics law in isotropic damage is considered [5-21] dO = dt
{HY
k
0
(5-271)
where the parameters B > 0, k > 0 in Eq.(5-271) are material constants that can be determined by experimental measurements in a similar procedure to that involving the parameters of A and n presented in [5-21], and Yd is the threshold value of the damage strain energy release rate at the start of damage o growth. Similarly, for the anisotropic damage, the rate equations can be represented in three anisotropic principal directions, ith (i = 1, 2, 3), as
Yi > Y di (i = 1,2,3) (5-272) in which parameters Bi and k i (i =1,2,3) are constants of anisotropic materials and may be determined by experimental measurements on anisotropic materials of interest. Yi is the damage strain energy release rate in the ith anisotropic principal direction and Ydi is the threshold value of Yi along the ith anisotropic principal damage direction at the start of damage Oi growth. The damage strain energy release rate Y can be determined by Eq.(363) and Eqs.(3-87) and (3-88) or Eqs.(3-146) and (3-147) for Y in isotropic damage, and Eqs.(5-98), Eq.(5-99) or (5-101) for Y (i = 1,2,3) in anisotropic damage. The integration of above damage kinetic equations of all types can be carried out using the Newmark time integration scheme. An accumulation rule of damage increment AD for each time step At at each Gaussian point in each element within the whole analyzed region should be taken into account for a description of the damage state and the damage growth process. The threshold characteristics in all types of damage kinetic equations provide a localization function of damage and damage growth, which makes the damage growth
310
5 Basis of Anisotropic Damage Mechanics
occur only at a point where the threshold condition is satisfied. Consequently, the damage increment AD will be accumulated based on the value of the previous damage state at that point and time where the threshold condition is satisfied. If the threshold condition is never satisfied at a point and time, then the damage would not grow and therefore materials may not be damaged at all if no initial damage exists. It is evident from these comments that the damage growth and propagation only take place at some possible points and in some possible directions where the threshold condition is satisfied, in other words where stress maybe is concentrated, deformation discontinued, material cracked or weakened and so on. This concept actually is based on a micropoint view. 5.10.1.3 Average Integration Scheme for Damage Evolution Equations
In the case of damage growth, the damage and stress distribution in an element are a function of time and coordinates, thus the integration of damage kinetic equations in F. E. dynamic analysis is hence more complex. The rules of time integration and accumulation of damage kinetic equations at each Gaussian point provide more accurate results of damage distribution, but we may spend more computational resources, such as CPU time and memory size as well as plotting capability especially for large dimension structural problems. In order to overcome this difficulty, it is necessary to introduce an average value and an average rate of damage in an element. The average integration scheme for overall damage evolution equations of an element may provide an economic method for integrating the damage rate equation, which will provide an overall mean damage state in the element at each time step. Then the damage growth law in an element can be approximately developed by an average value. Generally, the material parameters A and n (or Band k) are considered as constant within a small element which usually is a localized damage area and is therefore to be discretized by a denser mesh. In a multiaxial stress state it is similar to Eq.(5-269) and Eq.(5-272)
(5-273)
where (5-274) where
v;,
is the volume of the element.
5.11 Verify Anisotropic Damage Model by Numerical Modeling
311
The integration of Eqs.(5-273) and (5-274) can be carried out using a Gaussian integration technique and accumulation of the damage increment. The average damage accumulation in an element at time t = tj+l with respect to the effective stress model can be calculated by the equation (5-275) in which
8 eqn (tj) =
1 ALL Wkz[O"eq(~k' rll, tj)]n H[O"eq(~k' rll, tj) - O"di] 33
e
k=ll=l
(5-276)
where ~k' rJl (k, l = 1,2,3) are Gaussian points, Wkl (k, l = 1,2,3) are weight factors. 0"eq (~k' rJl, t j) is the equivalent stress at the Gaussian point (~k' rJl) and time t = t j . The average damage accumulation in an element at time t = t j +l with respect to the damage strain energy release model can be calculated by the equation
(5-277) in which
Yi(tj) =
1 ALL Wkz[Yi(~k' rJl, tj) ] ki H[ Yi(~k' 33
e
k=ll=l
rJl,
tj) - Ydi ] , (i=1,2,3)
(5-278) where Yi(~k' rJl, tj) are the ith components if the strain energy release rate at the Gaussian point (~k' rJl) and time t = ti.
5.11 Verify Anisotropic Damage Model by Numerical Modeling 5.11.1 Stiffness Matrix of Anisotropic Elastic Damage Model in F.E.M. The general virtual work theorem for the Cauchy stress and strain may be written in the following form, (5-279)
312
5 Basis of Anisotropic Damage Mechanics
where {c}, {u} are the virtual strain and displacement vectors, respectively, {Q} and {F} are the traction and body force vectors, respectively. For anisotropic damage materials, the constitutive equations derived previously in Eqs.(5-111)rv(5-119) must be substituted into Eq.(5-279) to obtain the corresponding stiffness matrix.
Consider finite element discretization in the standard manner, {u}
=
[N]{U}
(5-281a)
{c}
=
[B]{U}
(5-281b)
where {U} is the nodal displacement vector; [ N] is the shape function matrix; [B] is the strain-displacement matrix. Substituting Eq.(5-281) into Eq.(5-280), the following expression is obtained, [K(n)]{U} = {P}
(5-282a)
where (5-282b) which is the required stiffness matrix for a damaged element. The vector {P} is given as, (5-282c) which is known as the general force vector. Whence the nodal displacement vector {U} is obtained by solving Eq.(5282a), and the Cauchy stress of the damaged element may be computed by (5-283) and the effective stress tensor can then be subsequently obtained by the Eqs.(5-27) or Eqs.(5-33)rv(5-35). 5.11.2 Numerical Verifying for Elastic Damage Constitutive Relationship
The first example was taken from an article of Kawamoto et al. [5-9], where the experimental results of the model were provided. The specimens used in the model were plaster and cement mortar, as shown in Fig.5-22. The specimens have regularly embedded cracks formed using steel strips of 0.2
5.11 Verify Anisotropic Damage Model by Numerical Modeling
313
mm. The number of cracks for both specimens varies from 28 to 52. The undamaged material properties for both specimens are taken from [5-9]. The finite element mesh used is shown in Fig.5-23. The results obtained from the method presented in this chapter are shown in Fig.5-24 and Fig.5-25 together with the experimental results from reference [5-9]. The dotted lines indicate the results of isotropic materials with orthotropic damage states in a zigag shape of cracks, whereas the solid lines correspond to the results of orthotropic materials with orthotropic damage states in a regular form of cracks. In the model containing a zigzag shape of cracks, the anisotropic material properties of E2/ El = (1 - Sl2)2 had reasonably been assumed. I
100
I
I
y/
0 0
'"
I
V/
/
/
~-}-.) ~
~ ~
~~ 20~
~~~~
~~~~
~~~~ 0 0
~~~~
~~~~
'"
~~~~
~~~~
~~~~ ~~~~
Applied load
100
~~~~ ~~~~
/
(a) Regular type
1/
(b) Zigzag type
Fig. 5-22 Specimens for uniaxial load ([5-9])
_________
I ________ L I
unit: mm
Fig. 5-23 F. E. mesh for Fig.5-20
J I _________
I __ L
~-
__
~~
I
en
-~ >-< ~---:-=.......,--=I
II
I
"t:I
I
I
I
I
I
I
I
I
I
~
~
(}o
---------r--------~---------r---------
]
~
~
---------r--------,---------r--------I
0 Experimental --Numerical
I
Fig. 5-24 Apparent Young's module of the plaster specimens (regular type)
314
5 Basis of Anisotropic Damage Mechanics o • --
--S1.2 ""0
~ 1.0
rI.l
"'OJ)
I
8d .0 8 --
~
"t:S ~
. :.E
§
~
0.6
...
-!
- -
I ...... I......
0
I ~
I
I I
I
I I
I I
I I
"
'" L ...__
___
I
__ _
~
~'
I
I
___
___
L ---
I
_~~1.
-
....... I
..0.. . .- - A
Exp.zigzag Zxp.regular Num.zigzag Num.regular
'0
...I _ _ _ _ L __ _
"I
- -1;'- ...... I
--
0.4 '-----"'----"'------'-'---' 45 90 0 0 Crack angle (") (a) 28 cracks
+ - --
-1- --I
I
45 90 0 Crack angle (") (b)36 cracks
45 90 Crack angle (") (c) 52 cracks
Fig. 5-25 Comparison of numerical and experimental effective Young's module of cement specimens. (a) 28 cracks; (b) 36 cracks; (c) 52 cracks
5.11.3 Numerical Verifying for Symmetrization Comments The comments about deviation due to symmetrisation schemes will be verified and illustrated by simulation of an experimental test, presented in a reference of Kawamoto et al. [5-9]. In this experiment, a direct shear test is shown schematically in Fig.5-26, where Pn and Pt are the normal force and shear force subject to the specimen, respectively. The simulation model as shown in Fig.5-23 is based on a modified Mohr-Coulomb criterion, which will be employed in the finite element analysis with parameters obtained from P
L
~I
.;
0 N
/
//
/
-:J.///
/
/
/'
///// /
/'
/% /
.;
/
/
/
/70
.;
40
///// /
~t
,-
40
,/
/' /' /' /'
.;
/ /
150
p.
/
~/// \: /
0 " 1, which is unreasonable. In order to overcome this, an energy equivalent method can be employed to modify the effect. The commonly used methods for measuring networks of rock structural planes are the statistical windows method and geodetic line method, and the applied conditions of the second one are much wider. In the traditional single geodetic line method, two marked lines should be drawn in parallel with 0.5 m interval from top to bottom, which may approximately determine the plane density (A) and the volumetric density (J) of rock joints. It has already been proved that, as long as the density value in the direction of a geodetic line is tested out, the density value in the normal direction would be determined. In the spatial right angle co-ordinate system as shown in Fig.544, for the geodetic line OD the trend angle is a, the prone angle is 13 and for the geodetic line OE the trend angle is a1, the prone angle is 131. If, along the OD geodetic line, the line density of the rock structural plane is A', the area density of the rock structural plane is A~, whereas, if along the OE geodetic line, the line density of the rock structural plane is A~, the area density of the rock structural plane is As. Then
As
=
" A cosB
=
A sinacosf3sina1 cosf31
+ A, cosacosf3cosa1 cosf31 + Asinf3sinf31
(5-292) From that, the density value of the normal directions of joints is the same as that of the area density. In the calculations of the damage tensor, the
332
5 Basis of Anisotropic Damage Mechanics 0° 330° ;
E
......
)-./
AO=A(COS a)
/
300
0
I I f
.,.,30°, \ .. 60°
.../
/
\
\
\ 90°
i-
270° -t
I \
\
\
f
>-
240° "
/
,
/
';t..
240° . . . (a)
/
r
I
I
120°
>(/
180°
; 150°
(b)
Fig. 5-44 The relationship of rock structural plane densities and arbitrary geodetic lines. (a) Relation of geodetic lines and normal direction of structural planes; (b) The density of structural planes versus directions of that maximum density value of each group of rock structural planes should be generally determined individually. Therefore, a rough scatter of rock structural planes should be carried out. Thus there is a need to plot a diagram of equal density lines (i.e. contours) for joint verteces, and the rough scatter should be divided into combined groups for engineering purposes. The value of the volumetric density J of the rock mass can be determined based mostly on the obtained value of area density. (5-293) The value of J can also be calculated approximately by the engineering quantity of RQD J = (115 - RQD)/3.3
(5-294)
Using the mean values from Eq.(5-16) or (5-291), we may obtain (5-295) where, J i is the volumetric density of the ith joints, (Xi is the average area of the ith set joints. If we assume the form of the joint plane is like a circular dish form, then -k
(X.
•
=
1 J2 -nd·
4
•
(5-296)
where di is the average trace length of the ith set joints. Assume the trend angle is (X, the prone angle is f3 for a set of joints, then the normal vector of these joints can be expressed as
5.12 Numerical Application to Analysis of Engineering Problems
nl = cos (90° - f3)sina } n2 = cos (90° - f3)cosa
333
(5-297)
n3 = sin;3 The volumetric density can be determined based on either Eq.(5-293) or Eq.(5294), and from Eq.(5-294) we have fli = 0.24l i cZT(115 - RQD)(n~
Q9
n~)
(5-298)
From Eq.(5-293) we have (5-299) where
ni Q9 ni =
n1n1 nln2 n1n3] [ n2nl n2n2 n2n 3 n3nl n3n2 n3n 3
(5-300)
Substituting Eq.(5-299) into Eq.(5-291), the effective damage tensor of the fracture-damaged joint rock mass thus can be obtained. The rock structural planes of this region have been measured in geological strata of T1dy3-1 , T 1d y 2-2, T1dy2-1, and T1dyl-l, then the statistical method for rock structural planes was applied to divide groups. The trend angles, prone angles, trace lengths, intervals and density quantity of each group of rock structural planes were statistically worked out. While computing the normal vector of joints based on Eq.(5-297), the relation between the direction of section planes and the production form of rock joints to be calculated should be considered since in a specific investigated region, the characteristics of rock structure development are determinate. Ifthe relation between the direction of section planes and the production form of rock joints is not taken into account, then not only is the calculated damage tensor the same but also their effect on the properties of the rock mass would be the same. In other words, if we do not consider the relationship between these two factors, the influence of the damage tensor on any cross section planes calculated in any directions would be the same in the analyzed region. Obviously it is not true, since the stability of the rock mass is strongly controlled by the structures of the rock mass. Therefore taking different calculated cross sections, the influence of rock structures on their stability has very significant differences. Considering this fact, for a particular calculated plane, Eq.(5-297) should be nl = cos (90° - f3)sin(a - aD) n2 = cos (90° - ;3)cos(a - aD) n3 = sin;3
}
(5-301)
334
5 Basis of Anisotropic Damage Mechanics
where 0: is the trend angle of the calculated cross section plane. The damage values calculated for each geological stratum correspondingly are presented in Table 5-5. Table 5-5 Damage tensor and geometrical parameters of rock structural planes Code of Production geologi- form of cal joints
Trace Int e rval length (m) (m)
T , d y3 T , dy2 T , dy2 T , dy'
1
2
1
Dll
D'2
D'
1.57
1.02 0.64
0.384
0.249
- 0.141 0.210
- 0.088 0.092
1. 39
0.72 0.050
0.049
0.012
0.188
- 0.007 0.005
0.93
0.98
1.50 197° L 87° 1.13 315 0 L 69 ° 0.45 65° L 85° 0. 3 1 280 0 L 67° 0.35
300° L 60 ° 2° L 76 °
Damage t e nsor
1m2
strata
215° L 63 ° 60° L 81 °
Area density (number)
0.35
2.96
0.18
5.56
3
D' 4
D' 5
D16
0.24
4 .17
0.001
- 0.005 0.002
0.026
- 0.011 0.005
0.50
0.62
0.110
- 0.079 0.069
0.139
- 0.092 0.063
0.1 4
0.06
1.61 16.67
Fig.5-45 to Fig.5-49 (the other diagrams are ignored) present a part of the numerical results obtained by a finite element program of damage mechanics. It can be seen from these figures that the principal stress at the crack tip and in the excavated coal seam appears as a significant stress concentration in the area of the crag rock mass with a rank III. The stress directions within the stress concentration area have more deflections. The stress field in the rock mass still presents the field of the natural stress state due to self-weight. Displacements are mostly in the vertical direction and vary within - 0.05rv - 0.40 m and the horizontal displacement increases slightly in some regions of the downstream slope body. The maximum principal stress (}1 is about 0.8rv - 6.6 MPa (tensile as positive, pressure as negative), and the minimum principal stress (}2 is about - 0.13rv - 12.0 MPa. The maximum shear stress T ma x is 0.2rv4.2 MPa. Near the crack tip (}1 is - 3.6 MPa, (}2 is - 7.4 MPa and T max
(unit: MPa)
Fig. 5-45 Contours of the maximum principal stress distributed in crag rock mass with rank III
5.12 Numerical Application to Analysis of Engineering Problems
335
(unit: MPa)
Fig. 5-46 Contours of the minimum principal stress distributed in crag rock mass with rank III (for damage analysis)
(unit: MPa)
Fig. 5-47 Contours of the maximum shear stress max distributed in crag rock mass of rank III (for damage analysis)
(unit: MPa)
Fig. 5-48 Distribution of tensile stress areas in crag rock mass of rank III (for damage analysis)
336
5 Basis of Anisotropic Damage Mechanics
Fig. 5-49 Distribution of failure zones in crag rock mass of rank III (for d amage analysis)
is 2.3 MPa. A larger region of tensile stress areas appears around cracks on the top of the slope. A certain region with quite a few damage failure areas appears on the free empty face and around cracks on the top of the slope. Also some damaged failure zones appear near the excavated area in the coal seams. From the above mentioned numerical results it can be found that a significant stress concentration appears at crack tips, where there exist some damaged failure elements in the crag rock mass of rank III under action of stress due to the self-weight. The results of stress concentration will make the cracks produce continuous tensile failures or shear failures, and hence cause the slope to have a continuous tensile deformation, which agrees with practical observed materials. In the excavated empty area of the coal seam, some obvious stress concentration is also produced Consequently, some damaged failure elements also exist as well as shear failure to occur here. All these results cause the crag rock mass to completely bed down. This then makes the upper part of the crag rock mass a shear slip failure. The empty area on the crest face of the crag rock mass produces some certain areas of tensile stress in the horizontal direction and damaged failure zone. The empty free area of the crag rock mass will be deformed in tension along with the direction of the empty free face under the action of tensile stress, which causes the rock mass to break down. From the above analysis it can be concluded that the crag rock mass of rank III and the crack tips as well as the region of cross boundaries between the empty free area and excavated coal seams are in an unstable situation in this case, which may cause rock mass failure in the form of a breaking down The shear plane slips and cracks are continuously deformed by tension.
5.12 Numerical Application to Analysis of Engineering Problems
337
5.12.2.4 Comparison of Results
From the calculation model and numerical results obtained by the above two kinds of finite element methods, it is found that the results obtained by the method of damage mechanics were changed due to an increase in the overall stress level because of the existence of damage in the rock mass This means the Cauchy stress tensor is replaced by the effective stress tensor within the damaged zone in the rock mass. This kind change is not a simple superposition, having a close relationship to the damage tensor in the rock mass. The obtained damage tensors that are calculated based on different developed situations of rock structural planes in the rock mass have different states, therefore the changed situation of the effective stress is not the same. Usually the effective stress in most damaged elements increases, but may decrease in a few special elements. This may explain why the Cauchy stress has been replaced by the effective stress within the damaged zone in the rock mass because we are really considering the existence of damage in the rock mass, which makes the results more realistic. From contours plotted in the figures of principal stress distribution (see Figs.5.45-Fig.5.47), it has been found that the results calculated by the damage mechanics method are more reasonable and the location of the stress concentration has a more definite response, which can easily explain the rule of deformation and failure in rock mass, and is in good agreement with the real situation. Fig.5-50 to Fig.5-52 show comparison between Cauchy stresses and the effective stresses in the element chosen as the location of the crack tip. From the figures it can be shown that the effective stress calculated by the finite element method of damage mechanics is significantly higher than the Cauchy stress calculated by the traditional finite element method (comparison in absolute quantities). The increased amount of effective stress corresponds with the softening quantity of the elastic matrix weakened by the damage tensor. The response of the effective stress concentration is more obvious at the crack tip. The incremental regulation for normal effective stresses 0'; and 0'; is basically concordant, but for the effective shear stresses of T;y and T;y, the incremental regulation is quite different the one from the other due to the unsymmetrical nature of effective shear stresses (i.e. unequal T;y and T;y). Fig.5-53 shows a comparison of displacements on the top plate of the coal seam at the nodal point 499 which is the cross point of boundaries of excavated and unexcavated coal seams. The results in the figure calculated by finite element method of damage mechanics imply that the roof of the coal seam has no uniform differential settlements under pressure of the blanketed rock mass on the top. The results calculated by the traditional finite element method give a uniform settlement. Generally speaking, different thicknesses and densities of the coal seam roof may cause different pressures in the coal seam, which produce different vertical displacements at different points. The results of displacements calculated based on damage mechanics aptly describe the changed tendency of vertical displacements. Meanwhile,both the displacement results
338
5 Basis of Anisotropic Damage Mechanics 111
109
107
105
103
101
99
0.00
-1.00
Fig. 5-50 Comparison between Cauchy stress 111
109
107
105
O'~
and effective stress
103
101
99
~.--~.---~-,t.;:~--".----..,----.--.----.-~--,
-Traditonal -o-Damaged
O'~
(unit: MPa)
0.00
-3.00 -4.00
Fig. 5-51 Comparison between Cauchy stress
0'; and effective stress 0'; (unit: MPa) 2.00
-
Traditonal
- 0 - Damaged
1.00
111
109
107
105
10 -1.00
Fig. 5-52 Comparison between Cauchy stress T;Y and effective stress T;Y (unit: MPa)
5.12 Numerical Application to Analysis of Engineering Problems 500
498
496
494
492
339
490
~~::;;::::o::=-""-l -0.06 -0.08
--0--
Traditonal Damaged
-0.10 -0.12 -0.14
Fig. 5-53 Comparison of displacements on top plate of the coal seam (unit: mm) and calculated stress results that are based on damage mechanics have a very good consistency, which are in much better agreement with practice than the results calculated by the traditional finite element method. Comparison of results obtained by these two kinds of calculations show that the overall distribution either for displacement or for stress is basically concordant, which illustrates that the traditional finite element method only can give an acceptable result in essential engineering analysis when the required accuracy is not too high. The results of effective stress and displacement that are calculated based on damage mechanics have very good parallelism with the structural characteristics of rock mass. Since the different structural characteristics of rock mass have different damage tensors, their influence on the distribution of Cauchy stress and displacement should be different. The analysis shows that the damage tensor is mostly related to the dimension and the production form of rock structural planes. The damage tensor is directly proportional to the dimension of rock structural planes, and is inversely proportional to the interval between rock structural planes. The more the group number is, the higher the value of the damage tensor. The production form of rock structural planes mainly influences the quantity of components in the damage tensor. From Table 5-5, it can be seen that the average size of geological strata from T 1d y3 to T 1dy3-1 increases. The interval of geological strata between T 1dy2-1 and T 1dy2-2 is in 17.5",,64 cm, but in geological strata of T 1dy2-1 there is only one set of joints to develop. In T1dyl the minimum interval of joints is about 5.6 cm. The interval in the geological strata of T 1dy2-1 is much wider, which means that their relative integrality is much better and corresponds with the property of the rock and the thickness of the rock seam. These structural characteristics of rock mass resolve damage tensors with different deviations, where the quantity of the damage tensor in the geological strata of T 1dy3-1 is the relative maximum and the quantity in T 1dy2-1 is the relative minimum. The influences of the damage tensor on Cauchy stress can be observed in Fig.5-54,where the quantity of the damage tensor for elements 379",,385 in geological strata of P 2 l is 0, which has no effect on the Cauchy stress. Therefore, the two curves coincide; whereas the quantity of the damage tensor for elements 386",,388 in geological strata of P 2 d is relatively bigger and the effects on Cauchy stress are much higher; Further more, elements 389",,392 are in geological strata of
340
5 Basis of Anisotropic Damage Mechanics
T 1d y 2-l, the damage tensor of which is very small, and the effect on Cauchy stress is very lowfor T 1 d y 3-2 geological strata, because the damage tensor is the relative maximum, the effect on Cauchy stress should be the maximum (see Fig.5-54). 391
389
387
385
383
381
379 -0.50
-Traditonal --0- Damaged
-1.50 -2.50
Fig. 5-54 Comparison between Cauchy stress CT; and effective stress CT; in damaged element with different damage tensors (unit: MPa)
5.12.3 Damage Mechanics Analysis for Koyna Dam due to Seismic Event 5.12.3.1 Introduction of Objective Statements The Koyna dam is a 103 m high gravity structure completed in 1963. The dam started impounding water in 1962 and experienced a magnitude 6.5 earthquake, probably reservoir induced, on 11 December 1967 when the reservoir elevation was only 11m below the dam crest. The accelerations of the ground at the site were 0.49 g in the stream direction, 0.63 g in the cross-stream direction and 0.34 g in the vertical direction. The most important structural damage consisted of horizontal cracking on both the upstream and downstream faces of a number of the non-overflow monoliths [5-42"-'43]. A number of 2-D linear analyses have been made to determine the dynamic response of this dam when subjected to the recorded accelerations, while others attempted to include the non-linear features of cracking. A seismic study [5-44] by the finite element method (FEM) considered cracking with stress release once the tensile stress reached a critical value which included a factor to account for strain rate effects. Another study [5-45] used fracture mechanics and a contact/impact model for crack closure within a finite element formulation to analyze the seismic performance of the Koyna dam. Both studies revealed that the formation of cracks on both faces was to be expected during the 1967 earthquake. Experiments have also been conducted to study the dynamic response and cracking pattern. One of these was conducted at the University of California at Berkeley [5-46]. A 1:150 scale model was constructed of a plaster material containing lead powder. During the test run at 1.21 g of the shaking table a crack was initiated on the downstream face at the point of slope change,
5.12 Numerical Application to Analysis of Engineering Problems
341
which then propagated through the dam to the upstream face. Even though the excitation applied to the model was not actual Koyna ground motion and improper gravity scaling for rupture similarity was employed, the results still gave valuable insight into the cracking pattern and location under the test conditions. In spite of the limited field measurements of the pattern of crack-damaging the Koyna dam experience has provided the most complete information todate on seismic crack-damage of concrete gravity dams. Due to the complexity of the problem, analyses made so far have been restricted to simplified models. More sophisticated mathematical models for dealing with the damage process of concrete structures are still needed and model tests for either verification of the mathematical models or simulation of the prototype performance remain imperative. Since the narrow damage zone of the Koyna dam occurred in the upper part of the dam, near the point of slope change where a high stress concentration is to be found, initial crack-damage would be expected to occur at this location even during the early stages of ground shaking during the 1967 earthquake. Once the initial crack-damage has been formed, it is evident that damage mechanics theory should be employed to evaluate the damage growth and damage propagation process and the resulting pattern of the damage state. Based on the above considerations, the authors developed a new procedure for evaluation of the damage evolution process of concrete gravity dams during strong earthquakes based on the article [5-47]. In this procedure the finite element technique, modal analysis and linear elastic damage mechanics theory were combined. The accuracy of the proposed procedure was verified by a bending test of a beam with varied cross section for the dam model presented in [5-47]. The acceptable good agreement obtained between the numerical predictions and test results indicated that the new procedure is relevant for evaluation of the seismic damage process in concrete dam structures. This section is followed by the experimental results for a model of the Koyna dam with initial crack-damage tested on a shaking table under artificial input motion, for which the foregoing procedure is applied to predict the model performance including damage growth and damage propagation. Finally, the damage process of the Koyna prototype dam during the 1967 earthquake is examined, in which the time histories of the dynamic stress distribution and the damage profile during the earthquake are obtained. The results are also consistent with the observed damage of the prototype in terms of damage elevation on both faces and the phenomenon of elemental average damage on the downstream face, the latter confirming the complete penetration of the damage behavior in the dam as predicted by the present analysis. The numerical procedure for seismic damage analysis of concrete structures presented in Reference [5-47] comprises three distinct parts: (1) finite element (FE) analysis of the dynamic response for elastic damage systems; (2) impact simulation in damaged structures and (3) linear elastic damage mechanics theory for simulating the damage extension process.
342
5 Basis of Anisotropic Damage Mechanics
5.12.3.2 Some Results from Model Test of Koyna Dam and Correlation Analysis The objectives of the dam model tests were: (1) to provide, in addition to the beam with varied section test for the dam model reported in reference [5-47], further verification of the proposed numerical procedure for seismic damage analysis and (2) to obtain a qualitative evaluation of the damage process in the Koyna dam under simplified loading conditions. The model scale of the Koyna dam section is 1:200; namely 515 mm in height; 351 mm wide at the base and 80 mm thick as shown in Fig.5-55. The density 'Y of the gypsum material is 480 kg/m 3 and the dynamic modulus of elasticity E = 600 MPa. Acceterometers were installed on the table and at the crest of the model and strain gauges were located on both sides of the model at 5 and 12 mm from the front boundary of the damaged zone. Thus, based on fracture mechanics with strain Cy measured in the y-direction the stress intensity factor KJ was determined from (5-302) where r represents the distance between the damaged tip and the strain gauge. As shown in Fig.5-55, the model was fixed to the shaking table, with no reservoir water included. Harmonic sweeping tests were first conducted to obtain the frequency components of the model. To cause the dam model to rupture, lead blocks with a total mass of 2.76 kg were attached at the crest. A very narrow initial damage zone about 10 mm in length was simulated by a set of small regularly arranged holes drilled through the thickness at the location of slope change on the downstream face. Because of the capacity limitation of the shaking table, the input excitation for the rupture test was comprised of a series of load pulses, which were approximately periodic but not harmonic. For the numerical predictions, 2 percent damping (~ = 0.02) was assumed for the six modes considered and Poisson's ratio v was assumed equal to 0.2. As shown in Fig.5-56, the test model was divided into a finite element mesh assuming the condition of a rigid foundation to simulate the shaking table. The material properties of the elastic modulus E = 1.0 X 10 5 MPa and equivalent density 'Y = 7680 kg/m 3 were specified in the same quantities for all elements. The initial crack-damage is considered as an initial damage zone, which is profiled by very small holes drilled through the thickness of the dam model within the elements surrounding the broken line on the downstream faces distributed at Gaussian points near the horizontal interface between the dam sub-regions I and II as shown in the circle detail in the sub-figure of Fig.5-56. An unequal distribution of elements, with a much denser mesh near the narrow initial damage zone and also at the slope change location on the downstream face, was employed in order to refine the calculation of stress concentration factors and to permit the damage zone to develop following the crack extension.
5.1 2 Numerical Application to Analysis of Engineering Problems
Fig. 5-55 Koyna dam model with lead block on crest 80mm
.....
II>
S
a ........
....
351 mm
Fig. 5-56 FE mesh model of the dam
343
344
5 Basis of Anisotropic Damage Mechanics
The failure process of the dam model was simulated by the damaged constitutive equations and the damage growth equations presented in this chapter. A step-by-step time integration scheme was employed at each Gaussian point to obtain the structural response and the damage growth as well as the damaged zone extension accumulated from the initial damage state. Time step llt = 0.001 sec was used in the calculation. Once the damaged failure criterion is reached at a Gaussian point, the micro-structure to be considered at this Gaussian point comes into the failure process and therefore the damage will grow and accumulate from the previous damage state. Damage zone propagation occurs perpendicular to the maximum circumferential strain direction with infinite velocity. The foregoing value of the failure damage state can also be implied equivalently by the quantity Kid, which represents the dynamic fracture toughness of the concrete of the dam model and can be obtained directly from the theory of fracture mechanics and the test measurements by
E
~
(5-303)
Kid = 4(1 _ v) Ee,cr
The plot shows in Fig.5-57(b) where sudden rupture of the model is seen to occur at time 0.728 sec. It should be noted that different kinds of plaster were used in the tests of the previous cantilever beam [5-47] and the current model dam, resulting in very different values of KId as well as modulus E (600 MPa for the model dam).
N"-"
'"
lc
9 6
.~
~.,
0; u u
< Time(a)(s)
(b) Time(b)(s) - - - Measured
0.8
······ Computed
Fig. 5-57 Rupture test time histories for model dam. (a) Input motion of shaking table; (b) Measured and computed stress intensity factor KJ
5.12 Numerical Applica tion to Analysis of Engineering Problems
345
For the initially crack-damaged model, the measured and calculated frequencies are listed in Table 5-6. The results from the test and calculation are seen to be close. For the rupture t est the table excitation frequency was 6.1 Hz , with the input acceleration exceeding the table capacity and causing a distortion of the intended harmonic motion as is evident in Fig.5-57(a). The time histories of the measured and calculated (force method and assuming no damage zone extension) stress intensity factor KJ at the front tip of the damaged zone are compared in Fig.5-57(b). It is noted that the finite element results and the tested measurements are in good agreement in general. Fig.557(b) also shows that rupture of the model dam occurred at t = 0.728 s, when the strain gauges suddenly broke during the test. Table 5-6 Natural frequencies (Hz) for different modes of the damaged dam model Nat ural frequencies (Hz) Model Measured Calculated
ii
51.2 51.I
252.3 273.9
366 .4 375. 9
604.2 670.2
i5
1119 1127
1233 1187
Fig.5-58 shows a comparison of the crack-damage profiles at rupture observed in the test [5-47] and those obtained by numerical simulation using damage localization techniques with finite element computation. The agreement between these results is outstanding, thereby confirming that the localization damage models for crack-damage simulation are applicable for practical problems.
(a)
(b)
Fig. 5-58 Comparison of rupture profile for model dam between test (a ) and numerical (b) simulation
346
5 Basis of Anisotropic Damage Mechanics
5.12.3.3 Damage Analysis for Practical Koyna Dam in 1967 Earthquake The Koyna dam cross-section and its FE discretization are shown in Fig.5-59. To improve accuracy of local damage behavior by limiting the difference in element size encountered over the domain, a transitional sub-region is introduced in the FE discretization as shown in the detailed subfigure enlarged from the circle in Fig.5-59. The characteristics of the Koyna dam are [542 rv 43]: E = 3.1 X 10 4 MPa, "( = 2640 kg/m 3 , v = 0.2 and ~ = 0.05. y
'V91.80m
Fig. 5-59 FE discretization of Koyna dam
The elastic modulus of the foundation rock (7 x 104 MPa) is approximately twice that of tile concrete. Considering the high stiffness of the foundation rock and the fact that the ground acceleration was obtained directly at the dam base, a rigid foundation and earthquake input applied directly at the base were assumed. The corresponding accelerograms of the ground motion are shown in Fig.5-60. The component in the cross-stream direction was assumed not to affect crack-damage development and was therefore neglected. In the results to follow , seismic loading refers only to the effect of components of the 1967 Koyna earthquake in both the horizontal stream-wise and the vertical directions, whereas static and dynamic loading includes the seismic loading, the dam's self-weight as well as the hydrostatic force.
5.12 Numerical Application to Analysis of Engineering Problems
347
O~ r------------------------------~
e:o
0.4
';;' 0.2 .~
~., oof.~~M'"
U
8-0.2
«
-0.4 - O.O ~--~~--~~--~~--~~--~~--J
o
2
3
4
5 6 Time (s) (a)
7
8
9
10
II
,..,0.4 ~
.g 0.2
t 0.0
-.;
g-0.2
« -0 .4
L...---L.. _ _ L...---L.. _ _ -'-----L.._ _ -'-----L.._ _ -'-----'-_ _ .L-....J
o
2
3
4
5 6 Time (s) (b)
7
8
9
10
11
Fig. 5-60 Ground acceleration of Koyna earthquake, 11/12/1967. (a) Stream direction component; (b) Vertical component
According to field measurements taken after the 1967 earthquake [5-43], most of the downstream cracks occurred at, or near, the location of slope change where the effect of stress concentration is expected to be significant. Similarly according to the test model dam, an initial narrow crack-damage zone was assumed to exist at elevation 66.5 m near the downstream face and the initial damage state of Do = 0.3 is assumed to be put closely at two rows of Gaussian points within the conjoint elements surrounding the broken line on the downstream faces as illustrated in Fig.5-59. This gives a modeling the initial damaged points distributed at these Gaussian points near the horizontal interface between sub-regions I and II of the dam and they are scaled up in a detailed way in the circle sub-figure of Fig.5-59. This has the function of expressing the behavior of damage localization and the narrow damaged zone propagating in order to simulate the crack enlarging. A record over time of the dam's response was computed taking into consideration contributions from different points of view (saturations). As the damage developed (growth and propagation), the frequencies and mode shapes of the dam structure should be changed accordingly. Damping ratio ~ = 0.05 for all elements was assumed. Based on the foregoing assumptions, the following response behavior was predicted for the Koyna dam prototype.
348
5 Basis of Anisotropic Damage Mechanics
The first four modal frequencies of the initially undamaged Koyna damreservoir system are 3.07, 7.98, 11.21 and 16.52 Hz, respectively, which are very close to the results from FE analysis [5-43] and thus serve to verify the accuracy of the present damage finite element discretization. For a load combination consisting of static and dynamic components, time step integration was performed with !1t = 0.005 s and an earthquake duration of 6.0 s. The response of the dam in terms of the crest displacement and acceleration, and also the stresses both on upstream and downstream faces, was examined. The results may be summarized as follows: The horizontal displacement of the dam crest reached 43.6 mm in the downstream direction. The maximum accelerations at the crest were 22.8 m/s 2 and 22.6 m/s [5-43] in the horizontal and vertical directions, respectively, with corresponding amplification factors of 4.8 and 6.6. The computed maximum tensile stress of 6.69 MPa occurred near the point of slope change on the downstream face and already far exceeds the tensile strength of the concrete, thus confirming that the first crack-damage zone is indeed to be expected at the point of slope change. Fig.5-61 shows a comparison of historical displacements at the corner point of upstream faces on the dam top obtained by damage analysis and undamaged elastic analysis. From the results of the damage analysis, it can be found that both the horizontal and vertical displacements ofthe observed point reach the maximum value at time t = 4.4 s and decrease after 6.5 s. Whereas the results of undamaged elastic analysis reach the maximum value at time t=7.0 s and after that the responses of undamaged elastic analysis are always lower than those of the damaged one. These facts state that when the crack-damage is perforated on top of the dam, where the action on the crack-damaged zone due to the earthquake becomes relatively lighter the frequencies of the horizontal and vertical responses are not quite changed. Fig.5-62 presents some numerical results obtained by a Gaussian points time integration scheme and plotted in the form of photo sketches for simulated damage profiles at different rupture times during the 1967 Koyna earthquake computed as damage distribution at all Gaussian points in the observed cross section based on an isotropic damage model for the Koyna dam. The power law equation of a damage strain energy release model presented in Eq.(5-268) was taken into account in the finite element analysis for modeling the damage growth simulation. Sketches (a), (b) and (c) in Fig.5-62 show the procedure of crack-damage growth and crack-damage zone propagation within a cross section of the Koyna dam at the rupture time t = 3.8, 4.0 and 4.5 s due to ground acceleration of the earthquake. It can evidently be seen that the damage localization phenomena was successfully carried out by the damage growth model and the initial Gaussian point damage model. The procedure of narrow damage zone extension presents the crack-rupture profiles due to material damage. As can be expected, the most serious damage zones appear at the tensile stress concentration zones accordingly near the downstream face and horizon-
5.12 Numerical Applica tion to Analysis of Engineering Problems 6.0
8' e
- - EL"'a she ana I"YSIS
4.0 . ~ ~::R\!\Jl\!g~J!I)~IYS1·· i~
:
't
:, i
~
.;t.
349
.
Ii :::~::1¥v~W:f~~~~;~,I ~~t;C · · · · · · · . · ~· [. i ~ .· ;. · ·; ; · I
,•
-2 .0 ............. . . . ... . . . ... . ... ............ .
o -4.0
.
..
................................ ............................................................ : 1........ . ....... , . -6 .0'--_ _-'-_ _--'-_ _---'_ _ _-'--_ _-' 02468 10 Time (s) (a)Horizontal displacement 1.5 - - Elastic analysis ~ 1 Occ~cRamageanalysis
e . -5 0.5
~~
. . ........ ......... ......... ......... ......... . ... ........ ......••. .. ... .......
.
'i' ... . . . . ...
·. 1
1_::~~~,~h~1!.jlt~~i~ir~v1y~~· .~
~
~ I,
J" : '
Cl -I.0 ,, ......'... "", '
I
I. . I
'.
" .... ;....... ............ ......... 1
-1.5 '-----'-----'------':--''---'--'-----' o 2 4 6 8 10 Time (s) (b) Vertical displacement
Fig. 5-61 Comparison of displacement responses for different analysis due to Koyna ea rthquake, 11 / 12/ 1967. (a) Horizontal displacement; (b) Vertical displacement
(a)
(b)
(c)
Fig. 5-62 Sketches of simulated damage profiles at different rupture times during Koyna earthquake computed for damage distribution at all Gaussian points in the observed cross section based on isotropic damage model for Koyna dam. (a) t =3.8 s; (b) t = 4.0 s; (c) t =4.5 s;
350
5 Basis of Anisotropic Damage Mechanics
tal interface between sub-regions I and II of the dam at elevation 66.5 m as well as the regions of the toe corner and heel corner surrounding the bottom of the dam. Fig.5-63 plots the distribution of displacement contours in the damaged Koyna dam due to the Koyna earthquake wherein Fig.5-63(a) shows the horizontal displacement contours, and (b) shows vertical displacement contours respectively. A significant denser gradient both for horizontal and vertical displacement contours appears near the toe corner region of the dam where the less free deformation is restrained at the bottom due to the foundation rock, but does not appear in the crack region, since the crack is opened with freer deformation, which is less restricted due to the crack opening and closing.
(a) Horizon tal di splacemen t
(b) Vertical dis placement
Fig. 5-63 Contours of displacement distribution in damaged Koyna dam due to Koyna earthquake. (a) Horizontal displacement; (b) Vertical displacement;
The plot in Fig.5-64 shows a vector sketch of deformational direction distribution in the cross section of the Koyna dam due to the Koyna earthquake. It can be seen that the tensile form of the deformational direction vector appears in all expected regions surrounding the damage-crack, the toe corner and the heel corner. These tension zones are the most seriously damaged areas in the Koyna dam due to the Koyna earthquake. The profile of the mean damage pattern in damaged elements of the Koyna dam during the Koyna earthquake also has been analysed by the damage finite element method for anisotropic (orthotropic) materials based on the average integration scheme of damage evolution equations expressed in Eqs.(5-
5.12 Numerical Application to Analysis of Engineering Problems
351
11 0 and C3 > 0, which generates Y2 > 0 and Y 3 > o. The corresponding damage is relative to the directions crosswise to the compression direction.
C1
6.3.4.3 Applied Examples for Composite Materials The theoretical model consisting of state Eqs.(6-103) and (6-108) and the second criterion of Eq.(6-113) above was applied to ceramic matrix composites SiC/SiC and C/SiC (with m = 2). In the case of SiC/SiC, the residual strain was considered negligible (with Co = 0). Fig.6-4 shows the results of uniaxial t ension in direction 0° for SiC/SiC and C/SiC. The t ension curves with unloading are correctly reproduced by the model, including the residual strain on C/SiC, and the closing effect is clearly visible (These curves were used for identification). The transverse strain (c2) is also correctly predicted in both cases. The element of compliance C 12 is constant for SiC/SiC and variable for C/SiC (in this case the Poisson's ratio varies with damage). 300
Experimental dita [6-33)
CT
Model dita
(a) Stree
Strain Experimental data [6-33)
(b)
Fig. 6-4 (a) Tensile-compression t est on SiC/ SiC; (b) Tensile-compression test on C / SiC
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
383
Fig.6-5 shows the predictions made for tension and compression tests in directions off the axes for C /SiC and their comparison with experimental data. The predictions are fairly good.
o·
15 '
30' :::====~45'
Experimental data [6-33)
45' 0'
15'
- - - - - 45'
Ii
o· Fig. 6-5 "Off-axis" tests on C/SiC
It is interesting to note that the applied model for composites is a particular case of a more general model of initially fakeisotropic damageable elastic materials such as concrete. The simplification is due to the scalar character of the state variables describing the principal damage, related to the fact that the micro-cracks are considered to be guided by the heterogeneous, oriented structure of materials.
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory 6.4.1 Introduction and Objective The mechanical response of solids to a large extent depends on the type, density, size, shape and orientations of defects in its micro-structure. This section focuses on the influence of many atomically sharp micro-cracks on the elastic parameters of materials. This problem attracted a great deal of attention as a result of both its intrinsic importance and its complexity. In general,
384
6 Brittle Damage Mechanics of Rock Mass
the complexities are inherent in the random geometry of the microstructuremicro-defect system and the fact that the in-homogeneities introduce a length scale, rendering the problem non-local. The character of the response essentially depends on the micro-crack concentration. Micro-crack concentration can be considered dilute if the distance separating adjacent micro-cracks exceeds the decay length of fluctuations that they introduce into the stress field. In this case, the direct interaction of the micro-crack has a second order effect on the macro-response. The external stress field of a micro-crack is influenced by the neighboring micro-cracks only indirectly through the contribution to the overall state (effective elastic modulus). The overall (macro) response is, therefore, a function of the orientation weighted micro-crack density and the solid is locally macro-homogeneous rendering local constitutive theories applicable. The effect of direct micro-crack interaction on the macro-response grows with the increase in the micro-crack density, i.e., shrinking distances between neighboring micro-defects. As the micro-crack density is increased further, the micro-cracks self-organize into clusters. The disorder attributed to microcracks randomly scattered over most of the volume decreases. Eventually, the largest micro-crack cluster transects the specimen into two or more fragments, reducing the macro-stiffness of t he specimen to zero. At the incipient failure in a load controlled test , mechanical response of the specimen is dominated by t he largest cluster. Within this phase, the stress and strain fields are strongly inhomogeneous (localized) and the volume averages cease to be meaningful measures of the corresponding random micro-fields. In strain controlled tests, brittle materials exhibit softening as a result of internal stress redistribution, grain (or aggregate) interlocks and bridging, etc. From the percolation point of view [6-34]' the critical state is defined as a state at which the transition from the short- to the long-range connectivity of the defect cluster occurs. In other words, a system percolates at the point at which a cluster of interconnected slits spans the specimen, causing its fragmentation into two or more finite fragments. Since the percolation belongs to the class of the second order phase transitions, these two definitions should define the same state. However, the slit density at which the tangent stiffness vanishes (KT = 0) is not necessarily identical to the slit density at which macro-rupture occurs. One of the objectives of this section is to shed some light on this apparent contradiction. Estimates of the elastic modulus of solids weakened by a large number of slits are commonly provided using Mean Field Theory (MFT) [6-35]. Neglecting spatial correlations (direct interactions) of micro-slits, the mean field theory results are inadequate beyond some undefined micro-slit density. However, as pointed out by Ma [6-36], Cleary et al. [6-37] and many others, significant improvements in mean field theory cannot come from within. At higher densities, the influence of random micro-crack morphology (position, shape, size, etc.) on local stress field fluctuations and macro-properties grows from being important to becoming dominant. The conventional (local) continuum
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
385
theories are, however, inherently unable to replicate the circumstances for which the spatial disorder on the micro-scale becomes the dominant feature of the macro-response. Hence, it seems both sensible and necessary to resort to the methods of statistical physics in order to shed some light on the underlying phenomena. This need is further emphasized by a well-recognized fact that the application of micromechanical models is limited to cracks of very special geometry (rectilinear slits and planar, penny-shaped cracks), periodic arrangements of cracks, etc. The narrow objective of this study is to provide percolation theory estimates of critical slit densities for several different slit configurations emphasizing influence of the loss of isotropy. In a wider sense, these results plotted on the same diagram with the MFT estimates, provide some indication as to whether a particular model exhibits a proper trend for larger slit densities. 6.4.2 Mean Field Theory of Micro-Mechanics 6.4.2.1 Aspects of Mean Field Theory for Brittle Damaged Materials
For low to moderate concentrations of micro-defects, a material is typically assumed to be locally macro-homogeneous. Consequently, the elastic paramet ers of a solid can be estimated using the mean field theories (effective continua). The mean field theories are based on the assumption of equivalence of the strain energy of the actual solid with a disordered microstructure and that of an appropriately defined effective continuum. These methods imply the existence of a small Representative Volume Element (RVE) containing a statistically representative sample of in-homogeneities that can be mapped on a material point of the effective continuum, preserving the equivalence of internal energy density. A configuration space is compiled in the process of mapping the transport properties of the representative volume element on the material point of the effective continuum. This configuration space, attached to each material point of the effective continuum, contains data related to volume averages of micro-in-homogeneities over t he representative volume element , which defines the structure of the material locally, its recorded history and, thus, the macro-response. The most frequently used mean field theories, known as the Self Consistent Method (SCM) and Differential Models (DM), are based on the following assumptions [6-38]: (a) the external field of each micro-defect, assumed to be equal to the ext ernal (far) field, applies to the ent ire representative volume elemen, and (b) the size of the largest defect is much smaller than the linear dimension of the representative volume element. Subject to these assumptions, the problem of many interacting micro-cracks within the actual solid is reduced to a superposition of simpler problems considering isolated cracks embedded in a homogeneous, effective continuum. In the absence of a length parameter and suppressed during volume averaging of in-homogeneities, the
386
6 Brittle Damage Mechanics of Rock Mass
ensuing theory is local. The overall (average) compliance [C*(x , D)] in a material point of the effective continuum is within this approximation and obtained by superimposing contributions of all n cracks within the representative volume element
[C*(x, D)] = [c(x) ] + [C* ([C(x , Dm
(6-114)
where [C(x) ] is the compliance of the virgin matrix, while D denotes a set of parameters used to record history (irreversible changes of the microstructure) , i.e. the damage variable. Also n
[C* (x , D)] =
L C(i) [C(x , D)]
(6-115)
i=1
is the compliance attributable to the presence of all active micro-cracks within the representative volume elemen. In Eq.(6-115), C(i) is the contribution of the ith micro-crack to the specimen compliance, which may be an implicit and/ or explicit function of the overall compliance [C]. Furthermore, depending on the desired degree of accuracy and the selection of the analytical model, the compliance C(i ) of a single crack can be det ermined as a function of the compliance of the virgin matrix [C]=[ C] (self-consistent and differential models) or on the effective compliance of the pristine matrix [O]=[C] (Taylor model for very dilute slit concentrations). Det ermination of the components of the overall (macro) compliance t ensor [C(x , D) ] of a solid weakened by a dilute concentration of micro-cracks proved to be a rather popular topic in the recent past. In general, the effective compliances [C] can be derived from the expressions for crack opening displacements C( i) = C( i) (u) or from the expressions for the stress intensity factors C (i) = C(i) (K). These two approaches were shown to be different only in form [6-39]. Since the stress intensity factors K i are typically more accessible [6-40] the second approach, based on Rice [6-41] has certain, if formal , advantages. The mean field estimates used in this study are derived from Sumarac et al. [6-34]. Thus, even a cursory discussion of the mean field theory seems to be redundant. 6.4.2.2 Dilute Concentration (or Taylor's) Model of Brittle Damage
The most rudimentary brittle damage model is formulated assuming that each defect totally ignores the presence of all other defects. In this case, every defect is assumed to be embedded in the original, undamaged matrix, which is usually assumed to be isotropic. Since the literature devoted to fracture mechanics provides all the necessary formulas for the stress intensity factors (Ki and the elastic energy release rate in the case of isotropic elastic solids containing a single defect of simple geometry) the dilute concentration model of brittle damage is in most cases amenable to a closed form , analytical solution. This
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
387
method provides lower bounds on tensors [C*] and [C] since the weakening effect of adjoining defects on the matrix stiffness is totally ignored. Substituting [G]=[ C] into Eqs.(6-114) and (6-115) and assuming that the original matrix is isotropic and homogeneous, the components of the fourth rank tensor [C* ([CJ) ] can in most cases be determined analytically.
6.4.2.3 Self-Consistent Model of Brittle Damage As the crack concentration increases it becomes both advisable and necessary to incorporate the effect of the interaction between the neighboring cracks into the model. The simplest way in which this can be done is to assume that each micro-defect is embedded in an effective continuum, the parameters of which reflect in some average (smoothed or homogenized) sense the presence of all other micro-cracks within the representative volume element. Thus, the self-consistent estimates for elastic parameters are obtained substituting [6] by [G] in Eqs.(6-114) and (6-115).
6.4.2.4 Differential Scheme of Self-consistent Model The differential method is a clever extension and modification of the selfconsistent scheme. Defects are introduced sequentially in small increments from zero to their final concentration. The overall state is interpreted as being a result of a sequence of dilute micro-crack concentrations. The state containing (,) micro-cracks evolves from the preceding state containing (, - 1) micro-cracks through the addition of a single new micro-crack. Consequently, by its very nature the results obtained using the differential method depend on the sequence in which the micro-cracks are introduced. The simplest method for obtaining the governing equation of the differential method is to follow the above described procedure. A self-consistent estimate for the overall compliance of a representative volume element containing (, - I) micro-defects is from Eqs.(6-114) and (6-115)
[G] = [C] +
r-1
L
j (i) C(i) ([GJ)
(6-116)
i= l
The ,th defect is subsequently introduced , assuming that for, » 1 the increment in the total defect concentration is infinitesimal. As in the selfconsistent method (SCM) the actual location into which the ,th defect is introduced is considered to be irrelevant within this scheme. Under these stipulations, from Eq.(2.3) it follows that r-1
[G]
+ d[G] = [C] + L j (i) C(i) ([GJ) + c(r) ([GJ) dj i= l
(6-117)
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6 Brittle Damage Mechanics of Rock Mass
It is further assumed that it is possible to write [c(r) ]=[O][H ] (where [H ] is a fourth rank tensor) as an explicit function of the overall modulus (for the effective continuum containing (r - 1) micro-defects). Subtracting Eq.(6-116) from Eq.(6-117) and pre-multiplying both sides of that expression by [0]-\ the original differential equation can be recast into a much simpler form as
d(ln [O]) = [H ]df
(6-118)
The overall compliance [0] can now be obtained as a solution of the differential Eq.(6-118), and the initial condition [O]=[C] when f = O. Analytical quadratures of the differential Eq.(6-118) are possible only if the components of the tensor [H ] are defined as simple, analytical functions of the overall compliances. In general, Eq.(6-118) represents a system of coupled ordinary differential equations, which may, or may not, admit a closed form analytical solution.
6.4.3 Strain Energy due to Presence of a Single Slit To determine the elastic parameters of a brittle solid containing an ensemble of micro-cracks, it is first necessary to derive the expression for the strain energy attributable to the presence of a single slit. The expressions for the strain energy release rate for an open rectilinear slit embedded in an arbitrarily loaded , infinitely extended anisotropic, homogeneous, elastic, two dimensional continuum was derived by Sih et al. [6-42] as
yl = _ KId 1m KI(A~ 2
K
2
YK
=
22
" K2 , TCll Im[K2(AI
+ A~) + K2 X1 X2 ,
(6-119) ,
+ A2 ) + K I AI A2]
yk
where Yk and are the strain energy release rates associated with the slit loading Mode I and Mode II respectively. Also, C;j are components of the anisotropic matrix [C;j] in the local (slit) coordinate system denoted by primes and selected as in Fig.6-6. Additionally
A~ = r~
+ is;
(i = 1,2 and
,\~ =
r; - is;
r; ~ O,s; ~ 0)
(6-120)
are the roots of the characteristic equation [6-42] [6-43] written in the slit coordinate system (6-121) The total strain energy release rate is from Eqs.(6-1 20) and (6-119)
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
389
Fig. 6-6 Rectilinear slit global and local (primed) coordinate system (6-122) The second order matrix Eq.(6-122) is defined as
[cal
(with superscript a standing for anisotropy) in
C~2 2 [ (r~)2
r~ s~
+ r~s~
+ (s~)2][(r;)2 + (S;)2 ] C '1 1 '
,
-2( s 2 + s 1 ) (6-123) For a general anisotropic mat erial, the fourth order algebraic Eq.(6-121) does not admit an explicit analytical solution. Thus, the analytical expressions for the parameters s' and r' in Eq.(6-123) are not available either. For an orthotropic material the compliance tensor in the principal coordinate system simplifies t o (6-124) where in a special case (6-125) The above [C] denotes the global (total) compliances of the solid, which are still to be determined. The corresponding characteristic Eq.(6-131) (in the case of an orthotropic material) is defined by Eqs.(6-124) and (6-125), and when written in the principal coordinates system, it reduces to the form (6-126) where
390
6 Brittle Damage Mechanics of Rock Mass
(6-127) The roots of the bi-quadratic Eq.(6-1 26) are either purely imaginary or complex.
(A 2 )1 ,2 = - 1 ±
VI -
(6-128)
m
For m < 1 the roots in Eq.(6-128) are purely imaginary: (6-129) For m > 1 (which will be considered in this section) the solution of the characteristic equation (6-126) is a complex conjugate (6-130) From the parameters rand s, derived by substituting Eq.(6-130) into Eq.(6-126), we have the following form
r1
=
r2
=r =
J~-
1 ,
Sl
=
s2
J
Vm2 + 1
=S=
(6-131)
The above parameters rand s would suffice for a slit aligned with one of the principal axes. For a slit subtending an angle B with the principal coordinate system of an orthotropic material, it is necessary to find the parameters rand s in the slit (primed) coordinate system for which the compliance matrix is full. Nevertheless, once the solutions of the characteristic equation in principal coordinates are known as Eqs.(6-1 30) and (6-131), the roots of the characteristic equation (6-121), written in an arbitrary (local, primed) coordinate system can be derived using the Lekhnitskii [6-43] transformation
A' = Ak cos B - sin B k cos B + Ak sin B
"\' _ )..k cos B - sin B
/\k -
cos B + Ak sin B
(6-132)
Consequently, the parameters r' and s' in the slit coordinate system can be written in the form [6-44]:
r'l = w1r(rsin2B + cos2B) , r'2 = w2r(rsin2B - cos2B) ,
S'l
,
S 2
= =
SW1
sW2
(6-133)
where W2 ,1
= (ymsin 2B + cos 2B ± rsin2B) - 1
Substituting Eqs.(6-1 33) and (6-134) into Eq.(6-123), the matrix comes
(6-134)
[eij] be-
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
(rm -
391
(rm -
C' 1)cos 2e + 1 C;2 1) sin 2e 221+(m-l)cos 4 e 2 1 + (m - l)cos 4 e
(rm -
(rm -
C~l 1) sin 2e C' 1)sin 2 e + 1 11 1 + (m - 1)sin 4 e 2 1 + (m - 1)sin4 e
(6-135)
The transformation rule for the compliances is
[C' ij] = [T' im] T [Cmn ] [T'jn ] where the transformation matrix
[T~j ]
(6-136)
is
cos2e sin 2 e 0.5 sin 2e
1
[ sin 2 e cos 2e - 0.5 sin 2e
[T;j ] =
- sin e sin 2e
(6-137)
cos 2e
The expressions for the compliances Cb, and Ci1' in the local coordinate system are from Eqs.(6-124), (6-125), (6-136) , and (6-137)
Substitution of Eq.(6-138) into Eq.(6-135) leads to the final expression for the matrix [Cij] a
_
[C pq ] -
Sell
[(rm - 1 )COs2e + l ~(rm - l)Sin2el ~(rm - 1) sin 2e (rm - 1)sin e + 1 2
(6-139)
The path independent M integral can be determined from the expression for the J integral (6-140) Finally, the strain energy due to the presence of a slit in an anisotropic body is
W*=
J~ o
da = 2
J
Jda = 2
0
In the isotropic case (m the solutions are r'l
J
{Kp}T [C;q]{Kq}da
(6-141)
0
= I), Eq.(6-126) (or Eq.(6-121)) is bi-quadratic, and = r' 2 = 0
and
S'l
= S' 2 = 1
(6-142)
To derive an approximate analytical solution for the unknown overall compliances of a macro-orthotropic solid, assume that Ci1 i- C~2 but keep the parameters r' and s' as if the material is isotropic in Eq.(6-142). In this approximation, to be referred to as quasi-isotropic, the matrix [Cij] is
392
6 Brittle Damage Mechanics of Rock Mass
(6-143) where 6ij is the Kronecker's delta operator. The ultimate simplicity is achieved assuming that the material is isotropic. In this case, the following equation is commonly used for diluting the concentration of slits (Taylor approximation)
C'l1 = C' 22 = 1/ E'
(6-144)
where (for plane stress) (for plane stress)
(6-145)
Matrix [Cij] in Eq.(6-143) reduces, in this case, to an even simpler expression (6-146) The stress intensity factors for a rectilinear slit embedded in anisotropic two-dimensional continuum (Fig.6-6) , written in the local coordinate system [6-42] are (6-147) The expression for the strain energy attributed to the presence of a crack is then derived substituting Eqs.(6-147) and (6-146) into Eq.(6-141) and performing integration (6-148) In Eqs.(6-147) and (6-148), it was found convenient to use the Voigt's notation
In general, anisotropy may occur as an intrinsic property of the matrix or be induced by slits. In the latter case, which is of interest in this study, the compliances and the elastic modulus are overall or effective properties. Therefore, the formulas derived in this section will be in the sequel used such that [O]=[C] and [O']=[C']
6.4 Micro-mechanics of Brittle Damage Based on Mea n Field Theory
393
6.4.4 Compliances of 2-D Elastic Continuum Containing Many Slits 6.4.4.1 Influence of Cracks Induced Anisotropy The influence of the cracks on the overall (effective) modulus of an elastic solid has been taken as the object of numerous studies in the past. Some of the existing results will be quoted in the sequel, while the expressions listed in the previous section will be used to derive new results needed to assess the influence of crack induced anisotropy on the overall elastic moduli. For convenience, it will be assumed that the undamaged (virgin) matrix is isotropic. Within the mean field theories approximation, the macro-stresses are mapped on macro-strains by means of the fourth rank effective compliance tensor as
{c} = [C]{O"} = ( [C]
+ [C*]){O"}
(6-150)
The expression relating t he overall compliance tensor and the derivatives of the complementary "elastic strain energy" is [6-39]
Since the stress intensity factors in Eq.(6-147) are linear homogeneous functions of stresses, the compliances due to the presence of a single slit embedded in a two-dimensional elast ic continuum can be derived substituting Eq.(6-141) into Eq.(6-151) and performing requisite differentiations
*(k) [Cij ] =
f
d2W *(k) a [d{Kp}T] a [d{Kq}] , ,T = 4 d{ '} [Cpq ] , T da (p,q = 1,2) d{O"Jd{O") 0 O"i d{O"j}
(6-152) The final expression for the compliance tensor attributable to a single slit in an orthotropic two-dimensional continuum is derived substituting Eqs.(6139) and (6-147) into Eq.(6-152) and performing necessary differentiations and integration
[C~(k) ]
= 27ra 2
[Crl {62i}{62j}T + Cr2 ({62i }{i 6j}T + {66i}{62j}T)
+C:fd66i }{66j}T]
(i , j
= 1, 2, 6)
(6-153) The coefficients Cij in Eq.(6-139) are both explicit and implicit (through m) functions of [Cij ]' Hence, they must be determined numerically by iteration.
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6 Brittle Damage Mechanics of Rock Mass
Within the quasi-orthotropic Eq.(6-143) and isotropic Eq.(6-146) approximations, the components of the compliance tensor could be estimated as
[c;?) ] = 27ra 2(C~2 [{ 02iH 02j} T] + C~l
[{ 06iH 06j} T])
(i, j = 1,2,6) (6-154)
and
[C;?) ] = 27ra 2([{02iH02j}T] + [{06iH06j}T])
jE' (i , j = 1, 2,6) (6-155) respectively. Primes in the expression of Eqs.(6-154) and (6-155) (except for E') indicate a reference to the local (slit) coordinate system. The corresponding expression for compliances in the global coordinate system is obtained using the transformation rule (6-156) The transformation matrix in Eq.(6-156) is " [Tij] = [
1
cos28 sin 2 8 sin 28 sin 2 8 cos 2 8 - sin 28 - 0.5 sin 28 0.5 sin 28 cos 28
(6-157)
In Eq.(6-157), 8 is the angle subtended by the axes Xl and X[. From Eqs.(6153) and (6-156) the final expression (in the global coordinate system) for the compliance attributable to a single rectilinear slit is
(6-158) In the quasi-orthotropic Eq.(6-1 43) and the isotropic Eq.(6-146), the approximations are (i ,j
= 1, 2, 6) (6-159)
and
jE' (i ,j= l ,2,6) (6-160) The compliances due to the presence of a single slit are derived substituting Eq.(6-157) into Eq.(6-158) and using Eq.(6-139) below
6.4 Micro-mechanics of Brittle Damage Based on Mea n Field Theory
395
. 2e ; C(k) * = 27m 2 s C- 11 sm 12 = C(k)* 21 =0 ci~) * = C~~) * = 27ra 2 SOlI sin e cos e
(k) * C 11
C~;) * = 27ra2sy'mOllCOS2e
(6-161)
C~~)* = cg)* = - 27ra2sy'm0ll sin e cos e
C~~)* = 27ra 2 SOlI [1 + (y'm - 1)sin2 e] In the quasi-orthotropic Eq.(6-143) , the compliances are estimated substituting Eq.(6-138) into Eqs.(6-159) and (6-160) and using the transformation rule Eq.(6-157)
ci~) * = 27ra 2 [011(1 - 2cos 2 e + 2cos4 e - cos 6 e) C~;)* = 27ra 2 [0 11 (cos 2e - cos8 e -
+ 0 22 (cos 2e - 2cos 4 e + cos6 e)] cos 2 esin 6 e) + 022(COS 8 e + sin 6 ecos 2 e)]
(6-162) The expression for the compliances in the isotropic case is recovered from Eqs.(6-161) and (6-162) setting m = 1 [6-44] (6-163) The effective (overall) compliances for a solid containing many cracks can be derived from Eq.(6-114) once Eqs.(4-161)rv(163) for a single crack are available.
6.4.4.2 Case Study for Two Systems of Aligned Slits Consider the two-dimensional case in which all slits are divided into two systems. Each of these two slit systems consists of N / 2 parallel (aligned) slits of equal length 2a. Slits in these two systems subtend angles +eo and _ eo, respectively, with xi-axis. The compliances attributable to both systems of slits is within the mean field theories approximation (see Eq.(6-114)) [C;j ]
=
~ ( [Ci~ )* (eO)] + [Ci~ )* ( - eo)])
(6-164)
Substituting Eq.(6-161) into Eq.(6-164) leads to
C;l = 27r N a 2 SOlI sin 2eo C;2 C*
22
C 66
= C;6 = C;6 = 0 r:::::: - cos 2 eo = 27rNa 2 symC 11 2 = 27r N a SOlI [1 + (y'm - 1)sin 2eo]
(6-165)
where m in Eq.(6-127) is a function of overall compliances as yet unknown. In the quasi-orthotropic approximation subst ituting Eq.(6-162) into Eq.(6164) gives
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6 Brittle Damage Mechanics of Rock Mass
The overall compliances of the effective continuum can then be det ermined substituting Eq.(6-165) into Eq.(6-114) and solving the system of algebraic equations for unknown effective compliances. In order to compare results of mean field theories estimated by different models of Taylor's, self-consistent and differential methods, superscripts ' tm ' , ' se' and 'dm' are used to denote estimations of Taylor's, self consistent and differential methods respectively. Thus, based on the self-consistent method of Budianski and O'Connell [6-45] we obtained
C SC *
_ C 22 22 - 1 _ 2nD A m
where
All
and
A22
22
(6-167)
are crack state parameters to be described later on, and (6-168)
is the micro-crack density, or damage parameter, which can be considered as the average damage parameter. In Eq.(6-168) , N is the number of slits per unit area. For isotropic matrix, C ll = C 22 , from Eq.(6-167) is obtained m - 1=
27rD(A22 -
All)
(6-169)
In quasi-orthotropic approximation, substituting Eq.(6-166) into Eq.(6-114) leads to the system of equations as
(1 - 27rDB ll )Cll - 27rDB 12 C22 = C ll = liE , - 27rDB 21 Cll + (1 - 27rDB 22 )C22 = C 22 = l i E
(6-170)
The parameters A(m, Bo) and B(Bo) in Eqs.(6-167) and (6-170) are (6-171) and
+ 2cos4Bo - cos6Bo 2COS4Bo + cos6Bo
Bll = 1 - 2cos2Bo B12 B21 B22
= cos2Bo = cos2Bo - cos8Bo - cos2Bosin6Bo = cos8Bo + sin6Bocos2Bo
(6-172)
The Taylor estimate for the compliances can be derived directly from Eq.(6-114) and Eq.(6-166) as C ll = C 22 , =l / E, such that
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
397
E Itm*
1 1 E 1 + 27r.f?(Bll + B 22 ) 1 + 27r.f?sin 2 eo (6-173) E 2tm* 1 1 E 1 + 27r .f?cos 2 eo 1 + 27r.f?( B21 + B 22 ) In a special case, where the two slit systems are mutually orthogonal, bisecting the angle between the global coordinate axes, eo = ± (7r/ 4) , the overall compliances in Xl and X2 directions are equal. In this case from Eq.(6-169) m - 1 = 27r.f? s ( Vm - 1)
(6-174)
The only solution of Eq.(6-174), m = 1 is independent of .f? Thus, from Eqs.(6-167) and (6-171) (7SC* 11
=
(7sc* 22
1
= E(l - 7r.f?)
(6-175)
Consequently, the solid remains macro-isotropic and its overall elastic modulus is jj;sc*
- - = 1 - 7r.f?
(6-176) E An identical result is obtained from Eq.(6-170). In the Taylor approximation for eo = 7r/4 from Eq.(6-173) 1
(6-177) E It can be shown that the isot ropy of the solid is not violated whenever two mutually orthogonal systems contain an equal number of rectilinear slits irrespective of the angle eo This conclusion agrees with the fact that Eqs.(6-174) and (6-176) are independent of eo. The differential method (DM) estimate of the elastic modulus can be obtained by solving the ordinary differential equation derived from Eqs.(6-118) and (6-176) (6-178) Subject to the initial condition that for .f? = 0, jj;* = E, the solution of Eq.(6-178) gives the estimated result by differential method as (6-179) The three different estimates of elastic modulus in a two-dimensional case containing two mutually orthogonal systems of aligned slits are plotted in Fig.6-7 for 0 < .f? < 0.5. Even though the differential method and the selfconsistent method are based on the same, or at least a similar set of approximations, the resulting estimates of the effective elastic modulus are substantially different. In fact , the self-consistent method predicts that jj;sc* = 0 for
398
6 Brittle Damage Mechanics of Rock Mass
[2 = 1/7r, while the differential method predicts that same micro-slit density.
Edm *=0. 368 for the
I~ idm/E i 7E S
0.2 0.0 0.0
0.1
0.2
n
0.3
0.4
0.5
Fig. 6-7 Effective elastic modulus for a two-dimensional continuum containing two orthogonal systems of aligned slits. Superscripts 'em ', 'sc' and 'dm' denote estimations by Taylor's, self-consistent and differential methods respectively
In polycrystalline solids the grain boundaries are often of inferior fracture strength. Consequently, most of the cracks are intergranular. Assuming an idealized two-dimensional case in which all grains are regular hexagons, it is often found desirable to study systems of slits subtending angles of = ± 7r /6 [6-46]. In this case it is difficult to derive the explicit expression for min Eq.(6127) in terms of [2 in Eq.(6-168), from the system of Eqs.(6-167) and (6-169). For [2 = 0.2, using an iterative procedure it is possible to compute
eo
m
= 2.22,
(6-180)
In quasi-orthotropic approximation, the elastic modulus is from Eq.(6-170) 1.85(1.42 - [2)(0.38 - [2) 1 - 97r[2/16 From Eq.(6-181) for [2 m
= 1.97,
= 0.2,
1.85(1.42 - [2) (0.38 - [2) 1 - 77r[2/16 (6-181)
(6-182)
which compares well with Eq.(6-180). Thus, the quasi-orthotropic approximation procedure seems to be reasonably accurate for damage state [2 « 1. Taking the Taylor approximation for Eq.(6-173) we have
6.4 Micro-mechanics of Brittle Damage Based on Mean Field Theory
Ei m * E When D
1
1 1 - 0.51dl '
399
(6-183)
= 0.2, Eq.(6-183) gives (6-184)
Thus, in contrast to the previous case, two systems of aligned slits intersecting at an angle of = ±1f / 6 present the effective orthotropic continuum. The quasi-orthotropic and Taylor estimates for the two effective elastic moduli are plotted in Fig.6-8. The Taylor approximation (upper bound on E) provides rather poor estimates in this case as well.
eo
0.8 0.6 ~
It<j 0.4
£,"IE
0.2
o. 0 '-:-~---:-'-:-~--:-'-:--~--:-'-:----'-''''-1._~-:-' 0.0
0.1
0.2
0.4
0.5
Fig. 6-8 Effective elastic modulus for a two-dimensional continuum containing two syst ems of aligned slits intersecting at an angle of 7f / 6 radians
In a special case, where all N slits are parallel to the Xl axis (and belong to the same system), the coefficients of Eq.(6-172) are Bll = I, Bl2 = B21 = B22 = o. The effective (overall) compliances and the elastic modulus E2 are 1 and 1 - 21fD
E Sc *
_ 2_
E
= 1 - 21fD
(6-185)
The elastic compliance and modulus in the direction of the axis X l remains unchanged. From Eq.(6-185), the estimations of elastic moduli by differential method are (6-186)
400
6 Brittle Damage Mecha nics of Rock Mass
Taking the Taylor approximation for Eq.(6-173) we have 1
E +
tm*
(6-187) =1 E 1 + 2nD In the cases of Eqs.(6-185) to (6-187) the material is orthotropic as deduced by Laws and Brockenbrough [6-46]. The estimations of the effective elastic modulus E2 in a two-dimensional continuum weakened by a system of aligned slits of equal length (parallel to X l axis) are plotted in Fig.6-9. The different discrepancies between the differential and the self-consist ent estimation method are again evident even for very small D. and
1.0 0.8
0.6 ~
I~
0.4
0.2
0.0 L-~_.l....--- + (1 _ D) dj(P, P) , dP dP
p] _l_2+sin_si_n_ rp R rp
t
=0
(6-252) Substituting Eqs.(6-250) and (6-251) gives
(6-253) Substituting Eq.(6-231) and rearranging gives 2 CJij d CJ eq D - DP + Dj(P P)Rt 1 + sin rp 1 - D d CJij ' 2 sin rp 1
_ 3(1 - D) D2 1 + sinrp R t dj(P, P) D 2 2 sin rp dP
(1 - J£ n)dj(P,P)R1 +sinrp]p' -_ (1 - D)2dCJeq Dijkl (dUI + -dUk) + [n J£ + t ----'2 dCJij dXk dXI dP 2sin rp (6-254) Taking out the common factor from the left side and moving the other terms to the right side of the equation, we have
426
6 Brittle Damage Mechanics of Rock Mass
D)D~ d!(P,P)]
2aij da eq + [!(P, P) _ 3(1{ 1 - D daij 2
dP
1 + .sin -.
•
o
Computed Monitored
•
0
..I*] as
(7-129)
7.7 Anisotropic Elasto-plastic Damage Equations for N umerical Analysis
507
and (7-131) -
-
where [Kf ] and {K f }
T
were defined by Eqs.(7-26) and (7-27).
dF
The vector {da} can be calculated for Hill's model and Hoffman's model in the following forms: For Hill's model, from Eqs.(7-120) and (7-121) , we have T
-
T
dF Td[F*] T Td[F;] d{ D} = {a} d{ D} {a} = {a} [To- ] d{ D} [To-]{ a}
(7-132)
In the case of plane stress 2
F12'I/J31
d[P; ]
- 1
2Ff'I/Jr'I/J2
dDI
0 0
- 1
2Ff'I/Jr'I/J2 0 0 - 1
2Ff'I/J§'l/J1
d[P; ]
- 1
dD2
2Ff'I/J§'l/J1
p,2'I/J3 2 2
0
0
- 1
- 1
2
0 0
(7-133a)
1
Ff2'I/Jr'I/J2 0 0
(7-133b)
1
Ff2'I/J§'l/J1
In the case of plane strain 2
F12'I/J31 - 1
d[P; ] dD I
2Ff'I/Jr'I/J2 - 1
2Ff'I/Jr'I/J2 0
2Ff'I/Jr'I/J2 2Ff'I/Jr'I/J2
0
0
0
0
0
0
0
0
0
1
Ff2'I/Jr'I/J2
(7-134a)
508
7 Anisotropic Elasto-plastic Damage Mechanics
0 - 1
- 1
- 1
2Ff1/J§1/JI 2
2Ff1/J§1/JI - 2F? + Ff 2F? Ff1/J~
p,21/J3 2Ff1/J§1/JI 2 2 - 1 - 2F? + Ff 2Ff1/J§1/JI 2F? Ff1/J~
a [p;] aD2
0
0 0 (7-134b)
0
0
0
0 1 Ff21/Jr1/J2
For Hoffman's model, from Eqs.(7-127) and (7-121), (7-128) we have
(7-135)
In the case of plane stress 2
- 1
a [p;]
F lt F le 1/Jr - 1
2Flt F 1e1/Jr 1/J2
aD I
2Flt F 1e 1/Jr1/J2
0
0 0
0 - 1
a [p;]
- 1
2Flt F 1e1/J§1/JI 2
aD2
2Flt F 1e1/J§1/JI
FltFle1/J~
0
0
a{]*} = a[tli]T {K} = { aDI
aDI
aD2
In the case of plane strain
0 Ff21/Jr 1/J2
0 0
(7-136b)
1 Ff21/J§1/JI
oo} F2t -F2e o} F2t F2e '
' (1 - D2)2
(7-136a)
1
Flt-Fle (1 - Dd 2FltFle' ,
a{]*} = a[tli]T{K} = {O aD2
0
(7-136c)
(7-136d)
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
d[P; ] dDI
509
2 F lt F lc 1/;{ - 1 2Flt F l c 1/;r1/;2 - 1 2Flt F l c 1/;r1/;2 0 - 1
0 - 1
d[P; ] dD 2
2Flt F l c 1/;~1/;1 F lt F lc 1/;5 - 1 2Flt F l c - F2t F2c 2Flt F l c 1/;~ 1/; 1 2FltFlcF2tF2c1/;5
- 1
o
0
o
2Flt F l c 1/;~1/;1 2Flt F l c - F2t F2c 2FltFlcF2t F2c1/;5
o
2
o
o
1
Ff2 1/;~ 1/;1 (7-137b)
(7-137c)
d{f*} = d[tli]T {in = dD2 dD2
{
0
F2t - F2c F2t - F2c 0} ' (1 - D2)2 F2t F2c ' (1 - D2)2 F2t F2c'
(7-137d)
dG
The vector {dY} for both models can be calculated from Eq.(7-55) by substituting Eqs.(7-49)'V(7-51).
dG
dY
In the case of no damage growth, we have {dCT} = {dCT}' In the case of
dG
damage growth, the vector {dY} for both models should be calculated in terms of Eq.(7-7S) or Eqs.(7-53) and (7-54) as
d~~} = d~~} + Hbeqfc? - E~) (ta dG
ro
[d*]{ CT}
(7-13S)
dF
Since {dR} = {dR}' these two vectors have been determined by Eqs.(7104) , (7-105) and (7-106) in t erms of Eq.(7-107) for Hill's model and Eq.(7lOS) for Hoffman's model. The calculation of quantities Req and A have been mentioned by Eqs.(7103) and (7-105), (7-115) and (7-116) , respectively in subsection 7.6. Thus, the numerical analysis of elasto-plastic kinetic damage problems using the finite element method can be carried out.
510
7 Anisotropic Elasto-plastic Damage Mechanics
7.8 Coupled Damage and Plasticity in General Effective Tensor Models A coupled model between damage and plasticity to be developed in this sect ion for anisotropic continuum damage mechanics is based on the finite-strain plasticity using different effective tensors. The formulation is given in spatial coordinates (Eulerian reference frame) and incorporates both isot ropic and kinematic hardening. The von Mises yield function is modified to include the effects of damage through the use of the hypothesis of elastic energy equivalence. A modified elasto-plastic stiffness tensor that includes the effects of damage is derived within the framework of the proposed model. It is also shown how the model can be used in conjunction with other damage-related yield criteria. In particular, Gurson's yield function [7-21] which was later modified by Tvergaard [7-22] is incorporated in the proposed theory. This yield function is derived based on the presence of spherical voids in the material and an evolut ion law for the void growth is also incorporated. It also shows how a modified Gurson's yield function can be related to the proposed model. Some interesting results are obtained in this case. For more details the reader is referred to Kattan and Voyiadjis [7-23], Krajcinovic [724'"'-'25], Voyiadjis and Kattan [7-26], Voyiadjis and Park ([7-27], and Wang et al. [7-28].
7.8.1 Stress Transformation Based on Configurations Consider a body in the initial undeformed and undamaged configuration Co Let C be the configuration of the body that is both deformed and damaged after a set of external agencies act on it. Next , consider a fictitious configuration of the body C obtained from C by removing all the damage that the body has undergone. In other words, C is the state of the body after it had only deformed without damage. Therefore, in defining a damage tensor [il], its components must vanish in the configuration C (see Fig.7-12). States of deformation and damage: (a ) deformed damage state; (b) fictitious deformed undamaged state; (c) elastically unloaded damaged state (unstressed state); (d) elastically unloaded fictitious undamaged state (fictitious unstressed undamaged state), Kattan and Voyiadjis, [7-29].
7.8.1.1 Effective Transformation Tensors In the formulation that follows, the Eulerian reference system is used , i. e. all the actual quantities are referred to in the configuration C while the effective quantities are referred to in C. As mentioned earlier, the transformation between the Cauchy stress tensor {a} and the effective stress tensor {a * } can be formulated in the tensorial form as a ij= lJr ijk l akl, in which lJr ijk l are the components of the fourth-order linear operator called t he damage effect tensor (see subsection 5.7 in Chapter 5).
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
(c)
511
(d)
Fig. 7-12 Illustration of plasticity and damage configurations The deviatoric part {S} of the Cauchy stress tensor can be written in the configuration C as S ij = a ija mmOij /3 , in which Oij are the components of the second-order identity tensor [1], whereas a similar relation exists in the effective configuration C between {S} and {S*} in the form of S'0 = a ir a;"m Oij / 3 where Oij is the same in both C and C. Thus, a transformation relation for the effective and Cauchy deviatoric stress tensors can be derived as (7-139) It is clear from Eq.(7-139) that a linear relation does not exist between {S *} and {S}. On the other hand, one might suspect that the last two terms on the right-hand side of the Eq.(7-139) cancel each other when they are written in expanded form. However, this possibility can be easily dismissed as follows: suppose one assumes Si j = tJrijklSkl. Using this with Eq.(7-139) one concludes that tJrijkWnn = tJrppqraqrOij. Now consider the case when i -I- j, it has Oij = o and therefore tJrijmmann = O. It is clear that this is a contradiction of the fact that generally tJr ijkl -I- 0 and ann -I- O. Therefore, the additional terms in the Eq.(7-139) are non-trivial and such a linear transformation cannot be assumed. Upon examining Eq.(7-1 39) in more detail, eliminating Ski by using S ij = a ij - a mmOij /3 and simplifying the resulting expression, the following transformation can be carried out
{S*kz}
= [if.ijkl ]{ akz}
where if.ijkl are components of the fourth-order tensor [if.] given by
(7-140)
512
7 Anisotropic Elasto-plastic Damage Mechanics
(7-141) Eq.(7-140) represents a linear transformation between the effective deviatoric stress tensor {S* } and the Cauchy stress tensor {a}. However, in this case the operator [Jt.] is not simply like the damage effective t ensor [tli] but is a function of [tli] as shown in Eq.(7-141). The tensors [tli] and [Jt.] are mappings S---+Q.. and S---+Q..dev respectively, where S is the stress space in the current configuration C and Q.. is the stress space in the fictitious undamaged configuration, with {a} E Sand {a* }EQ... Next, we consider the effective stress invariants and their transformation in the configuration C. It is seen from Eq.(7-140) that the first effective deviatoric stress invariant J{ = Sii is given as (7-142) since Jt.iikl = 0 by direct contraction in Eq.(7-141). Therefore, one obtains Sii = Sii = O. The problem becomes more involved when considering the effective stress invariant {SijV {Sij }. Using Eq.(7-139) along with Sij = a ij- ammOij/3, one obtains (7-143) where (7-144) (7-145) (7-146) Substituting for {S} from Sij=aij-ammOij/3 into Eq.(7-143) (or more directly using Eq.(7-140) along with Eq.(7-141)), one obtains
{S*;j }T{S*;j } = {akt}T [Hklmnl{a mn }
(7-147)
where the component of the fourth-order tensor [H ] is given by (7-148) and the tensor [Jt.] is given by Eq.(7-141). The transformation Eq.(7-147) will be used in the next sections to transform the von Mises yield criterion into the configuration C.
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
513
7.8.1.2 Concept of Effective Back-Stress Tensor In the theory of plasticity, kinematic hardening is modeled by the motion of the yield surface in the stress space. This is implemented mathematically by the evolution of the shift or back-stress tensor {E}. The back-stress tensor {E} denotes the position of the center of the yield surface in the stress space. For this purpose, we now need to study the transformation of this tensor in the configuration C and C. Let {E'} be the deviatoric part of the back-stress tensor {E}. Therefore, one has the following relation Eij
=
o
Eij - E mm ij /3
(7-149)
where both {E} and {E'} are referred to in the configuration C. Let their effective counterparts {E*} and {E*'} be referred to in the configuration C. Similarly to Eq.(7-149), we have: 17* ij
=
17* ij
-
17* mmOij /3
(7-150)
Assuming a linear transformation (based on the same argument used for stresses) similar to {aij} = [tliijkl]{akt} between the effective back-stress tensor Eij and the back-stress tensor Eij (7-151) and following the same procedure in the derivation of Eq.(7-140) , we obtain the following linear transformation between {E*~j} and {Eij} (7-152) The first effective back-stress invariants has a similar form to that of the effective stress invariants, mainly, 17* ~i = E~i = 0 and (7-153a) In addition, one more transformation equation needs to be given before we can proceed to the constitutive model. By following the same procedure for the other invariants, the mixed invariant {a ij } T {E ij} in the configuration C is transformed to {Sij V {E*~j} as follows , (7-153b) and a similar relation holds for the invariant {E* ~j V {S7j }. The stress and back-stress transformation equations will be used later in the constitutive model.
514
7 Anisotropic Elasto-plastic Damage Mechanics
7.8.2 Strain State and Strain Transformation In the general elasto-plastic analysis of deforming bodies, the spatial strain increment tensor {s} in the configuration C is decomposed additively as follows, (7-154) where {se } and {sP } denote the elastic and plastic parts of {s}, respectively. In Eq.(7-154) the assumption of small elastic strains is made. However, finite plastic deformations are allowed. On the other hand , the decomposition in Eq.(7-154) will be true for any amount of elastic strain if the physics of elastoplasticity is invoked, for example, the case of single crystals. In the next two subsections, the necessary transformation equations between the configurations C and C will be derived for the elastic strain and plastic strain increment tensors. In this derivation, it is assumed that the elastic strains are small compared with the plastic strains and, consequently, the elastic strain tensor is taken to be the usual engineering elastic strain tensor {se }. In addition, it is assumed that an elastic strain energy function exists such that a linear relation can be used between the Cauchy stress tensor {a} and the engineering elastic strain t ensor {se }. The tensor {se } is defined here as the linear term of the elastic part of the spatial strain tensor where second-order terms are neglected. For more details, see the work by Kattan and Voyiadjis [7-29]. 7.8.2.1 Concept of Effective Elastic Strain The elastic constitutive equation to be used is based on one of the assumptions different from the previous chapters and is presented by a linear elastic relation in the configuration Cas {aij } = [D ijkl]{ski }, where the components in [D ] are the fourth-rank elasticity tensor represented by Lame's constants A and f-L in the form of Dijkl = AOij Okl + f-L(Oik Ojl + OilOjk)' Based on the linear elastic constitutive relation, the elastic strain energy function We({ se } ,D) in the configuration C is given by (7-155) One can now define the complementary elastic energy function 11: ({ a} ,D) based on a Legendre transformation as follows (7-156) By taking the partial derivative of Eq.(7-156) with respect to the stress tensor {a} , one obtains
{ e} = dll; ({a},D) s d{a}
(7-157)
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
515
Substituting Eq.(7-155) into Eq.(7-156) in the configuration C , one obtains the following expression for II( {CT }57) in the configuration C as follows , (7-158) The hypothesis of elastic energy equivalence, which was initially proposed by Sidoroff. [7-1], is now used to obtain the required relation between {Ee } and {E*e }. In this hypothesis, one assumes that the elastic energy IIe({CT}57) in the configuration Cis equivalent in form to II e ( {CT*}, 0) in the configuration C. Therefore, one writes (7-159) where II; ({CT} ,57) is the complementary elastic energy in C and is given by (7-160) where the superscript - 1 indicates the inverse of the tensor. In the Eq.(7-160), the effective elasticity modulus [D*(57) ] is a function of t he damage tensor {57} and is no longer a constant. Using Eq.(7-159) along with expressions of Eqs.(7-158) and (7-160), one obtains the following relation between [D] and [D*(57) ]
[D k1mn ] = [i]lij~I(57)][Dijpq][i]I;;!nn (57) ]T
(7-161)
where the superscript -T indicates the transpose of the inverse of the tensor. The tensor [tliij~I(57) ] can be defined by Eqs.(5-24), (5-148) and Eqs.(5144)"-'(5-146) . It should be pointed again that as mentioned in Chapter 5 for the unsymmetrized model [i]lij~l ][i]lmnktl is not an unite tensor ( ~ [ID , and only for symmetrized models [tli ij ktl- 1 [tli mnktl = [I ] is an unite tensor. Finally, using Eq.(7-157) along with Eqs.(7-158) , (7-159) , and (7-161) , one obtains the desired linear relation between the elastic strain tensor {Ee } and its effective counterpart {E*e } (7-162) The two transformation Eqs.(7-161) and (7-162) will be incorporated later in this chapter in the general inelastic constitutive model that will be developed. 7.8.2.2 Effective Plastic Strain Increment
The constitutive model to be developed here is based on a von Mises type yield func t ion F( {S},{ E'},wp, 57) in the configuration C that involves both isotropic and kinematic hardening through the evolution of the plastic work
516
7 Anisotropic Elasto-plastic Damage Mechanics
wp and the back-stress tensor {Ell, respectively. The corresponding yield function F({S*},{EI* l,w; , O) in the configuration C is given by
(7-163) where a y and c are material parameters denoting the uniaxial yield strength and isotropic hardening, respectively. The plastic work is a scalar function and its evolution in the configuration C is taken here to be in the following form
w;
dw; =
{defn T { defn
(7-164)
where {def; } is the plastic part of t he spatial strain increment tensor {de}. Isotropic hardening is described by the evolution of the plastic work wp as given above. In order to describe kinematic hardening, the rule of PragerZiegler evolution law is used here in the configuration C as follows (7-165) where df.L* is a scalar function to be det ermined shortly. The plastic flow in the configuration C is described by the associated flow rule in the form
P* } * dF { deij = dAp d{ atj }
(7-166)
where dA; is a scalar function introduced as a Lagrange multiplier in the constraint thermodynamic equations (see subsection 7.3) that is still to be determined. In t he present formulation, it is assumed t hat the associated flow of plasticity will still be held in the configuration C , that is
{ p}
dF
deij = dAPd{a ij }
(7-167)
where dA p is another scalar function that is to be determined. Substituting t he yield function F of Eq.(7-163) into Eq.(7-166) and using the transformation Eqs.(7-140) and (7-152) , one obtains (7-168) On the other hand, substituting the yield function F of Eq.(7-163) into Eq.(7167) and noting the appropriate transformation Eqs.(7-1 47) and (7-153), one obtains (7-169)
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
517
It is noticed that plastic incompressibility exists in the configuration C as seen from Eq.(7-168) where {dE;',fm } = 0 since [ltmmkd = O. However, this is not true in the configuration C since {dE~m } does not vanish depending on [Hmmkd as shown in Eq.(7-169). In order to derive the transformation equation between {dEP} and {dE*P } , one first notices that
(7-170)
where [Wpqij ]
.
IS
defined as
[a{U;q} ] Using the above relation along with a{Uij } .
Eqs.(7-166) and (7-167), one obtains
*p _ d~; -1 P {dEij} - d~ [Wjmn J {dEmn }
(7-171)
p
The above equation represents the desired relation, except that the expression d~ *
d/ needs to be determined. This is done by finding explicit expressions for P
both d~; and d~p using the consisting conditions. The rest of this section is devoted to this task. But first one needs to determine an appropriate expression for dJ.l* that appears in Eq.(7-165), since it plays an essential role in the determination of d~;. In order to determine an expression for dJ.l*, one assumes that the projection of {dE*' } on the gradient of the yield surface F in the stress space is equal to ;3{dE*P } in the configuration C , where ;3 is a material paramet er to be determined from the uniaxial tension test ([7-30], [7-29]) This assumption is written as follows ,
[ aF aF] ;3{dE*P} = {dE*' } a{u:'nn } a{u kl } kl mn ( aF ) T aF a{u;q}
(7-172)
a{u;q}
Substituting for {dE*'} and {dE*P} from Eqs.(7-165) and (7-166) , respectively,
aF
into Eq.(7-172) and post-multiplying the resulting equation by a{ U kl }' one
obtains the required expression for dJ.l*
(7-173)
Using the elastic linear relationship and taking its increment, one obtains
518
7 Anisotropic Elasto-plastic Da mage Mechanics
(7-174) where {dE*e } is assumed to be equal to d{ E*e } based on the assumption of small elastic strains as discussed earlier. Eliminating {dE*e } from Eq. (7-174) through the use of expressions in Eqs.(7-154) and (7-166) , one obtains (7-175) The scalar multiplier dA; is obtained from the consisting condition dF( {Skl} {E' ~d, 0, w;) = 0 such that
"dF ) ( "d{Skl}
T {
S*}
d kl
+
(
"dF ) "d{Ek;}
T {
*' }
dE kl
"dF
*
+ "dw; dwp = 0
(7-176)
Using Eqs.(7-154) , (7-164) , (7-166) , (7-173) , and (7-175) into Eq.(7-176), one obtains the following expression for dA; * 1 "dF [ 1{ * } dAp = Q* "d{Skl} D klmn dEmn
(7-177)
where Q* is the scalar and given by
(7-178)
Assuming that the kinematic hardening rule of Prager-Ziegler is held in the configuration C along with the projection assumption of Eq.(7-150), one can derive a similar equation to Eq.(7-177) in the following form (7-179) where Q is given by
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
519
(7-180)
In contrast to the method used by Voyiadjis and Kattan [7-31] where the two yield functions in the configurations C and C are assumed to be equal, a more consistent approach is adopted here. This approach is based on the assumptions used to derive Eq.(7-179). It is clear that in this method the two yield functions in the configurations C and C are treated separately and two separate consistency conditions are thus invoked. In the opinion of [7-29] this emphasizes a more consistent approach than the method used by Voyiadjis and Kattan [7-31]. One is now left with the tedious algebraic manipulations of Eqs. (7-177) and dA* (7-179) in order to derive an appropriate form for the ratio \ p . First, Eq.(7dAp
179) is rewritten in the following form , where the appropriate transformations [D*]----+[D] and {CY } ----+ { CY* } are used
dF
-1
T
QdAp = d{ S7j } [Dijpq ][tlipqmn ] {de mn }
(7-181)
Then one expands Eq.(7-177) by using the appropriate transformations {de*e } ----+ { dee } and {ds*P} ----+ { dsP} to obtain
(7-182) where the derivative d [iJl mnpq ] is defined in the next section. The last major step in the derivation is to substitute the term on the right-hand side of Eq.(7181) for the results using the transformations {CY} ----+ {CY* }, {cy} ----+ {S* } , [D] ----+ [D*] and others. Once this is done, the following relation is obtained
520
7 Anisotropic Elasto-plastic Damage Mechanics
The above equation is rewritten in the form: (7-184) where
(7-185)
It is noticed that al and a2 are the last two terms on the right-hand side of Eqs.(7-178) and (7-180), respectively. It should be noted that when the material undergoes only plastic deformation without damage, that is when the configurations C and C coincide, dA* then al = a2 and a3 = 0 since d[tli] vanishes in this case, thus leading to d/' P
The relation (7-184) is now substituted into Eq.(7-171) along with Eq.(7169) to obtain the following nonlinear transformation equation for the plastic part of the spatial strain increment tensor: {de;!} = [Zijkt] {dePij}
+ {dzij }
(7-186)
where the tensors [Z] and {dz} are given by (7-187) (7-188) The transformation Eq.(7-186) will be used later in the derivation of the constitutive equations. 7.8.3 Coupled Constitutive Model In this section, a coupled constitutive model will be derived incorporating both elasto-plasticity and damage. This section is divided into three subsections detailing the derivation starting with the equations then proceeding to the desired coupling.
7.8.3.1 Evolutional Equations of Damaged Materials In this section, an inelastic constitutive model is derived in conjunction with the damage transformation equations presented in the previous sections. An elasto-plasticity stuffiness tensor that involves damage effects is derived in
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
521
the Eulerian reference system. In this formulation, the rate-dependent effects are neglected and isothermal conditions are assumed. The damage evolution criterion to be used here is proposed by Lee et al. [7-32] and is given by (7-189) where J ijkl are the components of a constant fourth-order tensor [J] that is symmetric and isotopic. This tensor is represented by the following matrix 0 1~~ 0 ~ 1~ 0 [J] = ~ ~ 1 o 0 0 2(1 000 0 000 0
~)
0 0 0 0 2(1 0
~)
0 0 0 0 0 2(1 -
(7-190) ~)
where ~ is a material constant satisfying 2~ ~ ~1. In Eq.(7-190), Do represents the initial damage threshold, L(D) is the increment of damage threshold, and D is the scalar variable that represents overall damage. During the process of plastic deformation and damage, the power of dissipation G D can be modified from results of Voyiadjis and Kattan, (1990) (7-191) In order to obtain the actual values of the parameters {a}, {D}, w p , and D, one needs to solve an extremization problem, i.e. the power of dissipation G D is to be extremized subject to two constraints, namely, F( {S}, {E'}w p , {D}) = 0 and G({ a*}, L) = O. Using the method used in the calculus of multi variable functions , it is better to introduce two Lagrange multipliers dAl, and dA2 , and to form a function P such that (7-192) The problem now reduces to that of extremum on the function P. Therefore,
dP
dP
it needs to employ the necessary conditions d{ a} = 0 and dL = 0 to obtain p
{dcij}
+ {dDij} -
dF
dG
dAld{aij} - dA2d{ai) = 0
dG -dO - dA2 dL = 0
(7-193) (7-194)
dG Consequently, from Eq.(7-189) we may obtain dL = -1 and substituting this
into Eq.(7-194) gives dA2 =dD. Thus the factor dA2 describes the evolution of the overall damage parameter D which will be determined shortly. Using
522
7 Anisotropic Elasto-plastic Damage Mechanics
Eq.(7-194) and assuming that damage and plastic deformation are two independent processes, the following two incremental equations for the plastic strain and damage tensor can be expressed
{deijp} = dAl d{dF a ij } - {dDij }
(7-195)
de
(7-196)
= - dO d{a ij }
The Eq.(7-195) is the associated flow rule for the plastic strain introduced earlier in Eq.(7-167), while Eq.(7-196) is the evolution of the damage tensor. It should be noted that dAl is exactly the same as the multiplier dAp used earlier. However, it is still necessary to obtain explicit expressions for the multipliers dAp and dD. The derivational expression of dAp will be studied in the next section when the inelastic constitutive model has been discussed. The procedure to derive the expression of dD can be done by invoking the consistency condition d e( {a },{ D} ,L) = O. Therefore, we obtain
de ) ( d{a ) i
T
{daij}
dF
(
+ d{D ij }
)
T
de
{dDij } + dL dL
=0
(7-197)
de
Substituting {dD} from Eq.(7-196) along with dL
dL dL = dD( dD) gives
dD
-
=
(dt:
})T{da'J}
'J
dL dD
= 1 and
(de d{D pq }
)T
de d{a pq }
(7-198)
Finally, by substituting Eq.(7-198) into Eq.(7-196) , we obtain the general evolution equation for the damage t ensor {D} as follows.
[~(~rJ {M,,} dL dD
(
de d{D pq }
)T
de d{a pq }
(7-199)
The evolutional Eq.(7-199) will be incorporated in the constitutive model in the next two sections. It will also be applied to the derivation of the elastoplastic stiffness tensor. It should be pointed out that Eq.(7-199) is based on the damage criterion of Eq.(7-189) which is applicable to anisotropic damage.
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
523
However , using the form for [J] given in Eq.(7-190) restricts the formulation to isotropy.
7.8.3.2 Co-rotational Plastic Deformation of Damaged Materials In the analysis of finite strain plasticity, one needs to define an appropriate co-rotational stress increment that is objective and frame-indifferent. Detailed discussions of this type of stress increments are available in the papers of Voyiadjis et al. [7-26 rv 27, 7-29 rv 31]. The co-rotational stress increment to be adopted in this model is given for {da A } in the following form: ,
,
{daij } = {daij } - d [Pip]{apj}
+ {aiq}
T
'
d [Pqj ]
(7-200)
where the modified spin tensor [P'] is given by [7-29] as follows. (7-201) In Eq.(7-201), [8 ] is the material spin tensor, which is the antisymmetric part of the velocity gradient and fw is an influence scalar function to be determined. The effect of f won the evolution of the stress and backstress was discussed in detail by [7-31]. The co-rotational increment do; ' has a similar expression as that in Eq.(7-200) keeping in mind that the modified spin tensor [P'] remains the same in both equations. The yield function F to be used in this model is given by Eq.(7-163) with both isotropic hardening and kinematic hardening. The Isotropic hardening is described by the evolution of the plastic work as given earlier by Eq.(7-164) , while the kinematic hardening is given by Eq.(7-165). Most of the necessary plasticity equations were already provided in subsections 7.8.1rv7.8.2 and the only one remaining is the derivation of the constitutive equation. Substituting dAp from Eq.(7-177) into Eq.(7-175) derives the general inelastic constitutive equation in the configuration C as follows.
{dakl} = [D7Jkl]{ dCkl}
(7-202)
where the elasto-plastic stiffness tensor [DeP] is given by
(7-203) The next step is to use the transformation equations developed in the previous sections in order to obtain a constitutive equation in the configuration C similar to Eq.(7-202).
524
7 Anisotropic Elasto-plastic Damage Mechanics
7.8.3.3 Coupling of Damage and Plastic Deformations In this section, the transformational equations developed in subsections 7.8.1 and 7.8.2 are used in the constitutive model provided in the previous section in order to transform the inelastic constitutive Eq.(7-202) in the configuration C, to set up a general constitutive equation in the configuration C that accounts for both damage and plastic deformation. Using Eq.(7-162) and taking its derivative, the transformation equation for {c-*e} is obtained as follows. {dc-;;} =
d [tliij'~ln] T {c-:'nn} + [tliij~l( {dc-%d
(7-204)
where d[tli]-T is obtained by taking the derivative of the identity [tli]T[tlirT = [I ] and noting that d [I ]=O. Thus, we have
_l ]T d [tlik1mn
- 1 ]T = - [_ tliijk1l ]T d [tliijpq ]T[tlipqmn
(7-205)
The derivative d[tli] can be obtained by using the chain rule as follows: (7-206) Also, the component of the co-rotational derivative d[tli' ] may be used defined by the following Lie derivative given by a general mathematical hand book.
d [tlii~mn] = d[tliijmn ]- [tlipjmn] d [Pi~]- [tliiqmn ]d[P;q ]- [tliijrn] d [P~r]- [tliijms ] d [P~s ] (7-207) The transformation Eq.(7-204) for the effective elastic strain increment tensor {dc-*e} represents a nonlinear relation. A similar nonlinear transformation Eq.(7-186) was previously derived for the effective plastic strain {dc-*P}. These two equations will be combined together to be used in the derivation of the constitutive model. Now we are ready to derive the inelastic constitutive relation in the configuration C. Starting from the constitutive Eq.(7-202) and substituting the effective transformation for {O";j} and {dc-k/ } respectively, along with Eqs.(7204) and (7-186), we obtain
+ [tliijpq]{dO"pq} + [ Zklmn]{dc-~n} + {dZ k1 })
d[tliijmn]{O"mn}
= [D:)kl ]([tliki;m( {dc-~m }
+ d [tliki~q(]{C-~q}
(7-208) Further substituting d[tli] and d[tli]T from Eqs.(7-205) and (7-206), respectively, and employing {dc-P } from Eq.(7-154) and a similar equation of {dc-e} from (7-174) , i.e. {dc-ij} = [D;jkl ]- l{dO"kl} into Eq.(7-208), the resultant expression can be carried out as
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
525
d1~~:}] {dDpq}{O'mn} + [tJiijpq]{dO'pq} = [D:Jktl( [tJikl~n]T [Dpqmnrl {dO'pq} -
[tJi;;;~l( ~f~::; {dDmn}[tJiuvpq ][Dcdmnr 1{O'cd}
+ [Zklmn]{dcmn } - [Zklmn][Dabmnr 1{dO'ab} + {dz kl })
(7-209) Finally, after substituting {dD} from Eq.(7-199) into Eq. (7-209) , {dO'} can be solved in t erms of {dc }. Under several algebraic manipulations, the desired inelastic constitutive relation in the configuration C is obtained as follows. (7-210) where the effective elasto-plastic stiffness tensor [D eP*] nand the additional tensor {dg} which is comparable to the plastic relaxation stress introduced by Simo and Ju [7-33] are given by
]-1[DeP pqmn ] [Z mnkl ]
(7-211)
{d9ij } = [S;:ij rl[D~~mn]{dzmn }
(7-212)
p *] [Deijkl -
[;:::*p
~pq ij
where the fourth-order tensor [S*P] is given by
d [tJiijmn] ~{O' }~ d{D uv } d{O'u v} mn d{O'pq} =[tJiijpq ]+ dL (dG )T dG dD d{D ab } d{O'ab} - [D:Jktl
([tJikl~n]T [D;qmnrl -
[Zklmn][D;qmnr 1
dG dG - 1 T d [tJi;t~v]T d{O'mn} d{O'pq} - 1 T -1 T ) - [tJigtktl d{Dmn } dL _ ( dG )T ~ [tJiuvcd ] [tJirscd ] {O'rs} dD
d{D ab }
d{O'ab}
(7-213) The effective elasto-plastic stiffness tensor [Dep *] in Eq.(7-211) is the stiffness t ensor including the effects of damage and plastic deformation . It is derived in the configuration of the deformed and damaged body. Eqs.(7-211) and (212) can now be used in finite element analysis. However, it should be noted that the constitutive relation in Eq.(7-21O) represents a nonlinear transformation that makes the numerical implementation of this model impractical. This is due to the additional term {dg ij }, which can be considered as the
526
7 Anisotropic Elasto-plastic Damage Mechanics
same residual stress due to the damaging process. Nevertheless, the constitutive equation becomes linear provided by {dg ij } = O. This is possible only when the term ({O"mn } - {Emn })[.~ijrr] vanishes as seen in Eqs.(7-212) and (7-188) and therefore (7-214) Upon investigation of the nonlinear constitutive Eq.(7-21O), it is seen that the extra term {dg ij } is due to the linear transformation of the effective stress {O"*} and {O"}. It was shown in Eq.(7-139) that this transformation leads to a nonlinear relation between {S* } and {S}. Voyiadjis et al. [7-5, 7-26] show that a linear constitutive equation similar to Eq.(7-214) can be obtained if a linear transformation is assumed between the deviatoric stresses {S* } and {S} in the form {S7j } = [iJlijkl ]{Skd For completeness, one can obtain an identity that may be helpful in the numerical calculations. This is done by using the plastic volumetric incompressibility condition (which results directly from Eq. (7-168)) (7-215) in the configuration C. Eq.(7-21 5) is commonly used in metal plasticity without damage [7-5, 7-29]. Using Eq.(7-168) along with the condition of Eq.(7215) , a useful identity is obtained: (7-216) Eq.(7-216) is consistent with the previous conclusion of Eq.(7-142) since it was shown earlier that [Jt.rrkd = O. In finite element calculations the critical state of damage is reached when the overall parameter [l reaches a critical value called [l er at least in one of the elements. This value determines the initiation of micro-cracks and other damaging defects. Alternatively, we can assign several critical values [l~~), [l~;), etc. for different damage effects. In order to determine these critical values, which may be considered as material parameters, a series of uniaxial extension tests are to be performed on tensile specimens and the stress-strain curves drawn. In order to determine [l~V (the value of [l at which damage initiation starts for a particular damage process "i th" ), the tensile specimen has to be sectioned at each load increment. The cross-section is to be examined for any cracks or cavities. The load step when cracks first appear in terms of the strain C l is to be recorded and compared with the graph of [l vs C l. The corresponding value of [l obtained in this way will be taken to be the critical value for [l er ' This value has been used in the finite element analysis of more complicated problems. For more details see the papers presented by Chow and Wang [7-34] and Voyiadjis et al. [7-5,7-29].
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
527
7.8.4 Application of Anisotropic Gurson Plastic Damage Model to Void Growth Gurson [7-21 ] proposed a yield function F({cr},w) for a porous solid with a randomly distributed volume fraction of voids. Gurson's model was used later ([7-22], [7-28]) to study necking and failure of damaged solids. Tvergaard et al. [7-22, 7-35] modified Gurson's yield function in order to account for incremental sensitivity and necking instabilities in plastically deforming solids. The modified yield function is used here in the following form (which includes kinematic hardening)
F = ({Sij} - {E'ij})T({Sij }- {E'ij }) - 2qICI]WCosh
(crk~) - cr](1 +Q2w2) = 0 2crf
(7-217) where cr f is the yield strengt h of the matrix material and Ql and Q2 are material parameters introduced by Tvergaard [7-35] to improve agreement between Gurson's model and other results. In Eq.(7-217) the variable w denotes the void volume fraction in the damaged material. In Gurson's model damage is characterized by void growth only. The void growth is described by the increment of change of w given by Voyiadjis, [7-30, 7-35]
dw = (1 - v )dE~k
(7-218)
In Gurson's model it is assumed that the voids remain spherical in shape through the whole process of deformation and damage. The change of shape of voids, their coalescence and nucleation of new voids are ignored in the model. Eq.(7-218) implies also that the plastic volumetric change dEkk, does not vanish for a material with voids. In the following it is shown how the proposed model outlined in t he previous sections of this chapter can be used to obtain the damage effective tensor [Ili] as applied to Gurson's yield function. It is also shown how certain expressions can be derived for the parameters ql and q2 in a consistent manner. That firstly starts with the yield function F in the configuration C. Therefore, use Eq.(7-163) in the form
(7-219)
cw;
in which the term was dropped since the isotropic hardening is not displayed by Gurson's function. Using the transformation Eqs.(7-147), (7-153) , and (7-154) and noting that crJ = 2cr~2 /3, Eq.(7-219) becomes
(7-220)
528
7 Anisotropic Elasto-plastic Damage Mechanics
It is noticed that Eq.(7-219) corresponds exactly to Gurson's function ofEq.(7217) with W = O. Using Eq.(7-149) to transform the total stresses in Eq.(7-220) into deviatoric stresses, one obtains
F
=({Sij} - {L~J) T [Hijkl l ({Skd - {L~l}) + (O"mm - L nnf {Oij}T [Hijk~{Okd - O"J (7-221)
or
2 2 +1 gHmnmn(O"pp - O"qq) - O"f
Eq.(7-221) represents the yield function F in the configuration C, which can now be compared to Gurson's yield function of Eq.(7-217). Thus, upon comparing Eq.(7-217) with Eq.(7-221), it is clear that the deviatoric parts of the two functions must be equal. Therefore, we obtain
({Sij } -
{L~j }) T[Hijkl l ({Skd - {L~l}) =
({Spq} -
{L~J) T ({Spq} - {L~J)
(7-222)
On the other hand , upon equating the remaining parts of the two functions , one obtains
(0"
1 2 2 2 -Hmnmn(O"pp - O"qq) 2 = 2qlO"fwcosh - kk ) - Q20"fw 9 20"f
(7-223)
The problem is now reduced to manipulating Eqs.(7-222) and (7-223). Rewriting Eq.(7-222) in the following form
({Sij } -
{L~J) T ( [Hijkt] -
[{Oik HOlj}TJ) ({Sij } -
{L~j }) = 0
(7-224) This concludes that the tensor [H l is constant for Gurson's model and can be expressed as follows: (7-225) It is clear that the deviatoric part of Gurson's yield function does not display any damage characteristics as given by Eq.(7-217). This is further
7.8 Coupled Damage and Plasticity in General Effective Tensor Models
529
supported by Eq.(7-225) where the damage effect tensor is independent of the damage variable {ill. Next, upon considering Eq.(7-225), it obtains that the tensor becomes a scalar constant as Hmmnn = 3. Substituting this value into Eq.(7-223) yields
~ (O"pp -
LqJ
2
= [2 q1 cosh
(;;~)
- q2W] wO"J
(7-226)
Eq.(7-226) must be satisfied for a possible relationship between Gurson's model and the proposal model. Eq.(7-226), as it stands, does not seem to merit an explicit relationship between parameters ql, Q2, and w. This is due to the presence of the term "cosh" function on the right-hand side. Therefore, it is clear that we cannot proceed further without making some assumptions. In particular, two assumptions are to be employed. The first assumption is valid for small values of (O"kk) , where only the first two terms in the "cosh"
20"f
series expansion are considered cosh (O"kk) ;::::; 1 + O"kk
20"f
80"J
(7-227)
The second assumption concerns the term Lqq which appears in the Eq.(788). For the following to be valid , it needs to consider a modified Gurson yield function where the volumetric stress O"kk is replaced by (O"kk- Lqq). Therefore, upon incorporating the above two assumptions into Eq.(7-226), it gives
(7-228) It is clear from Eq.(7-228) that the following two expressions of ql and q2 in terms of W need to be satisfied ql
= 4j(3w) ,
(7-229)
The relations in Eq.(7-229) ' represent variable expressions of parameters ql and q2 in terms of the void volume fraction , in contrast to the constant values. The relations (7-229) are consistently derived and although they are approximate, in the mentioned opinion, they form a basis for more sophisticated expressions. As it stands, Gurson's function of Eq.(7-217) is a modified version of the general Gurson's function containing the term cosh instead of cosh
[(;;~) ]
[(O"~;~)q)]
that is used in the derivation of Eq.(7-228). This
point should be pursued and the proposed modified Gurson function explored further.
530
7 Anisotropic Elasto-plastic Damage Mechanics
7.8 .5 Cor otational E ffective Spin Tensor In this section, a formal derivation is presented for the transformation equation of the modified spin tensor that is used in the co-rotational incremental equations. In the configuration C, the co-rotational derivative of the effective stress tensor is given by (7-230) where [dP'*] is the effective modified spin t ensor. The problem now reduces to finding a relation between [dP"] and [dP"*] This should keep in mind , however, that Eq.(7-230) is valid only when a Cartesian coordinate syst em is adopted. The same remark applies to Eq.(7-200) in the configuration C. In order to derive the required relation, lets us firstly start with the transformation {CTtj} = [iJlijkl ]{CTkd. Taking the co-rotational derivative of this equation and rearranging the t erms, it becomes (7-231) Substituting {dCT:n from Eq.(7-230) int o Eq.(7-231) and using the material derivative {dCTtj} = d [iJlij kl]{ CT kl} + [iJlijpq ]{ dCTpq}, the derivative det ail list s below
{dCT~d = [iJlij~l ] ({dCT;j} - [dPi~ ]{CT;j}
+ {CT;j}[dPi~*]-
{dCT~d = ( [iJlij~l ][diJIijrs]{CTrs } + [iJlij~Ll[iJlijpq]{dCTpq}) + ( [iJlij~Ll {CT;j][dPi~*] )
( [iJlij~Ll[dPi~ ]{CT;j})
( [iJlij~l ][diJIi~mn]{CTmn}) = [iJlijkl ]([diJIij mn] - [diJIij mn ]) { CT mn } + {dCTkd - [iJlij~l ][dPi';][iJlpjab]{ CTab} + [iJlij~l ][iJliqcd]{ CTcd}[dP;;J
" -1
{dCTkd
-
[diJIi~mn]{CTmn})
-
"
finely we obtain
(7-232) Comparing the two co-rotational derivatives appearing in Eq. (7-200) {dCT: j } + {CTiq}[dP;j ] and Eq.(7-232) , and after performing some t edious algebraic manipulations, we can finally obtain a relation between [dP"] and [dP"*] in the following form
{dCT ij } - [dPi~]{CTpj}
[dP';n] = [Amnpq ] [dP;q ] + [dBmn] where the t ensors [Amnpq ] and [dBmn] are given by
(7-233)
7.9 Elasto-plastic Damage for Finite-Strain
[dBmnl = -[C;"nkl rl[tli~ll l ( [dtlipqrsl - [dtli;qrs])[o-rsl
531
(7-235)
and the tensor [C;"'nkll is given by (7-236)
With the availability of the transformation equation of the spin tensor, the theory presented in this section is now complete.
7.9 Elasto-plastic Damage for Finite-Strain In this section, the damage for finite strain elasto-plastic deformation is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. The proposed approach provides a general description of damage applicable to finite strains. This is accomplished by directly considering the kinematics of the deformation field and, furthermore, noting that it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. In this section the damage is described in both the elastic domain and the plastic domain using the fourth-order damage effect tensor, which is a function of the second-order damage tensor. Two kinds of secondorder damage tensor representations are used with respect to two reference configurations. The finite elasto-plastic deformation behavior with damage is also viewed here within the framework of thermodynamics with internal state variables. 7.9.1 Configuration of Deformation and Damage
A continuous body in an initial undeformed configuration that consists of the material volume Vo is denoted by Co , while the elasto-plastic damaged deformed configuration at time t after the body is subjected to a set of external agencies is denoted by Ct. The corresponding material volume at time tis denoted by Vi, Upon elastic unloading from the configuration C t an intermediate stress-free configuration is denoted by Cdp' In the framework of continuum damage mechanics, a number of fictitious configurations, based on the effective stress concept, are assumed that are obtained by fictitiously removing all the damage that the body has undergone. Thus, the fictitious configuration of the body denoted by C t is obtained from C t by fictitiously removing all the damage that the body has undergone at Ct. Also, the fictitious configuration denoted by C p is assumed, which is obtained from Cdp by fictitiously removing all the damage that the body has undergone at Cdp' While the
532
7 Anisotropic Elasto-plastic Damage Mechanics
configuration C p is the intermediate configuration upon unloading from the configuration C t , the initial undeformed body may have a pre-existing damage state. The initial fictitious effective configuration denoted by Co is defined by removing the initial damage from the initial undeformed configuration of the body. In the case of no initial damage existing in the undeformed body, the initial fictitious effective configuration is identical to the initial un deformed configuration. Both formulation of Cartesian tensors and matrixes are used in this section and the tensorial index notation is employed in all equations. The matrix form of t ensors used in texts and formulations are denoted by the matrix name with subscriptions in a square bracket. However, some superscripts in the notation do not indicate tensorial indices but merely stand for the corresponding deformation configurations such as "e" for elastic, "p" for plastic, and "d" for damage, etc. The below barred and tilted notations refer to the fictitious effective configurations.
7.9.2 Description of Damage Tensors The damage state can be described using an even order t ensor ([7-38], [7-37], [7-36]) pointed out that even for isotropic damage one should employ a damage tensor (not a scalar damage variable) to characterize the state of damage in materials. However, the damage generally is anisotropic due to the external agency condition or the material nature itself. Although the fourth-order damage tensor can be used directly as a linear transformation tensor to define the effective stress tensor, it is not easy to characterize physically the fourthorder damage tensor compared with the second-order damage tensor. In this section, the damage is considered as a symmetric second-order tensor. However, the damage tensor for finite elasto-plastic deformation can be defined in two reference systems [7-39]. The first one is the damage tensor denoted by [il] representing the damage state with respect to the current damage configuration C t Another one is denoted by [il/\ ] and is representing the damage state with respect to the elastically unloaded damage configuration C dp ' Both are given by Murakami [7-40] as follows , 3
sum over k)
(7-237)
ilij = ~ ilkmi mj (no sum over k)
(7-238)
n Jtij =
'""' A k k ~ Jtkni nj A
A
( no
k=l ,
3 ,"", "AkAk
k=l
where {iLk} and {rhk} are eigenvectors corresponding to the eigen-values Dk and D~ of the damage tensors [ill and [il/\] respectively. Eqs.(7-237) and (7-238) can be written alternately as follows (7-239)
7.9 Elasto-plastic Damage for Finite-Strain
[l:j = eikej/2~l
533
(7-240)
The damage tensors in the coordinate system that coincides with the three orthogonal principal directions of the damage tensors n kl and n~l' in Eqs.(7239) and (7-240), are obviously of diagonal form and are given by (7-241)
[~~o ~; n;~ 1
(7-242)
0
and the second-order transformation tensors [b] and [e] are given by
[b ij ] = [:: :: ::], [eij ] = [:: :: ::]
nr
n~
n~
mr
m~
(7-243)
m~
These proper orthogonal transformation tensors require that (7-244) where 6ij is the Kronecker delta and the determinants of the matrices [b] and [e] are given by
Ibl = lei = 1
(7-245)
The relation between the damage tensors [[l] and [[l/\] will be shown in subsection 7.9.4. 7.9.3 Corresponding Damage Effective Tensor for Symmetrized Model II In a general state of deformation and damage, the effective stress tensor {CY*} is related to the Cauchy stress tensor {CY} by {CYij} = [lliijkl]{CYkL} which is a linear transformation shown in previous chapters. Some linear transformation tensors called the symmetrized damage effective tensors are applied in much literature using symmetrization methods (detail to be seen in subsections 5.7 and 5.9). In this section, the fourth-order damage effect tensor defined in Eq.(5-130) and represented by tP[j/i).. = (6ik-[lik)1/2(6jl-[ljl)1/2 will be used because of its geometrical symmetrization of the effective stress. However, it is very difficult to obtain the explicit representation of (6ik - [lik) 1/2. The explicit representation of the fourth order damage effective tensor [[tP] using
534
7 Anisotropic Elasto-plastic Damage Mechanics
the second-order damage tensor [S?] is more convenient in the implementation of the constitutive modeling of damage mechanics. Therefore, the damage effective tensor [[tP"] applied in this section should be obtained using the coordinate transformation of the principal damage direction coordinate system. Thus, the applied fourth-order damage effective tensor [tP"] can be written as follows, -
= bmi bnjbpkbqiJ!mnpq -~
iJ!ikj l
(7-246)
where [iJ!~] is a fourth-order damage effect tensor with reference to the principal damage direction coordinate system. The fourth-order damage effective tensor [iJ! ~] an be written as follows, -
~
iJ!mnpq
= a mpa nq ~
~
(7-247)
where the second order tensor [a~] in the principal damage direction coordinate system is given by 1
0
)1 - Dl ~
[aij]
~ -1/2
= [(Oij - Dij )
]
=
0
0
1
0
)1 - D3
0
0
(7-248)
1 )1 - D3
Substituting Eq.(7-247) into Eq.(7-246), the following relation can be obtained (7-249a) or -
iJ!ijkl
= bmi bnjbpkbqla mpa nq = aikajl ~
~
(7-249b)
Using Eq.(7-249), a second-order tensor [a] is defined as follows, [aik]
= [bmi]T [a~ mp][bpk]
(7-250)
The matrix form of Eq.(7-18) is given as follows , [a] = [b] T [a~][b] ..1>.u..".u..+~+~ ~+~+~~+~+~ v',-D, v"-D 2 v" - D2 v',-D, v"-D 2 v" - D2 v',-D, v" - D2 v" - D2 ~+~+~~+~+~ ~+~+~ v',-D, v"-D v" - D v',-D, v"-D v" - D v',-D, v" - D v" - D
2
2
2
2
2
2
~+~+~~+~+~ ~+~+~ v',-D, v"-D 2 v" - D2 v',-D, v"-D 2 v" - D2 v',-D, v" - D2 v" - D2
(7-251)
7.9 Elasto-plastic Damage for Finite-Strain
535
7.9.4 Elasto-plastic Damage Behavior with Finite Strains 7.9.4.1 Deformation Gradient and Finite Strain
A position of a particle in configuration Co at to is denoted by {X} and can be defined at its corresponding position in configuration C t at t , denoted by {x }. Furthermore, assuming that the deformation is smooth regardless of damage, one can assume a one-to-one mapping such that
{x} = {Xk({X},t)} or Xk = Xk({X},t)
(7-252)
or (7-253)
The corresponding deformation gradient is expressed as follows:
d{x} dX i [F ] = d{X} or Fij = dX j
(7-254)
and the change in the squared length of a material element {dX} is used as a measure of deformation such that (ds)2 - (dS)2 = {dx}T{dx} - {dX}T{dX} = 2({dXi }f [ci j]{dXj} (7-255) or (7-256)
where (ds)2 and (dS)2 are the squared lengths of the material elements in the deformation with damage configuration C t , and the initial undeformed configuration Co , respectively, while [10] and are the Lagrangian and Eulerian strain tensors, respectively, and [E] are given by 1
1
T
[10] = 2{ [F ] [F ]- [I]} = 2( [G]- [ID or Cij
[E] = Eij =
1
1
T
(7-257)
= 2(Fik ) (Fik - 6ij) = 2(G ij - 6ij )
~{ [I] - [F rT[Fr l} = ~( [I] - [E rl ) or 2 1
1
2
(7-258)
2(6ij - FikT Fi,/ ) = 2(6ij - Ei/ )
where [G] and [E] are the right Cauchy-Green and the left Cauchy-Green tensors, respectively.
536
7 Anisotropic Elasto-plastic Damage Mechanics
The velocity vector field in the current configuration at time t is given by
{V-} = d{xd ,
(7-259)
dt
Thus the velocity gradient in the current configuration at time t is given by
[L] = d{v} = ~ ( d{ x } ) = ~ (d{ X} d{X}) = ~ ( [F ] [FrT ) dt d{x} T dt d{X} d{x}T dt d{x}
1:]
= d L
[FrT
- dVi _
'J -
~
dx - - dt J
+ [F]
d [~~-T
= [d] + [w]
(dX i dX k ) _ ~ dXk dx - dt J
= dij + Wij
(F,k F-kj
1) _
-
F
dFik F- 1 dFk-:/ dt kj + ,k dt (7-260)
where d [F] indicates the material time derivative and where [d] and [w] are dt the rate of deformation (stretching) and the vorticity respectively. The rate of deformation [d] is equal to the symmetric part of the velocity gradient [LJ, while the vorticity [w] is the antisymmetric part of the velocity gradient [L ] such that 1
[d] =
~
([F] + [F] T) or dij = "2(Fij
+ Fji )
(7-261)
[w] =
~
([F] - [F]T ) or wij = "2(Fij - Fji )
(7-262)
1
Strain rate measures are obtained by differentiating Eqs.(7-255) and (7256) such that
or
7.9 Elasto-plastic Damage for Finite-Strain
537
(7-264) Thus
(7-265)
By comparing Eq. (7-263) and (7-265), it obtains the rate of the Lagrangian strain that is the projection of [d] onto the reference frame as follows , (7-266) while the deformation rate [d] is equal to the Cotter-Rivin connected rate of the Eulerian strain as follows:
The convective derivative shown in Eq.(7-267) can also be interpreted as the Lie derivative of the Eulerian Strain (Lubarda and Krajcinovic, [7-41]). 7.9.4.2 Damage Configurations of Deformation Gradient and Finite Strain
A schematic drawing representing the kinematics of elasto-plastic damage deformation is shown in Fig.7-13. In the figure, Co is the initial undeformed configuration of the body which may have initial damage in the material. C t represents the current elasto-plastically deformed and damaged configuration of the body. The configuration of Co represents the initial configuration of the body that is obtained by fictitiously removing the initial damage from
538
7 Anisotropic Elasto-plastic Damage Mechanics
the Co configuration. If the initial configuration is undamaged , there is consequently no difference in the configurations Co and Co. The configuration C t is obtained by fictitiously removing the damage from the configuration Ct. Configuration Cdp is an intermediate configuration upon elastic unloading. In the most general case of large deformation processes, damage may be involved due to void and micro-crack development because of external agencies. Although damage at the micro-level is a material discontinuity, damage can be considered as an irreversible deformation process in the framework of continuum damage mechanics. Furthermore, one assumes that upon unloading from the elasto-plastic damage state, the elastic part of the deformation can be completely recovered while no additional plastic deformation and damage takes place. Thus, upon unloading the elasto-plastic damage deformed body from the current configuration, C t will elastically unload to an intermediate stress free configuration denoted by C dp as shown in Fig.7-1 3. Although the damage process is an irreversible deformation thermodynamically, the deformation due to damage itself can, however, be partially or completely recovered upon unloading due to closure of micro-cracks or contraction of micro-voids. Nevertheless, recovery of damage deformation does not mean the healing of damage. The deformation gradient tensor and the Green deformation tensor of the elasto-plastic damage deformation can be obtained through Path I, Path II, or Path III as shown in Fig.7-13. Consider Path I - the deformation gradient referred to the un deformed configuration Co, denoted by [P] and is polarly decomposed into the elastic deformation gradient denoted by [pe] and the damage-plastic deformation gradient denoted by [pdp] such that (7-268)
c.
__~Cd-,-P___ . ___ . ___ .~ . _,_ . _,_ .
I
" ~: !I "
Path! : - Path II: - - - - Path 111 : - '-- '-
IF
6'::----!~ -----c:0 ----~ ------cb I
i
...
I
Fig. 7-13 Schematic representation of elasto-plastic damage deformation configurations
7.9 Elasto-plastic Damage for Finite-Strain
539
The elastic deformation gradient is given by
[pe] = d{ x } or pe = dXi d{ x dp } 'J dx dp
(7-269)
J
The corresponding damage-plastic deformation gradient is given by
dp pdp = dxfP [pdp] = d{ x } d{X} or 'J dXj
(7-270)
The right Cauchy-Green deformation tensor [G] is given by (7-271) The finite deformation damage models emphasize that "added flexibility", due to the existence of micro-cracks or micro-voids, is already embedded in the deformation gradient implicitly. Murakami [7-40] presented the damage deformation using the second-order damage tensor. However, the lack of an explicit formulation for finite deformation with damage leads to failure in obtaining an explicit derivation of the damage deformation. Although most finite strain elasto-plastic deformation processes involve damage such as micro-voids, nucleation and micro-crack development due to external agencies, only the elastic and plastic deformation processes, however, are considered due to the complexity in the development of damage deformation. In this section, the kinematics of damage will be explicitly characterized based on continuum damage mechanics. The elastic deformation gradient corresponds to elastic stretching and rigid body rotations due to both internal and external constraints. The plastic deformation gradient arises from purely irreversible processes due to dislocations in the material. Damage may initiate and evolve in both the elastic and plastic deformation processes. In particular, damage in the elastic deformation state is termed elastic damage, which is the case for most brittle materials, while damage in the plastic deformation state is termed plastic damage, which is mainly for ductile materials. Additional deformation due to damage consists of damage itself with additional deformation due to elastic and plastic stiffness. In this section, kinematics of damage deformation is completely described for both damage and the coupling of damage with elasto-plastic deformation. The total Lagrangian strain tensor is expressed as follows ,
[E] =
~ ([FdP([Fd P]- [I ]) + ~ [PdP(([pe]T[pe]_ [1]) [pd p]
= [Edp] + [pdp([Ee][pd p] = [E dp ] + [Ee] or
(7-272a)
540
7 Anisotropic Elasto-plastic Damage Mechanics _ 1 (Fdp Fdp
Cij -
2"
ki
kj -
s;) + 2"1 Fdp(Fe Fe mi km kn
Vij
-
S;
Vm n
)Fdp _ nj -
dp
Cij
+ Fdp mi
e
Emn
Fdp _ nj -
dp
Cij
+ Cije
(7-272b) where [e dp ] and [eel are the Lagrangian damage-plastic strain tensor and the Lagrangian elastic strain tensor respectively, measured with respect to the reference configuration Co , is the Lagrangian elastic strain tensor measured with respect to the intermediate configuration C dp ' Similarly, the Eulerian strains corresponding to deformation gradients [Fe] and [Fdp] are given
dp = -1(5:v·· - F dpl dp- 1 ) or kt· Fk J. 2 tJ
e·· 'J
e
e·· tJ
1 = -1(5: - Fket·- Fke-J· ') 2 v·· tJ
(7-273)
The Eulerian strain tensor can be expressed as follows,
(7-274) The strain tensor [e dp] is referred to the intermediat e configuration Cdp , while the train tensors [EJ, [Ee] and [Edp] are defined relative to the current configuration as a reference. The relationship between the Lagrangian and Eulerian strains is obtained directly in the form of (7-275) The change in the squared length of a material element deformed elastically from C t to C dp is given by
(ds)2 - (dS)2 = {dx} T {dx} - {dxdp{{dx dp } = {dx i}T {dx i} - {dx idP} T {dx idp} = 2{dXdT [eeij]{dXj}
(7-276)
However, the change in the squared length of a material element due to damage and plastic deform ation from Cdp to Co is given by (7-277) The kinematics of finite strain elasto-plastic deformation including damage is completely described in Path 1. In order to describe the kinematics of damage and plastic deformation, the deformation gradient given by Eq.(7-268) may be further decomposed into
[F ] = [Fe] [Fd ][FP ] or Fij = FtkF~mF~j
(7-278)
However, it is very difficult to characterize physically only the kinematics of deformation due to damage in spite of its obvious physical phenomena.
7.9 Elasto-plastic Damage for Finite-Strain
541
The damage, however, may be defined through the effective stress concept. Similarly, the kinematics of damage can be described using the kinematic configuration. Considering Path II, the deformation gradient can be alternatively expressed as follows , (7-279) where [Fd*] is the fictitious damage deformation gradient from configuration C t to C t and is given by
[Fd*] = d{ x } or F d* = dX i d{x* } tJ dX;
(7-280)
The elastic deform ation gradient in the effective configuration is given by
[Fn] = d{ X* } or F e* = dX; d{ x p *} tJ dXr
(7-281)
The corresponding plastic deformation gradient in the effective configuration is given by :\ p*
[FP*] = d{x P* } FP*=~ d{X*} or tJ dX*
(7-282)
J
while the fictitious initial damage deformation gradient from configuration Co to Co is given by
[Fdo*] = d{X* } or F do * = dX; tJ dX. d{X} J
(7-283)
Similar to Path I, the right Cauchy Green deformation tensor, [G], is given by
(7-284) The Lagrangian damage strain tensor measured with respect to the fictitious configuration C t is given by
(7-285) and the corresponding Lagrangian effective elastic strain tensor measured with respect to the fictitious configuration C P is given by
542
7 Anisotropic Elasto-plastic Damage Mechanics
(7-286)
The Lagrangian effective plastic strain tensor measured with respect to t he fictitious undamaged initial configuration Co is given by (7-287)
The total Lagrangian strain tensor is therefore expressed as follows:
[c:] = ~ ([FdO*]T [Fdo*] - [1]) + ~ [F dO*( ( [FP*] T [FP*]- [1]) [Fdo*]
+ ~ ([FdO*]T[Fp*]T ( [Fe*]T[F e*] - [1]) [FP*] [Fdo*] + ~ [FdO*]T[FP*]T[Fe*]T ( [Fd*]T [Fd*] - [1]) [Fe*] [FP*] [Fdo*] (7-288)
or C:ij
=21 (Fkido *Fkjdo * -
(j) ij
1 pdo* (FP* FP* +2 mi km kn
+ ~Fdo* pp*(Fn F e* _ 2 nt rn qr q s
(j
rs
(j
mn
) pdo* nj
)FP* pdo* sm mJ
d*F d* _ + ~pdo* pp* pe*(Fqr 2 nw rn qs Wt
-
(j
rs
(7-289)
e* FP* F do * )Fsm mk kJ
The Lagrangian init ial damage strain tensor measured with respect to the reference configuration Co is denoted by (7-290)
The Lagrangian plastic strain tensor measured with respect to t he reference configuration Co is denoted by (7-291)
One now defines the Lagrangian elastic strain tensor measured with respect to the reference configuration Co as follows,
[c: e *] = [F do*( [FP*]T[Ee*][FP*][F do *] or c:f; = F~f* F;Z E%::n F!~F:r (7-292) and the corresponding Lagrangian damage strain t ensor measured with respect to the reference configuration Co is given by
[c: d*] = [Fdo*] T [FP*] T[Fn] T [Ed*][Fe*][FP*][Fdo*] or = pdo*pp* pe* Ed* pe* PP*Fdo* c: d* Wt wn nk km mr rs SJ t)
(7-293)
7.9 Elasto-plastic Damage for Finite-Strain
543
The total Lagrangian strain is now given as follows through the additive decomposition of the corresponding strains (7-294)
The change in the squared length of a material element deformed due to fictitiously removing damage from C t to C t is given by
(ds)2 - (ds* )2 = {dx }T {dx } - {dx* }T {dx* } = dx jdx j - dXj* dx j*
= 2{dX;}T[c d *;j]{ dXj}
(7-295)
The change in the squared length of a material element deformed elastically from C t to C p is given by
The change in the squared length of a material element deformed plastically from Co to C p is then given by
while the change in the squared length of a material element deformed due to fictitious removing of the initial damage from Co to Co is given by (7-298)
Finally, Path III gives the deformation gradient as follows, (7-299)
where [FdA] is the fictitious damage deformation gradient from configuration to C dp and is given by
C;
[F d']
__
d{ x dp } d' dX dp or F·· = - 'd{ XP' } 'J dXr
(7-300)
and the corresponding plastic deformation gradient in the effective configuration is given by
[FP'] =
~~X;} or Fl;' = ~~{
(7-301)
Similar to P ath II, the Right Cauchy Green deformation tensor [G] is given by
544
7 Anisotropic Elasto-plastic Damage Mechanics
[G] = [F do*( [FP'( [Fd'( [Fef[Fe*][Fd'][FP'][F do *] G ij
do *F P' Fd' Fe Fe Fd' FP' F do* = Fmk kq qp pi mn nl ls sj
(7-302)
The Lagrangian damage strain tensor measured with respect to the fictitious intermediate configuration C; is given by
(7-303) The total Lagrangian strain tensor is expressed as follows , * Cij = 2"1 (Fdo* ki Fdo* kj - J) ij
d + 2"1 Fdo*FP' ni Tn (Fd'F qT qs'
-
+ 2"1 FdO*(FP' mi km FP' kn
- Jmn )FdO* nj
do* sm mj TS)FP'F
s:
U
+ 2"1 Fdo*FP' wi nw Fd' Tn (Fd qT Fdqs -
(7-304)
)F d' FP' Fdo* TS 8m mk kj
s:
U
The Lagrangian damage strain tensor measured with respect to the reference configuration Co is denoted by
[c d*] = [F do*( [FP'( [Ed'][FP'][F do *] or cfj* = Fft* F:r,'k
Ec;;,n FX~F:r
(7-305) The Lagrangian elastic strain tensor measured with respect to the reference configuration Co is denoted by
e
_
Cij -
Fdo*FP'Fd' e Fd'Fp'Fdo* li kl mk Emn nq qT Tj
(7-306)
The corresponding total Lagrangian strain is now given by
The change in the squared length of a material element deformed by fictitious removal of damage from Cdp to C; is given by
The change in the squared length of a material element deformed plastically from Co to C; is then given by
7.9 Elasto-plastic Damage for Finite-Strain
545
The total Lagrangian strain tensors obtained by considering the three paths are given by Eqs.(7-272), (7-294) , and (7-307). From the equivalence of these total strains, we obtain the following explicit presentations of the kinematics of damage. With the assumption of the equivalence between the elastic strain tensors given by Eqs.(7-272) and (7-307), the damage-plastic deformation gradient given by Eq.(7-270) and the Lagrangian damage plastic strain tensor can be expressed as follows, (7-310) and (7-311) Furthermore, one obtains the following expression from Eqs.(7-294) and (7-307) as follows , (7-312) which concludes that C; and C p are the same. Substituting Eqs.(7-293), (7305), and (7-306) into Eq.(7-312), the effective Lagrangian elastic tensor is
e* _ Cij -
Fdo*FP* (d ' ki mk Emn
-
Fe* qm
e* + F ' d* Frn qm
Eqr
e Eqr
Fd')FP*Fdo* rn ns sj (7-313)
Using Eqs.(7-292) and (7-313), we can now express { E} as follows,
(7-314) This expression gives a general relation of the effective elastic strain for finite strains of elasto-plastic damage deformation. In the special case when we assume that
[E d'] _ [Fe*]T [Ed*] [Fe*] = 0 or
d' Eij
e* Emn e* = 0 Fmi d* Fnj
(7-315)
Eq.(7-314) can be reduced to the following expression
[E e*] = [Fd']T[ E e][Fd'] or
e* Eij
= Fd' ki
e Ekl
Fd' lj
(7-316)
This relation is similar to that obtained without consideration of the kinematics of damage and only utilizing the hypothesis of elastic energy equivalence. However, Eq.(7-316) in the case of finite strains is given by relation in
546
7 Anisotropic Elasto-plastic Damage Mechanics
Eq.(7-314) which cannot be obtained through the hypothesis of elastic energy equivalence. Eq.(7-315) may be valid only in some special case of small strain theory. 7.9.4.3 De formation Gradients Due to Fictitious Damage
The two fictitious deformation gradients given by Eqs.(7-280) and (7-300) may be used to define the damage t ensor in order to describe the damage behavior of solids. Since t he fictitious effective deformed configuration denoted by G t is obtained by removing damage from the real deformed configuration denoted by Gt , the differential volume of the fictitious effective deformed volumes denoted by d V t is therefore obtained as follows ,
d~* = d~
- dVd = )(1 - Sll )(l - Sl2)(1 -
Sl3)d~
(7-317)
dVt = Jd*d1/t *
(7-318)
where Vd is the volume of damage in the configuration Gt and Jd* is t ermed the Jacobian of the damage deformation which is the determinant of the fictitious damage deformation gradient. Thus, the J acobian of the damage deformation can be written as follows ,
Jd*
-IFd*1 _ J(l -
ij
-
1
(7-319)
Sll)(l - Sl2)(1 - Sl3)
The determinant of the matrix [a] in Eq.(7-251) is given by (7-320) Thus, one assumes the following relation wit hout loss of generality (7-321) Although the identity is esta blished between Jd* and Iai, this is not, however, sufficient to demonstrate the validity of Eq.(7-321). This relation is assumed here to be based on the physics of the geometrically symmetrized effective stress concept. Similarly, the fictitious damage deformation gradient [F dA] can be written as follows ,
[Fd'] = ([1]-
[Sl]) - ~
or
Fi~' = (Oij - Slij) - ~
(7-322)
Finally, assuming that {x* }= {x A } based on Eq.(7-312) , the relations between [FdA] and [Fd*] and [SlA] and [Sl] are given by
[F d'] = [F e][Fd*][F er 1 or Fi~' = Fki Ft t F lj -
1
(7-323) (7-324)
7.9 Elasto-plastic Damage for Finite-Strain
547
7.9.4.4 ecomposition of Elasto-plastic Finite Strains Coupled with Damage The kinematics of finite deformation are described here based on the polar decomposition by considering three paths as indicated in the previous section. In order to proceed further, one assumes a homogeneous state of deformation such that the completely unloaded stress-free configuration C dp has opened cracks and micro-cavities. Furthermore, one assumes that these cracks and micro-cavities can be completely closed by subjecting them to certain additional stress. The configuration that is subjected to the additional stresses is denoted by Cp and it is assumed that this configuration has only deformed plastically. The additional stress which can close all micro-cracks and microcavities is assumed as follows , (7-325) If no initial damage is assumed in the configuration Co, it can be assumed that Cp = C;. The total displacement vector {u( {X}, t)} can be decomposed in the Cartesian reference frame in the absence of rigid body displacement such that (7-326) (7-327) (7-328) (7-329) (7-330) where {xd} is a point in the intermediate unloaded configuration Cdp and {x P} is a point in the configuration C;. Recalling that {u} = {x} - {X} and using the notation
ui,j
dX
= dX2, the corresponding total Lagrangian strain tensor {E J
given by Eq.(7-257) can be written in the usual form as follows:
[]= E
~2 1
(d{U} d{ X}
(d{U}) T
+ d{ X}
d{U} (d{U}) T) =
+ d{ X} d{ X} 1
E·· 2J = 2 (U 2,J. + UJ,t. + Ut, kUk ,J.) = -2 (J2J
~2 ( [J ] + [J ]T + [J ][J ]T )
+ JJt + JkJkJ·) 2
(7-331) Substituting Eq.(7-326) into Eq.(7-331) gives
548
7 Anisotropic Elasto-plastic Damage Mechanics
[e] = [E P ] e lj
P
=E ij
+ [Ed] + [Ee] + [E pe ] + [E de ] + [E Pd ] + Eijd + Eije + Eijpe + Eijde + Eijpd
or
(7-332)
where [EP] termed the pure plastic strain is given by EP
'J
= ~2
(up.
t,J
+ up. + uP" kU~ ,J.) J,t
(7-333)
[Ed] termed the pure damage strain is given by
(7-334) [Ee] termed the pure elastic strain is given by Ee
'J
= ~2
e . + uekuek .) (u et ,J. + u J,t " ,J
(7-335)
[Ede] termed the coupled elastic-damage strain is given by
(e
de. = -1 uk ·Uk· d E·'J 2 ,' , J
+ u·dkUk e) . ,J
t,
(7-336)
[Epe] termed the coupled elastic-plastic strain is given by E·pe. = 'J
-21
(uk e ·u p pe) ,' k ,J. + U" kUk ,J.
(7-337)
and [EPd] termed the coupled plastic-damage strain is given by pd_ 1 (d E·tJ . - -2 uk ,t·U Pk ,J.
+ UP" kUkd) . ,J
(7-338)
One defines the Lagrangian elastic strain as follows , [eel = [Ee]
+ [E pe ] + [Ede]
or eL =E~j
+ EfJ + EfJ
(7-339)
the the Lagrangian damage strain as follows: [cd] = [Ed] or e1j =Efj
and the Lagrangian plastic strain as follows, (7-340)
The coupled term of elastic-damage and plastic-damage strains are linked, respectively, with the elastic and plastic strains since they directly influence the stresses acting on the body. Consequently, the total Lagrangian strain can be written as follows: (7-341)
7.9 Elasto-plastic Damage for Finite-Strain
549
The differential displacement is given by
{duJ = {Xi (t
+ dtn - {xi (tn
(7-342)
Then the corresponding differential total displacement can be decomposed into an elastic, plastic and damage parts as follows:
{du i } = {dun
+ {dun + {dut}
(7-343)
Evidently, one obtains the following decomposition of the velocity tensor field
{v({x }, tn {Vi(Xi, t)} = {Vf(Xi, t)}
+ {vt(Xi,tn + {Vf(Xi,tn
(7-344)
where {v e } is the velocity vector field due to elastic stretching and rigid body rotations, {v d } is the velocity vector field due to the damage process, and {v P } is the velocity vector field arising from the plastic deformations due to dislocation motion. According to Eqs.(7-260)rv(7-263) the gradient of the frame {x } is given by the following relation (7-345) (7-346) (7-347)
7.9.5 Thermodynamic Description of Finite Strain Damage Finite elasto-plastic deformation behavior with damage can be viewed within the framework of thermodynamics with internal state variables. The Helmholtz free energy per unit mass in an isothermal deformation process at the current state of the deformation and material damage is assumed as follows, (7-348) where U is the strain energy which is a purely reversible stored energy, while if> is the energy associated with specific micro-structural changes produced by damage and plastic yielding. Conceptionally, the energy if> is assumed to be an irreversible energy. In general, an explicit presentation of the energy if> and its rate ~~ is limited by the complexities of the internal micro-structural changes. However, only two internal variables which are associated with damage and plastic hardening, respectively, are considered in this work. For the sake of a schematic description of the above stated concepts, the uniaxial stress-strain curves shown in Fig.7-14 are used. In Fig.7-14, Eo is the initial undamaged Young's modulus, E* is the damaged (effective) Young's modulus, {E} is the second Piola-Kirchhof stress, and {E} is the Lagrangian strain. Even though
550
7 Anisotropic Elasto-plastic Damage Mechanics
these notations are for the case of a uniaxial state, they can be used in indicial tensor notation in the equations below without loss of generality. Referring to the solid curve in Fig.7-14, the total Lagrangian strain t ensor {c} is rewritten from Eq.(7-741) in vector form (7-349) where {cP } is the plastic strain tensor , {ce } is the elastic strain tensor , and {cd} is the additional strain t ensor due to damage. Comparing Eqs.(7-272) and (7-349) gives (7-350)
a
- -Elasto-plastic-damage a -I: curve - - - -Elasto-plasticea-I: curve
E'
E
Fig. 7-14 Schematic representation of elasto-plastic damage stress-strain curves for uniaxial state of stress Furthermore, the additional strain tensor due to damage can be decomposed as follows, (7-351) where { c~;} is the irrecoverable damage strain tensor due to lack of closure of the micro-cracks and micro-voids during unloading, while {ct;} is the elastic damage strain tensor due to the reduction of the elastic stiffness tensor. Thus, the purely reversible strain tensor {c E } due to unloading, can be obtained by (7-352) The strain energy U which is shown as the shaded triangular area in Fig.7-14 is assumed as follows ,
7.9 Elasto-plastic Damage for Finite-Strain
551
(7-353) where p is the specific density. Furthermore, this strain energy can be decomposed into the elastic strain energy Ue and the damage strain energy Ud as follows, (7-354) The elastic strain energy Ue is given by
Ue = 21p {ce }T[ D]{c e } or Ue = 21p cTj Dljklckl
(7-355)
and the corresponding damage strain energy Ud is given by U
d
= ~{cE}T[D*]{cE} - ~{ce }T [D]{ce } 2p
2p
or l
E
*
E
I
e
e
Ue = 2p cijDljkl Ckl - 2p cijDljklCkl
(7-356)
where [D] (i.e. , [DO]) and [D*] are matrices of the initial undamaged elastic stiffness and the damaged elastic stiffness, respectively. These matrices of stiffness are defined such that
cPU or DOkl = cPU [DO ] _ - d{ce}d{ce}T tJ dcij dckl
(7-357)
d2 U D* [D *] _ - d{cE }d{cE}T or ijkl
(7-358)
d2 U
= dcfy dc~
The damaged elastic stiffness matrix in the case of finite deformation is given by Eq.(7-249) in subsection 7.9.3 or by using Eq.(5-145) in subsection 5.7.1 as follows: or D7jkl
= 'l'..tmJn . D':nnpq'l'. pkql
(7-359)
where
['l'. ] = [tJrr
1
= [a]-T[a] or 'l'.*ij kl = tJr~kll = a-:-k1a-:-J 11 tJ t
(7-360)
The elastic damage stiffness matrix given by Eq.(7-359) is symmetric. This is in line with the classic sense of continuum mechanics which is violated by using the different hypothesis of continuum mechanics. Using the similar relation between the Lagrangian and the Eulerian strain tensors given by Eq.(7-275) , the corresponding strain energy given by Eq.(7-353) can be written as follows ,
552
7 Anisotropic Elasto-plastic Damage Mechanics
or
(7-361b) where {Efy} is the Eulerian strain vector corresponding to the Lagrangian strain vector shown in Eq.(7-352), and [K*] is termed the Eulerian elastic stiffness matrix which is given by (7-362) The second Piola-Kirchhof stress tensor {ElI} is defined as follows,
au aUe { L II } = Pa{c E } = Pa{ce}
or
The second Piola-Kirchho stress tensor stress tensor {a} as follows,
{"II} = J[F] ~
-1
II
au
aUe
L Ij = Pacfy = PacTj {2:II}
"II
(7-363)
is related to the Cauchy
_1
{a }[F] or ~Ij = JFik akmFjm
(7-364)
The first Kirchhof stress tensor { EI} is related to the Cauchy stress tensor
{a} as follows:
I
"~ ij = Ja·2J
(7-365)
The rate of Helmholtz free energy is then given as follows: dW
= dU +d4>
(7-366)
where d4> is the rate of 4> associated with the two neighboring constrained equilibrium states with two different sets of internal variables {il} and {'Y} (see subsections 3.5 and 5.5). Using Eq.(7-353) or Eq.(7-354), the rate form of the strain energy can be given as follows since d U = O.
or
7.9 Elasto-plastic Damage for Finite-Strain
553
(7-367b) and
(7-368)
and
pdUd
= ~{eE}T[dD*]{eE } + {eE} T[D*]{de E }
- {ee}T [D]{de e}
(7-369a)
_ dp ({ eE} T [D *]{ eE} _ {ee }T [D]{ ee })
2p
or (7-369b)
E e D e) - dp 2p (ED* eij ijklekl - eij ijklekl
If the deformation process is assumed to be isothermal with negligible temperature non-uniformities, the rate of the Helmholtz free energy can be written using the first law of thermodynamics (balance of energy) based on theories in Chapter 5 section 5.5 as follows , (7-370) . where T is the temperature and S
dS
= ill is the irreversible entropy produc-
tion rate. The product TS represents the energy dissipation rate associated with both the damage and plastic deformation processes. The energy of the dissipation rate is given as follows,
.
TS =
{
L
II }
T
{de
d"
}+ {L
II
T
.
II
d"
II
} {deP} - d
. is a scalar factor of proportionality. Due to the plastic incompressibility of the matrix material, the growth rate of the void volume fraction j can be related to the macroscopic plastic dilatational strain rate i~k as (7-413) The equivalence of the macroscopic plastic work rate and microscopic plastic dissipation rate gives
{erij }T {ifj} = (1 - f)er Mi*/l
(7-414)
Eq.(7-414) can be rewritten as (7-415) where h
der*M
= -d-. c*P M
The consistency condition is expressed as
7.11 Numerical Analysis for Anisotropic Gurson's Plastic Damage Model
571
(7-416) Combining the above equations, the macroscopic stress rates can be expressed in terms of the macroscopic strain rates. For uniaxial tension applied at 0° from the rolling direction, (T22 = (T33 = O. Therefore, (T; f = (T11 and (Tkk = 3(TM = (T11. We first solve the initial value of (T11 from the macroscopic yield criterion in Eq.(7-405) for a given f and (TM(= (Ty). With the initial conditions rate equations are needed to determine the evolution of the macroscopic stresses (T ij, the matrix flow stress (T M, and the void volume fraction f. From Eq.(7-412) , the macroscopic plastic strain rates t'~\, t'~2 and t'~3 are obtained as (7-417)
(7-418)
(7-419) where Q2 = q2J1
+ 2Re/ [6 (1 + Re) ]· For a prescribed t'~\, the scalar factor >.
can be solved from Eq.(7-417). Once>. is solved, t'~2 and t'~3 can be obtained. Then the macroscopic plastic dilatational strain rate t'~k can be determined as .p
2 ckk
. dF = Ap . [6ql smh . ( Q2-* (T 11) = Ap--.-Q2] *-
O(Tkk
(TM
(TM
(7-420)
Substituting Eq.(7-420) into Eq.(7-413) gives the growth rate of the void volume fraction , j. For uniaxial tension at 0° from the rolling direction, Eq.(7415) becomes (7-421) From Eq.(7-421), CrM can be obtained. The consistency condition in Eq.(7416) becomes
(7-422)
572
7 Anisotropic Elasto-plastic Damage Mechanics
Now all can be solved from Eq.(7-422). Once all is determined, i h can be determined by Eq.(7-403). Then ill can be determined by Eq.(7-402) and , based on the rate equations discussed earlier, the evolution of the macroscopic stress 0"11' the matrix flow stress 0" M and the void volume fraction f can be obtained incrementally as a function of Cll with the initial conditions of f and O"'M( = O"y). For the uniaxial tensile load applied at 45° with respect to the rolling direction, denote the macroscopic tensile stresses as 0"'11 and 0"'12 , the macroscopic strains as C~l and C~2' Note that 0"~2 = C~2 = O. Based on the stress transformation, 0"11 = 0"~d2 + 0"~2 ' 0"22 = 0"~d2 + 0"~2 ' 0"21 = 0"~d2, 0"23 = 0"31 = 0"33 = 0 as well as on the strain transformation, C'12 = - 0"11 /2 + c22/2, the macroscopic plastic strain rate ifj can be obtained from the associated flow rule as c.pll = /\;, -dF- = /\;, [ 20"11 p dO"
11
p
+ 2Ro(0"1l (1 + R 0 )0"*2 M
0"22)
. h (Q 2 0"11 + 0"22) + 2q1 f sm 0"* M
-Q2 ] 0"* M (7-423)
(7-425)
i~k = '\p -:.dF
OO"kk
= ,\p [6qd sinh (Q2 0" 11
~ 0"22) ~2 ]
O"M
O"M
(7-426)
Manipulate the imposed macroscopic stress, strain conditions and the stress, strain transformations, 0"'12 can be expressed as 0"'12(R90 - RO)/2(R+4R90RO + R90)' Therefore, O"~f = Q30"'1l, where
Q3 = ,-------------------------------~-----------------------
1
-
2
2RoR90(3 + 2R45 ) + (1 + 2R45)(Ro + R90)2 + SRoR9o(1 R 90 (1 + Ro)(Ro + 4RoR90 + R 90 )
+ R 45 )(Ro + R 90 )
(7-427) and O"kk = 0"11 + 0"22. We first solve the initial value of 0"11 from the macroscopic yield criterion in Eq.(7-409) for a given f and O"'M( = O"y). With the initial conditions, rate equations are needed to determine the evolution of the macroscopic plastic strain rates ifj' the matrix flow stress O"'M and the void volume fraction f.
7.11 Numerical Analysis for Anisotropic Gurson's Plastic Damage Model
573
For a prescribed if) , the scalar factor ;\ can be solved from Eq.(7-423). Once ;\ is solved, i~2 ' if2 and ifk can be obtained. Once ifk is determined , Eq.(7-413) gives the growth rate of the void volume fraction, j. Eq.(7-415) becomes .p
all cll
+ a22 c.p22 + 2a 12c.p12 =
1
h(l -
* .*
J)aMa M
(7-428)
Eq.(7-428) can be used to determine a M' The consistency condition in Eq.(7-416) becomes
(7-429) Now all can be solved from Eq.(7-429) and the stress transformation, all = e can th en I /2 - a I 12, 0'22 = all I /2 + 0'12 I I / 2 . cll' 'e ' e an d' all , 0'12 = all C22 C12 be det ermined by Eq.(7-403) when all' a22 and a12 is determined. Then ill i22 and i 12 can be determined by Eq.(7-402). Based on the rate equations discussed earlier, the evolution of the macroscopic stress all' a22 and a 12' the matrix flow stress a M and the void volume fraction f can be obtained incrementally as a function of if1 with the initial conditions of f and a M (= 0' y). For the uniaxial tension applied at 90° from the rolling direction case, 0'31 = 0'33 = O. Therefore, a:f = Q4a12, where
Ro(l + Rgo ) Rgo (1 + Ro)
(7-430)
and akk = 3a m = 0'22. The numerical procedure is the same as that of the uniaxial tension at 0° from the rolling direction except that we first solve the initial value of 0'22 from Eq.(7-405) for a given f and aM(= a y). However, Eqs.(7-417)rv(7-419) should be modified as
(7-431)
574
7 Anisotropic Elasto-plastic Damage Mechanics (7-432)
(7-433) and Eq.(7-420) becomes .p . dF = Ap . [6qIJsmh . ( Q2-*CT22) -Q2] ckk = Ap--:-,*OCT kk CT M CT M
(7-434)
Eq.(7-415) becomes (7-435)
(7-436)
+ [2 qIJ cosh ( Q2 ;~)
-
2q3i ] j
=0
For the equal biaxial tension case, CTn = CT22 and CT33 = O. In this case, CT;f = ((1/(1 + Ro)(Ro + R90)R90)1/2 CTCT11 =;f= ((1 / (1 + Ro)(Ro + R90)R90)1 /2 CT22 and CTkk = 2CT11. The numerical procedure is the same as that of the uniaxial tension at 0° from the rolling direction. However, Eq.(7-417)rv(7-419) should be modified as
(7-437)
(7-438)
and Eq.(7-420) becomes
7.11 Numerical Analysis for Anisotropic Gurson's Plastic Damage Model
11) -Q2] *-
.p . ClF = Ap . [6qd smh . ( Q22a*Ekk = Ap-:-.oa kk aM
aM
575
(7-440)
Eq.(7-415) becomes (7-441) Eq.(7-416) becomes
(7-442)
7.11.4 Finite Element Analysis for Voids Growth of Gurson's Plastic Model 7.11.4.1 Finite Element Modeling for Voids Growth A porous ductile material containing a triply periodic array of spherical voids is considered here to investigate the plastic behavior of porous ductile materials. Because of the regular arrangement of the voids, the porous ductile material containing a triply periodic array of spherical voids can be modeled by considering a unit cell of the cube with a spherical void at its center, as shown in Fig.7-33(a). The Cartesian coordinates Xl, X 2 , and X3 perpendicular to the cube faces are also shown in the figure. The Cartesian coordinates coincide with the material orthotropic symmetry axes. Note that the relative dimensions of a unit cell in the three directions can affect the plastic behavior of the unit cell ( [7-50]). In this study we concentrate on the effects of plastic anisotropy of the matrix and therefore a unit cell of a cube is taken for consideration. For demonstration of the finite element mesh, only one eighth of a finite element mesh used for computations is shown in Fig.7-33(b). Note that unlike the one-sixteenth cube model used in Hom and McMeeking [7-51 ] and Jeong and Pan [7-52]' we adopt the entire cell model to properly take account of the planar plastic anisotropy. The void surface is specified to have zero traction. Macroscopically uniform displacements are applied on the faces so that the outer faces of the unit cell remain planes during the deformation. To take the planar anisotropy into account, we consider three different loading scenarios with the principal loading direction at 0°, 45° and 90° from the rolling direction of the sheet metals. Uniform normal displacements M I, ~X 2, and
576
7 An isotropic Elasto-plastic Dam age Mechanics
,A '\:J
r'~------I ~,
(a)
(bl
Fi g. 7-33 (a) A voided un it cell ; (b) one eighth of a finite eleme nt mesh of t he un it cell. Note t ha t a full un it cell is used for com p utation s
in the X l , X 2 , and X 3 directions are applied on t he cell faces perpendicular to the X l, X 2 , and X 3 directions, respectively. For the principal loading dir ection at 0° (degrees) from the rolling dir ection of t he sheet met al , t he relative uniform normal displ acem ent s applied to t he faces of t he unit cell are list ed in Table7-1 F ive st raining con dit ions wit h different displ acem ent ra t ios are cons ide red : equal-t riax ia l, equa l-biax ial, plan e strain, nearl y uni axial (~X 2 /~XI = 1/ 2) and nearly pure shear (~X 2 /~XI = -1) . The displ acem ent ratios are ass igned accord ing to the small strain rigid isot ropic plast icity conventi on . In t his table, "not pr escrib ed" mean s t hat t he surface remains plan ar wit ho ut any specified nod al for ce or displ acem ent . For t he princip al loading dir ection at 90 ° from the rolling dir ection of the sheet met al, t he relative uniform normal displ acem ents applied to the faces of t he unit cell are list ed in Tabl e7-2. For t he prin cip al loading di rection at 45° from t he rolling direction of the sheet met al , t he mesh of t he unit cell is rotated 45° with resp ect to t he X 3 dir ecti on , while t he plasti c ort hotropic symmet ry planes remain un changed . For t his load ing directi on , t he relati ve un iform normal disp lacem ent s applied to the faces of t he unit cell are the sa me as t hose in the cases with t he principal loading di rection at 0° from t he rolling direction. In all loading cases at different princip al loading directions, the symmet ry planes of plasti c orthotropy rem ai n un chan ged. ~X3
Table 7- 1 Rela t ive uniform normal d isplaceme nts applied t o the faces of the unit cell for different loading cond it ions with the majo r pr incipal loading at 0° from the rolling d irect ion EqualEqual-biaxia l Pl an e Nearly Nearly P ure triaxial St rai n Un iaxial Shea r 1 1 1 1 1 1 - 1 1 0 - 1/2 Not prescribed Not presc ribe d Not prescribed No t p rescribed 1
7.11 Numerical Analysis for Anisotropic Gurson's Plastic Damage Model
577
Table 7-2 Relative uniform normal displacements applied to the faces of the unit cell for different loading conditions with the major principal loading at 90° from the rolling direction . Nearly Nearly Pure Equal- biaxial Plane EqualStrain Uniaxial Shear triaxial - 1 1 1 o - 1/2 1
1
1
1
Not prescribed Not prescribed Not prescribed Not prescribed
Table 7-3 Material properties of the steel conference) Young's Modulus Poisson's Ratio (GPa) 0.3 Steel 206 0.33 Aluminum 71
1
1
and the aluminum (from Numisheet'93 Yield Stress (MPa) 269.5 137.0
R 1.73 0.71
1.34 0.58
2.24 0.70
The matrix material is assumed to be perfectly elasto-plastic. We consider a high strength steel and aluminum to be used as benchmark materials. The material properties of the steel and aluminum are listed in Table7-3. Several initial void volume fractions (J = 0.01,0.04,0.09 and 0.12) are considered here to examine the applicability of the proposed yield criterion in Eq.(7-409). Hill's quadratic anisotropic yield criterion in Eq.(7-400) is used to describe the matrix material with planar anisotropy. Wang et al.[7-28] had used the commercial finite element program ABAQUS to perform the computations of this problem. Under different loading conditions, the macroscopic stresses are calculated by averaging the surface tractions acting on the faces of the unit cell. The macroscopic yield point is defined as the limited stress state where massive plastic deformation occurs. The corresponding macroscopic effective stress CJ~f in Eq.(7-411) and macroscopic mean stress CJ m in Eq.(7-407) are then calculated and compared with those based on the anisotropic Gurson 's yield criterion in Eq.(7-409). In addition to the elastic perfectly plastic material model employed to calculate the fully plastic limits, the macroscopic plastic flow characteristics due to matrix strain hardening are investigated under proportional nearly uniaxial and equal-biaxial tensile loading conditions. The relative uniform normal displacements applied to the faces of the unit cell are based on the normality flow rule and the yield criterion for the matrix as in Eq.(7-400) under uniaxial and equal-biaxial conditions. The ratios of the normal displacement applied to the faces of the unit cell are listed in Table7-4. In the application, the matrix effective tensile stress CJ M is a function of the effective tensile strain Eil and can be expressed as (7-443)
578
7 Anisotropic Elasto-plastic Damage Mechanics
Table 7-4 Relative uniform normal displacements applied to the faces of the unit cell for nearly uniaxial and nearly equal-biaxial conditions when the matrix hardening is considered . Nearly Uniaxial 'O;oo, - - - - - - -4"5"0;--"------;c90""0, ,------- Nearly Equal- biaxial 0 Not prescribed 1 1 1 Ro/ R90 Not prescribed Not prescribed 1 Not prescribed Not prescribed Not prescribed Not prescribed
where C 1 = 677 MPa, C 2 = 0.1129, and C3 = 0.186 for the high strength steel, and C 1 = 5700 MPa, C 2 = 0.01502 and C3 = 0.469 for the aluminum. These material constants are based on the tensile stress- strain relation in the rolling direction as specified by article [7-28]. 7.11.4.2 Numerical Results
Finite element computational results are used to evaluate the applicability of the Gurson's anisotropic yield criterion in Eq.(7-409) to model the macroscopic anisotropic plastic behavior of porous materials. The computational results are examined for porous materials under elastic and perfectly plastic conditions with different void volume fractions (f = 0.01 , 0.04,0.09, and 0.12). Figs.7-34(a)rv(c) shows the comparison between the computational results, represented by symbols, for the steel with principal loading directions at 0°, 45° and 90° from the rolling direction, respectively. In these figures , both the macroscopic mean stresses and the macroscopic effective stresses are normalized by the matrix yield stress CJ y in the rolling direction. For comparison, various forms of curves based on the unmodified anisotropic Gurson yield criterion (ql = q2 = q3 = 1) in Eq.(7-409) are also shown for different void volume fractions. As shown in these figures, when the void volume fraction is small, the computed finite element results are in agreement with those based on the unmodified anisotropic Gurson yield criterion. However, when the void volume fraction is large, the yield contours based on the unmodified anisotropic Gurson's yield criterion are much larger than those of the finite element computations when the normalized mean stress CJm/CJ y is low. But when the normalized mean stress CJm/CJ y is high, under equal-triaxial loading conditions, the unmodified anisotropic Gurson's yield criterion underestimates the yield behavior for the steel, whereas the unmodified anisotropic Gurson's yield criterion overestimates the yield behavior for the aluminum. Therefore, three fitting parameters q1, q2, and q3 are applicable in the anisotropic Gurson's yield criterion as suggested by Liao et al. [7-46]. Figs.7-35(a)rv(c) shows computational results (represented by symbols) and results obtained based on the modified anisotropic Gurson's yield criterion (represented by various curves) with the selections of fitting parameters q1 = 0.45, q2 = 0.95 and q3 = 1.6 for the aluminum, respectively. The values of ql, q2 and q3 for the aluminum are the same as those suggested by Chien
7.11 Numerical Analysis for Anisotropic Gurson's Plastic Damage Model
:;~f:'~"o""" . .....!.. ~~. ?~~~>< 0.75 ~. 1.00
J:t b" 0.50
579
-.
······················rJ.··
;::-~:~1: ":':'\ :\\\ ,
-·-F=0.09
--F=0.12 0.25 a
o • o
, ', '
'.; 01), 1 .... .... .... .... .... , . .... . . .......... +.............. 04), !. 09): '\ 12): 1.0 1.5 2.0 2.5
fI:lM(F=O FEM(F=O FEM(F=O FEM(F=O
0.5
'. '
,,~ :
1 +............
.'i
I
3.5
::: ~;~ij~,;"
45'
,: , \1
-I -F=O .O I
b~
, \ .,
" ... " \ 'r-' F=9·09 " .... : ~-- F=O.12 '., ' v FEM(F=O.OI)" '"
~ 0.50 ·,· •••F=9·0 4
0.25
• FEM(P;:;O~04): \
a
FEM(F=0~09)' 'i
• FEM(F=Oi 12) :
OLL__~~·~~~__~'~~~w-~~__~u--£~
o
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1 .OOI.:f.~~;~~~o=~TI·~~=~~~,'=·~~J·=·===:-----,~-,-,---,--~ 0.75 b~
~ 0.50
0.25
, ".
'.
\ \
...
.
:
i~ ~~' ?' \\,--\ 1 - --p=O.12 'i v FEM(F=O.OIJ, ..... \
·· oFEM(P;:;d:04):
D FEM(F=O.09)' • FEM(F=O:,J 2) :
t
\
: :': .'. :j
OLL____L-__~__~,~'~__L-~-L--~L-~~
o
0.5
1.0
3.0
3.5
Fig. 7-34 Comparison of finite element results (symbols) and results of unmodified anisotropic Gurson 's yield criterion (curves) for steel materials under different loadings in the major principal direction at (a) 0°; (b) 45°; (c) 90°
580
7 Anisotropic Elasto-plastic Damage Mechanics 1.00 r.~r~=~~=~~".=~~:r}4".,=.=::::::=:--:---r-:p-;-ri=-n=cTip=a=)=-d'' r""1 - e:-ct-io-n-(J"
0"
,
.
0.75
'-;
", \.'.
,- -F=O.OI " , ·····F=O.04 \ \ --1=0.09 --- F=O.12 . , FEM(F=O,.OI? 0.25 , •0 FEM(F=O;04) o FEM(F=O,09j • FErv'(F- OP2):
~0.50
0W---~--~--4-~--~--~--~~~
o
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1. 00 Iv'.=rs=,.::==:::::---,----,-;-;::::::T::'::=--,-------,
.·
o~··· · · 6· ~ .Q.
f~-~~ ·r
0"
0.75 f- , ............. :: .... ;., . ......:.,.kk){Oij }/3, {eij} = {cij } - (l:)'kk){Oij}/3 and O"kk = v(l D)Eckk/(l + v)/(l - 2v) , Eq.(8-131) can be written as
{O"ij(t)} = {Rij(t)}
J
v(l - D)E
"
+ (1 + v)(l - 2v) (L.,ckk (t)){Oij}
t
+
exp{ -[Ln(t) -
Ln(~)]}
1; = 0
x
(2G*d{Ci j } _ 2G*d(L ckk){O} _ d{R ij } _ dD{Rij})d~ 3
d~
d~
d~
'J
d~ 1 - D
(8-132) where, from Eq.(8-132)
L n (t)
=
J t
(
1;=0
dD
KvT=7? [vT=7? 1
3G* (d p ) dl; l iN
(~) l
+ (1 - D)Qr + vT=7?k
)
+ ~l - D d~ (8-133) Moreover, the evolution of the kinematic hardening variables given by Eq.(8113) can be transformed as (8-134) with (8-135) and integrated to obtain (8-136) where
bt(O)} = 0 and
J akvT=7? d~d~ t
Ak(t) =
d
(8-137)
1;=0
The same kind of manipulation can be applied to the Eq.(8-114) governing the isotropic hardening to get
617
8.3 Asymptotic Expansion of Visco-plastic Damage Mechanics
1 f exp{ -[H(t) - H(~)]} v'1=7? d~d~ ~
t
r(t) =
(8-138)
~=O
with
H(t) =
J bv'1=7?~~d~
(8-139)
~=O
The damage variable [! is involved in the integral Eqs.(8-132) , (8-133), (8136)"-'(8-139) , and can be evaluated by integrating the damage constitutive Eq.(8-120) or Eq.(8-121). Using Eq.(8-115) for example, [! is given by
[!(t) =
~L t
[(Y) S(1 _1ill dd~P ] d~ S
(8-140)
It can be then considered as a new paramet er [!(t), as similar as Ln(t) , Ak(t) and H(t) given by Eqs.(8-133), (8-137) and (8-139). Consequently, the integral form of each state variable is related to the new scalar parameters Ln , Ak and H associated respectively to Cauchy stress, kinematic and isotropic hardening (the compilation of the integral equations can be expressed based on asymptotic expansion for a non homogeneous integral in the next section). The damage variable [! given by Eq.(8-140) is added as a supplementary parameter. It is worth noting that, in the case of the present constitutive equations, the parameters Ak and H are linearly dependent and can be expressed in terms of the accumulated plastic strain P which is given by (8-141)
8.3.3 Recursive Integration Method for Visco-plastic Damage 8.3.3.1 Basis of Asymptotic Expansion for Non-homogeneous Integral Considering a nonhomogeneous integral is the Laplace integral with the following form
J(f>t) =
f
t +~t
exp{ -[Ln(t + f» - Ln(O]}
d
~~~) d~
(8-142)
~=t
where Ln(t
+ f>t)
is a monotonically increasing function of t
+ f>t
and
d~~~)
is assumed to be a function which has an evanescent memory in the integral If
618
8 Theory of Visco-elasto-plastic Damage Mechanics
Ln(O is written as Ln(t + b.t - (t + b.t - 0), it may be expanded by Taylor's theorem in the form
Ln(O = Ln(t+b.t) - (t+b.t -~)
dLn(t
+ b.t)
dt
1
+ 2"(t + b.t -
02
d 2 Ln(t + b.t) dt 2 + ... (8-143)
Hence, the integral (8-142) becomes
f
t+M
J(M) =
exp
(
- (t
+M
e=t
-
) d
1
(~)
~)Ln(~) + 2"(t + b.t - ~)2 Ln(~) Td~ ~
(8-144)
A change of variable,
~=t-(,
allows the integral to be written as
f exp ( -[(b.t - ()Ln(() M
J(b.t) =
~(b.t -
()2 L n (() ])
d~~() d(
(8-145)
(= 0
By introducing a new variable z = M - (, allows the integral to be written as
f exp tJ.t
J(b.t) =
1.. )dm(b.t -Z ) -[zLn(z) - 2"z2 Ln(z) ] dz dz
(.
(8-146)
z=o
with the help of Taylor's theorem (8-147)
so that
dm(b.t - z) dm(b.t - z) . dz =dt =-m
..
1
+ zm - 2" z
2'"
m
+ ...
(8-148)
and
f exp(-ZLn(Z)) exp(~z2Ln(z)) [m -zm + ~z2m + .]dz M
J(b.t) =
z=o
Expanding exp that
(8-149) (
z 2"Ln )
-2-
in series gives exp
(
z 2 Ln " )
-2-
.. = 1 + z2 Ln / 2 + ... , so
8.3 Asymptotic Expansion of Visco-plastic Damage Mechanics
619
f exp( -zLn(z)) [1 + ~z2Ln(z) + ... ] [m -zm + ~z2m + .]dz M
J(l1t)
=
z=o
(8-150)
Integration with respect to z gives J(!'>.t) =
:n (1 - exp( - MLn»
Ln
+ ~
Ln
[M exp( - !'>.tLn) - J(1 Ln
1·· [ 2 • - - .- (inLn - iii) !'>.t exp( - MLn) 2Ln
- exp( - MLn))]
exp( - MLn))]
. + -.2!'>.t - exp( - !'>.tLn) Ln
2 -. - (1 L~
+ ...
where the derivatives of m and Ln are evaluated at time t
+ I1t.
(8-151)
8.3.3.2 Recursive Integration Method for Visco-plastic Damage In the following, the numerical time integration is performed by an asymptotic integration algorithm initially proposed by Walker [8-65]; Chulya and Walker [8-67]; Freed and Walker [8-66] and extended in Nesnas [8-31] to the damaged visco-plastic model described in the previous section. The basic idea behind the algorithm is to solve approximately the set of integral Eqs.(8-132) , (8133) , (8-136)rv(8-139) using a recursive relationship. In order to evaluate these integrals, an asymptotic expansion of the related integrand is performed at about the upper limit of the time interval [t, t + M J, resulting in an implicit integration scheme. The main advantage of this method is that only a 2x2 (or for the uncoupled visco-plastic model1x1) matrix need be solved during the iteration process. Within a typical time step [t, t + I1tJ, one can cast the integral equations written over the time interval [0, t + I1t], for instance the one related to the stress in Eq.(8-132), into a recursive relation by splitting t he interval of integration into two parts
(8-152)
620
8 Theory of Visco-elasto-plastic Damage Mechanics
Substituting the identity 1 = exp(Ln(t))exp( - Ln(t)) into the first integral {Iij (t)} of this equation results in
v(l - D)E '" ] {Iij(t)} = exp( - ~Ln) [{O"ij(t)} - {Rij(t)} - (1 + v)(l - 2v) (L ..,c·kk(t)){6ij} (8-153) which simplifies Eq.(8-152) to the desired recursive integral equation
(8-154)
This relation is practical when a solution exists for evaluating the integral which appears in it. This latter is referred to as non-homogeneous integral because it represents the non-homogeneous contribution in the solution to the first-order ordinary differential equation. In the case of visco-plastic constitutive equations treated in this work, Ln is a monotonically increasing function of time and then the non-homogeneous integral is the Laplace integral of the form as expressed in subsection 8.3.3 (A) as
I(~t) =
f
HM
exp{ -[Ln(t
d
+ ~t) - Ln(~)]} ~i~) d~
(8-155)
E,=t
where
d~i~)
has an evanescent memory in the integral, therefore the inte-
grand has its largest value at the upper limit, t + ~t. This fading memory means that the solution will depend mainly on the recent values of the forcing function. Several different solution strategies can be used for this integraL They differ in their approach (implicit and explicit Taylor or implicit Euler-Maclaurin) and in their accuracy of approximation (i. e. the number of terms kept in their series expansions). In this work, the implicit solution is adopted. The integrand is then expanded in Taylor series at about the upper limit, t + ~t , where the integrand has it largest value. By retaining just a few terms in the Taylor series expansion (first-order terms), the integrand is accurately approximated where it is largest, and the neglect of the higher order terms is only felt near the lower limit, t, where the integrand contributes only a
8.3 Asymptotic Expansion of Visco-plastic Damage Mechanics
621
small amount to the integral because of its exponential decay from the upper limit. An approximation of the integral Eq.(8-155) is obtained by expanding the arguments of the exponential and the forcing function into Taylor series Ln(t + /':,.t - z ) = Ln(t + /':,.t) - Ln(t + /':,.t) z + H.O.T, where H.O.T means higher order terms, and a similar expansion for m( t + M - z). Thus, this integral can be rewritten from Eq.(8-151) (see subsection 8.3.3 (A) as
m(t + /':,.t) . (1 - exp{ - Ln(t + M)M}) J(M) = . Ln(t + /':,.t)
(8-156)
where the derivatives of Ln and m are approximated by the values at the beginning and end of the current step
L
~
n-
Ln(t + /':,.t) - Ln(t)
(8-157)
M
Eq.(8-156) is an implicit representation of the integral Eq.(8-155) because Ln and m are both evaluated at a future time and are therefore unknown. Applying this result to the recursive integral Eq.(8-154) leads to the desired approximation
(8-158) which is the linear implicit asymptotic solution of the ODE (8-129) with
ALn __
ti
Jt
KV 1 -
3G* /':,.p [l [vT=l](/':,.p/ M) ]l/N
+ Q*r + vT=l]k
+
/':,.[l 1 - [l
(8-159)
Remark: It is worth noting that the asymptotic solution of the Eq.(8-129) is obtained as: (2G*{/':,.cij} - 2G*(/':,.LCkk ){6ij }/3 - {/':,.Rij} - /':,.[l{Rij } / (1 [l)) / /':,.Ln. This can be easily derived from Eq.(8-158) when the time step is very large (t --+ (0), the exponential term of Eq.(8-156) becomes small compared with unity, and the asymptotic expansions of the relation Eq.(8-158) lead to
622
8 Theory of Visco-elasto-plastic Damage Mechanics
{O"ij(t + M)} - {Rij(t
1( *
rv = ~LJ! 2G
+ ~t)} -
(1 ~lV~lD2~v) (L ckk(t
+ ~t)){Oij}
2G* ' " {~Cij} - -3-~(L...Ekk ){Oij} - {~Rij} - 1 ~D) _ D {Rij}
(8-160) Therefore, the asymptotic solution of Eq.(8-129) is contained within the implicit asymptotic integration method. A procedure similar to that for Cauchy stress is used to obtain the final asymptotic recursive forms of the remaining state variables, namely kinematic hardening, isotropic hardening and damage variable. For kinematic hardening, the relationship is
(8-161) with (8-162) (8-163) and (8-164) for isotropic hardening,
r(t
+ ~t) = exp( - ~H)r(t) +
[
1 - exp( - ~H)] ~H
~p
(8-165)
with (8-166) For the damage variable, two kind of approximation are possible: first-order approximation
D(t + ~t) = D (t)
+ ~D = D (t) + n(t + ~t)~t
(8-167)
second-order approximation
D(t + M) = D (t)
+ ~D = D (t) + 0.5 [n(t) + n(t + M) ]M
(8-168)
Recursive relationships, given by Eqs.(8-158), (8-161) , (8-165) and (8167) or (8-168) for determining {O"ij ( HM)}, bij(H~t)} , r(H~t} and {D( t +~t)} , involve the calculation of the parameters ~LJ! , ~Ak, ~H and ~D.
8.3 Asymptotic Expansion of Visco-plastic Damage Mechanics
623
These parameters, in turn, via Eqs.(8-159), (8-164) and (8-166) , require a knowledge of {() ij (t+~t)}, hij (t+~t)} , r( t+~t) and {J?( t+~t)} for their evaluation. These equations are then recursive or implicit in nature. Therefore, the recursive relationships comprise a set of four implicit equations which can be resolved by Newton- R aphson iterations. However, the parameters ~Ak and ~H are linearly dependent and are related to the cumulated plastic strain ~p = p(t + ~t)~t. One can evaluate these parameters by computing the second invariant of the plastic strain rate. The plastic strain increment is written as follows , (8-169) The unknown parameters are then reduced to ~Ln and ~J?, which can be determined by the resolution of the following implicit nonlinear system
gl(IlLn , M?) = IlLn -
3G*llp KvT=7? [v 1 - S?(llp/ llt) ]1/N + Q*r+ vT=7?k
M?
+ -l - S?
= 0
(8-170) and (8-171) or (8-172) If ~Ln = Xl and written as
~J?
=
X2,
then Eqs.(8-170) and (8-171) or (8-172) can be
(8-173) The resolution is based on the Taylor expansion of the functional gi composing the system Eq.(8-173), which leads to the following linearized system
[d{g~~~~~})} l {oxj} = - {gi({xj}]}
(8-174)
Each step of the iteration process requires the solution of Eq.(8-174). For
{oxj} , which defines a new intermediate solution {x;+l } (8-175) being the basis of the next iteration step. This continues until convergence toward the suitable solution, when the following convergence criteria are satisfied
624
8 Theory of Visco-elasto-plastic Damage Mechanics
(8-176) where Cl and C2 are tolerance limits ( cl = C2 = 10- 4 ) and 11*11 designates a Euclidean norm. The apparent advantage of these numerical schemes involves naturally the solution of only 2 x 2 matrix equations as opposed to 15 x 15 (for 2D problems) matrix equations in the case of other implicit schemes, such as the classical trapezoidal scheme. From a computational standpoint, the asymptotic integration algorithm appears then to be quite appealing. In Eq.(8174), the coefficients matrix denoted by [J] =
[~{g;}] o{ Xj }
(is a 2x2 Jacobian
matrix), may be derived analytically or determined numerically. Numerically, the matrix components of [J] can be evaluated by finite difference perturbation techniques and placed in the following form:
{J } = {gi (~Ln
+ dLn,M?)}
dLn
t1
- {{gi (~Ln , M?)}
{J } = {gi (~Ln,M? + dD)} - {{gi (~Ln , ~D)} dD
t2
where i = 1,2, dLn = Remarks:
O.Ol~Ln
and dD =
(8-177) (8-178)
O.Ol~D.
1. In the case of time-independent plasticity, the system is reduced to three equations. In fact, an additional equation has to be added to determine the plastic multiplier - '\. With the help of the consistency condition together with the yield function , one can obtain the following system [8-31]
(8-180) g3(~Ln , ~A , ~D)
=
~D
- ~A
1 ( y)S S !3 = 0 {1 - D)
(8-181)
2. It is noted that the proposed method still works for a sum of more than two kinematic hardening variables according to the equation {Rij} = L:dRfj}. The size of the reduced system remains the same. 8.3.4 Outline of Visco-plastic Damage Equations and Algorithm
The compilation of all the system equations can be summarized in different schemas as follows.
8.3 Asymptotic Expansion of Visco-plastic Damage Mechanics
625
8.3.4.1 Summary of System Equations in Differential Form
{ifj} = ~
1 {Sij} - {Rij } ). 2 ~ J 2 ({Sij} - {Rij})
ht} =
h~}
-
(8-112) (8-113)
adl't }).
1
.
(8-114)
i=(~-br) ..
st
1-
. - (Y) S
st .
)..*
= p=
).
~
1
S
(1 - st) i3
=
)...
2
(8-115) T
-3 {if) {if)
(8-120)
8.3.4.2 Summary of System Equations in Integral Form
{aij(t)} = {Rij (t)}
+
(1
v(l - n)E ' " + v)(l _ 2v) (L../,kk(t)){6ij}
+ It
exp{ -[Ln(t) - Ln(On
~ =O
x (2C* d{ Ei j } _ 2C* d(L: Ekk) {6} _ d{R ij } _ dn {R ij }) d~ 3 d~ 'J d~ d~ 1 - n
d~ (8-132) (8-136)
ret) =
1 ~ f~=O exp{ -[H(t) - H(~)]} ~dtd~ 1 st
(8-138)
S(1 _1st) i3 dd~P ] d~ EL [(Y) S
(8-140)
t
=O
where !!.. and J.L are exponents of the stress deviator [S] and the second-rank tensor [tli] with components tli,j = (Oij - Dij ) -l given by the damage tensor [D]. Now, the main problem is to determine the scalar coefficients 'ifJ( v ,/1» as functions of the integrity basis and experimental data. In order to solve this problem we suggest the following procedure, which may be useful for practical applications. A representation with the same tensor generators as contained in Eq.(8-223) can be found by separating the tensor variables [S*] and [tli] in the following way:
Eij = f ij ([S], [tli]) = where the isotropic tensor functions
1
2(XikYkj + YikXkj)
(8-224)
636
8 Theory of Visco-elasto-plastic Damage Mechanics
Xij_= X i j( [S]) = .TlOOi j
+ TlISij + Tl2 S;j}
Tlv - Tlv(I 1 , J 2 ,J3 , i(,n) Yij Pp,
= Yij([l]i]) ~ POOij + Pll]iij + P2(l]iij )2} = pp,(tr [l]i ] )) = pp,(l]ij, I]iII, I]iIII)
(8-225)
(8-226)
(/J, v = 0, 1,2 and A = 1,2,3) are used. Thus, we arrive at the representation of Eq. (8-223) with scalar coefficients: /J , V = 0,1,2
(8-227)
where scalars Tlv have been determined in Ref. [8-79] by utilizing a tensorial interpolation method. The coefficients Pp" in Eq.(8-227) can be found by solving the following system of linear equations Po Po Po
= = =
r
+ (I]iI )2P2 = (I]iI I]iIIPI + (I]iII )2P2 = (I]iII fI I]iIIIPI + (I]iIII )2P2 = (I]iIII )mffI I]iIP 1
J
r
}
(8-228)
The exponents mI , mIl, mIll in Eq.(8-228) are det ermined by using the creep law (Eq.(8-182) or Eq.(8-190)) in t ests on specimens cut in mutually perpendicular directions, Xl, X2 , X3 · Because of Eq. (8-202) and l]iij = diag {a, ,6, /' }, the principle values in Eq.(8-228) can be expressed through l]ir
= l/a,
I]iII
= 1/,6,
l]im
= 111'
(8-229)
where the parameters a, ,6, /' are fractions which represent the net cross section elements of Cauchy's tetrahedron perpendicular to the coordinate axes [8-12]. In the case of two equal parameters, for instance a f= ,6 = /" the scalars Pp, in Eqs.(8-226) and (8-227) can be determined by using the tensorial interpolation method as has been described in Refs. [8-16,8-78]. As can be seen from the Eqs.(8-223) 0 : (¢(x)) = 1 when ¢(x) < 0 : (¢(x)) = 0 which means that
< ¢n (F*) > = 1 if yield has occurred < ¢n (F*) > = 0 if yield has not occurred
(8-251)
Regarding the suggestion in [8-83], the model of the visco-elasto-plastic yield function can be chosen as one of following forms,
t
¢(F*) = (F* - Fo oT' ¢(F*) = e M ( P*;"Po) - 1 (8-252) Fo where M, N are material constants measured from the specified visco-elastoplastic test. If employing different plastic yield models in F* of Eq.(8-252), different visco-elasto-plastic constitutive models with corresponding plastic yield function can be used generally. The failure process of visco-elasto-plastic damage is actually a rheological damage flow and coupled evolutions of elastic-damage and visco-plastic damage, hence the application of the principle of minimum dissipative energy in the study of visco-elasto-plastic damage theory may have more advantages and significant effects.
644
8 Theory of Visco-elasto-plastic Damage Mechanics
8.5.2.3 Evolutionary Model of Visco-elasto-plastic Damage Mechanics Substituting Eqs.(8-242) and (8-244) into Eq.(8-247) in subsection 8.5.2.1, the visco-plastic strain rate, the damage evolution rate and accumulative rate of the harden(ing)-softening parameter of visco-elasto-plastic damaged materials can be formul ated as
{i vp } =
- )11
[{O"} - {h - (1 - D)(fc - It)}]
il = - A2Du exp( -'"'~B) "y
=
(8-253)
- A3~B
Comparing the first expression of Eq.(8-247) with the right-hand second term of Eq.(8-248) , the first proportional constant Al can be determined as (8-254) Substituting Eqs.(8-243) and (8-254) into Eq.(8-247), the components of the irreversible visco-plastic rheological strain rate can be arranged in
dF* {ivP}n = 1] (¢n (F*))d{O"}n or ~VPl
= - 1] (¢ (F: ))[20"1 -
{ EVP2 = - 1] (¢ (F
))[20"2 -
i VP3 = - 1] (¢ (F*)) [20"3 -
0"2 0" 1 0"2 -
+ (1 0"3 + (1 0"1 + (1 -
0"3
D)(fc - i t )] D)(fc - it) ] D)(fc - it) ]
(8-255)
Consequently substituting Eq.(8-244) into Eq.(8-247) gives an exponential function for damage variable D and a power function for accumulative strain harden(ing)-softening parameter "( (8-256) According to the test result in reference [8-84]' the relationship of damage D and accumulative strain ~D can be fitted by the ultimate damage value Du shown as D = Du - Du exp( - '"'~B)
(8-257)
Differentiating Eq.(8-257) with respect to time, it gives (8-258) where the rate ~D can be determined by differentiating Eq.(8-245) with respect to time. - 1
~D( {Evp} T {Ev p}) ""2 {Evp}T {i vp} = ~L/{Evp}T {ivp}
(8-259)
8.5 Visco-elasto-plastic Damage Mechanics Based on Minimum Dissipative Energy Principle
645
Substituting Eqs.(8-258), (8-259) and (8-255) into Eq.(8-258) , then comparing it with the expression in Eq.(8-256) , the second proportional constant ),2 can be obtained as
-1](¢n (F* ))K;m~B'-2{cvp}T d~:;n
),2
=
),2
= - 1](¢ (F*))K;m~B' - 2{cvp}T[3{0'} - {h - (1 - [2)(fc - it)} ]
or
(8-260)
Consequently, the model of the damage evolution rate is formulated as
n = 1] (A.'Pn (F *)) K;m~Dm{ cvp }T d{dF* dF2 a} n dY or
Jt
tl = 1] (¢n (F*))K;m[2u~'B exp( -K;~'B){ cvp} T d~:; n tl = 1](¢n (F* ) )K;m[2u~'B exp( - K;~'B){cvp}T [3 {0'} - {h - (1 - [2)(fc - it)} ] (8-261) The accumulative harden-softening function of studied material can be given by fitting t est data in a power function form as I
=
(8-262)
K;~'B
Differentiating Eq.(8-262) with respect to time, we have "y
= mK;~B' - l~D = - ),3~'B
(8-263)
Substituting the expression in Eqs.(8-256), (8-259) and (8-255) into Eq.(8263), then comparing it with the expression in Eq.(8-256), the third proportional constant can be obtained as *
-2
T dF*
),3
= - 1](¢ (F
),3
= - 1] (¢ (F* ) ) mK;~D2{cvp}T[3{0'} - h + (1 - [2)(fc - it) ]
) ) mK;~D {cvp} d{O'}
or
(8-264)
Sequentially, the model of the accumulative harden(ing)-softening parameter evolution rate is formulated as
(8-265)
646
8 Theory of Visco-elasto-plastic Damage Mechanics
Assembling Eqs.(8-255), (8-261) and (8-265) together, the generalized evolution equations of the visco-plastic rheological strain rate, the visco-plastic damage development rate and the accumulative harden(ing)-softening rate in visco-elasto-plastic damage mechanics can be theoretically modeled by
dF* {svP}n = 7](¢ (F*))d{a}n fin =
7](¢(F*))K;mDu~D exp( -K;~D){cvp}T d~:;n
1n =
7](¢(F* ))mK;~B-2{cvp}T d~:;n
(8-266)
8.5.3 Numerical Modeling of Visco-elasto-plastic Damage Mechanics 8.5.3.1 Finite Element Model of Visco-elasto-plastic Dynamic Damage Problems
Assembling the visco-plastic damage yield model Eq.(8-242), the evolution equation Eq.(8-266) and the constitutive equation shown in Eqs.(8-248) (J e, the solution of Eq.(8-315) using the initial condition fl = flo at t = 0 reads _1_ = _1_ (1 _ ~) 1 - fl 1 - flo te
1
(1 +x)
(8-316)
where X is a material constant te is a normalizing constant given by
_ (1 - flo) HX te - -'-----'-"--l +X
(_A_)
x
(J - (Je
(8-317)
From Eqs.(8-311) and (8-316) , it follows t hat a t heoretical value for fl at failure is fle = 1 which occurs at t = te. In practice, however, materials typically fail at values of fl < 1 (e.g. , Lemaitre and Chaboche, [8-6], p. 364) corresponding to times t < te. Denoting the value of fl corresponding to failure of the material by fl f and the corresponding time-to-failure by t f ' then Eq.(8-316) yields 1 tf = [ 1 - ( 1
=fl~
fl)
l+X] te
(8-318)
To complete the uniaxial characterization of the material, values for flo, fl f ' X and A are needed. If flo is known then creep-rupture tests can be performed at different stress levels to experimentally determine tf as a function of stress,
664
8 Theory of Visco-elasto-plastic Damage Mechanics
and subsequently Eqs.(8-317) and (8-318) can be used to determine the values of [2j , X and A. Consider the instantaneous damage [20 that occurs during rapid ramp loading to a stress level CT. We note that in the case of coupled elastic/damage behavior, the ratio of the unloading compliance to the loading compliance provides a measure for the level of damage. In particular, it is well-known ([8-59]) that 1 1 - [20
So
(8-319) So where So is the unloading compliance. To determine [20, "spike" tests each consisting of a constant stress rate loading-unloading cycle, were performed up to the stress levels CT = 55.69, and 83.0 MPa corresponding, respectively, to approximately 40, 50, and 60 percent of the ultimate tensile strength. The rate of loading used was a = 5.6 MPa/s, which is sufficiently high to keep the visco-elastic effects to a negligible level so that the response is essentially elastic.
8.7.1.2 Description of Experimental Results From these tests, values for SO were determined. Values for So were also determined to eliminate the effect of sample to sample variability in So. Fig. 8-4, which shows the results of these three tests, clearly suggesting a linear correlation between SO / So and CT. Since CT < CT c no damage occurs, then using Eq. (8-319) it is possible to write 1
(CT-CT )
c - - = 1 + -'----::--::":'"
(8-320) 1 - [20 C where C is a normalizing constant determined from Fig. 8-4 as C = 510 MPa. With the above expression for [20 at hand , it is now possible to evaluate [2j , X and A from time-to-failure data. For the material under consideration, preliminary creep-rupture tests at temperature T = 75° F and 50 percent relative humidity are available [8-95] as shown in Fig. 8-5. Clearly, the data exhibit a large amount of scatter which, again, is attributable to inhomogeneity and randomness of the swirl-mat polymeric composite material. Nevertheless, these data can be used to provide some lifetime estimates for the material. The constants [2 j, A and X were determined by fitting the experimental data to the expression for tj given by Eqs.(8-317) and (8-318), and using Eq.(8-320) for [20. Based on the best fit curve, depicted in Fig. 8-5 by the solid line, the following values were det ermined n -Jtj
0.67, X -- 7.1, A -- 260 MP a. h our 1/ 7.1
(8-321)
In view of the large scatter in the creep-rupture data, the above values should be considered as preliminary estimates rather than conclusive material properties.
8 .7 N umerical Studies on Visco-elasto-plastic Damage Behaviors
665
1.12 r - - - - - - - - - - - - - - - - - ,
Expeliment Linear cnrve fit
1.10
1.08
S"o I D o=1+ (a- a,YC 1.02
C=5IOMPa
1.00 l...----''-----'-_----'-_--'--_.L..----''----'-_-' 70 50 60 80 90 a(MPa)
Fig. 8-4 Ratio between the uniaxial unloading and loading compliances as a function of stress. 1400
g ~
]
.9
~
P
Experiment (T = 75 ° F)
...
1200
- - Mode l
1000
n , =0.67 X=7. 1
800
A
=260 MPa.hollf
ll71
600
400 200 0 95
... lOS
11 5 125 a(MPa)
135
145
Fig. 8-5 Ti me to failure as a function of stress in uniaxial creep-rupture tests
In addition to the aforementioned inherent variability in properties of the swirl-mat polymeric composite, a factor that significantly contributes to the scatter in creep-rupture data is the high sensitivity of the behavior of the considered material to fluctuations in the ambient environment (temperature and relative humidity). Typically creep-rupture tests require long durations, where the effects of uncontrollable fluctuations in a laboratory environment accumulate and may change results significantly. It is apparent that the large scatter in the limited experimental data in Fig. 8-5 undermines the reliability of long-term predictions based upon the values listed in Eq.(8-321). We therefore restrict attention only to times that are short in comparison with the time-to-failure ttl so that the effects of environmental fluctuations can be neglected.
666
8 T heory of Visco-elasto-plastic Damage Mechanics
Short-term creep tests (approximately 170 hours) were performed under stress levels of 55, 69 and 83 MPa at temperature T = 75° F and 50 percent relative humidity. The above stresses were ramped at the same rate as that for the t ests in Fig. 8-4 so that the expression for no in Eq.(8-320) remains applicable. Under creep conditions CT = const. , and substitution ofEqs.(8-312) , (8-314) and (8-316) into Eq.(8-311) yields
E = So_CT_ 1-
T
n
+ SI_CT_ 1-
f (t - T)"'-dTd ( I- '=' ) t
n 00 -
-l/(1+X)
tc
dT
(8-322)
Upon integrating by parts and changing the integration variable from can be solved [8-96] so that
It, the integral in Eq.(8-322) E=
where
CT - + S It -CT- ( 1 + ~r(K: + l)r(p + n)(t)n) S0 L.,.. I - n I - no n=l r(K: + l +n)r(p) tc K,
r
T
to
(8-323)
is the Gamma Function and
p = I/(I +X)
(8-324)
If simply denoting
(8-325) the function F(I , p, 1+K:; t l t c ) is a hyper-geometric series that converges for t ~ tf
< to·
Experimental data along with model predictions according to Eq.(8-323) can be found in Ref.[8-30]. A good agreement between model predictions and the experimental results was shown in [8-30]. Using Eqs.(8-318) and (8-320) and the values for the creep-rupture constants in Eq.(8-321), the times-to-failure are estimated as tf ~ 1.07 X 107 , 1.72 X 105 and 1.14 x 10 4 hours for the stress levels 55, 69 and 83 MPa respectively. Obviously a creep duration of 170 hours represents only a very small fraction of the lifetime, during which the damage remains essentially constant at its initial value no, as can be verified from Eq.(8-316). This is in agreement with the experimental observations [8-30] that after creep periods ranging from 0.5 to 170 hours the unloading compliance S{) remains essentially unchanged , implying that damage remains practically constant. Thus, for the short-time creep-damage behavior t « t f < t c , a good approximation to the constitutive Eq.(8-311), or equivalently Eq.(8-323), is
n
CT
E ~S(t)--
1 - no
(8-326)
8 .7 Numerical Studies on Visco-elasto-plastic Damage Behaviors
667
Indeed, using Eq.(8-320), the difference between this approximation and the exact calculation of the integral in Eq.(8-311) may be neglected for the creep durations shown in Fig. 8-6. 0.030.---------------, (}"= 83 MPa 0.025
(T=75°F)
0.020
0.010 0.005 0.000 '--.l...--'----'--1...----'----'---'----'_'--.l...--'--".1 o 4000 8000 12000 Time (h)
Fig. 8-6 Results from model prediction for creep-damage behavior up to failure
For illustration purposes, the model prediction for the long-term creepdamage behavior up to failure as obtained from Eq.(8-311) for CJ = 83 MPa is shown in Fig. 8-6. This figure demonstrates that, similarly to metals, the material at hand exhibits a significant amount of tertiary creep prior to failure. Features of such behavior were observed in creep-rupture tests, especially at high temperatures [8-95]. The fact that tertiary creep can be significant for the material considered herein is also consistent with the relatively large value found for nf. It is also noted that the estimated strain to failure of about 2.3% (see Fig. 8-6) is consistent with experimental data [8-95]. Such value is relatively small and, hence, the small strain formulation adopted in this work can be utilized for life-time assessment of structural components made of the swirl-mat polymeric composite considered in this section.
8.7.1.3 Discussion of Remarks The experimental application of a coupled visco-elastic damage model was proposed to verify the theoretical framework presented in subsection 8.2.3 that accommodates visco-elasticity, continuum damage and permanent viscous deformation. Using scalar damage, it was shown that when the proposed model is applied to swirl-mat polymeric composites subjected to uniaxial tensile stresses, a complete identification of all parameters in the model is obtainable from creep data and from time-to-failure information. In this work, damage evolution was related by the empirical forms suggested by Kachanov. These relations contain fixed stress parameters that serve
668
8 Theory of Visco-elasto-plastic Damage Mechanics
as thresholds for the onset of damage, and are best suited for monotonic creep loadings. For more complex loading histories involving, for instance, complete or partial stress removals, the onset of damage is usually not related to a specific threshold stress. In these cases, the concept of damage surfaces ([8-8]) offers a more versatile approach to damage evolution. It appears, however, that these damage surfaces are more suitably expressed in strain space than in stress space. Finally, while emphasis in the work of subsection 8.7.1 was placed on the visco-elastic part of the deformation, several deformation mechanisms and modeling approaches to permanent deformation can be readily accommodated within the proposed thermodynamics framework.
8.7.2 Observation of Asymptotic Integration for Visco-plastic Damage Problems 8.7.2.1 Behavior of Numerical Results at Gauss Point Level The above integral constitutive equations derived in subsection 8.3 and the corresponding implicit. Asymptotic Integration scheme (AI) have been implemented in the general purpose finite element code by Nesnas and Saanouni [8-31]. The differential constitutive equations with two local integration schemes, namely the explicit Runge-Kut t a scheme (RK) and the implicit Euler Cauchy scheme (EC) , are used for comparison with the proposed formulation. Moreover, the derivation of the consistent tangent matrix from the integration scheme is necessary to preserve the quadratic convergence of the global Newton type equilibrium iteration scheme [8-97]. However, since several algorithms have been used , the matrix has been calculated, in the work [8-31]' by means of a numerical perturbation method. Despite its expensive cost, this technique has an advantage when used with a large class of integration schemes. This allows one to treat similarly the different algorithms and to focus the comparison on the efficiency of the local integration scheme. Numerical examples, in both Gauss point and structural levels, are given to demonstrate t he utility of the proposed numerical scheme. Several aspects are illustrated through these examples namely accuracy, convergence, cost and applicability in structural analysis. The adopted material parameters are compiled in Table.8-1. Computations are carried out with both isochoric and non-proportional biaxial mechanical loading paths. All examples are systematically computed using both uncoupled and coupled damage models.
8.7.2.2 Numerical Observation of Uncoupled Model The accuracy analysis of the integration scheme is performed by specifying a cyclic strain history and employing the implicit asymptotic algorithm to
8.7 Numerical Studies on Visco-elasto-plastic Damage Behaviors Table 8-1 Used material parameters Parameter Value E 144,000 Isoreopic elasticity v 0.3 N 10 Viscosity K 2000 Yield stress 211 k 3000 Q Isotropic hardening b 10 10,000 C Isotropic hardening 20 a S 10 1 s Ductile damage (3 1 0.3 'Y r 10 Fatigue damage 15 rJ
669
Unit MPa
MPa MPa MPa MPa
obtain the stress response. Three selected cyclic strain histories namely I, II and III are used (Fig. 8-7 and 8-8). Each history is composed of straight segments, connected to define a closed cycle in the biaxial Cll - C22 space. Simulations are carried out at a constant strain rate of O.OOl/s with different values of a constant strain increment size (0.0005 , 0.001 , 0.005 and 0.01). The resulting stress-plastic strain behavior is given in Figs. 8-9, 8-10 and 8-11 It is noteworthy that the stress plastic behavior is extremely non-linear in large parts of the hysteresis loops and thus, the selected strain histories are believed to constit ute a realistic test for the integration algorithm. In fact, these imposed strain paths are characterized by a "severe" rotation of the outward normal to the yield surface. As shown
~
.---.----.-!----,i~..-...-...-.... , :'. -::-: -::-:,:I. :. -::-: -:.~,
:: ::1: :: 0.03
o .. _._.3. ! !
-0.01 .. - .... _..
j. . _. ..-
.02 .. - ·_· ....
1"......·..
--~141~
.
:
+....·-
l. ·. ·. .,·. ·. ·. . 1"'..·. ·..l. ·. ·. ·-
-0.03 '----'----'-----''---''''''----'----' -0.03 -0.02 -O.Q1 o 0.01 0.02 0.03 s train en History I
Fig. 8-7 Cyclic strain histories I
670
8 T heory of Visco-elasto-plastic Damage Mechanics
0.Q3 ,---,---,-----,,---,..----r----,
+.. . . . . +.. . . . . .... . . . .+.. . . . . ·1i-·····_-··· ~
0.02 ..- ....... : 001
~
--+--1 --1-+--1---3·to-o-oo-to-o-OO-I'·o-oo-ot°-o*ot··+··-···
0 ..- ....
~: =:~r :I: :~ : l::I:::: !
!
!
!
!
0
0.01
0.02
- 0.03 '----'-----' 0.'0.'- --'------'---'-----'
.03 -0.02
. trai n En
0.Q3
Hi tory II
...........1 ....-.... !
-. -- ~- .-- .-----
is
'"
!:
0 -·······5
2·--····
- 0.01
o
strain &11 History ill
0.01
0.03
Fig. 8-8 (a) Cyclic stra in histories III; (b) Cyclic strain histories III
Fig. 8-9 Behavior of plastic strain-stress relation at different step size ilt in strain history I (for uncoupled calculations)
8.7 Numerical Studies on Visco-elasto-plastic Damage Behaviors
671
o
o .....
,~ ,. ~"""'f;EI&-
-2 .
.
:
:
.
.
-4 ..... ,... :....... ,.~ .. ,..... ,' ....
-0- 0.5 x 10 ' -O- I.0 x 10 ' .. . ----.1>- 5.0 x 10 ' ___ IO.O x 10'
-6 L-~--~--~--~~~~
-3
-2
-I
Sue
0
0"11
2
3
( x 10 ' MPa)
Fig. 8-10 Behavior of plastic strain-stress relation at different step size ilt in strain history II (for uncoupled calculations)
.
,
,
6 1F.::;..:.c""-""=~fi ······:······, ' ··· ' ·· '· · ' ····
.. .. .. .
4
o
~
~.
2
~
__IBOliF-fo,....·····,··············· .
... L ....
-0- 0.5
x
10 '
-o- 1.0 x 10 ·
----.1>- 5.0 x 10 ' ___ 10.0 x 10 '
_ 8L-~~--~~~--~~~
-4 - 3 - 2 - I
0
234
Stress 0"11 ( x 10 ' MPa)
Fig. 8-11 Behavior of plastic strain-stress relation at different step size ilt in strain history III (for uncoupled calculations)
in Figs. 8-9, 8-10 and 8-11 , good accuracy is obtained. The results are quite also reasonable for the large increments and in particular, accurate solutions are achieved in the linear parts of the hysteresis loop where the normal to the yield surface has a fixed direction. In the nonlinear parts, the predicted stresses are correctly estimated. Nonetheless, the robustness of the asymptotic algorithm is clearly demonstrated by these results.
672
8 T heory of Visco-elasto-plastic Damage Mechanics
8.7.2.3 Numerical Observation in Coupled Case This purpose is to study the stability, the accuracy and the convergence speed of the asymptotic algorithm in the coupled case. A comparison has been performed on the AI, RK and EC algorithms. The Gauss point is subjected to simple elongation (plane strain) at a constant strain rate of O.OOl /s. Fig. 8-12 is plotted to show t he stability and the accuracy of the AI scheme as the strain increment size is increased. It appears that the numerical solut ion is stable, even in the softening stage where the non-linearity due to the damage is dominant. The accuracy is also satisfactory. However, the difference between the responses appears in the softening stage when the stain increment size increases. This generated error is acceptable in this stage where the evolution of the variables is highly non-linear. Moreover, CPU time and errors, taken at an elongation of 7%, are quite different for the different schemes. Table.8-2 shows the results for both uncoupled and coupled cases. The error is evaluated for a considered variable by determining a relative error in comparison to a reference numerical value calculated by the RK scheme with a strain increment size of 0.00001. One may see t he diminishing of the error when the increment size is decreasing particularly for the AI scheme which demonst rates its convergence. This latter case gives an error more important in comparison to EC and RK schemes, corresponding respectively to secondand fourth-order integration schemes. This seems to be due to the accuracy of the approximation (first-order Taylor series) used in the development of the AI scheme. However, the RK scheme, despite its high accuracy, finds it difficult to integrat e the constitutive equations when the material coefficients are chosen to represent the time-independent plasticity. In fact, these equations become stiff. This mathematical stiffness requires a very small strain increment in order to integrate the constitutive models without loss of stability. As a result , the computation time of the RK scheme becomes enormous and under complex loading, solving the problems often becomes impossible. The AI scheme, on the other hand, keeps t he same efficiency to integrate these kinds of equations. Details about integrating the stiff differential equations can be found in [8-31]. The accuracy may be definitely improved by using higher order Taylor expansion so that it leads to more accurate evaluation of the non-homogeneous integrals. In addition, the error produced by the AI scheme is more stable in the sense that it does not vary significantly when the increment size increases (10 - 6 rv 10- 5 for uncoupled case and 10- 4 rv 10- 2 for coupled case). This result may be justified by the fact that the AI scheme becomes accurate when the increment size increases, since it tends towards t he asymptotic solution corresponding to the exact asymptotic solution of the resolved constitutive equations. The results of the coupled case show, on the other hand , the same conclusions, except that the errors are more important than in the uncoupled case due to the high non-linearity of the damage evolution. Comparison of CPU time indicates that the AI scheme is more computationally
8.7 Numerical Studies on Visco-elasto-plastic Da mage Behaviors
673
2.0 .-------.,.------,---....,-----:-----,
0= 1.0·· ··· ··· · .... .;.... .. ........ :
i
.. ~ --
-0- 0.5 x Hr'
- o - l.O x la"' ........ S.Ox l a"'
Q
.
" 0.5 .. ... ... .. __ 10.0 x 10"' .... . ... ... ... ; ..
~
·3
S" o
0.1 0.15 0.2 Accumu lated plastic strain s'
0.05
0.25
Fig. 8-12 Equivalent stress 0"11 versus accumulated plastic strain accuracy of the AI scheme (coupled calculations) Table 8-2 Comparison of CPU time and errors Strain Increment Scheme EC Uncoupled case 0.0005 C PU time 2.04 2.98xlO - 8 Stress error 0.001 CPU time 1.10 1.15xl0- 7 Stress error 0.005 CPU time 0.33 2.38x 10- 6 Stress error Coupled case CPU time 2.33 0.0005 6.48 xlO - 6 Stress error 2.56xl0 - 6 Damage error 0.001 CPU time 1.34 1.12x 10- 4 Stress error 5.45x 10- 5 Damage error 0.005 CPU time 0.38 4.41xlO- 3 Stress error 2.26xl0- 3 Damage error
EP
for stability and
for different schemes
RK
AI
2.28 6.44 x 10- 12 1.23 4.91xlO- 12 0.41 3.22x 10- 10
1.91 2.20xlO- 6 1.01 4.26x 10- 6 0.34 1.95x 10- 5
2.37 2.34 xlO- 8 2.24xl0- 8 1.32 5.51x10- 7 1.04xl0- 7 0.42 4.46 x 10- 5 7.02xl0- 6
2.05 3.02x 10- 3 9.89x 10- 4 1.09 7.66x 10- 3 1.72x 10- 3 0.37 3. 21x10- 2 1.04x 10- 2
efficient , despite its iterative nature. This results from the number of the resolved equations which are reduced to 2 (and 1 in the uncoupled case). The CPU time can be also improved if an adaptive increment control is used. 8.7.3 Numerical Studies of Visco-plastic Damage Behavior in Simple Structures
To demonstrate the numerical behavior of the algorithms with finite element analysis, both the asymptotic integration and the fourth-order RK algorithms are used for comparison efficiency. Although the constitutive equations incor-
674
8 Theory of Visco-elasto-plastic Damage Mechanics
porated in the program can be used for any general three-dimensional state of stress, the problems considered here are merely two-dimensional.
8.7.3.1 Application to Simplified Three Bars Structure In order to examine the stress distribution due to the damage effect and the related solution with the studied algorithms, the example of a three bars structure shown in Fig. 8-1 3 is used. The three bars are constrained to follow the same displacement under cyclic strain control (cycled between ± O.016 total strain within a period of 64s). A severe stress concentration can be obtained with this simple structure despite the homogeneous stress field inside each bar. The simulation is performed in order to compare the solution of the asymptotic algorithm versus the RK algorithm. The local responses are presented in Fig. 8-14, where the maximum equivalent stress and the damage versus the reduced cycle number (NR is the lifetime of the structure) are plotted for each element with both AI and RK algorithms. It can be shown that correspondence between the two algorithms is fairly good. The lifetimes of the structure corresponding to the failure of the three bars are 718 and 698 cycles respectively for RK and AI schemes. One may say, however, that AI is more computationally attractive since the CPU time of its calculation is 8334s in comparison to RK which needs 14327s of CPU time. The calculation with the AI scheme may be also improved significantly by its association with an adaptive time step size control technique.
-
Fig. 8-1 3 A simplified three bars structure
8.7.3.2 Application to Plate with a Central Circular Hole This example concerns a rectangular plate with a centered circular hole as shown in Fig. 8-1 5. The material coefficients are those of Table.8-1 , except
8.8 Effects of Localization Approach to Creep Fracture Damage
675
I----!--+--+-+--+.
1.0
r-. I. I
--e--Element I, RK ---a-- Eleme nt 2, RK _.- -----6-- Element 3, RK - - . --Eleme nt I, Al - -.- - Eleme nt 2, Al - -. - - Element 3, Al
!
!
+i _._0 . I I I I I I ·-·T-·_"-"t- _·t--·-t T--·-·1 ._.+._00_ +_ _00+_'- 100_._00
oo
_'_
OO
0.2
o
o The maximum equivalent stress (x 10' MPa)
Fig. 8-14 Behavior of three bars structure subjected to cyclic loading with RK and AI schemes (coupled calculations)
for the damage law where the coefficients I and R are taken respectively as 0.35 and 10. The two opposite ends of the plate are subjected to uniform displacements with no lateral constraints. A complete loading unloading cycle at a constant strain rate of O.OOl/s is applied to the plate within a time period of 28s. By taking advantage of symmetry, only a quarter of the plate was modeled by 288 eight-nodal plane strain elements. Calculations are conducted also with both AI and RK schemes. Local responses are represented in Fig. 8-16 at the Gauss points A, Band C belonging respectively to elements 277, 217 and 145. The first broken Gauss point belongs to element 277 (Gauss point A in Fig. 8-15). It turned out that both algorithms give comparable results at different points. The lifetime of the first broken Gauss point, obtained from the two calculations, totals 159 and 154 cycles respectively with RK and AI. Variation between the two values is weak (about 3.1 %). Although no adaptive time step control is performed, the time calculation is less important for the AI scheme (34632s) in comparison to the RK scheme (with an adaptive time step technique (39985s)), giving about 15% of difference.
676
8 Theory of Visco-elasto-plastic Damage Mechanics
u
§ 00
......
Fig. 8-15 A plane strain sheet with a central circular hole n)() P~I=q:::;:::::=t==::::lLI~l -e- Ekn~lIl I, RK
- G - Ekn~lIl 2, RK ----0-- Berrlellt 3, RK
1(XX)
- - . - - Ekr'llClI(
----~
L At
r --. --Berrlell! 2. At Ekr'lICII( 3,
At
-OO ~--+---1---~---+--~·---1
00~"""'0"L.:2;-'-'''''''''0~ .4~''''0:-l:.6'''''''''''''''''0J,; .8~~''''''''~1.2 Redyced cycle nun"er (NINR)
0.8 f----t---t--f-----'f----i-if----l 0.6 ~--+---1----+---+----;-l--____l --e-- Ekme~ I , RK 0.4 f- =!=~~::i:~~ -+---1--+--1 - - . - - Ekmelll I , .'\1
0.2 f- --·-I- ~~::r~;
.-.~
o Ot..........Ob.2=.:::O±.4::;;:::::;:OI.6:::::.0J:8.........---..J--'-
0.6
~ 0.4 ~
a
I
I'
~
0.2 0.0 ---~~=----.---.--....---..-----, o 3 6 9 12 15 18 21 T ime t (s)
Fig. 8-51 The damage evolution curve
8.9.4.5 Safety Assessment of Longtan Gravity Dam under Earthquake After duration of the earthquake, the maximum residual horizontal displacement at the crest of the dam is 8.65 cm. Because the hight of concrete gravity dam is 190.5 m, this residual deformation (0.05 %) of dam body is not enough to cause instability of the dam structure and loss of the reservoir. So it is still in the safety range of deformation of the dam body. The result of stress analysis shows that stress concentration appears at the dam heel and dam toe, but the maximum stress is still on up leg of stressstrain curve of concrete. In the duration of the earthquake, when t = 16.88s, the maximum vertical t ensile stress at a local area in the dam heel reaches -3.223 MPa, and when t = 16.90s, the maximum horizontal t ensile stress at a
References
715
local area in the dam heel reaches - 2.214 MPa. Since these two tensile stress values may exceed the ultimate dynamic tensile strength of concrete. So some cracking may occur at the dam heel. The damage of the concrete behaves in the form of limited cracking, the situation of which is illustrated by the moderate damage state in the dam. It can be shown from results that the damage value at the dam heel and the dam toe is more significant, and damage values of these two sites reach 0.4 due to the first actions of the earthquake (initial shock), and reach 0.629 and 0.583 after sequential actions of the earthquake (second shock) respectively. The result of dynamic damage analysis shows that the damage of these two values in Longtan gravity dam heel and dam toe is considerable higher, but the zone with high damage values is still very small relative to the dam body. So some cracks may appear and exit in somewhat zone of the dam structure, cracking not extensive enough zone to cause instability of the dam structure and loss the reservoir. So, after duration of the violent earthquake, the Longtan rolled concrete gravity dam may come in some damage state, but it is still possible to retain the water stored in the reservoir and the stability of the dam structure. It should be pointed out, however, all local damages in the dam and foundation after the earthquake should be seriously taken into concerning for reinforcing them in order to protect another seismic actions.
References [8-1] Kachanov L.M., Time of the rupture process under creep conditions. TVZ Akad Nauk S.S.R Otd Tech. Nauk, 8(1-4), 26-31 (1958). [8-2] Murakami S. , Ohno N., A continuum theory of creep damage. In: Proceedings of 3rd IUTAM Symposium on Creep in Structures. Springer, Berlin, pp.422-444 (1981). [8-3] Schapery RA ., On viscoelastic deformation and failure behavior of composite materials with distributed flaws . ASME J . Adv . Aero. Struct. Mater., 23(2) , 5-20 (1981). [8-4] Schapery RA ., A theory of nonlinear thermoviscoelasticity based on irreversible thermodynamics. In: Cardon A.H ., Fukuda H. , Reifsnider K. (eds.) Progress in durability analysis of composite systems.) Balkema Publication, Rotterdam, The Netherlands, pp.21-38 (1996) . [8-5] Weitsman Y., A continuum damage model for viscoelastic materials. J . Appl. Mech., 55(4), 773-780 (1988). [8-6] Lemaitre J ., Chaboche J ., Mechanics of Solid Materials. English translation, translated by Shrivastava B. Cambridge University Press, New York (1985). [8-7] Lemaitre J., A Course on Damage Mechanics. Springer, New York (1992). [8-8] Krajcinovic D., Damage Mechanics. Elsevier, New York (1996). [8-9] Betten J ., Creep theory of anisotropic solids. J . Rheol. , 25(6) , 565-581 (1981). [8-10] Rabotnov LN., Creep Problems in Structural Members. North-Holland, Amsterdam , London (1969) . [8-11] Betten J ., The classical plastic potential theory in comparison with the tensor function theory. Eng. Fract. Mech., 21(4) , 641-652 (1985).
716
8 Theory of Visco-elasto-plastic Damage Mechanics
[8-12] Betten J ., Damage tensors in continuum mechanics. J . Mecan. Theor. AppL , 2(11) , 13-32, (1983). [8-13] Betten J., Net-stress analysis in creep mechanics. In: Proceedings of the 2nd German-Polish Symposium on Inelastic Solids and Structures, Bad Honnef, Germany. Ingen. Arch., 52, 405-419 (1982) . [8-14] Kawai M ., Constitutive modeling of creep and damage behaviors of the nonMises type for a class of polycrystalline metals. Int. J. Dam. Mech. , 11(3) , 223-245 (2002). [8-15] Zhen C., A model-based simulation procedure for the evolution of t ertiary creep with combined damage diffusion. Int . J. Dam. Mech., 14(2) , 149-163 (2005). [8-16] Betten J. , Tensorrechnung fur Ingenieure. Teubner , Stuttgart , B .G ., in German (1987) . [8-17] Betten J ., Recent advances in mathematical modeling of materials behavior. In: Proceedings of the 7 th International Conference on Mathematical and Computer Modelling, Chicago. Math. Comput. ModeL, 14, 37-51 (1989) . [8-18] Bodner S.R. , Hashin Z. , Mechanics of Damage and Fatigue. Pergamon Press, New York, Toronto (1986). [8-19] Krajcinovic D. , Lemaitre J ., Continuum Da mage Mechanics. Springer, Wien , New York (1987). [8-20] Gittus J., Creep, Viscoelasticity and Creep Fracture in Solids. Applied Science Publishers Ltd. , London, pp.567-569 (1975). [8-21] Lagneborg R. , Creep: Mechanisms and theories . In: Bressers, J . (ed .) Creep and Fatigue in High Temperature Alloys. Applied Science Publishers Ltd. , London, pp.41-71 (1981) . [8-22] Evans H.E. , Mecha nisms of Creep Fracture. Elsevier Applied Science Publishers, London, New York, pp.66-96 (1984). [8-23] Davies P .W ., Dutton R ., Cavity growth mechanisms during creep. Acta MetaIL, 14(99), 1138-1140 (1966). [8-24] Courtney T .H., Mechanical behavior of materials. McGraw-Hili, Boston (1990). [8-25] Kawai M ., Constitutive model for coupled inelasticity and damage. JSME Int ., Ser. A, 39(4) , 508-516 (1996). [8-26] Kawai M ., History-dependent coupled growth of creep damage under variable stress conditions. Int . J . Metals Mater., 4(4) , 782-788 (1998) . [8-27] Kachanov L., Crack growth under conditions of creep and damage. In: Proceedings of the 3rd IUTAM Symposium on Creep in Structures. Springer, Berlin, pp.520-525 (1981) . [8-28] Betten J. , Applications of tensor functions in continuum damage mechanics. Int . J . Dam. Mech., 1(1) , 47-59 (1992) . [8-29] Betten J ., EI-Magd E ., Meydanli S.C ., Anisotropic damage growth under multi-axial stress: Theory and experiments. in preparation [8-30] Abdel-Tawab K. , Weitsman Y .J. , A coupled viscoelasticity/da mage model with application to swirl-mat composites. Int . J . Dam. Mech., 7(4), 351-380 (1998). [8-31] Nesnas K. , Saa nouni K., Integral formulation of coupled damage and viscoplastic constitutive equations: Formulation and computational issues. Int . J . Dam. Mech., 11(4) , 367-398 (2002).
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9
Dynamic Damage Problems of Damaged Materials
9.1 Introduction When a structural component is subjected to impact or dynamic loading, its response can cause an elevation in the stress level especially in a damaged zone or in the region surrounding cracks or defects. In particular, the microstructure of the material within the damaged zone is significantly changed compared to its undamaged state, due to the activation and growth of the damage [9-1 , 9-2]. The dynamic response of a damaged structural component is considerably different to the corresponding undamaged one due to the change in the micro-structure. For example, the frequency decreases and both the damping ratio and the amplitude increase. During damage evolution, the macroscopic properties of the material change too [9-3, 9-4]. In most cases, the deviation from the elastic response derives from the nucleation of new micro-cracks and the growth of existing micro-cracks. So it can be said that the non-linear behaviour of such materials arises as a consequence of the irreversible changes in the micro-structure, which is what happens in a damage process [9-5, 9-6]. It is of paramount importance in civil engineering field to be able to predict the effects of these damages on the frequency and dynamic characteristics of structures especially the ones subjected to long-term dynamic loading. The dynamic response of a damaged structural component and the dynamic behavior of damaged materials are dealt with in this study within a continuum approach using the concept of damage mechanics that will be discussed in this chapter. Hence, when analysing damage-mechanics problems, not only the damage initiation, growth and failure of a structure need to be taken into consideration, but a number of other mechanical properties of the material also need to be looked at [9-7, 9-8]. These properties may include elastic modulus, ultimate strength, yield stress, fatigue limit, creep rate, damping ratio and heat conductivity. The effects on these properties may be even more significant in cases of anisotropic damage [9-9, 9-10].
W. Zhang et al., Continuum Damage Mechanics and Numerical Applications © Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2010
724
9 Dynamic Damage Problems of Damaged Materials
From the numerical examples presented in this chapter, it was found that the dynamic loading applied to a damaged structure leads to significant growth and propagation of the damage, to a reduction of the natural frequencies of the system and to a state of resonance due to damage growth. In studying the properties of the damaged mat erials, it was found that the damping ratio increased significantly, whereas the equivalent viscous damping and critical damping decreased, owing to damage growth. In the present study, Audoin and Baste [9-11] developed a specific ult rasonic device by evaluat ion of stiffness t ensor changes due to anisotropic damage in a ceramic matrix composite in order to identify damage in a material. Pande and Biswas [9-12] developed an analytical model for detecting and locating damage in structures using changes in the flexibility matrix. Gamby et al. [9-21] presented a model to predict the non-uniform development of damage induced by a kinetic wave in composite laminates.
9.2 Fundamentals of Dynamic Damage Mechanics 9.2.1 Basic Equations of Dynamic Evolutional System The mathematical description involves the following equations, which can be represented in the form of incremental vectors and matrixes for isotropic damage cases as (1) Dynamical equilibrium equations
[d]{dO"}
+ {dF} = p{du}
(9-1)
(2) Geometrical equations (Strain-displacement relations) {de}
= [d]{du}
(9-2)
(3) Constitutive equations (Stress-strain relations) {de}
= {dee } + {de P } + {d'"{ }
(9-3)
= [D*]{ dee }
(9-4)
{dO"}
{de P }
= )..P
d~~}
F({O"* } , {e P }) )..P { __ >
~
(9-5)
0
(9-6)
00 for plastic loading or nuture load exchanging (when F = 0) for elastic loading or any type unloading (when F < 0) (9-7)
9.2 Fundamentals of Dynamic Damage Mechanics
725
(4) Kinematical evolution equations of internal state (such as damage development equations, et al.)
f ( } = h({a} , D)
(9-8)
D =f ({a},D)
(9-9)
(5) Boundary, initial and critical conditions (Solution determined conditions) (9-10)
{du} = {du} (on Su)
(9-11)
{u} lt=to = {uo} , {u} lt=to = {uo}
(9-12) (9-13)
D lt=to = 0 /1J *
=~
min
2 {u(t)} EU
{11( {du(t)} , {dX(t)}, t)M}
(9-14) (9-15)
tE[t,t+M]
In the above equations {a} is the stress vector {a x, a y, a z , T xy, T yz, T zx }T ; {c:} is the total strain vector {c:x, C: y, c: z , "Ixy , "Iyz, "Izx }T ; {u} is the displacement vector {u, v, w} T; D is the isotropic damage variable; {c: e } is the elastic strain vector; {c: P } is the plastic strain vector; {"Ie } is an internal nonlinear state vector (for example the creep strain vector, accumulative hardening vector , and so on) ; [D* ] is the effective constitutive property matrix of damaged materials; F is the yield function of the damaged material; G is the plastic potential function of the damaged material; )...P is the proportional plastic flow factor; {F} is the body force vector; {Q} is the boundary surface force vector; h is the internal evolution tensor function; f is the damage develop function; Dc is the critical value of damage (0 :::; Dc < 1); V is the space domain of the body; Sp is the surface domain of boundary force; Su is the surface domain of boundary displacement. The matrix [a] in Eq.(9-1) is the partial differential operator matrix. If only the small deformations of the solid continua are considered, the partial differential operator matrix is in (3x6) rank detail as
726
9 Dynamic Damage Problems of Da maged Materials
d d d 0 0 0 dy dx dz d d d 0 --0 [dl = 0 dy dx dz 0
(9-16)
O~O~~ dz
dy dx
The matrix [Ta l in Eq.(9-10) is the coordinate transformation matrix corresponding to the direction cosines {I, m, n} of the boundary with (3x6) rank as
l 0 0m 0
n]
[Ta l = [ 0 mOL n 0
(9-17)
OOnOml
The effective constitutive property matrix [D*l of damaged materials can be expressed correspondingly by the anisotropic effective elastic matrix [D:l of elastic damaged materials for anisotropic (elastic or brittle) damage analysis as Eq.(5-110) and Eq.(5-111) presented in Chapter 5 and by the anisotropic effective elasto-plastic matrix [D:pl of plastic damaged materials for coupled elasto-plastic-damage analysis as Eq.(7-69) and Eq.(7-85) presented in Chapter 7 or by the effective visco-elasto-plastic matrix [D~pl of visco-elastoplastic damaged materials for coupled visco-plastic-damage analysis discussed in Chapter 8. In the isotropic elastic or brittle damage case, the effective elastic matrix can be given alternatively either by a single scale model or by a double scale model in terms of the relationship [D:l = [tliElT[ De][tliEJ, where the detail of the undamaged isotropic elastic co-efficient matrix [D el is
[D el
=
Dl D2 D2 0 D2 Dl D2 0 D2 D2 Dl 0 0 0 o D3 0 0 0 o 0 0 0 0
0 0 0 0
0 0 0 0 D3 0
o
(9-18)
D3
where
D = 1
(1 - v) E- D = v E- D = 1 E (1 + v)(1 - 2v) , 2 (1 + v)(1 - 2v) ' 3 2(1 + v)
(9 19) -
in which E and v are Yang's modulus and Poisson's Ratio of undamaged materials. The above descriptions of a damaged solid mechanic system are different from the classical solid mechanic system based on the point view of its timedependent evolutional system, the nature of which is represented in the form of partial differential equations with spatial and time variation co-efficients.
9.2 Fundamentals of Dynamic Damage Mechanics
727
9.2.2 Variation Principle of Dynamic Evolutional Continuous System It is known from the classical variation principle that if the purpose is to establish the objective function of a system based on the principle of mechanical potential energy, the displacement vector {u} or strain vector {E} of the system can be taken into account as independent variables whereas, if the purpose is to establish the objective property function based on the principle of complementary mechanical energy, t he stress vector {a} of the system should be taken into account as independent variables. In order to optimally control an evolutional dynamic system, one needs firstly to know the kinematical rule of the target system, which means one needs to carry out mathematical modeling of this kinematical rule. For herein the studied dynamical damage evolution problem and the mathematical modeling of the kinematical system mean setting up the internal state evolution equation and the damage development equation as
{"( } = h({a},D)
(9-20)
n = f({a}, D)
(9-21 )
During the evolutionary process of the system, since the stress and deformation of solid materials should satisfy the constitutive equation from beginning to end, the constitutive relations must thus restrict the system. The independent variables {u} and {E} must firstly satisfy the strain-displacement relations (the stress {a} must satisfy equilibrium conditions); therefore, the constitutive restraint equations and the restrained conditions of independent variables consist of all the restrained conditions of the system. The constitutive restraint equations are defined by the constitutive relationships of Eqs.(9-3) to (9-7) in solid mechanics. Based on the principle of the mechanical potential energy of independent variables {u} (or {E}) , the incremental quantities of {du} (or {dE}) should satisfy the strain-displacement relationship in Eq.(9-2) and the boundary conditions of displacement thus
{du(t)}
E
U: ({du} l{dE} = [ClJ{du}, {du} = {du} on Su)
(9-22)
However , based on the principle of the complementary mechanical energy of independent variables {a}, the incremental quantities of {da} should satisfy the equilibrium equations and boundary conditions of forces , thus
{da(t)}
E
Uc : ({ da} 1[Cl]T {da}
+ {dF} - p{dii.} , [TaJ{ da} = {dQ}
on Sp) (9-23)
728
9 Dynamic Damage Problems of Damaged Materials
Therefore, Eqs.(9-3) and (9-4) with Eq.(9-2) provide the restrained conditions of independent variables {d u} and {dO"}. The objective property function needs to define the functional function of strain energy W( {du} ,t) or the system firstly within the time interval [t , t+~t] as
W({du},t) = f [A({du},t) - {dF} T {du} ]dV - f {dQ}T{du}dS v ~
(9-24)
where A( {du},t) is the functional function of the strain energy at time t , the specific expression of which is determined according to different problems that enable one to carry out equilibrium equations and force boundary conditions in the process of variational calculus for Eq.(9-24). In a similar manner, the functional function of the complementary strain energy II c ( { dO" },t) is given by
IIc({dO"},t) = f Ac({dO"} ,t)dV - f ([TO"]{dO"}) T{du}dS v Suo
(9-25)
where Ac( {dO"},t) is the functional function of the complementary strain energy at time t , the specific expression of which is determined according to different problems that enable one to carry out strain-displacement equations and displacement boundary conditions in the process of variational calculus for Eq.(9-25). Based on optimal control theory, the real independent variables {du (tn and {dO"(tn must make the following objective property functions possess the minimum quantities in the overall time process [to , t f ], where t f is the time of solid material failure. tf
J({u}) = min fW({du},t)dt
(9-26)
to
or tf
Jc({O"}) = min f IIc({dO"},t)dt
(9-27)
to
9.2.3 Unified Description of Dynamic Evolutionary Continuous System Since the constitutive relationship of a dynamic evolutionary continuous system is strong non-linearity and complexity, so one needs to manipulate it in order to obtain system equations in a standardized form. Thus, the yield function F ( { 0" }, { c P }) should be extended in the first order form as
9.2 Fundamentals of Dynamic Damage Mechanics
F({17}, {e: P }) = FO
(d~~}) T {d17} + (d~~}) T {de:
+
729
(9-28)
P}
where the superscription "0" indicates the previous state before increment. From Eqs.(9-3) to (9-7) we have
{d17}
= [D *] ({de:} - {de:P }
-
{d-{ })
= [D *]{de:} - AP d~~} {de:P }
-
(9-29)
where ).C = {d,C }, Substituting Eq.(9-29) into Eq.(9-28) gives
dF F({17}, {e:P }) =Fo + ( d{17}
)T [D *]{de:} 1
dF ) T dG (dF) T * dG P + [( d{e:p} d{17} - d{17} [D ]d{17} ). _
[D*] )'c
(9-30)
(~)T [D*] )'C d{17}
In order to obtain the loading condition repressed in Eqs.(9-6) and (9-7) , a required non-negative compensatory factor 1'0, should be introduced herein making
{
F({17}, {e: P }) I'O,).P = 0,1'0, ?:
+ 1'0, = 0 O,).P
?: 0
(9-31 )
Eq. (9-31) can be rewritten in the form of a more general type as
{
0.0003
(9-193)
When the spectrum of amplitude and phase of the incident transient wave was calculated by Eq.(9-176), the relationship between the amplitude vector {Hd of the incident transient wave at point A and the amplitude vector {HT} of the transmission transient wave at the end point B can be obtained by Eq.(9169) Consequently, the response of the transient wave can be expressed by Eq.(9-177). Model of Linear Damage State Function: The linear damage state function in the region AB can be considered again as a more simple transient wave model, the expression of the linear damage state model D(x) is chosen rather simply as
D(x) = 2x, 0:::; x :::; I
(9-194)
Based on the computational algorithm provide previously, considering N = 30, the enveloped figure of the frequency spectrum of the incident transient wave at point A and the transmission transient wave at point B are shown in Fig. 9-24, as well as the transient response of the transmission wave in the time domain shown in Fig. 9-25. 0.040
E
..s ~
0.035
- - Incident wave
0.030
----- Transmis ion wave
0.025 0.020
0.015 0.010
0.005 20000
40000 (J)
6000 (radls)
80000
100000
Fig. 9-24 Amplitude sp ectrums of incidence and transmission wave in linear changing d amage model
Fig. 9-24 and Fig. 9-25 show that the weakened phenomenon of the stiffness in concrete materials cause the displacement amplitude of the transmission transient wave in the frequency domain to increase more than that of the incident transient wave, but the variation tendency is unchanged. The calculation process also shows that the phase of frequency spectrum has changed more due to the inhomogeneous effect of mat erials, and the phase variation in
774
9 Dynamic Damage Problems of Da maged Materials 0.14 0.12 0.10 0.08 :::: 0.06 0.04 0.02 0.00 -0.02
~
0
l.0
t (ms)
2.0
3.0
Fig. 9-25 Receiving response curve of linear changing damage model
the propagation process of different harmonic wave components with different frequencies is more significant, which means a little increase in the signal amplitude in the time domain, but the model form of waves has a certain anamorphosis. Second Order Polynomial Model of Damage State Function: Considering another second order polynomial model for the damage state function in the region AB , the expression of the different model D(x ) is chosen simply for the transient wave as D(x )
= - 10x2 + 6x, 0:::; x:::; I
(9-195)
Since the variation rat e of the damage state function changes with x varying, when choosing the discrete number N = 300, similarly to the linear transient model, the frequency spectrum of the incident transient wave and the transmission transient wave are shown in Fig. 9-26, as well as the transient response curve in the time domain of the transmission wave received at the end point B as shown in Fig. 9-27. 0.050 0.044 0.040 0.035 E 0.030 2: 0.025 S 0.020 0.QI5 0.010 0.005 0
- - Incident wave ---- - Transmission wave
0
20000
40000 (j)
6000
80000
100000
(radls)
Fig. 9-26 Amplitude spectrums of incidence and transmission wave in conic changing damage model
9.5 Wave Propagation in Damaged Media and Damage Wave
775
0.20 0.15
~ 0.10 ::::
0.05 0.00 -0.05 +----,---,-------,----.-----,----r 1.0 2.0 3.0 o t (ms)
Fig. 9-27 Receiving response curve of conic changing damage model
The variation characteristics of the signal illustrated in Fig. 9-26 and Fig. 9-27 are similar to those of the linear model, but the increasing size of the transmission wave amplitude in the time domain is greater than that of the model of damage linear variation, since the stiffness of the material is weakened even more severely. Opposite Test in Experimental Model: Considering the opposite test model as that where the signal emissive point A and receiving point B are put on the two sides of the damaged structure, the damage state of the material appears as an inhomogeneous variation along with the testing path. Assuming the material density is still a constant (similar to before), the inhomogeneous character is determined by the damage state function f?(x) , the distribution form of which is illustrated in Fig. 9-28.
c:
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0
0.05
0.10
0.15
0.20
0.25
0.30
x(m)
Fig. 9-28 Sketch of damage variety in opposite test model
Launching a transient input wave from point A, in order to apply the combination method, both of the frequency domain analysis and transfer matrix analysis discussed previously, into the stress boundary value problem , here an
776
9 Dynamic Damage Problems of Damaged Materials
alternative boundary condition of the emissive wave is given differently to the displacement boundary condition presented in the previous subsection and is considered as an emissive boundary condition of the stress wave.
f(t) = {
O.IMPa,
o ~ t ~ 0.00003 t < 0 or t > 0.00003
0,
(9-196)
According to the located state of the signal source point and the received point, we understand that the stress wave supplied at the boundary is a longitudinal wave, i. e. 0"1 (O ,t) = f(t) , 0"1 is the normal stress along the propagation direction. Transforming the problem into the frequency domain, we have
_* (
0"1
) _ E* (X)dU(X ,W) dX
(9-197)
x,w -
where 0-1' (x, w) and U (x, w) are the images in Fourier transformation of effect ive stress O"i (x, t) and displacement U(x, t) with respect to time t. Obviously, the co-efficients of the wave spectrum still satisfy Eq.(9-169), and the emissive condition of Eq.(9-196) can be represented in the frequency domain as (9-198) where subscript "I" indicates the paramet ers corresponding to the first layer of the inhomogeneous region after discretization; F(w) is the Fourier transformation image of f(t). Using a similar algorithm to calculate, the frequency spectrum and displacement response of the received signal at point B in the time domain are plotted as shown in Fig. 9-29 and Fig. 9-30.
E:::t
~ 0.02
0.01
o o
100000
2()()()()0
w(rad/s)
Fig. 9-29 Amplitude spectrum of transmission wave in opposite test model From Fig. 9-30 it can be seen that a significant phenomenon of additional oscillation in the received signal occurs. This phenomenon makes it difficult
9.5 Wave Propagation in Damaged Media and Damage Wave
777
2.0 1.8
I
::
1.6 1.4
1.2 1.0 0.8 0.6 1.0
I
(m )
2.0
3.0
Fig. 9-30 Receiving signals in time doorman of opposite test model
to solve the problem in the inverse analysis. In order to eliminate the additional oscillation in the inverse analysis, a Laplace transformation should be applied to the analysis for the complex frequency domain that may efficiently overcome the difficulty caused by this phenomenon. Discussion of Conclusion: Using frequency domain analysis and the transfer matrix of wave spectrum co-efficients, the impulse response in the time domain after the wave has passed through the inhomogeneous damaged region can be analyzed transiently. Due to the existence of damage, both amplitude and phase frequency spectrums of the output wave have been changed therefore, which causes the wave amplitude to increase and the wave model form to anamorphosize. From numerical examples, it can be found that the greater the damage, the more significant the increase in amplitude and the change in the anamorphosis of the wave model form. Some phenomena of additional oscillation happened under certain boundary conditions. Research into all these phenomena provides an important basis for the method of inverse analysis for identification of a medium 's properties in the further application of the transient wave response field. 9.5.4 Kinematic Wave Applied to Crack Tips 9.5.4.1 Illustration of Damage Development Due to Wave Experiments reveal that transverse cracks always start at the free edges of specimens, then slowly multiply and propagate toward the center of specimens, thus building up a non-homogeneous damage distribution across the width of specimens. At a given number of cycles N, damage depends on the distance x from a free edge. Our objective will be first to propose a model leading to an equation governing a pseudo-damage variable J), which as a function of x and N is not the same as the traditional defined damage, and then to compare predictions of the model with the experimental results [9-21 ]. The structure under consideration is assumed to be a three-layered laminate. As shown in Fig. 9-31, transverse cracks are located in the central off-axis
778
9 Dynamic Damage Problems of Damaged Materials
layer. They start from a free edge and occupy a part of the width of the specimens (in x direction), in contrast to the situation observed in quasi-static tests or in comparatively thick 90° layers. Their propagation is constrained by the presence of the outer 0° layers.
Fig. 9-31 Cracks in a constrained layer transverse to the load direction In the early life of specimens, cracks are short and few in number and they occupy narrow strips along each specimen edge. As the number of cycles increases, these cracks become longer and closer to each other, as shown in Fig. 9-32. However, during a significant part of the specimen's life, there is no interaction between the two crack arrays emanating from the free edges.
I -.....--.-+-- x
t-
-
f-
-
N= 2000 cycle
N= 5000 cycles
N =50000 cycles
Fig. 9-32 Transverse crack development during fatigue Here, damage f}( x, t) at position x and time t is defined as the crack density in the y direction (i. e., the load direction), that is the ratio f} = n ell , where ne is the number of cracks crossing an element of length I along the y direction, as shown in Fig. 9-33. In the special case of interest here, this definition is consistent with the general framework introduced by Talreja [9-66] and Allen, Harris and Groves [9-67, 9-68].
9.5 Wave Propagation in Damaged Media and Damage Wave I I
::------1..~ In.=
I
779
I
I
f+ 11I,= 41 •
I
11,= 2
~11I,= 3 1
I
x
= 0"":. 0 - - - - -____01-1••- - - - -_ _ x = 21 7.5 Illm
I
7.5 mill
n=II/1 Fig. 9-33 The pseudo-damage definition
9.5.4.2 Modeling Kinematical Waves In order to model the phenomenon, we resort to the concept of kinematic waves introduced by Lighthill and Whitham [9-69] to describe traffic flow on highways. Conservation Equation: The basic assumption is that a new crack tip can appear only at the specimen edges, which is true as long as the matrix does not contain sizeable defects in its original state. Then a conservation equation for crack tips can be written, which connects the number of crack tips entering and leaving a rectangle of dimensions (X2 - Xl, I) in the specimen to those contained in the same rectangle, as shown in Fig. 9-34.
q(x,. I) :~~~!§ , ~,~H~,!==.1
q(x" f)
.1 : I
.' I
x-x,
Fig. 9-34 Crack tip conservation
More precisely, let ne(x, t) denote the density of crack tips at X, i.e., the local number of crack tips per unit length in a strip of height I, and let q(x , t) denote the flux per unit time; i.e. the number of crack tips crossing an element of height I at position X per unit time. The rate of change in the total number of crack tips in the above rectangle of finite dimensions (X2 - Xl, l) is balanced by the net in-flow across Xl and X2 d
dt
f ne(x, t)dx + X2
q(XI '
t) -
q(X2 '
t) = 0
x,
Taking the limit as
X2 ---+ Xl,
we obtain the conservation equation
(9-199)
780
9 Dynamic Damage Problems of Damaged Materials
(9-200) If we introduce the velocity function v(x, t) of the crack tips at x, then q and n e are connected by q = n eV . As illustrated in Fig. 9-35, from the definitions of damage (crack density along y) and crack tip density (along x ), nand n e are related through
dx
11, =1,2,3,4,5
J
l"-
II
q
-
I 1
'N II
q
1M2 = - II,dx I
'-
Fig. 9-35 Illustration of ne versus n
(9-201)
Relationship between q and f): All equations written so far are purely geometrical and do not rely on any specific assumption concerning the behavior of the material. The simplest description of the damage propagation phenomenon is obtained when we stipulate a functional relationship between v and f) in the form v = ±Q(f)), the sign ± accounting for damage propagating both ways. It means that each crack adjusts its velocity to the crack density in its immediate vicinity, as suggested in Fig. 9-36.
Free Edge
Load
Fig. 9-36 Interaction of a crack tip with neighboring cracks Indeed, it has been observed by Boniface and Ogin [9-70] and Lafarie Frenot and Henaff-Gardin [9-71] that the interaction between crack tips could
9.5 Wave Propagation in Damaged Media and Damage Wave
781
be viewed as if they were hindering each other's progression: when a crack tip has close neighbors, it tends to reduce its velocity. It would be a better approximation to suppose that v is a function of
aD a D as well as oX ax
2 ~ , -2
the first and second density gradients
A
D, which means A
that crack tips will reduce their velocity to account for an increasing density ahead. This introduces a certain amount of non-local damage (see, [9-72] in the constitutive relationship between v and D, taking into account D , and
2D aa2~
A
A
at x is a restricted way of considering the values of
fJ
aD
~ oX
in a small
vicinity of x(x - x < x < x + x). Then, in the spirit of [9-73], the simplest form of the corresponding relationship is
a (aD) ax ax
v=C(D)-K -ln A
_
A
(9-202)
or
aD
q(x, t) = CW) a~
+K
a2 D ax;
(9-203)
when attention is restricted to crack tips, moving in the positive direction. Partial Differential Equation for f]: Eliminating q and n e from Eq.(9200) through Eq.(9-203) results in
afJ = Ka2fJ _ C(D/ fJ
(9-204) ax 2 ax When C(fJ) = 0, Eq.(9-204) reduces to the well-known one-dimensional diffusion equation. This approximation applies during the early stage of the specimen's life, and the corresponding damage development was studied by Gamby [9-74]. When K = 0, Eqs.(9-200) and (9-201) together with the former assumption q = ± QW) lead to
at
A
afJ ± C(D/ fJ = 0 or
at
A
ax
(9-205)
(9-206) Eq.(9-206) is the simplest form of a second-order quasi-linear hyperbolic equation known as a nonlinear wave equation. It describes many dispersive wave propagation phenomena with the possible formation of shock waves. From now on, the analysis will be restricted to situations where Eq.(9-206) applies.
782
9 Dynamic Damage Problems of Damaged Materials
Characteristic Curves-simple Wave Solution: Let us assume that the solution [) function of x and t is known. The two families of curves (3s (x =
x (s) ,t = t(s)), paramet erized by arc-length s, such that
dx
dt = ± C([)) ,
are
the characteristic curves associated with this solution. If one of the families
dx
of characteristic curves (31, say dt = ± C( [)), consists of straight lines, the
aD +
corresponding solution is called a simple wave [9-75] then, in view of at
C([))(~~)
= 0, one has
~~
=
(~~) (~:)
+
(aa;)
(~:)
= 0 along
(31. Thus, [) is constant along each curve of this family. Similarly, in view
aD + C([)) (aD) a~
of at
= 0, [) would be constant along each straight line dx
characteristic of a family such that dt = - C([)). Simple wave solutions are encountered in several important cases, e.g. for the initial boundary value problem in the quarter plane (x > 0, t > 0) with the conditions ([)= o and
aD
at ~ 0 for t = 0) and prescribed values G(t) = [)(O,t) on the axis x = O. In this case, the solution is such that [)(x, t) = G(t - x/CW)). When circle loading is of interest, the time t has to be replaced with the number of cycles N. Simple solutions also prevail when the domain of interest is the half-strip (0 < x < 2L, N > 0) where 2L is the width specimen, which is precisely the case here, for numbers of cycles less than N f' N f being the number of cycles when the characteristic curves emanating from the points (x = 0, N = 0) and (x = 2L, N = 0) intersect. If the model applies, the
dx
characteristic lines of slope dN = C([)) [respectively -C ([)) ], along which [) is constant, should be straight lines in the left (respectively right) part of the specimen, as long as the crack arrays emanating from both edges do not interact, as shown in Fig. 9-37. tor N
tor N
Edge
x=o
Load
Fig. 9-37 Simple wave solution.
~ = Ox 2
0
9.5 Wave Propagation in Damaged Media and Damage Wave
783
9.5.4.3 Ogin's Model In order to be able to assess the consistency of the above theory with experimental results, we need some more assumptions concerning the mean crack da velocity v(x, N) = dN ' where a = a(x, N) is the mean crack length at position x after N cycles. From a simple shear-lag analysis, Ogin, Smith, and Beaumont [9-76] showed that the stress-intensity factor pertaining to a crack tip located between two neighboring cracks spaced 2s apart has the form Kmax = B 1 CJMV2s, where 2s = l/ Q , CJM is the maximum applied stress and Bl is a constant; they also assumed that the crack growth rate obeyed some da . . form of Pans law such that v(x, N) = dN = B 2(Kmax )m, where B2 IS another constant. Then, the co-efficient C(Q) of the wave equation has the form da / dN = C(Q) = F(CJ M)Q-m 2, which predicts that C varies with Q according to a power law. Experimental Results: Circle loading tests were performed on T300/914 carbon/epoxy laminates by Lafarie Frenot and Henaff-Gardin [9-71]. Two stacking sequences were investigated: [03 , 90, 04]s- (A) and [0 7 , 90]s- (B). In laminate (B), where damage propagation is easier ([9-77]) , an early interaction between cracks emanating from both edges is observed, which prevents the simple wave solution from prevailing for a large number of cycles ([9-71]). In this study, we only use the results pertaining to laminate (A). The load ratio is 0.1, the maximum applied stress being CJ M = 0.6CJR where CJR = 1440 MPa is the static failure stress of the laminate. For several values of the number of cycles, the damage distribution across the specimen width was recorded through X-ray pictures (see [9-78]). Experiments versus Theory: In order to assess the correspondence of the model with the experimental results, we plotted the iso-damage contour lines in the (x, N) plane, for the left part of the specimen only (0 < x < L). As can be seen in Fig. 9-38, each curve is indeed a straight line of slope C(Q) , as predicted by the model. This allows the value of C to be determined for each value of Q, as illustrated in Fig. 9-39. It is remarkable that the so-obtained experimental curve fits in with a power law curve, as predicted by the above theory; its exponent m/2 is close to 1. It is interesting to point out that Boniface and Ogin ([9-70]) arrived at a similar value of m for a slightly different material and by a completely different procedure. In summary, according to the proposed model, the co-efficient C of the wave equation is related to damage variable Q through a power law whose exponent has been denoted - m/2. Plotting the contour lines of equal damage results in an experimental curve such that m/2 is close to unity, for the particular material and laminate investigated. It would be informative to perform the same verification for other laminates, for instance laminate (A) mentioned
784
9 Dynamic Damage Problems of Da maged Materials 60000
// / / / V/ / / / V/ / / / 1/ / / '
50000
// //
/
/"
on
/ / / / ' / ' /""'" / / ' / / ./' ~ ~ 20000 '7Y- v./ /"': /"" ...../J. R'/ ~ ~.....-:10000 l/.M ~ ,t:?""'" ;..--:
17~
oI
2
4
3
-5
-
6
0.0 =0. 1 • .0 = 0.2 • .0 = 0.3 • .0 = 0.4 • .0 = 0.5 = 0.6 • .0 = 0.7 . Q = 0. 8 . Q=0.9 • .0 = 1.0 In = 1. 1
7
x( mm )
Fig. 9-38 Iso-damage curves . [9-E6] 8.0 X 10' -.j- ·--t. -
7 OX 10' -..1--. .
!
. -t. . ·_. . ·t-..·_+-..--j-..·-i. ·_. ·t.
l_....._L. _ .....L .._ ..L . .-1._ ........l_.. -L ii
I
I
!
Iii
!
-1- ,-+-+--!---I----\---+ ....... SOX I0' --1-- J iii i ---1.---1..l a. '() 4.0X 10' -..1 - - ,._+._.-!--. --i--t lOX 10' -J---l ---1..-. 1---1----+ Ii 6.0X 10'
! !
2.0X 10'
I.OX 10'
+--1 -"1'''1--1-~I I"
o-·1-·---t.. o
0.2
:
I
I
I
I
1 "- ___
--f--L1
-·--t-·-"-r-"-"·r-"~---r----
0.4 0.6 0.8 I 1.2 C(Q)=8.5X 10' XQ ....
.
1.4
Fig. 9-39 C(SJ) versus $] [9-66]
earlier, provided the only part of t he specimen life for which characteristics are still straight lines is used. The influences of the loading level and ratio should be investigated. The validity of the model also needs to be assessed for other materials and an extension to two-dimensional damage distributions (as encountered in notched laminates, for instance) would be an important step toward construction of a general damage growth law, incorporating the transition to the next degradation mechanism and an ultimate failure criterion. 9.5.5 Damage Wave in Elastic-Brittle Materials 9.5.5.1 Essential Aspects of Damage Wave This study aims to address the non-local effect arising from damage evolution, t he spatial fluc t uat ion of damage measure, and the micro-structural interactions and to present a unified theory to describe the propagation of damage waves. We introduce a scalar damage variable, and take it and its gradient as internal state variables. A nonlinear partial differential equation for the kinetics of damage evolution is formulated within the framework of non-equilibrium
9.5 Wave Propagation in Damaged Media and Damage Wave
785
thermodynamics. The traveling-wave solutions of this equation are sought in the form of solitary waves of the kink type, based on the assumption of free energy including the nonlinear energy attenuation and spatial energy fluctuation caused by damage wave propagation. Although a number of models [9-36, 9-79,9-51] have been proposed to simulate the damage wave phenomena and the anti-kink behavior of damage evolution through a damage diffusion model, there are no attempts to describe explicitly the solitary wave-related behaviors of damage wave propagation except [9-49]. This study is an extension of the work of Zhang and Mai [9-49] and the results obtained can provide some guidance for future experimental and theoretical studies on impact dynamic behaviors of elastic-brittle materials. Furthermore, the physics-based model developed here can be used as a benchmark for the development of a unified computational-mechanics technology for engineering applications. This section is organized as follows: A statement is given of wave propagation in an infinite medium. The basic principles of thermodynamics based on internal state variables are briefly discussed and the corresponding equations of damage wave motion are derived. In order to establish the governing equations, a specific case is studied for elastic-brittle materials with specified free energy density. It is found that the resulting damage wave is a solitary wave without energy dispersion. The analytical solution for the one dimensional case is derived in detail. The features of analytical solutions are discussed in comparison, and asymptotic analysis for stored energy in the damage wave is carried out. The validation of the developed model is presented by comparison between experimental data from the literature and the issued analytical solutions
9.5.5.2 Thermodynamics Basis of Damage Wave Consider an infinite solid in which damage evolves in time and space, the damage ext ent is attached to material points based on phenomenological description. At each material point, its location and motion at time t are decided by the coordinates {x;} and {Vi (Xi, t)} , respectively. For the sake of simplicity, a scalar function D(x, t) within the range [0, 1] is introduced as an independent damage variable to describe the effect of micro-defects and their neighborhood on material degradation. We recognize that the isotropic scalar damage variable is an approximation to the first order, even if homogeneous isotropic material exhibits severe anisotropic damage [9-50]. Consider that the effective stress {O'*}, which acts on the damaged materials, is related to the applied Cauchy stress {O'} in the general form ([9-80])
{ *} 0'
{O'}
= f(D)
(9-207)
in which D is the traditional damage variable dealing with an average measure of the reduction in the cross-sectional area to sustain the applied stress.
786
9 Dynamic Damage Problems of Damaged Materials
Like elastic wave theory, the distribution of D(x, t) in the medium is called the damage field and its time-dependent variation is denoted by the rate of damage tJ and the spatial-dependent variation by the gradient vector of
aD
damage a{ xd . The rate of damage has been studied extensively in much of the published literature. Interested readers associating with this section can refer to Lemaitre and Chaboche [9-79], Lemaitre [9-81]' Lu [9-46], Hild et al. [9-50], and Bai et al. [9-47, 9-48]. Here, some important physical background information related to the gradient of damage is summarized. It is well known that many studies on damage and/or fracture, for example Peerlings et al. [9-82]' Chen et al. [9-51 ]' De Borst and Schipperen [9-83], and Bazant [9-84]' reveal that a length scale is required in the characterization of materials and structures. The classical one-variable form of damage description indicates that the damage evolution should be around the front of the stress wave. In addition, the damage is an averaging quantity and it can increase or decrease due to the interaction of micro-cracks. The radiation and attenuation of energy caused by changes in microstructures cannot be explored, and neither is the spatial energy fluctuation due to inherent non-homogeneity. Many remedial methods have been put forward , such as non-locality, strain gradient and Cosserat (or micro-polar) medium to solve the issue. In this section we introduce the gradient of damage as an additional internal state variable in the thermodynamic description of the dissipative process. This quantity is clearly related to the spatial fluctuation of the mean damage field. The same technique has been widely used in the nearest neighbor models of statistical mechanics, for example the Ginzburg-Landau theory for phase transition ([9-85]), [9-86] for surface instability of thin film , and [9-87] for the mechanics of earthquakes ([9-85]'"'-' [9-87]). Therefore, the resulting phenomena from micro-structural evolution can be dealt with within the general framework of non-equilibrium thermodynamics, as stat ed by Lemaitre and Chaboche [9-80]. The reversible energy E for damaged elastic-brittle mat erials is defined by
(9-208) in which p is the mass density, {vd the velocity vector, PT the generalized mass density related to tJ, and E is the free energy density rate. The physical meaning of the above equation can be further interpreted as follows , (1) The first term inside the brackets is the kinetic energy defined in continuum mechanics. (2) The second term is the additional kinetic energy associated with time dependent damage evolution. It must be emphasized that damage can
9.5 Wave Propagation in Damaged Media and Damage Wave
787
evolve by itself. For example, fast propagation of a micro-crack cannot be stopped by unloading the applied stress. The energy radiation from the point asperity ahead of a fast-moving crack front has been experiment ally observed by Sharon et al. [9-54] and theoretically studied by Willis and Movchan [9-53]. The emission of kinetic energy from a fast-moving shock wave front should be taken into account in the construction of the reversible energy. Based on the dimensional analysis, PT is a general mass density associated with the mean density of emitted kinetic energy by micro-cracks. (3) The third t erm is the free energy as defined in CDM [9-80]. Within the framework of thermodynamics, the free energy contains the mean strain energy to account for energy attenuation with increasing damage, and the spatial fluctuation about this mean strain energy. The first part has been delineated in CDM by Lemaitre and Chaboche [9-80]. On the other hand, the internal energy of randomly distributed microstructures would give rise to small fluctuations about the spatial- and temporal- dependent mean strain energy. If the wavelength of the shock waves is compatible with this micro-structural size, the energy fluctuation becomes non-negligible. It is assumed that the free energy density is a function of the strain tensor {Si j} , strain rate tensor {iij }' entropy S and entropy flux {gil ,
dD
damage field D, and its gradient d{ Xi }' The first gradient of damage variable is used commonly to capture the fluctuation, as is done in statistical mechanics by Rundle et al. [9-87] and Muller and Grant [9-86]. The idea of considering its gradient as an internal varia ble in CDM can be found in [9-88] and [9-89]. Therefore, the free energy density can be expressed as a function of a set of thermodynamic state variables
({Si
w=W
j},
{iij }' s, {gi} , D,
d~~})
(9-209)
If the heat flux is denoted as {qd , the flux of entropy is given by
(9-210) in which T is the current absolute t emperature. It is evident that the dissipated energy is associated with the interaction of micro-cracks, for example crack coalescence or frictional sliding of crack surfaces. Thus, there is an increasing trend towards energy dissipation with an increase in damage extent. The dissipative energy can be defined as
cp =
If t
(
TS
+ (d{dT Xi })
T
{gd
+ AD )
dVdt
(9-211)
in which A denotes t he work-conjugated force associated with the damage, to reflect the energy dissipation by the micro-crack coalescence per unit volume.
788
9 Dynamic Damage Problems of Damaged Materials
The problem studied here is related to the deformation and evolution of damage in an infinite medium, so that the effect of boundary conditions can be ignored. It is also reasonable not to include the external forces, heat flux and damage source on the boundary in deriving the equations of motion and constitutive relations. To obtain the motion equation and the evolution equations for the thermal state variables, we can construct the energy functional
U = E + W* + W
(9-212)
in which W is the external work caused by the applied force, heat flux and damage source on the boundary. The Lagrangian equilibrium equations are: (1) Equations of motion
d{ v;} dt
d{ O"ij } d{ xj }
p-- = - -
..
Pr n
d{H;}
= d{ xd - Y + A
(9-213) (9-214)
(2) Evolution equations of state variables:
dW {O"ij } = Pd{Cij } Y
dW
= P dn
(9-215) (9-216)
dn
in which Y and {Hd are the generalized forces associated with nand d{ Xi } ' respectively. For simplicity, we make a small deformation assumption and let {cij } be small too. Also, it is noted that because the time derivative of n is not necessarily continuous, we can choose the left derivative as follows
n=
lim n(t) - n(t - M) M-.O
/).t
(9-217)
As far as they are concerned, all the time derivatives in this section are left derivatives. The time derivative of the functional U may produce kinematics laws of damage evolution. The time-related variation of U leads to
(9-218)
To obtain the constitutive equations, we resort to the restrictions of thermodynamics. As the material is assumed to be elastic-brittle, the specific energy due to elastic deformation vanishes. The first law states
9.5 Wave Propagation in Damaged Media and Damage Wave
.
TaT'
W = {O'ij} {Cij } - a{xJ [T{gi}) - {Hd D]
789
(9-219)
in which the extra entropy flux is {Hi }D and internal energy sources vanish since there is no plastic dissipation as stated above. Besides, the constitutive laws also obey the second law of thermodynamics, that is the Clausius-Duhem inequality,
T(PS + a{xJ a{9i}) ?: 0
(9-220)
Combining Eqs.(9-218)
