Albrecht Bertram Elasticity and Plasticity of Large Deformations
Albrecht Bertram
Elasticity and Plasticity of Large Deformations An Introduction
fyA Springer
Professor Dr. Albrecht Bertram Otto -von- Guerieke- Universitat Institut fur Mechanik Universitatsplatz 2 Gebaude N 39106 Magdeburg Germany
[email protected] Library of Congress Control Number: 2005921096
ISBN 3-540-24033-0 Springer Berlin Heidelberg New York ISBN 978-3-540-24033-4 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science + Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready copy from author Cover-Design: deblik Berlin Production: medionet AG, Berlin Printed on acid-free paper
62/3141
5 4 3 2 10
V
Preface This book is based on the lecture notes of courses given by the author over the last decade at the Otto-von-Guericke University of Magdeburg and the Technical University of Berlin. Since the author is concerned with researching material theory and, in particular, elasto-plasticity, these courses were intended to bring the students close to the frontiers of today's knowledge in this particular field, an opportunity now offered also to the reader. The reader should be familiar with vectors and matrices, and with the basics of calculus and analysis. Concerning mechanics, the book starts right from the beginning without assuming much knowledge of the subject. Hence, the text should be generally comprehensible to all engineers, physicists, mathematicians, and others. At the beginning of each new section, a brief Comment on the Literature contains recommendations for further reading. Throughout the text we quote only the important contributions to the subject matter. We are far from being complete or exhaustive in our references, and we apologise to any colleagues not mentioned in spite of their important contributions to the particular items. It is intended to indicate any corrections to this text on our website http://www.uni-magdeburg.de/ifme/l-festjgkeit/elastoplasti.html along with remarks from the readers, who are encouraged to send their frank criticisms, comments and suggestions to
[email protected].
All the author's royalties from this issue will be donated to charitable organisations like Terres des Hommes. Acknowledgment. The author would like to thank his teachers RUDOLF TROSTEL, ARNOLD KRAWIETZ, and PETER HAUPT who taught him Continuum Mechanics in the early seventies, and since then have continued to give much helpful advice. Many colleagues and friends also made useful comments and suggestions to improve this book, including ENRICO BROSCHE, CARINA BRUGGEMANN, SAMUEL FOREST, SVEN KASSBOHM, THOMAS KLETSCHKOWSKI, JOHN KINGSTON, WOLFGANG LENZ, GERRIT RISY, MANUELA SCHILDT, MICHAEL SCHURIG, GABRIELE SCHUSTER, BOB SVENDSEN and, most of all, THOMAS BOHLKE and ARNOLD KRAWIETZ, who gave countless valuable comments. The author is grateful to all of them.
VI
Contents List of Frequently Used Symbols
VIII
List of Selected Scientists
XIII
Introduction
1
1
3
2
3
4
Mathematical Preparation 1.1
Repetitions from Vector Algebra
1.2
Tensor Algebra
5 10
1.2.1
Tensor Product and Tensor Components
17
1.2.2
The Eigenvalue Problem
21
1.2.3
Special Tensors
29
1.2.4
Tensors of Higher Order
35
1.2.5
Isotropic Tensor Functions
46
1.3
Tensor Analysis
53
1.4
The EUCLIDean Point Space
71
1.4.1
The Co variant Derivative
76
1.4.2
Integral Theorems
87
Kinematics
91
2.1
Placements of Bodies
91
2.2
Time Derivatives
94
2.3
Spatial Derivatives
95
Balance Laws
121
3.1
Mass
121
3.2
The General Balance Equation
122
3.3
Observer-Dependent Laws of Motion
130
3.4
Stress Analysis
136
3.5
The Thermodynamical Balances
149
The Principles of Material Theory
153
4.1
Determinism
153
4.2
Local Action
154
4.3
EUCLIDean Invariances
155
VII 4.4 5
Extension of the Principles to Thermodynamics
Internal Constraints
159 166
5.1
Mechanical Internal Constraints
166
5.2
Thermo-Mechanical Internal Constraints
171
6
Elasticity
175
6.1
Reduced Elastic Forms
176
6.2
Thermo-Elasticity
177
6.3
Change of the Reference Placement
178
6.4
Elastic Isomorphy
180
6.5
Elastic Symmetry
183
6.6
Isotropic Elasticity
193
6.7
Incremental Elastic Laws
198
6.8
Symmetries in Thermo-Elasticity
203
7
Hyperelasticity
206
7.1
Thermodynamical Restrictions
206
7.2
Hyperelastic Materials
208
7.3
Hyperelastic Isomorphy and Symmetry
213
7.4
Isotropic Hyperelasticity
215
8
Solutions
225
8.1
Boundary Value Problems
225
8.2
Universal Solutions
232
9
Inelasticity
244
Plasticity
248
10 10.1
Elastic Ranges
249
10.2
Thermoplasticity
275
10.3
Viscoplasticity
282
10.4
Plasticity Theories with Intermediate Placements
284
10.5
Crystal Plasticity
294
References
307
Index
322
VIII
List of Frequently Used Symbols Sets 08 38o 38t .
directional vector of a slip system in Chap. 10.5 stretching tensor, rate of deformation tensor element of area in current placement
da0
element of area in reference placement
da
vectorial element of area in current placement vectorial element of area in reference placement
Div, div
material and spatial divergence operator
dm
element of mass
dv
element of volume in current placement
dv0
element of volume in reference placement
dx
vectorial line element in current placement
dx0 e* E = sym(H)
vectorial line element in reference placement basis vector of ONB linear strain tensor ALMANSI's strain tensor spatial BIOT's strain tensor material BIOT's strain tensor GREEN's strain tensor spatial generalised strain tensor
7
Eb=I-V~7 E B =U-I E G = y2 ( c
-1)
E gen E Gen
E h = In V E f F = Grad %
g/.B1' g = grad 0 go= GradO Grad, grad Gik,Gik H = Grad u I
y y
material generalised strain tensor spatial HENCKY's strain tensor material HENCKY's strain tensor internal energy resulting force acting on the material body deformation gradient basis vectors of dual bases spatial temperature gradient material temperature gradient material and spatial gradient operator metric coefficients displacement gradient unit tensor of 2nd-order
IX
X unit tensor of 4th-order symmetriser of 4th-order antimetriser of 4th-order JACOBIan, determinant of the deformation gradient
/ /
s
IA J = det¥
G
K= Vi | v2 dm SB, t
G
K = Grad(KQ KO~J) #
L = grad v = F F /= p T• L
7
G G
y G& G y G y
m n 0
0 ONB
G
P
G
=
f v dm
G
kinetic energy local change of the reference placement velocity gradient specific stress power
G
G
111(9
D
m
y
moment with respect to O mass normal vector zero vector zero tensor orthonormal basis pressure linear momentum
G «>t plastic transformation in Chap. 10 Principle of form invariance Principle of invariance under superimposed rigid body motions Principle of material objectivity G y spatial heat flux vector G y material heat flux vector G& heat supply G 0*&
which assign to each tensor its three principal invariants (where we did not distinguish between functions and values in the notation). The first and the third invariant have names: IA
=
III A
tr(A) =
trace of a tensor A
det(A)
determinant of a tensor A .
The trace is a linear mapping tr(A + aB) = tr(A) + atr(B)
V A , B e i , , a e 9$.
It obeys the following rules. tr(A) = tr{AT) = tr(sym(A))
VA
tr(skw(A)) = 0
VA
tr(A B) = fr
HA = W ( A ) - f r ( A 2 ) } = '/2{(I-A)2-I.A'}
(1.13)
7//A = VtffrJ(A)- 72 *r(A) *r(A2) + V, tr(A3 ) = '/« (I • A)3- '/2 (I • A) (I • A2) + !/31 • AJ.
For an invertible tensor A we obtain /(A ')
= //A
////A
//(A ') = 4 ////A ///(A') =
1/IIIA.
For the deviatoric part Af of a tensor A we obtain 7A. = 0 = 7/ In particular we have for the unit tensor I ///A-
h = Ih = 3
Mi = 1
and for the zero tensor 0 h = Ih = Hh = 0. There are also non-zero tensors with all principal invariants zero. The invertible tensors form the general linear group ^ C ^ . The sign of the determinant indicates if the tensor changes the orientation of a basis or keeps it. Therefore, we call a tensor with positive determinant orientation preserving. The orientation preserving tensors form the special linear group <jC^+. Tensors with determinant equal to ±1 are called unimodular. Their set is denoted by °^c^, the general unimodular group. Tensors with determinant +7 form the special unimodular group
TENSOR ALGEBRA
17
Theorem 1.10. Every invertible tensor F e j C ^ can be multiplicatively decomposed
into J
•
its (unique) unimodularpart F : =/// F ~
•
and its (unique) spherical part Fo : = Ilh
1.2.1
Tensor Product and Tensor Components
F 3
I
The tensor product or dyadic product is a mapping
® : Tx Y-t ^L
| (u, v) H> u ® v
defined by is action on an arbitrary vector
a e f
u ® v (a) : = (v • a) u . Obviously, this constitutes a linear mapping of vectors into vectors. The (simple) dyad u (8) v therefore is a (specific) tensor, which maps every vector in the direction of u . The tensor product is linear in both factors (u + &. As we have already seen, we can represent 2nd-order tensors after choosing a basis as
TENSOR ALGEBRA {2)
_
/
37
k
with generally 32 = 9 independent components, and the application to two vectors u , v e ^ i s defined by {2)
k
i k
A (u , v) = Aik (g • u) (g • v) = Aik u v . We have shown before in (1.8) that the trace induces an inner product between such tensors, which gives A (u , v) = A • (u (E) v ) . For the components of a dyad we obtain after (1.15) {2)
k
Aik = A-(g/®g*) {2)
ik
A
i
k
i
= A • (g ® g )
{2)
k
At = A-(g/®g ) {2)
^ ^ = A-(g
i
®g^).
Triads For ^ = J we have a trilinear mapping
Such a triad is the generalisation of the dyadic product to three factors A - AMgl0
g k® g l
and has 33 = 27 independent components. The application of such triad to three vectors u , v , w e ^ i s defined by (3)
A (u , v , w) = Aikl(gl • u) (g* • v) (g ; • w) = Amulvk
w\
which is automatically linear in the three inputs. This gives rise to introduce an inner product between triads by generalizing A(g,-,gjfc,g/) = : A • (g/® g*®g/) = Am or A( '
k
') =• A
'® *® '
= AM
to arbitrary linear combinations of triads in the right factor. We obtain analogously A B = AmBw
= Aikl Baa
38
MATHEMATICAL PREPARATION
with respect to dual bases. One can also interpret such a triad as a linear mapping of a 2nd-order tensor onto a vector by assigning A (T) = AU Tmn (g* • gm) (g' • g«) g' = AU Tkl g'
s Y
(double contraction), or vice versa by A(u) = ^ ( g ' - u j g ^ g * = AMul
gl®gk
(simple contraction).
W
An example for a triad gives the 8 -tensor, which has the permutation symbols as components with respect to any ONB {e,} 8 = 8lJ
ei®ej®ek.
It stands for a tensorial form of the triple product of t , u , v e Y 8 (t,u,v) = ^ ( e r t ) ( e r u ) ( e ^ v ) =
smtiUjvk
= [t, u , v] = t • (u x v ) . s can also describe the cross product of two vectors (3)
8 (u , v) : = elJ (tj - u) fa • v) ez- = s11 uj vk e,- = u x v .
Tetrads For K = 4 we have a quadrilinear mapping
M V
sty/
"4/
'V/
sty/
flfi
Such tetrads are quite important in mechanics. Therefore we introduce a distinct notation as A = A . A tetrad can be represented by tensor products of 4th-order A =
Aikbngi®gk®gl®gm
with 34 = 81 independent components. The application to four vectors u , v , w , x e I^is defined by A(u , v , w , x) = Aiklm (gf"• u) (g* • v) ( g l . w) (g m • x) (fourfold contraction). This gives again rise to introduce an inner product on tetrads by generalizing
TENSOR ALGEBRA
A (g,-, gj ,gk,gi)
39
=: A- (g/® g,-
or 4 ( g \ g 7 , g* , g') = : A • ( g ' 0 g>® g* g') = ^ * ' to arbitrary linear combinations of tetrads in the right factor. This gives analogously the inner product of two arbitrary 4th-order tensors A.B
= AmBijU
=
AmBm
with respect to dual bases. We can also use a tetrad for linearly mapping a 2nd-order tensor
into another one A\T] : = AMm g1'® g ^ g ' • g0) (g m • gp) Top
(double contraction). In this way, the tetrads are the general linear mappings between 2nd-order tensors A :^
-> A[T] .
and in this role they are very important in mechanics. The composition of tetrads is defined as it is for functions. We can now generalise many concepts of 2nd-order tensors to 4th-order tensors. In doing so, we can pose the eigenvalue problem for tetrads, leading to eigentensors, to a characteristic polynomial, and, as its coefficients, to principal invariants. Examples for Tetrads. The ¥th-order identity tensor maps each tensor into itself / :T
H-* T
VTGJI.
It has a simple representation with respect to an arbitrary basis {g/} and its dual {g1} / = g/< The 4th-order transposer maps all tensors into their transposed ones : 1 h-> 1
VT
It has the representation (1.28)
f
= p f(x) p -^) a1 (x) p
The symmetriser maps all tensors into their symmetric part Jpw
I
T h-> sym(T).
G A .
40
MATHEMATICAL PREPARATION
It has the representations (1-29)
/ s = y2 (I + 7 ) = 'A (S\ SJ) + Sjk 8li) gt® gj ® gk® gl. The anti(sym)metriser maps all tensors into their skew part
lA \-&>-* crfL*
|
T h*skw(T).
It has the representations (1-30)
IA = V2(l -T) = Y2 (gi®gk-gk®gi)®gi®gk =
1
/2(SikSjl-SjkSil)gi®gj®gk®gl.
This leads to the following decomposition of the 4th-order identity tensor / = / s + /*. The deviatoriser maps all tensors into their deviatoric part D : T h-> T?
V T e ^ .
It has the representation (1.31)
D = I - 7 5 I<x)I. All these tetrads are subspace projectors on ^ ^ . We can decompose the identity tensor now into three subspace projectors / = '/si®! + Dls +IA mapping every tensor into the sum of its spherical part, its symmetric deviator, and its skew deviator.
The transposed AT of a tetrad A is defined in analogy to (1.4) by the bilinear form of tensors ( L32 )
S • AT[T] = T • A[S]
V S ,
A tetrad has the (major) symmetry if AT = A . For its components this leads to Aiklm ~ ^Imik -
The identity tensor is symmetric, just same as / , / , and D . Other symmetries are also often used. The right subsymmetry of a tetrad A is given, if the following equivalent conditions are valid: A[TT] = A[T]
TENSOR ALGEBRA
o
A = AT
o
A = A Is
o
AIA = 0
Aikm = Aikm.
41
A tetrad A has the left subsymmetry if its transposed has the right subsymmetry. This is equivalent to all of the following conditions: (A[T])T = A[T] «•
A = TA
A = ISA
o
I AA = O
Aik = Aki .
VTej4
The following theorem contains immediate results of these definitions. Theorem 1.22. 1) A tetrad with the left subsymmetry maps all tensors into symmetric ones. 2) For a symmetric tetrad, the left and the right subsymmetry are equivalent. 3) The identity tensor is symmetric, but has no subsymmetry. 4) The symmetriser is symmetric and has both subsymmetries. 5) A tetrad with any subsymmetry is singular (non-invertible). 6) A tetrad has 34 = 81 independent components. 7) A tetrad with the two subsymmetries has 6 x6 = 36 independent components. 8) A symmetric tetrad has 9 (9+1) / 2 = 45 independent components. 9) A symmetric tetrad with the two subsymmetries has 6 (6+1) / 2 — 21 independent components. For the linear mappings between symmetric tensors the two subsymmetries can be generally assumed. By doing so, we reduce the number of independent components from 81 to 36. In this case, it is convenient to consider symmetric 2nd-order tensors as 6 dimensional vectors according to T =
Tjjej^ej
+ T22i2i®
> e 2 + TT33 ® e5
+ T23 (e 2 ® e3 + e^ 0 e2) + T31 (e5 ® e ; + e7C + T12 (e ; 0 e2 + e2CSe 7 ) =
TJJ
eV] + T22 eV2 + T33 eV3
+ V2 T23 eV4 + ^2 T3li eV5 with the coefficients
+ ^2 T}2
eV6
42
MATHEMATICAL PREPARATION
{Tn,T22,T33,^T23,
^2T31) ti T12]
with respect to the tensorial ONB in ^ ^ ^ eV} = e7 ® e7
eV2 = e2 ® e2
e^
]
(1.33)
eV4 = U (e2 ® e3 + e 3 ® e 2 ) ej/j = 7 / ^ (e 7 ® e 3 + e3 ® e 7 )
e^
= 7
/^ (e2 0 e ; + e7 ® e 2 )
where the factor 1/^2 results from the normalisation of the basis {eVa} like, e.g. ]
/y2 (e7 0 e2 + e2 W The general linear mapping of a ATth-order tensor B into an Zth-order tensor C (L+K)
is represented by a (AT+L)th-order tensor A as (L)
(L+K)
(*)
C = A [B]
(K-fold contraction). Eigenvalue problems can only be posed for endomorphisms, i.e., for mappings of some linear space into itself. This interpretation is always possible for tensors of even order 2K (K > 0), since we can use them as mappings between ATm-order tensors (2K)
(K)
(K)
T : A h^ B . Their eigenvalue problem is then (2K)
(2K)
(K)
(K)
( T - Xr I ) [ P r ] = 0 (2K)
(K-fo\d contraction) with the 2^th-order identity tensor
I , the ATth-order zero
tensor 0 , the ATth-order eigentensor P r and the corresponding eigenvalue Xr. By analogy to 2nd-order tensors, this again leads to a characteristic polynomial, (2K)
the coefficients of which are the principal invariants of
T .
By analogy to (1.23), a projector decomposition of an arbitrary symmetric tensor (2K)
of even order T is of the form (2K)
T
K
(2K)
= S Xr P,
44
MATHEMATICAL PREPARATION (2K)
with K being the number of different eigenvalues Xr of
T . The projectors
(2K)
Pr
again fulfil the projector rules (1.24).
We should finally remark that one can generalise the above tensor concept to linear mappings between different inner-product spaces "V and W of arbitrary dimensions
A representation of such mapping can be obtained by introducing the dyadic product between vectors of the corresponding spaces w (8) v
with v G *Vy w G W
which maps any vector a G y^into ^according to w (8) v (a) : = (v • a) w . By using the bases {fl} a ^ and {g;} 0) and A a 2ndorder tensor. Then the RAYLEIGH14 product of T by A is defined as ., , A * T - A * (fK-'
1 35
C- )
g/ ® gk 0 ... (8) g/)
: = 7 * - l (A & ) (8) (A ft) (8)... (8) (A & ) .
Of course, the result does not depend on the choice of the basis. If A is a versor, then the product is a rotation of T . The R A Y L E I G H product does not commute, but it is associative in the left factor
14
John William Strutt, since 1873 Lord Rayleigh (1842-1919)
TENSOR ALGEBRA
A * ( B * T ) = (AB)* T
45
VA,B G ^ .
Thus we can omit the brackets. In this product, the identity tensor also gives the identity mapping I* T = T and the inversion is done by A * (A"; * T ) = T
VA
G
JL,.
Moreover, if S and T are tensors of arbitrary order, then we have A * (S ® T) = (A * S) ® (A * T)
VA
G ^
.
This would not hold, if we replace the tensor product by the composition or an arbitrary contraction, unless A is orthogonal. For K = 0 (scalars) the RAYLEIGH product coincides with the identity. For K = 1 the RAYLEIGH product coincides with a linear mapping A * t = At
VA
and fox K =2 we have
For invertible 2nd-order tensors we obtain (A * T)" ; = A~T * T"7
V A , T G <JL>.
A tensor of arbitrary rank T is called an isotropic tensor if it is invariant under the RAYLEIGH product with any orthogonal tensor: (K)
(K)
Q* T = T
VQG
&U.
Examples. All spherical tensors are isotropic 2nd-order tensors, the same as all scalar multiples of even-order unit tensors. The triad £ is also isotropic, as well as the following 4th-order tensor with respect to an ONB {e/}
with a, J3, y e & .
46
MATHEMATICAL PREPARATION
1.2.5
Isotropic Tensor Functions
An application of such RAYLEIGH products is the specification of symmetry properties for functions between tensor spaces of arbitrary order (K)
f : X
(M)
H> M
K,M
>0.
We call a 2nd-order tensor A a symmetry transformation of / , if for all X in its domain the following holds / ( A * X) = A * / ( X ) . However, this is only one possibility, and there are others depending on the nature of the argument and the image, as we will see later on. If all proper orthogonal tensors (versors) are symmetry transformations, then we call / an isotropic tensor function. Some authors also include improper orthogonal tensors in this definition. However, if K and M are even, then these two definitions of isotropy coincide. Isotropy of tensor functions plays an important role in material theory, because it provides possibilities to find representations for such functions. We will give two examples. Definition 1.23. A function q> : *j&m,-± &
I
C i-> 0 h
= Urn — (f(v + Aadv)-f(v)) = a Urn - ( f ( v + tdv)-f(v)) = adf(\ , dv). We next show the additivity in the second argument of the differential. Let u , v , w e Y, then di(y , u + w) = Urn - (f (v + A(u + w)) - f (v)) h->0 h
- Urn - (f (v + h u + h w) - f (v + h u) + f (v + h u) - f (v)) A0 A
= Urn — (f (v + A u + h w) - f (v + h u)) h0
h
+ Urn - (f (v + h u) - f (v)) /^o A = rff (v , w) + df(\ , u ) . By interpreting linear functions between vectors as tensors, we introduce the gradient or the FRECHET18 derivative of a differentiable function f(v) with respect to v as a tensor, for which different notations are used like df(v)
dv 18
= gradiiy) = f'(v).
Maurice Rene Frechet (1878-1973)
TENSOR ANALYSIS
55
It performs f "(v) dv : = diiy , dv)
Vv,dveX
In general, the derivative has no unique representation as a tensor, as we will see in the sequel. The derivative obeys the chain rule for the composition of functions. Namely, let ^ , y, and W be normed vector spaces, and g :
t-.y
Y
-> w
differentiable functions. Then the derivative of the composition equals the composition of the derivatives (fg)'(v) = f'(g(v))g'(v). Consequently, for an invertible function, the gradient of the inverse is equal to the inverse of the gradient (which therefore exists) 7 ;
(r r = (fr .
The derivative is (pointwise) linear (f;(v) + a f 2 (v))' = f; r(v) + af 2 '(v)
VaeJ.
For arbitrary products 0 between tensor functions S(v) and T(v) of arbitrary order, the product rule or LEIBNIZ19 rule holds (S(v) 0 T(v))' = S '(v) 0 T(v) + S(v) 0 T '(v). Herein, 0 can be a scalar, tensor, cross product or any reasonable contraction. We will next specify the concept of the differential for mappings between linear spaces of different order. •
Let / be a real-valued real function
f :&-*&. Then df(x ,dx) : = Urn - (f(x + h dx) -f(xj) h->0
= —f(x dh
h
+ hdx) \h=0
and for h dx = Ax we obtain the usual form from real analysis df(x , dx): =
19
Urn — (f(x + Ax) -f(x)) Ax^>o Ax
Gottfried Wilhelm Leibniz (1646-1716)
=^-dx. dx
56
MATHEMATICAL PREPARATION
As the real axis has only one direction, the differential is independent of the direction. •
Let f be a real vector function
f: T -> &. Then df(\ , dv) is linear in the second vectorial argument
This can always be represented by the inner product with a (co-) vector — — , dv so that df(\ , dv) = - ^ ^ • dv . dv For achieving a component form of the gradient, we choose a vector basis {g;} c Xand represent v and dv as v =
v1 g;
and
dv = dv1 g ,-.
We now take the differential in the direction of the basis vectors
df(y,gj) = Hm l(f(v
+
hgj)-f(v))
h->0 h
= jim l- (f((v' + A^)g,)-f(v' g / )). As the function f does not depend on the (constant) vector basis, we can substitute the vector function by a real function of the components v1 f(v'gr) = : f(v\ v\ ..., vN)
with N: = dim V,
and the above limit becomes the partial derivative of this function with respect to the z-th component
d/(vV, .- ,v") The gradient can be expressed by the dual basis {g*} (L36)
*(v) d\
df{v\v\
v")
ii
dv'
as Jf V x ( ) Of J , g/) = — j - 1 • g/ = T T7 g • g/ T >r>T
>rfry
e^.
If / is defined only on symmetric tensors, then also the derivative acts only on symmetric tensors dT e ^ , dT) = $- • dT = sym{^-) • dT G ^. J y dl dT and the skew part remains indeterminate. In this case one can always symmetrise the derivative. As an example of a real tensor function, we consider the trace of a power of the tensor /(T)
= tr(Tk) = T
k
l
for k > 0 . By applying the definition of the differential, we obtain for, e.g., k = 3 after the product rule
= — {tr((T + h dT) T T) + tr (T (T + h dT) T)
+ tr(TT(T+hdT))} 2
\iH)
r
= 3 ^r(T T dT) = 3 (T ) • dT . In general, we have
J/(T,dT) = ktrij*-1 dT) = kTk'J
dTT = k(Tk~])T-dT
and therefore (1.38)
/ ' ( T ) = jfc(T*"y. In particular we get for A: = 7 d/ T = I • dT = fr(dT) = 7 dT . With the representation for the second principal invariant (1.12)
MATHEMATICAL PREPARATION
60
77T = V2 {tr\T) - tr{T2)} we obtain by the same formula (1.38)
dIIT = {tr(T)l - TT} dT. In order to compute the derivative of the third principal invariant, the determinant, we first take the trace of the CAYLEY-HAMILTON equation of Theorem 1.13. Ilh tr(l) - 77T HJ) + 7T tr(T2) - tr(T3) = 0 , which can be used to express the determinant in terms of traces (1.13). By again applying formula (1.38), the differential at T in the direction of dT turns out to be
dIIIT = {T
[ - tr(T) T } • dT . For invertible tensors T this is simpler = 777T T
dllh
r
d
We thus obtained the following rules.
dIT _ dT (1.39)
dIIT _ dT dllh dT
ll
I
Tr
_ T 2T-IT T r + / 7 I T
777TT~r
Later, we will need the differential of a real isotropic tensor function. By the representation Theorem 1.24. this can be given the form
/(T) =fUh,Ih,in?). We determine the gradient by the chain rule r
m
=
(1.40)
S7 T
$-=
dT
dfis
°
dlT
dIT dT
I
9fiso
d I h
dIIT dT
l
dfiso
dIIIT dT
df r ™J ISO {T T T ^ \ - + T;— (7T I - T ) -
= a01 + ctj TT + a2 Tn with the three real functions of principle invariants of T ^(7 T ,77 T ,777 T ) : -
^Jiso
dIIIT
J
TENSOR ANALYSIS
a,(/T, Ih JIh) : = - % - dIIT
61
^
dIIIT
If T is invertible, we obtain
= pol
(/TI-Tr)
dIT
dIIT
+ PJTT
+ p_i T~T
dIIIT
with the three real functions R (T
TT
TTT \
P,(hJhJIh)
-—
:=-
P^(hJITJIh) :=
Jiso
i
OJiso
j
dIIT
J^///T-
If a real function depends on several real, vectorial, and tensorial arguments of arbitrary order T 7 , . . . , T^ the differential of such function is because of its linearity rf/(T;,T2,...,TK,dT,,dT2,...,dT =
rf/(T7>T2>...,TJC,dT7,0,...,0)
+ 4r(T;,T 2 > ... > T J C ,0,0,...,dT A ) = :-^.dT, ST.with the symbol for the partial derivative -^— of / with respect to T;, which are defined by the differential
62
MATHEMATICAL PREPARATION
(no sum). The dot • stands for the scalar product of tensors of arbitrary, but equal, order. Analogous forms can be obtained for tensorial functions, which we will consider next. •
Let F be a tensorial tensor function F
'. 32 + P
2
M
with P/J i: = p/ ® p y (E) pfc ® p / . The derivatives with respect to the right eigenvectors are
and the others analogously. The inverse differentials can be written as
-f-
= (AA)-'(p/
7
+ p " )+ ( A A ) - 7 ( P /
dP2
—f- = dA P/7,)
;
+ P, P 77J ;J)
MATHEMATICAL PREPARATION
70
which can be verified by the condition dA
= /
(double contraction between the last terms). Here / is the 4th-order identity tensor with the respresentation
/ = i P*'V k,l=l
We obtain the following rules. —
= ]
dA
Jl / p
rr+J
+
p r+lr
rr+2
r+2r
,
(1.45)
dA
dp; _ dA
-1
r
r+l)
r + (Ar-zr+2y'(pr
r4
r "2rr+2
+
Vr+2rr+2r)
for r = 1,2,3 modulo 3 For a symmetric tensor A we again obtain the results of the foregoing example (1.44).
THE EUCLIDEAN POINT SPACE
1.4
71
The EUCLIDean Point Space
A Comment on the Literature. In some books on tensor calculus one finds introductions to the EUCLIDean space, like in ERICKSEN (1960), McCONNELL (1957), SOKOLNIKOFF (1951, 1964), TROSTEL (1997). An axiomatic introduction to the EUCLIDean space can be found in NOLL (1964) and BERTRAM (1989, p. 36-42). In the present context, however, we will proceed more pragmatically. We describe the classical, i.e., non-relativistic physical space as a point space, where we can perform EUCLIDean geometry like describing points, straight lines, planes, volumetric regions, etc., and where we can measure distances and angles. First of all, the EUCLIDean space & consists of (spatial) points x , y , z , ... € &. For each pair of points we can measure their distance d:
(fx
#->
&
| (x,y)
h-> d(x,y).
This distance or EUCLIDean metric makes a metric space out of -> the notation xy .Accordingly, (x 9y), (x\yr) e xy = x'yr . The EUCLIDean shifters form a vector space in a natural way. The sum of two of them is defined by the parallelogram rule. We choose two appropriate point
72
MATHEMATICAL PREPARATION
pairs for the two shifters, such that the endpoint of one coincides with the initial point of the other like (x , y) and (y , z). The sum is then -» -» ->
xy
+ yz = xz .
The multiplication of a vector xy by a scalar a e & is executed by an elongation of the vector by the factor a . For negative a we additionally reverse the sense of the vector, like
(-l)xy = yx . With these operations, the set *V of all EUCLIDean shifters becomes a vector space, which is associated with the EUCLIDean point space Y I x K> ox = : ro(x) which uniquely assigns to each point x the position vector ro(x) . As it is often easier to handle vectors than points, one uses this as an identification between & and Y, which evidently depends on the chosen reference point. In fact, if we choose another one o' e
- > - > - >
ro-(x) = o'x = o'o + ox = do + ro(x). All position vectors with respect to o' differ from those with respect to o by the constant vector do . However, in {cp\ cp2, cp3}
THE EUCLIDEAN POINT SPACE
73
being a continuous and injective mapping. If we compose (p with the z-th projection pl of & into & , then we obtain the i-th coordinate x h-> (p
i = I 2, 3.
Here, we did not distinguish between function and the value of the function, in order to not complicate the notation. In principle it does not matter whether we put the index of a coordinate in the upper or in the lower position. However, if we calculate the differentials, the upper indices match the sum convention better, as we will see. Therefore, we will prefer the upper position, although there may rise the risk of confusing it with powers. If for some COOS ^ coincides with the entire space #, we call it a global COOS. For a cylinder COOS {r , 0, z} , one usually leaves points with r = 0 and 0=0 out of °fc in order to avoid ambiguities. Such COOS are not global. InNEUTSCH (1995) one finds specific COOS.
We will generally use only such COOS, for which the change is differentiable. That means, if { ^ , cp} and { ^ , \j/} are two COOS, then
is assumed to be continuously differentiable. Then all the partial derivatives do exist Pl
d
t
_j
k
_7
i,k=
1,2,3
i,k=
1,2,3.
and
ccp
ocp
The JACOBI2U matrix of the COOS change 20
Carl Gustav Jacob Jacobi (1804-1851)
MATHEMATICAL PREPARATION
74
dy/k
exists everywhere in are mutually inverse
dcp1
dcp1
dy/1 dcp2
dy/2 dcp2
dy/3 dcp2
dy/1 dcp3
dy/2 dcp3
dy/3 dcp3
dy/1
dy/2
dy/3
and is non-singular. The two JACOBI matrices
n
dy/1 dcpk dcpk dy/1
dcp1
i
=
i,1=1,2,3.
'
After the sum convention, we sum over all repeated indices in a counter position. The upper position in the denominator then corresponds to the lower position in the numerator, and vice versa. We have the choice to identify the EUCLIDean space 2, 3,
oy/ i.e., by the transposed JACOBI matrix. Because of the duality r Stt = rwii • ry
dy/
rk • j r^k d
i9r
=1,2,
3,
the gradient vectors transform as r-1
2 ? ' ' '
different from the tangent vectors. The tangent and the gradient bases are called the natural bases of the COOS. They generally depend on the point (or on its coordinates), and are neither orthogonal nor normed. As long as we use only one COOS, we can omit the COOS index, as, e.g., r' instead of rq/. Let W be a tensor space of arbitrary order. A function
f:
