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On the Evolution of Simple Material Structures

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The evolution of a distribution of material inhomogeneities is investigated by analyzing the evolution of the corresponding material connections. Some general geometric relations governing such evolutions are derived. These relations are then analyzed by looking at the restrictions imposed by the material symmetry group.
TECHNISCHE MECHANIK, Band 22, Heft 1, (2002), 36-42
Manuskripteingang: 13. Dezember 2001
On the Evolution of Simple Material Structures 1
M. El˙zanowski, E. Binz
The evolution of a distribution of material inhomogeneities is investigated by analyzing the evolution
of the corresponding material connections. Some general geometric relations governing such evolutions
are derived. These relations are then analyzed by looking at the restrictions imposed by the material
symmetry group.
1 Introduction
Laws of evolution are integral part of theories such as plasticity and visco-plasticity. Since plasticity is
often viewed as a process of re-arrangement of patterns of defects it seems natural to discuss the issue of
evolution of inhomogeneities within the framework of the geometric theory of uniformity, cf., El˙zanowski
(1995), Epstein & de Leon (1998), Wang & Truesdell (1973).
The structure of the law of material evolution has been already investigated within this realm both in
the context of simple materials as well as those of the second-order, see e.g., Epstein & Maugin (1996),
Maugin & Epstein (1998), and Epstein (1999). In the approach presented there material evolution was
modeled by a first order differential equation for the uniformity maps. Postulating the principles of
covariance and actual evolution, and assuming the uniformity of evolution (laws of evolution are material
point independent), the geometric methods were used to investigate the form and structure of such a
differential equation.
In this note we look at the evolution of material from yet another perspective by investigating how
the time dependent deformations (evolutions) of a uniform reference configuration show through the
evolution of the corresponding material connection. Our objective is to determine if there may be point-
wise evolutions of the uniformity maps, which although non-trivial and constitutively admissible, produce
no measurable change of the underlying pattern of inhomogeneities. We believe that such evolutions of
material structures my account for these non-elastic deformations which do not change ”defectiveness”
of the material body as measured by the torsion of its material connection. A somewhat similar problem
was investigated by Parry (see Parry (2001) and references therein) within the context of the structurally
based theory of defects. Using purely kinematic considerations he was able to show that there exists
a non-trivial class of inelastic deformations between states possessing the same elastic invariants. Such
deformations are akin to the classical slip mechanism of the phenomenological plasticity and represent
rearrangement of material points while preserving the local lattice structure.
Our presentation is divided into a number of short sections. After a brief review of the concepts of material
uniformity, homogeneity and that of a material connection in Section 2 the mathematical aspects of the
evolution of connections are discussed in Section 3. This is followed in Section 4 by the discussion of the
role of the Principle of covariance and the Principle of actual evolution as pertaining to the choice of a
particular material symmetry group.
2 Uniformity and Material Connections
The material body B is a continuum having the structure of an orientable differentiable manifold. We
assume that it can be covered by a single (global) coordinate chart. In other words, we postulate that
the body B possesses a global reference configuration. Although fairly general theory can be, and has
been, developed without this simplifying assumption such considerations are beyond the scope of this
note. The interested reader may consult Wang & Truesdell (1973) and El˙zanowski (1995).
1This paper is dedicated to Professor Wolfgang Muschik on the occasion of his 65th birthday.
36
The mechanical response of a simple material body is determined point-wise by the value of the deforma-
tion gradient at each material point xB. Adopting an Euclidean vector space Vas a reference crystal
(an archetype of a material element) the local configuration of the body at the point occupied by xis
given by a linear map K(x) from Vinto the tangent space of B at x. Specifically, selecting a frame eA
in Vthe local configuration at xis represented by the induced basis
fj(x) = KA
j(x)eA(1)
where the matrix valued functions KA
j(x) represent the linear transformation K(x). The deformation
gradient Fcan now be viewed as a linear automorphism of the tangent space of B at x. Namely,
ˆ
fi(x) = Fj
ifj(x)(2)
where the frame ˆ
firepresents the deformed configuration at x. The density of the stored energy per unit
reference volume is given in pure elasticity by a function W(F;x). We say that the body B is materially
uniform if there exists a selection of linear isomorphisms P(x) from Vinto the tangent space of B and
a function ˆ
Wsuch that
W(F;x) = ˆ
W(FP(x)) (3)
at any xand for all deformation gradients F.ˆ
Wrepresents here the density of stored elastic energy of
B per unit volume of the reference crystal while the collection of the induced bases uj(x) = PA
j(x)eA
defines the uniform reference configuration.
An automorphism Gof the reference crystal Vis said to be a material symmetry (of a reference crystal)
if given a uniformity map P(x)
H(x) = P(x)G(4)
is also a uniformity map, i.e.,
W(F;x) = ˆ
W(FP(x)) = ˆ
W(FH(x)) (5)
for all deformation gradients F. If ˆ
eAis a basis of the reference crystal Vsuch that
eA=GB
Aˆ
eB,(6)
where the matrix GB
Arepresents the symmetry element G, then
ˆ
uj=HB
jˆ
eB=PA
jGB
AeB(7)
defines yet another uniform reference configuration. Two different uniform frames, say ˆ
ujand uj, are
then related by
ˆ
ul=PA
lGB
A(P1)j
Buj=Gj
luj(8)
where Gj
lrepresents an element of the symmetry group of the material point xcorresponding to GB
A
symmetry of the reference crystal.
A smooth collection of uniform configurations over the body B represents a hypothetical re-arrangement of
local configurations of material points so that the relative mechanical response becomes point independent.
Being materially uniform is the mathematical way of saying that the body B is made of the same material
at all points. The uniform configuration does not necessarily represent any true physical state of the body
B as a whole as it may not necessarily come from any global configuration, even if globally defined.
A uniform configuration may or may not come from a global configuration of the body B. However, if it
does, the material body B is said to be homogeneous. In other words, the materially uniform body B is
homogeneous if among all its uniform configurations there exists at least one which is also a global (truly
physical) configuration.
37
Given the uniform reference configuration ujor equivalently the corresponding uniformity maps PA
j(x)
one is able to define the concept of material parallelism. Indeed, consider a vector field wdefined at least
in some open neighborhood of the material point xB. Let w= ˜wjujwhere, in general, both ˜
wjand
ujare material point dependent. We say that the vector field wis parallel relative to the uniform frame
ujif its coordinates ˜
wj, are constant functions of position. In other words, if for any direction, say fα,
the directional derivative ˜
wj
= 0. To this end let wA:= ˜
wjPA
j. Then
˜wj
=wA
(P1)j
AwA(P1)l
APB
l,α(P1)j
B.(9)
where the relation
(P1)j
A,α =(P1)l
APB
l,α(P1)j
B(10)
has been utilized. It is now easy to observe that the vector field w= ˜wjuj= ˜wjPA
jeA=wAeAis parallel
in ujif and only if
wA
=wD(P1)l
DPC
l,α.(11)
Note also that
ΓC
(P) := (P1)l
CPC
l,α (12)
are, as shown in Wang & Truesdell (1973), the Christoffel symbols of a linear connection on B pulled
back to the reference crystal. In the reference configuration of the body B these functions become
Γk
(P) = PC
l,α(P1)k
C,(13)
and
˜wj
=wA(P1)j
A,α + ˜wlΓj
(P).(14)
If another uniform reference configuration, say as given by (7), is considered where gauging (modifying)
by the symmetry group of the reference crystal may as well be material point dependent the corresponding
Christoffel symbols are
ΓC
(H) = (G1)D
AΓB
(P)GC
B+ (G1)B
AGC
B,α (15)
as it can easily be seen from (12) and (7). A vector field wwhich is parallel relative to ujmay not be
parallel relative to ˆ
uj. However, we say that wis materially uniform (or materially parallel ) if there
exists a uniform reference configuration wis parallel in.
Summarizing the above presentation we may say that any uniform reference configuration of the ma-
terial body B, given by some assignment of the uniformity maps P(x), defines a parallelism and the
corresponding material connection represented by the Christoffel symbols ΓA
(P). A smooth gauging
by the symmetry elements of the reference crystal induces in turn the whole family of material connec-
tions. It can be shown by a straightforward calculation that all material connections have zero curvature,
El˙zanowski at al (1990). They may, however, have a nonzero torsion. On the other hand, if a given
material connection, which already has no curvature, is symmetric then there exists a reference config-
uration such that the corresponding Christoffel symbols vanish, and the underlying uniform reference
configuration uiis integrable i.e., it comes from a global placement of the body B. Inversely, if any global
configuration is a uniform reference configuration then the corresponding uniformity maps PA
k(x) can
be selected as point independent making the corresponding Christoffel symbols vanish. In short, we say
that the body B is homogeneous if and only if there exists a torsion free material connection, cf., Wang
& Truesdell (1973).
3 Time Evolution of Material Connections
We shall investigate now how the material connections evolve under the gauging by the elements of the
group of linear automorphisms of the reference crystal GL(V). As any material connection is uniquely
38
defined by a uniform reference configuration we look first at the gauging of these configurations. Hence,
let us consider some uniform reference configuration uj(x) = PA
j(x)eAand let GB
A(x, t) represent a one-
parameter family of linear automorphisms of the reference crystal V, and such that GB
A(x, 0) = δB
Afor any
x. Superposing one operation onto the other we get a one-parameter family of reference configurations
uj(x, t) := PB
j(x, t)eB=PA
j(x)GB
A(x, t)eB.(16)
These configurations are not necessarily uniform configurations unless the automorphisms GB
A(x, t) are the
symmetries of the reference crystal V. Note also that given any other evolution of reference configurations,
say ˆ
PB
j(x, t), there always exists a smooth one-parameter family of linear automorphisms ˆ
GB
A(x, t) such
that ˆ
PB
j(x, t) = PA
j(x, t)ˆ
GB
A(x, t).
Let us now restrict our analysis to a single material point and consider two different evolutions, say PA
j(t)
and HA
j(t). We say that these evolutions are parallel if there exists a non-trivial automorphism GB
Asuch
that
HB
j(t) = PA
j(t)GB
A.(17)
It seems natural to expect that parallel evolutions are somewhat ”equivalent”. To this end let us compare
the ”time” derivatives of the corresponding induced frames uj(t) := PA
j(t)eAand wi(t) := HA
i(t)eA.
First
˙
wi=˙
HA
ieA=˙
HB
i(H1)l
Bwl=˙
PC
iGB
C(G1)D
B(P1)k
Dwk=˙
PC
i(P1)k
Cwk(18)
while
˙
uj=˙
PA
jeA=˙
PB
j(P1)k
Buk.(19)
Let L(P) := ˙
PP1, or in coordinates, Lk
j(P) := ˙
PB
j(P1)k
B. Therefore, given a family of uniformity
maps, say PA
j(t), the time rate of change of the family of the induced uniform frames uj(t) = PA
j(t)eAis
˙
uj=Lk
j(P)uk.(20)
In fact, as easily confirmed by equations (18) and (19) the following is true:
Proposition 1 Two evolutions PA
j(t)and HB
k(t)of the uniformity maps are parallel if and only if the
corresponding mappings L(P)and L(H)are identical.
Following Epstein & Maugin (1996) we shall call L(P) = ˙
PP1the inhomogeneity velocity gradient.
As we have argued earlier any uniform reference configuration defines a material connection. Also, any two
material connections differ by a point-wise action of the symmetry group (a collection of all symmetries)
of the reference crystal. In fact, it is easy to show that any two zero-curvature linear connections on B
differ by the deformation (gauging) of the elements of GL3(IR), isomorphic to GL(V), cf., Kobayashi &
Nomizu (1963).
Let us therefore consider two parallel evolutions of the uniformity maps, namely HB
k(t) = PA
k(t)GB
A. As
the automorphism GB
Ais time independent the time derivatives of the corresponding material connections
are related by
˙
ΓC
(H) = (G1)D
A˙
ΓB
(P)GC
B(21)
as implied by (15). Moreover, a straightforward computations show that
HA
j(t)˙
ΓC
(H)(H1)p
C(t) = PD
j(t)˙
ΓB
(P)(P1)p
B(t).(22)
The induced connection velocity
£p
(P) := PD
j(t)˙
ΓB
(P)(P1)p
B(t) (23)
39
becomes the connection counterpart of the inhomogeneity velocity gradient Lp
j(P).
Proposition 2 For any two parallel evolutions of the uniform reference configurations the corresponding
induced connection velocities are identical.
In contrast to Proposition 1 the converse to Proposition 2 is not obvious at all. Hence let us look at the
gauging of material connections not only by the symmetry group of the reference crystal but rather the
whole group of isomorphisms GL(V). To this end let GB
A(x, t) represent a deformation of the reference
crystal. Let us also assume that the material connection ΓA
- generated by the uniformity maps PA
j(x)
- is given. Applying the family GB
A(x, t) introduces the family of reference configurations (not necessarily
uniform) HA
j(x, t)eA=PB
j(x)GA
B(x, t)eAand the family of linear connections ΓC
(H)(t). Evaluating
the time derivative of these Christoffel symbols one gets that
˙
ΓC
(H) = (G1)D
AΓB
(P)˙
GC
B(G1)F
A˙
GE
F(G1)D
EΓB
(P)GC
B+˙
ΓC
(G).(24)
Moreover, for any collection of uniformity maps PB
j(x, t)
Ll
j,α(P) = ˙
PB
j,α(P1)l
B+˙
PB
j(P1)l
B,α =˙
PB
j,α(P1)l
B˙
PB
j(P1)r
BPC
r,α(P1)l
C(25)
which is simply identical to £l
(P). Consequently, the definition of the induced connection velocity and
(24) imply that
(P1)k
C˙
ΓC
(H)PA
j=£k
(G(x, t)) + [ΓB
(P), Lk
j(G(x, t))] (26)
where [·,·] denotes the Lie algebra commutator. This finally leads to the following conclusion:
Proposition 3 Given material connection ΓB
(P)and the family of gauge transformations GA
B(x, t)the
family of connection forms ΓB
(HA
j(x, t)), where HA
j(x, t) = PB
j(x)GA
B(x, t), will not evolve if and only
if
£k
(GA
B(x, t)) = [Lk
j(GA
B(x, t)),ΓB
(P)].(27)
When the gauge transformations GA
B(x, t) are material point independent the relation (27) reduces to
[Lk
j(GA
B(t)),ΓB
(P)] = 0.(28)
4 Material Evolution
As long as a (uniform) body remains elastic its material structure (a collection of all uniform reference
configurations) , as determined by the density of its stored energy function W, remains unchanged. How-
ever, if we allow the body to experience other than elastic deformations while assuming that the strain
energy is still measurable the underlying geometric structure may change. For example, it is traditionally
accepted that plasticity involves a mechanism which modifies the distribution of inhomogeneities, defects
in particular. Mathematically, such a re-arrangement of defect patterns can only be observed if the un-
derlying material structure evolves, i.e., the set of uniform reference configurations and the corresponding
material connections evolve outside of the symmetry group.
The exact form of the law of evolution of any particular material can only be determined through
constitutive modeling. There are, however, some general principles we would like any ”reasonable” law of
evolution to satisfy. In particular, we postulate that any such law satisfies the following two fundamental
principles:
Principle of covariance: A law of evolution must be independent of the particular reference
configuration chosen.
Principle of actual evolution: A law of evolution must at all times select the inhomogeneity
velocity gradient L(P)outside of the algebra of the instantaneous symmetry group.
40
These principles were originally postulated by Epstein & Maugin (1996) where it was also suggested
that the evolution of a material is governed by a first order differential equation for the uniformity maps
with the Eshelby tensor as the driving force. In this work, consistent with our view on the evolution of
structures, we assume the evolution law of the form:
˙
ΓA
=fA
D
, P D
j,· · ·) (29)
where the functionals fA
may still depend on other objects like for example the Eshelby tensor or the
deformation gradient. Note that the law of evolution is taken material point independent to parallel the
uniformity of the body. According to the Principle of covariance such evolution law must be invariant
under the change of the global reference configuration. In fact it can be shown, see Binz & El˙zanowski
(2001), that this postulate prohibits any functional fA
from being dependent explicitly on the uniformity
maps PA
j.
We proceed now to investigate the role of the material symmetry group in the context of the Principle
of actual evolution. In contrast to what was done in Epstein (1999) we shall not investigate the form
of the evolution law. Rather, looking at different symmetry groups and their algebras, we shall try to
determine the restrictions which the Principle of actual evolution imposes on the choice of the allowed
evolutions (gauging). In other words, what is a proper evolution i.e., the evolution changing the essential
characteristics of a distribution of material inhomogeneities. Aided by Proposition 3 we shall try to
determine the sets of solutions to the relations (27) and (28). We assume that any allowable gauge
transformation GA
B(x, t) is unimodular at any xand any t, and that we only consider unimodular material
symmetries. We also assume that the symmetry group remains unchanged during the evolution. The
much more difficult case of the evolution process in which not only the uniform reference configuration
but also the structure group may change is left for future research.
To start our analysis let us look closer at sl 3(IR), the Lie algebra of the special linear group SL3(IR),
that is the space of all trace-less 3 ×3 matrices. Let so3denote the algebra of the special orthogonal
group SO3, namely the set of all skew-symmetric 3 ×3 matrices. Furthermore let sym3be the space of
all trace-less symmetric 3 ×3 matrices while so2,3stands for the Lie algebra of the group of all rotations
about a fixed axis. This is a one-dimensional subalgebra of so3. Thus we have
sl3(IR) = so3sym 3.(30)
and it is elementary to observe that so3,so2,3, and the set of all trace-less diagonal 3 ×3 matrices
diag{a, b, (a+b)}, are all abelian subalgebras of sl 3(IR) while sym3is only a vector subspace.
First, let us consider the relation (28) where the gauge transformations GA
B(x, t) are assumed material
point independent. Supposing that the connection form ΓA
(P) takes value in a non-trivial subalgebra
hsl3(IR) and accepting the Principle of actual evolution we look for the deformations GA
B(t)SL3(IR)
such that L(G)6∈ hand [L(G), X] = 0 for every Xh. In other words, given the subalgebra hsl3(IR),
we look for the set
c(h) := {Ysl3(IR)/h: [Y,X] = 0for all Xh}(31)
where sl3(IR)/hdenotes the complement of hin sl3(IR). Note that, in general, sl 3(IR)/his not a Lie
algebra. Consequently, c(h) is not a Lie algebra either, cf., Carter at al. (1995). It is now a matter of
simple calculations to show that:
full isotropy:
c(so3) = {0}.(32)
transversal isotropy:
c(so2,3) = {cij }where c12 =c22 =1
2c33 ,c12 =c21 ,c13 =c23 =c32 =c31 =0.(33)
41
We may also add that in the case of the simple elastic fluid, where the material symmetries are all
unimodular transformations, every evolution is trivial. On the other hand every evolution of the triclinic
crystal - which has no symmetries - is non-trivial.
In conclusion; we have shown that as far as the material point independent deformations of material
structures are concerned every proper deformation (i.e., obeying the principle of actual evolution) of the
isotropic material structure yields a change in the material connection, see (32). However, there are
some nontrivial proper evolutions of transversely isotropic structure which while deforming the structure
will not alter the corresponding material connection, (33). More detailed analysis of this as well as the
material point dependent case will be presented in Binz & El˙zanowski (2001).
Acknowledgments: The work presented in this note was partially supported be grants from the Ministry of
Education of Baden-W¨urttemberg and the University of Mannheim. The writing of this paper was completed
during the second author’s sabbatical stay at the University of Calgary. This paper was presented at CIMRF-2001
held at Technical University of Berlin, September 3-6, 2001. Travel support for both visits was provided by the
Faculty Development Fund of Portland State University.
Literature
1. Binz, E., and El˙zanowski, M., Another look at the evolution of material structures, (2001), preprint.
2. Carter, R., Segal, G., and Macdonald, I.G., Lectures on Lie Groups and Lie Algebras, London
Mathematical Society Student Text 32, Cambridge University Press, Cambridge, 1995.
3. El˙zanowski, M., Mathematical Theory of Uniform Material Structures, Kielce University od Tech-
nology (Politechnika ´
Swietokrzyska), Kielce, 1995.
4. El˙zanowski, M., Epstein, M., and ´
Sniatycki, J., G-Structures and Material Homogeneity, J. Elas-
ticity, 23(2-3), (1990), 167–180.
5. Epstein, M., Towards a complete second-order evolution laws, Math. Mech. Solids, 4(2), (1999),
251–266.
6. Epstein, M., and de Leon, M., Geometrical theory of uniform Cosserat media, J. Geometry and
Physics, 26, (1998), 127–170.
7. Epstein, M., and Maugin, G.A., On the geometrical material structure of anelasticity, Acta Me-
chanica, 115, (1996), 119–134.
8. Kobayashi, S., and Nomizu, K., Foundations of Differential Geometry, Vol.I, Wiley, New York,
1963.
9. Maugin, G.A., and Epstein, M., Geometrical material structure of elastoplasticity, International J.
PLasticity, 14, 1-3, (1998), 109–115.
10. Parry, G.P., The moving frame and defects in crystals, Int.J. Solids Struct. 38, (2001), 1071–1087.
11. Wang, C.-C., and Truesdell, C., Introduction to Rational Elasticity, Nordhoff, Leyden, 1973.
Address: Ernst Binz, Department of Mathematics, University of Mannheim, Mannheim, Germany, e-
mail: binz@euler.math.uni-mannheim.de. Marek Z. El˙zanowski, Department of Mathematical Sciences,
Portland State University, Portland, Oregon, U.S.A., e-mail: marek@mth.pdx.edu.
42
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  • S Kobayashi
  • K Nomizu
Kobayashi, S., and Nomizu, K., Foundations of Differential Geometry, Vol.I, Wiley, New York, 1963.