Uniform materials and the multiplicative decomposition of the deformation gradient in finite elasto-plasticity

0.33 MB·English

by  V. Ciancio

UNIFORM MATERIALS AND THE MULTIPLICATIVE
8 DECOMPOSITION OF THE DEFORMATION GRADIENT IN
0 FINITE ELASTO-PLASTICITY
0
2

n V.CIANCIO,M.DOLFIN,M.FRANCAVIGLIA,ANDS.PRESTON
a
J
Abstract. In this work we analyze the relation between the multiplicative
1 decomposition F = FeFp of the deformation gradient as a product of the

elasticandplasticfactors([12,17,18,26])andthetheoryofuniformmaterials
]
i ([33, 40, 41]). We prove that postulating such a decomposition is equivalent
c tohavingauniformmaterialmodelwithtwoconfigurations -total φandthe
-s inelasticφ1.
l We introduce strain tensors characterizing different types of evolutions of
r thematerialanddiscusstheformoftheinternalenergyandthatofthedissi-
t
m pative potential. The evolution equations areobtained for the configurations
. (φ,φ1)andthematerialmetricg.
t Finallythedissipativeinequalityforthematerialsofthistypeispresented.
a
It is shown that the conditions of positivity of the internal dissipation terms
m
relatedtotheprocessesofplasticandmetricevolutionprovidetheanisotropic
- yieldcriteria.
d
n
o
c February 2, 2008
[

1 1. Introduction.
v
7 The objective of this work is to investigate the relationbetween the geometrical
1 theory of uniform materials and the multiplicative elasto-plastic decomposition of
3
the deformation gradient of Bilby-Kroner-Lee (BKL-decomposition) and Nemat-
0
Nasser (see [2, 17, 18, 26]).
.
1
Such a relation was first studied in [29]. In particular, the relation between
0
the inhomogeneity velocity gradient L (see below) and the plastic distortion rate
8 P
0 L¯˙ = F¯˙p ·(F¯p)−1 was introduced. In this paper we study the geometrical form of
: the relation introduced in [29].
v
i In Section 2 we introduce the basic concepts and review properties of uniform
X
materials. InSection3abijectivecorrespondencebetweentheBKLdecompositions
r ofthegradientofaconfigurationφofanelasto-plasticsolidandthetriple(φ,φ ,P)
a 1
is established. Here P represents the uniform material structure and φ and φ
1
are respectively, total and inelastic (intermediate) material configurations.
In Section 4 we introduce the natural strain tensors measuring the relations
between the Cauchy-Green deformation tensors C(φ) and C(φ ) and the material
1
metricginducedbytheuniformstructureP. Inthesamesectionthecombinations
ofthesetensors,materialmetricanditscurvaturecharacteristicindependentonthe
decomposition of plastic deformation gradient Fp = φ ◦D are determined and
1∗
the strain rate tensors are introduced.
In Section 5 the form of internal energy u depending on variables (φ,φ ,g) and
1
their derivatives is postulated and the dissipative potential D is introduced. We
1
2 V.CIANCIO,M.DOLFIN,M.FRANCAVIGLIA,ANDS.PRESTON
also formulate the system of equations describing evolution of dynamical variables
(φ,φ ,g). In the same section different stress tensors present in our scheme are
1
defined and relations between them are discussed.
In section 6 we write down the dissipative inequality for the suggested scheme
and separate the terms corresponding to the internal dissipation related to the
processes of integrable inelastic and uniform structure evolutions. We show that
the conditions of positivity of the corresponding terms in dissipative inequality
provide the anisotropic yield criteria for initiating the corresponding processes.
Another form ofa relationbetweenthe finite elasto-plasticitybasedonthe mul-
tiplicativedecompositionandtheuniformitystructuresusingthesecond-ordercon-
nection was suggested by S. Cleja-Tigoui, see [7].
2. Uniform Materials: material connections and material metrics.
Uniformmaterialsenterthesceneofmaterialscienceabout1952whenK.Kondo
introducedthe materialconnectionandthe materialmetricasthe tools tomodela
properties of materials. Later development in the works by K.Kondo, B.Bilby and
his collaborators, W.Kroner, W.Noll ([33]) and C.C. Wang (see [40, 41]) establish
the basis ofthis theory. Inthe worksof1980-presentby M.Elzanowski,M.Epstein,
M. De Leon, G. Maugin different aspects of this theory: models of higher grade
uniform materials, dynamics of material properties, thermodynamical properties
of such materials, role of Eshelby stress tensor, geometry of functionally graduate
material, etc., were further developed.
In this Section we present the basic geometrical structures of the theory of uni-
form materials that will be used in latter parts of the paper. Our presentation is
based on [9, 11, 12, 29, 34].
2.1. Material and physical spaces. A material body (material manifold) is
usuallyrepresentedbyaconnected3-dimensionalsmoothorientedmanifoldM with
a piecewise smooth boundary ∂M. Constructions of this paper are local, so it is
sufficient to consider M as a connected open domain in R3 with local coordinates
XI,I =1,2,3 .
As the physical space our body is placed in we consider the 3-dimensional Eu-
clidean vector space (E3,h), h being the (flat) Euclidean metric. We introduce a
global Cartesian coordinates xi in R3. In these coordinates the metric h takes the
form h=h dxidxj.
ij
We will also use the concept of ”archetype” ([9],[25]), a 3-dimensional vector
space V endowed with a standard Euclidean metric and the orthonormal basis
e = {e , i = 1,2,3}. For convenience we identify the ”archetype” space V (see
0 i
[12, 29]) with the tangent space at the originO of the physical space: V =T (R3)
O
and its metric with the metric h at the origin.
2.2. Configurations and the Cauchy metric. Configuration of the body M
is a (diffeomorphic) embedding φ : M → E3 into the physical space E3. To each
configurationφ there correspondsthe deformation gradient - the mapping from
the tangent space T (M) at the point X ∈ M to the tangent space T (E3) at
X φ(X)
the point φ(X)∈E3, [22],
F(X)=φ :T (M)→T (E3),
∗X X φ(X)
UNIFORM MATERIALS AND THE MULTIPLICATIVE DECOMPOSITION 3
given, in coordinates XA,xi, by the matrix of partial derivatives
F(X)i =φi .
I ,I
Here and below we will use notation φi = ∂φi (X) for the partial derivatives of
,I ∂XI
configuration components φi(X).
To a configurationφ(X) there correspondsthe rightCauchy-Greendeformation
tensor - the flat metric C(φ) = φ∗h in M obtained as the pullback of Euclidian
metric h in physical space by the configuration mapping φ. In coordinates (XI)
tensor C(φ) has the form
C(φ) =h φiφj. (2.1)
IJ ij I J
We will fix a specific configurationφ and callit the reference configuration.
o
Usuallyitpresentsthestateofthematerialbodythatisfreefromloadsandstresses
(see [39, 25]), although it might happen that such a configuration does not exist
and one has to choose a reference configuration differently. The body M is often
identified with its image under the embedding φ .
o
To the reference configuration φ there corresponds its Cauchy-Green tensor
o
called the reference metric in M:
g =C(φ ), g =h φi φj , (2.2)
o o oIJ ij o,I o,J
and the corresponding reference volume form v (X)= |g |dX1∧...∧dXn.
o o
Using the mapping inverse to the reference configuration φ : M → E3 one can
p
define the frame p in M by the rule
o
∂φ−1 I ∂
p (X)=φ−1 (e ), (p ) = o , i=1,2,3.
o o,∗X 0 o i ∂xi ∂XI
1.
Fromnow on we assume that the coordinatesXI are introducedin the material
manifold M using the reference configuration, i.e. XI(X) = φI(X). Then the
o
vectors of the frame p take the form (p ) = ∂ , I =1,2,3.
o o I ∂XI
Finally we define a history of deformation as a time parameterizedfamily of
smooth configurations: φ(t,X):M ×R→E3.
2.3. Uniform materials, I. Recall([40,33])thatamaterialiscalledhyperelas-
tic if its constitutive response (to a loading conditions) at any configuration φ is
completely characterized by two scalar functions:
(1) The elastic energy density function (per unit of reference volume v )
o
W(X,F(X)) depending on a material point X ∈ M and the deformation
gradient F(X) at this point; and
(2) The mass density function ρ (X)>0 in the reference configuration φ .
ref o
Nextweintroducethebasicnotionofauniform material (body). Intuitively
speaking, a uniform body is one that is made of the same material at all its
points. Thepropertyofuniformity ischaracterizedintermsofaparallelism KY in
X
thebody M ([40,39,9]). Morespecifically,ahyperelasticmaterialbody (M,W)is
1 Here and bellow for a differentiable mapping ψ :M →N between manifolds M and N we
denote by ψ∗X :TX(M) →Tψ(X)(N) the linear mapping of tangent spaces at a point X ∈M.
In coordinates (XI,xi) mapping ψ∗X is given by the matrix FIi =φi,I. Corresponding mapping
ofthetangentbundleswillbedenotedbyψ∗: ψ∗:T(M)→T(N),see[22],Ch.1.,SMK
4 V.CIANCIO,M.DOLFIN,M.FRANCAVIGLIA,ANDS.PRESTON
calleduniformifforanytwomaterialpointsX,Y thereexistsalinearisomorphism
KY :T (M)→T (M) between tangent spaces at these points such that
X X Y
KY∗(W(Y,F(Y))dv (Y))=W(X,F(Y)◦KY)dv (X) (2.3)
X 0 X 0
for all values of deformationgradientsF(Y) atY. Here KY∗ is the pullback of the
X
n-form of energy density by the mapping KY.
X
Introduce the scalar factor λY, characterizing the behavior of the reference vol-
X
ume formunder the parallelismKY: KY∗v (Y)=λYv (X).Then, in terms ofthe
X X o X o
energy density function W condition (2.3) takes the form
λYW(Y,F(Y))=W(X,F(Y)◦KY) (2.4)
X X
for all points X,Y in M and for all values of deformation gradient F(Y) at the
point Y.
2.4. Material connections. The localization of the definition of uniform mate-
rials given above leads to the introduction of a linear connection (material con-
nection) ω in M having vanishing curvature (an absolute parallelism, see ([16])).
Having such a connection available, the mappings KY are defined by the parallel
X
translation defined by connection ω from the point X to the point Y along any
curve connecting X and Y (result of such translation is independent on the choice
ofacurveduetothevanishingofthecurvature). ThetorsiontensorT ofconnection
ω provides the measure of non-homogeneity of the material, see [8, 9].
It is known (see [16], Ch.2) that in a simply connected body M which admits
a global tangent frame, a zero curvature connection is determined by a choice of a
global tangent frame parallel with respect to the connection ω
p(X)={p =pI(X)∂ , k =1,...,3, ∇ωp =0}.
k k XI k
Remark 1. A choice of such a frame is unique up to the (natural) right action of
the group GL(n,R) on the tangent frames and the left action of the symmetry
gauge group GM of the connection ω (see ([11, 34])).
A global frame p may also be defined by the uniformity mapping smoothly
depending on the point X
P :V →T (M), P (e =(P )I∂ , i=1,2,3. (2.5)
X X X i X i XI
Mapping P defines the linear isomorphism of the archetype space V with the
X
tangent space at each point X ∈ M. Section p and the uniformity map P are
related by
∂
p(X)=P (e )⇔p (X)=PJ . (2.6)
X 0 I I ∂XJ
Parallel translation KY defined by the connection ω can be written in terms of
X
the uniformity mapping as the composition
KY =P ◦P−1.
X Y X
Using the referenceframep (see above)andthe frame{e }inthe spaceV,one
o i
can associate to a material frame p two other geometrical objects:
(1) A smooth mapping k : M → GL(V),X → k(X) (an element of the gauge
group GL(V)M) such that for all X ∈M
p (X)=p (X)·k(X)⇔pI(X)=(pL k(X)I, I,J =1,2,3,
J o J J 0 J L
UNIFORM MATERIALS AND THE MULTIPLICATIVE DECOMPOSITION 5
hereGL(V)isthegroupofinvertiblelineartransformationsofthearchetype
space V;
(2) A non-degenerate (1,1)-tensor field DI(X) such that
J
D(X)p (X)=p(X),ı.e. pI(X)=D(X)I(p )J(X)=DI(X), i,I =1,...,3,
o i J 0 i i
last equality being true due to (p )I(X)=δI.
0 i i
Non-degeneracy of the (1,1)-tensor D(X) means that D(X)∈GL(T (M)).
X
Using the relation between the frame p and the corresponding gauge mapping
k :M →GL(V) we get the relation between k and the uniformity mapping P cor-
responding to the frame p, namely, p (X)=P (e )=(p ) k(X)=P (e )k(X),
i X i 0 i o,X i
so that
P =P ◦k(X).
X o,X
These considerations are summarized in the following
Proposition 1. Let M be a simply connected parallelizable (i.e. admitting a global
frame) manifold . With a choice of a reference configuration φ and a frame e in
o i
the archetype space V there is a bijection between the following objects:
(1) Global frames p in M (global smooth sections of the frame bundle F(M));
(2) Smooth uniformity mappings P :V →T (M);
X X
(3) Smooth mappings k : M → GL(V),X → k(X) (elements of the gauge
group GL(V)M) such that for all X ∈M
p(X)=p (X)k(X);
o
(4) Non-degenerate smooth (1,1)-tensor fields DI(X) in M such that
J
D(X)p (X)=p(X),
o
or, in terms of uniformity mappings P and P
o
D(X)=P ◦P−1.
X o
Remark2. Itisthebijectionbetweenthefirsttwoandthelasttypesofgeometrical
objects(non-degenerate(1,1)-tensorfields)thatwillbeprimarilyusedinthispaper.
2.5. Uniform materials, II. A uniformity mapping P determines its own vol-
ume form by translating to the material the Euclidian volume element from the
archetype: v (X)=P−1 ∗(e ∧e ∧e ). Denote by J (X) the factor relating two
P X 1 2 3 P
volume forms v and v
o P
v (X)=J (X)v (X)−
P P 0
-Jacobianof the mapping P−1.
Comparing definition of the factor λY in (2.4) with the definition of the factor
X
J (X) we get, for a uniform material following relation between these factors:
P
J (X)
λY = P . (2.7)
X J (Y)
P
In terms of the volume factor µ uniformity condition (2.4) takes the form
P
J−1(Y)W(Y,F(Y))=J−1(X)W(X,F(Y)◦P ◦P−1) (2.8)
P P Y X
Combining the deformationgradientF(X) and the uniformity mapping P one
X
gets the linear automorphism of the archetype space A = F(X)◦P ∈ GL(V).
X X
Comparing (2.4) with (2.8) we rewrite the condition (2.8) as follows
J−1(X)W(X,F(Y)◦P ◦P−1)=J−1(Y)W(Y,F(Y))=J−1(Y)W(Y,F(Y)◦P(Y)◦P(Y)−1)
P Y X P P
6 V.CIANCIO,M.DOLFIN,M.FRANCAVIGLIA,ANDS.PRESTON
for arbitrary points X,Y ∈ M and an arbitrary value of the deformation gradient
F(Y) at the point Y.
Define a function Wˆ of a point X ∈ M and a linear mapping A ∈ GL(V) by
setting
Wˆ(X,A)=J−1(X)W(X,A◦P−1). (2.9)
P X
IntermsofthefunctionWˆ definition ofuniform material (2.8)takes verysimple
form
Wˆ(X,A)=Wˆ(Y,A). (2.10)
Thus,theuniformitycondition(2.4)forthestrainenergyfunctionW is
equivalenttothestatementthatthefunctionWˆ(X,A)), X ∈M,A∈GL(V)
does not depend on the point X ∈M. Asaresult,Wˆ(X,A)itisafunctionon
the linear groupGL(V) only. This result is the central point of the theory of (first
grade) uniform hyperelastic materials. It reduces the study of material properties
of body M and the evolution of those to the study of the uniformity mapping P
X
and the function Wˆ on the linear group GL(V).
Additionalphysicalrequirements(e.g. materialframeindifference,presenceofa
nontrivialmaterialsymmetrygroup,etc.) leadtoadditionalrestrictionsontheform
of the energy function W. For instance, material frame indifference requirement
leads to the conclusion that Wˆ(A) is a function of invariants of matrix A. If a
uniform material is isotropic, function W(A) is left invariant with respect to the
multiplication by elements of SO(3) ([40, 11, 34]).
Returning tothe the strainenergydensity functionW(X,F(X))we seethatfor
auniformmaterialwiththeuniformitymappingP thestrainfunctionW takesthe
form ([9, 12])
W(X,F(X))=J (X)Wˆ(F(X)◦P(X)). (2.11)
P
2.6. Material metric of a uniform structure. As it was already known to E.
Cartan (see [3]), to a zero curvature linear connection ω (absolute parallelism)
determined by a frame p (or by the corresponding uniformity map P) there cor-
responds the material metric g defined as the pullback of Euclidian metric h by
the mapping P−1
X
g(X)=P−1h. (2.12)
X∗
This definition is equivalent to the declaring the frame p g-orthonormal at each
point X ∈M. In local coordinates XI the metric g has the form
g (X)=(P−1)i(P−1)jh =(D(X)−1)M(D(X)−1)Ng ,
IJ X I X J ij I J 0 MN
the first expression being given in terms of the uniformity mapping P while the
second is in terms of the corresponding (1,1)-tensor field D.
The curvature of the metric g is then defined by the torsionofthe connection ω
(see [22, 39]).
2.7. Examples. Elastic strain tensor of a body in a configuration φ is defined by
1 1
Eel = ln(g−1C(φ))≈ (g−1C(φ)−I),
c 2 0 2 0
wheresecondexpressionisthelinearapproximationofthefirstone,[22,39]. Recall
that the strain energy function of an isotropic material in linear elasticity has the
UNIFORM MATERIALS AND THE MULTIPLICATIVE DECOMPOSITION 7
form
W(φ)=λ[Tr(Eel)]2+µTr[(Eel 2)],
where λ,µ are Lam´e coefficients (see [39]).
Using the same function Wˆ = λ[Tr(A)]2+µTr[A2] on the linear group GL(V)
butanontrivialuniformitymappingP,wecometothemodelofaquasi-isotropic
material. Uniformity mapping P defined the material metric g as above. This
allows to redefine the elastic strain tensor using metric g instead of the reference
metric g :
0
1 1
Eel = ln(g−1C(φ))≈ (g−1C(φ)−I). (2.13)
2 2 0
Strain energy of a quasi-isotropic material in linear elasticity is defied as follows
W (X,F(X))=µ (X)[λ(Tr(Eel))2+µTr(Eel 2)]. (2.14)
P P
Itis easyto seethatthe strainenergyisthe quadraticfunctionofthe conventional
elastic strain tensor Eel with the tensor of elastic moduli depending on material
c
point X.
Another example is providedby a quasi-Hookeanmaterial(see [22], p.11),i.e.
the uniform analog of the neo-Hookean material with
W(φ)=α[Tr(Eel 2)−3].
c
The quasi-Hookean material corresponding to a uniformity structure P is defined
by the same strain energy function but with the redefined strain tensor Eel =
1ln(g−1C(φ)) where material metric g is used instead of the reference metric g
2 0
W (X,F(X))=α(Tr(Eel 2)−3)]. (2.15)
P
In the case of a homogeneous uniformity structure last expression reduces to the
strain energy of standard neo-Hookean material.
2.8. Evolution of the uniform structure. Evolutionof the properties of a uni-
form material is characterized by the time-dependence of the uniformity mapping
P and that of the function Wˆ. An appropriate characteristic of the evolution of
uniform structure P is the material velocity L(X) that was studied by different
authors, see for instance [29, 12, 1].
The material velocity of the uniformity structureP is defined as the material
point and time dependent linear mapping
∂P
L (X)=P−1◦ X :V →V.
t X ∂t
Under a loading both the uniform structure P and the deformation mapping φ
areevolving. As aresultthe couple(P (t),φ(t,X)) (or (g(t,X),φ(t,X))describes
X
boththe(total)deformationofamaterialandtheevolutionofitsproperties(elastic
moduli, reference density, etc.). The rate of change of this couple is given by
(L (X),V(t,X)), where V(t,X)= ∂φ is the physical velocity.
t ∂t
3. Elasto-plastic multiplicative decompositions of the deformation
gradient
At the end of 1950s B.Bilby, E.Kroner ([17] ) and later on E. Lee ([18]) pro-
posedthe following multiplicative decomposition of the deformation gradi-
ent (BKL-decomposition)
F=FeFp (3.1)
8 V.CIANCIO,M.DOLFIN,M.FRANCAVIGLIA,ANDS.PRESTON
asthe productoftwosmooth (1,1)-tensor fieldsofelastic and plastic defor-
mations, respectively. To provide a geometrical illustration of this decomposition
an intermediate configuration C∗ was introduced between the material body
t
M and the current configuration C =φ (M).
t t
DecompositionF=FeFp isusedtostudythebehaviorexemplifiedbyanelasto-
plastic behavior of a materialwhich undergoes deformationunder a slowly applied
load beyond the elastic range and then, after unloading, preserves some ”perma-
nent”strain(deformation). Wereferthethemonograph[26]formoreexamplesand
references concerning multiplicative decompositions of the deformation gradient F
and their applications.
3.1. Relationbetweenthe BKL-decompositionandthetheoryofuniform
materials. RecallthatthedeformationgradientF (X)ofaconfigurationφ:M →
t
E3 is the two-point (1,1)-tensor field in M defined by the linear isomorphism of
the tangent spaces φ ) : T(M) → T(φ (M)) at X ∈ M. Here C = φ (M) is the
∗ t t t
configuration of the body at the time t.
Thedecomposition(3.1)canbehardlyinterpretedotherthenasthecomposition
of tangent bundle mappings over some mappings of corresponding base manifolds
T(M) −−−−→ T(C∗) −−−−→ T(φ (M))
t t
π π π
M −−−−→ Ct∗ −−−−→ φt(M)
y y y
since the tensor fields Fe and Fp should be strictly anchored at some manifolds
(domain and target of each). Moreover the first mapping Fp should define a map-
pingfromthetangentspaceT (M)atapointX ∈M tothetangentspaceatsome
X
pointY ofintermediateconfigurationC∗. ThecorrespondenceX →Y shouldbe
X t X
one-to one, otherwise the composition (3.1) cannot be an isomorphism of the tan-
gent bundles. Therefore, there exists a unique one-to one mapping φ : M → C∗
1 t
underlying the tangent bundle mapping Fp. Mapping φ can be assumed to be
1
differentiable.
In the same way Fe can be viewed as a mapping of tangent bundles T(C∗) →
t
T(C ) over the differentiable mapping φ :C∗ →T(C ) of basis manifolds.
t 2 t t
Weobviouslyhaveφ=φ ◦φ . Therefore,φ isonto. Restricting,ifnecessarythe
2 1 2
intermediateconfigurationmanifoldonemayassume,withoutloosingofgenerality,
that φ is onto and φ is one to one. Thus, both φ and φ can be considered as
1 2 1 2
diffeomorphisms.
Remark 3. Definingthedecomposition(3.1)someauthorspresumethatthemap-
pings Fe and Fp are nonsmooth or even noncontinuous, reflecting microdefects
densities in the manifold M. Translating this into the language of tensor fields
and using the derivatives of these tensor fields one should however assume some
smoothness. Usuallyitisdonebyconsideringthesetensorfieldsassmoothaveraged
characteristics of the structural state of the material.
Remark4. Mappingφ presentstheintermediateconfiguration introducedin60th
1
byavarietyofresearchers,see[18,37,35]. Itwasusedfortheconstructionofplastic
deformationgradientFp andtheelasto-plasticdecompositionsoftotaldeformation
gradientF butasfarasweknow,wasnotconsideredpreviouslyasanindependent
dynamical variable.
UNIFORM MATERIALS AND THE MULTIPLICATIVE DECOMPOSITION 9
Now we are ready to make the next step. Consider the tangent mapping φ :
1t∗
T(M) → T(C∗) and compare it with the mapping Fp(t,X) : T(M) → T(C∗).
t t
Since mapping φ is linear isomorphism at each point X ∈M, one can write, for
1t∗
all tangent vectors ξ ∈T (M),
X
Fp(t;(X,ξ))=φ ◦D (X)·ξ
1t∗X t
where D (X) is uniquely defined smooth (1,1)-tensor field in M.
t
In exactly the same way one can present
Fe(t,Y,η)=φ ◦Fe∗(t,Y)·ξ
2t∗Y
for the uniquely defined smooth (1,1)-tensor field Fe∗(t,Y) in C∗.
t
If we pull back the (spacial index of) tensor field Fe∗(t,Y) from C∗ onto M by
t
the differential φ ofthe pointmapping φ we getanother(1,1)-tensorDe onM.
1t∗ 1
Sinceφ(t,X)=φ ◦φ foralltand,therefore,φ(t,X) =φ ◦φ ,
t2 t1 ∗X t2 ∗φt1(X) t1 ∗X
combining this with the decomposition (3.1) we get
φ =Fe◦Fp =(φ ◦De)◦(φ ◦D)=(φ ◦φ )◦(φ−1◦De◦φ )◦D=
∗ 2∗ 1∗ 2∗ 1∗ 1∗ 1∗
=(φ ◦φ ) ◦(De◦D)=φ ◦(De◦D),
2 1 ∗ ∗
so that
De(t,X)·D(t,X)=id .
T(M)
As a result, being transferred to the material manifold M, the (1,1)-tensor
fieldsconnectingintegrablemappingsφ (i=1,2)tothetangentbundlesmappings
i∗
FpandFe areinverseto oneother. Thisishardlyasurprisesinceinthephysical
literature only one of these tensors was considered as an independent dynamical
variable; see [6, 26].
In the same way, φ = φ◦φ−1 would be also redundant. As a result, the only
2 1
independentdynamicalvariablesinthisschemearediffeomorphicembeddingsφ,φ
1
and the material (1,1)-tensor field D.
Remark 5. One can of course choose another triple of variables as independent
dynamicalquantities,for instance onemay use (φ ,φ ,D) ifit is preferableto deal
1 2
with the elastic deformation φ explicitly.
2
Remark 6. We consider here only the decomposition FeFp, but the same argu-
mentswouldproduceageometricalrepresentationofthereverseFpFe-decomposition
as well.
Remark 7. Noticethatthe choiceofanintermediateconfiguration(C∗,φ )par-
t 1 t
ticipating in the decomposition (3.1) is far from being unique. In particular, let
us show that we may formally choose the image C∗ = φ (M) as the intermedi-
t o
ate configuration with φ = φ being time independent. To do this denote by
1 t o
ψ : C∗ → C the diffeomorphism ψ = φ ◦φ−1. Transfer the tensor Fe to C
t o o 1 o
as follows: Fe 0 = ψ (Fe ∗), where we are using the diffeomorphism ψ together
∗
withits inverseto push forwardthe (1,1)-tensorFe ∗. Thus we getthe mapping of
tangent bundles
FˆFe =χ ◦Fe o =χ ◦ψ ◦Fe ∗◦ψ−1.
t∗ t∗ t∗ t∗
10 V.CIANCIO,M.DOLFIN,M.FRANCAVIGLIA,ANDS.PRESTON
Define also the diffeomorphism χ : C → C as χ = φ◦φ−1. and define the
t o t t o
mapping of the tangent bundles by setting:
Fˆp =φ ◦Fp M.
o∗
Then we have φ=χ◦φ , and, as is easy to check, Fˆe◦Fˆp =φ as required.
o ∗
As a resultwe get a simplified scheme of the elastic-plastic FeFp-decomposition
of the deformation gradient F = φ of a uniform material. Notice that the in-
∗
tegrable part φ of the plastic deformation gradient Fp is lost in this simplified
1 t
scheme. That is why it is preferable to work with the previous scheme where the
intermediate configuration is different from the image of the reference embedding
φ .
o
Remark 8. Notice that the couple (D,φ ) represents another model of evolution
1
of the material of the same type with the same uniformity structure. This
model of pure inelastic evolution is related to the model (φ,φ ,D) by the elastic
1
deformation φ .
2
IfwestartwithatimedependentuniformitymappingP andtwoconfigurations
t
φ,φ : M → R3, then one can (reversing the arguments above) construct the
1
”elastic deformation” φ = φ ◦ φ−1 and the mappings of tangent bundles Fp :
2 1
T(M) → T(C∗ = Im(φ ), Fe : T(C∗ = Im(φ ) → T(C = Im(φ )), such that
1,t t 1,t t t
the construction above returns us to the triple (P ,φ,φ ).
t 1
Finally,thereisafreedominthechoiceofthedecompositionFp =φ ◦Dgiven
1∗
by an arbitrary diffeomorphism ψ ∈Diff(M):
Fp =φ ◦D=(φ ◦ψ−1) ◦(ψ ◦D). (3.2)
1∗ 1 ∗ ∗
Thus, we can introduce the following equivalence relation between the pairs
(φ ,D) of the (time dependent) mappings φ :M →Rn and nondegenerate (1,1)-
1 1
tensorfields DinM. We saythat twopairs(φ ,D),(χ ,K)areequivalentif there
1 1
is a diffeomorphism ψ ∈Diff(M) such that
χ =φ ◦ψ−1,K =ψ ◦D. (3.3)
1 1 ∗
Collecting the considerations presented in this section we get to the following
conclusions
(1) BKL-decomposition F = FpFe of the deformation gradient F = φ of a
∗
(total) configuration φ : M → E3 presupposes the existence of (inter-
t
mediate) inelastic configuration φ : M → E3 and of the non-degenerate
1 t
(1,1)-tensor field D in the material space M such that
t
Fp =φ ◦D, (3.4)
1∗
Fe =φ ◦D−1◦φ−1. (3.5)
∗ 1∗
(2) Configuration φ is the mapping φ : M → E3 defining the integrable
1 1 t
part of inelastic (plastic!) deformation gradient Fp,
(3) (1,1)-tensor D is equal to
D =φ−1◦Fp(t,X).
t 1∗