The Mechanics and Thermodynamics of Finite Gradient ... · A. Bertram, Preprint The Mechanics and Thermodynamics of Finite Gradient Elasticity and Plasticity Oct. 2013 2 There are

A. Bertram, Preprint The Mechanics and Thermodynamics of Finite Gradient Elasticity and Plasticity Oct. 2013

1

Preprint

The Mechanics and Thermodynamics

of Finite Gradient Elasticity and Plasticity

by

Albrecht Bertram

Institut für Mechanik, Otto-von-Guericke Universität Magdeburg [email protected]

Magdeburg, Oct. 2013

Introduction

There is some evidence that for particular materials an internal length scale exists which influences both its elastic and its plastic behaviour1. Such an internal length can be introduced into the material model by taking into account higher gradients than the usual first deformation gradient appropriate for simple materials. As a first step towards such an extension of the theory of simple materials, a second gradient theory is needed. In the range of small deformations, such a general theory has been given by BERTRAM/ FOREST (2013). In many applications, however, the deformations and the gradient of the deformations are not small, but can be rather large. For such cases, a generalization of this linear theory into the non-linear range is needed. Such an inclusion is by no means trivial or straight-forward. In particular, many principal questions from finite plasticity arise again, such as the following ones.

What are appropriate kinematical and dynamical variables for modelling of second order materials?

How do they transform under Euclidean transformations and changes of the reference placement?

What are reduced forms for gradient elasticity?

How can the concept of isomorphisms be translated to a gradient format?

What are the symmetry transformations for constitutive equations defined on such higher-order spaces?

How can we introduce elastic and plastic internal variables?

Does there exist a decomposition of the kinematical variables, either in a multiplicative or in an additive form?

How does the elastic law evolve during yielding?

Which are the restrictions imposed by the second law of thermodynamics for such models?

In this paper all these questions are addressed. The methodology is oriented along the lines which have been suggested for simple elastoplastic materials in, e.g., BERTRAM (2005). In the second part dedicated to the thermodynamics of such materials, we follow the lines of BERTRAM/ KRAWIETZ (2012) for classical thermoplasticity, and BERTRAM/ FOREST (2013), which contains the geometrically linear gradient plasticity.

1 see, e.g., FLECK/ HUTCHINSON (2001)


2

There are not many publications yet on gradient plasticity for large deformations.

HWANG et al. (2002) use a symmetrised gradient of Green´s strain tensor as a higher order material variable. Their analysis becomes rather complicated although they apply it only for the isotropic case.

In SVENDSEN/ NEFF/ MENZEL (2009) the elasto-plastic theory of gradient materials is suggested where the plastic variables are introduced as primitive concepts. This differs from our approach in many aspects. In particular they choose the gradient of the right Cauchy-Green tensor as the second kinematical variable, the same as TOUPIN (1962) already did. However, although principally equivalent to our procedure, it turns out that this choice leads to rather complicated expressions in elasto-plasticity.

Many authors like, e.g., LUSCHER et al. (2010) and CLEJA-TIGOIU (2011) use the multiplicative decomposition and obtain a second order plastic variable as the gradient of Fp . In present theory, however, we consider these variables as independent of each other.

In WANG (1973), TESTA/ VIANELLO (2005) and PODIO-GUIDUGLI/ VIANELLO (2013) the symmetry properties of gradient materials are also investigated in the context of elastic fluids. These authors, however, apply them to variables which are neither spatial nor material and, thus, inappropriate for material modelling of solids.

Notations

We tried to follow the standard notations in the field of tensor calculus within continuum mechanics. However, by the inclusion of third-order tensors (triads), many more operations have to be introduced if one prefers a direct notation. In any case, the author did his best to use a notation which is as customary, direct, and uncomplicated as possible.

We write for vectors small bold letters like v, w, x V etc. and for dyads or second-order tensors T, U, V etc. For

triads or third-order tensors we use A, B, C etc. k-th order tensors with k > 3 are notated like

k

C . R denotes the

real numbers, R + the positive reals, Orth is the orthogonal group within the dyads.

We denote an arbitrary basis by {ri} and its dual by {ri} . In particular, such bases occur as the natural bases induced by a coordinate system {

i} and then written as {ri} and {ri} . An orthonormal vector basis is written

as {ei} .

The standard scalar products between tensors of equal order are indicated by a dot “ ”. The simple contraction of vectors and tensors is written without a product dot like T v, T U, T A etc. The tensor product is written as .

While a second order-tensor T has a unique transpose TT, a third-order tensor has more than one. We will mainly need the right sub-transpose At which gives for the components with respect to an orthonormal vectorbasis (At)ijk

= Aikj . If a triad is symmetric with respect to this particular transposition, we call it right subsymmetric.

For two triads we obtain then

(1) At B = A B t.

Very helpful for higher order tensors is the Rayleigh product. It maps all basis vectors of a tensor simultaneously

without changing its components. To be more precise, let

k

C be a tensor of kth-order (k 1) and T a dyad. Then the Rayleigh product between them is defined as

(2) T

k

C = T (C ik ... l ri rk

... rl) : = C ik ... l (T ri) (T rk)

... (T rl) .

Of course, the result does not depend on the choice of the basis. If T is orthogonal, then the product is a rotation of k

C .

For k 1 the Rayleigh product coincides with a linear mapping

(3) T c = T c ,


3

and for k 2 we obtain

(4) T C = T C TT.

The Rayleigh product acts on a simple triad like

(5) T a b c = (Ta) (Tb) (Tc) = (Ta) (Tb) c TT = T (a c b TT ) t TT

or analogously for a triad A

(6) T A = T [At TT ] t TT.

The Rayleigh product is associative in the left factor

(7) S (T k

C ) = (S T) k

C

and in the right one. In fact, if k

C and m

D are tensors of arbitrary order, then we have

(8) T (k

C m

D ) = (T k

C ) (T m

D )

for all dyads T. This would not hold, if we replace the tensor product by the composition or an arbitrary contraction, unless T is orthogonal.

In this product, the second order identity tensor also gives the identity mapping

(9) I k

C = k

C and the inversion is done by

(10) T (T –1 k

C ) = k

C . The Rayleigh product commutes with the composition with the inverse in the following sense

(11) T –1 (T k

C ) = T (T –1 k

C ) .

For two second order tensors A (invertible) and B and a higher order tensor k

C we obtain the rule

(12) B A–1 (A k

C ) = A (A–1 B k

C ) . For the scalar product of arbitrary tensors we get

(13) (T k

C ) k

D = k

C (TT k

D ) .

Besides the Rayleigh product, we will need another product between an invertible dyad T and a triad A denoted by

(14) T ○ A : = ijk T –T ei) T ej) T ek)

T –T [At TT ] t T T

with (6) = T (T –1 T –T A)

with (12) = T –T T –1 (T A) .

The following rules hold for this product.

(15) (T ○ A) B = A (TT ○ B)

for all dyads T and all triads A and B . The second order identity tensor also gives the identity mapping

(16) I ○ A = A

and the inversion is done by

(17) T ○ (T –1 ○ A) = A .

Furthermore, the product is associative


4

(18) S ○ (T ○ A) = (S T) ○ A

for all dyads S and T and triads A .

For the case of T being orthogonal, this transformation coincides with the Rayleigh product.

As an overview over the different sets, spaces, and groups we give the following list.

R the real numbers

R + the positive reals V three-dimensional Euclidean vector space

Orth orthogonal group within the dyads

Lin : = {(T, T) T dyad, T triad with right subsymmetry}

Conf : = {(C, K) Lin C positive-definite and symmetric dyad, K triad with right subsymmetry}

Inv : = {(P, P) Lin P invertible dyad, P triad with right subsymmetry}

Unim : = {(P, P) Lin P unimodular dyad (det P = 1), P triad with right subsymmetry}

B0 region occupied by the body in the reference placement

Bt region occupied by the body in the current placement

B0 region occupied by the boundary of the body in the reference placement

Bt region occupied by the boundary of the body in the current placement

I. Mechanical Theory

Kinematics

We will denote the region occupied by the body in the reference placement by B0 and its boundary by B0 , and the region in the current placement by Bt and by Bt its boundary there.

Let be the motion of the body and

(19) F = Grad = L with determinant J = det F

the deformation gradient. Here L denotes the Lagrangean nabla. With respect to the natural bases of a spatial coordinate system { i} and a material one {

i} it can be calculated as

(20) F = i

k

r k ri.

We will later-on need the differential of the inverse of the deformation gradient

(21) d(F –1) = Grad F –1 dx0 = F –1 dF F –1

= F –1 (Grad F dx0) F –1

= F –1 [(Grad F) F –1] t dx0 .

Thus

(22) Grad F –1 = F –1 [(Grad F) F –1] t.

The spatial velocity gradient is

(23) L = grad = v E = D + W = F F –1


5

where the suffix E stands for the Eulerean or spatial derivative. The dot denotes the material time derivative. Its symmetric part is D and its skew one W . We also have the relation with the right Cauchy-Green tensor C= FT F

(24) C= FT 2 D .

For two second-order differentiable tensor fields in the Lagrangean description S and T we obtain for the gradient of the product

(25) Grad (S T) = S Grad T + [(Grad S) t T] t.

This can be verified by the following calculation

(26) Grad (S T) = (S T) L = S (T L) + S T L = S Grad T +

S T L

where the arrows indicate the term to which nabla has to be applied. The last term is then with respect to an orthonormal basis

(27) S T L = Sij ,k Tjm ei em ek

while

(28) [(Grad S) t T] t = [(Sij ,k ei ej ek) t T] t = [Sij ,k ei ek ej T] t = [Sij ,k Tjm ei ek em] t

= Sij ,k Tjm ei em ek

gives the same. An analogous result holds also for the gradient in the Eulerean description.

For any tensor field we have by the chain rule

(29) Grad = grad F and grad = Grad F –1

where it is understood that is in the Lagrangean description if Grad is applied, and in the Eulerian one if grad is.

We will also need the second deformation gradient

(30) Grad F = Grad Grad = L L

which is a triad (field) with the right subsymmetry by definition.

Stress Power

The starting point for our gradient theory is the global stress power after BERTRAM/ FOREST (2007)2 of a body

(31) P = tB

1/ [T grad v + S grad grad v] dm

with Cauchy´s stress tensor T and a spatial hyperstress tensor of third order S .

Cauchy´s equations (local balances of linear and moment of momentum) are for second gradient materials3

(32) div (T div S) + b = v

(33) T = TT.

T is symmetric because of the balance of moment of momentum, and the first term (31) can be substituted by T D.

grad grad v has the right subsymmetry by definition. So the same symmetry can be imposed on S without loss of

generality within the present frame. The balance of moment of momentum does not impose any restriction on S .

We will next bring the stress power in a material form which is invariant under Euclidean transformations. For this purpose we use (23)

2 see also TROSTEL (1985) 3 see, e.g., MINDLIN/ ESHEL (1968), where also the higher-order boundary conditions are shown.


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(34) grad grad v

by (29) = Grad (F F –1) F –1

by (25) = F (Grad F –1) F –1 + [(Grad F) t F –1] t F –1

by (22) = F F –1 [(Grad F) F –1] t F –1 + [(Grad F) F –1] t F –1

by (6) = F –T [ F T F F –1 Grad F + F T Grad F]

= F –T ○ [ F –1 F F –1 Grad F + F –1 Grad F]

= F –T ○ K

with

(35) KF –1 Grad F .

This triad has been used by CHAMBON/ CAILLERIE/ TAMAGNINI (2001), FOREST/ SIEVERT (2003), CLEJA-TIGOIU (2011) and other authors4. It is sometimes called the connection. However, we prefer the name curvature tensor, although this might lead to confusion with the well-known Riemannean curvature tensor. The product ○ in (34) can be interpreted as the push-forward from the reference placement to the current placement taking into account the different transformation behaviour of tangent and cotangent vectors.

These fields can be calculated with respect to the natural bases of the coordinate systems { i} and { i}

(36) Grad F = (i

k

r k ri)

j

r

j

= [2 k

i j

+

l

i

m

j

(

lm

rr

k) +k

l

(

l

j

rr i)] r k r

i rj

and

(37) K = p

k

[

2 k

i j

+

l

i

m

j

(

lm

rr

k) + k

l

(

l

j

rr i)] r p r

i rj.

It can be shown (KRAWIETZ 1993, see also HWANG et al. 2002) that the curvature tensor K can be determined by the right Piola-Kirchhoff tensor C and its gradient according to

(38) K = C –1 Sym Grad C

with the following symmetrisation of a triad

(39) Sym Tijk : = ½ (Tijk + Tikj Tkji) .

We obtain for the global stress power (31) with (34), (35), and (15)

(46) P = 0B

1/0 J { T (F –T ½ C) + S (F –T ○ K)} dm

= 0B

1/0 {½ S C + SK K} dm

with two material stress tensors, namely the second Piola-Kirchhoff tensor

(47) S : = J F –1 T F –T = F –1 J T

and a third order material hyperstress tensor defined as

(48) SK : = F –1 ○ J S .

The product ○ in (48) can be interpreted as the pull-back of S from the current placement to the reference placement.

4 see also NOLL (1967)


7

Example. We consider the bending and tension of a ring segment. The motion is given by the coordinate transformation

r = 1(R , , Z ) = a R

(40) = 2(R , , Z ) = b

z = 3(R , , Z ) = Z

with respect to a cylindrical COOS {R , , Z} for the initial or reference placement, and another cylindrical COOS {r , , z} for the current placement.

a = 1, b = 1.7

a < 1, b = 1

The deformation gradient with respect to the natural bases of these coordinates and the normed ones is

(41) F =

a 0 0

0 b 0

0 0 1

rk ri =

a 0 0

0 ab 0

0 0 1

ek ei

which describes a state of plane strain. The right Cauchy-Green tensor is

(42) C = FT F =

2

2 2

a 0 0

0 b r 0

0 0 1

ri r

j =

2

2 2

a 0 0

0 a b 0

0 0 1

ei ej .

The determinant of F is J = a2 b. Incompressibility would be characterized by a2 = 1/b . In this case we would obtain

(43) Finc =

a 0 0

10 0

a0 0 1

ek ei .

Further

(44) Grad F = R a (1 b2) r1 r2 r

2 =

1

R (a ab2) e1 e2 e2

and

(45) K R (1 b2) r1 r2 r

2 =

21 b

R

e1 e2 e2

which is independent of a , and for b 1 vanishes completely.


8

Gradient Elasticity

Before we start with gradient plasticity, it is necessary to introduce gradient elasticity. This will be done by extending the definition of a simple elastic material to a second-order one by enlarging the set of independent variables by the second deformation gradient.

Definition 1. We will call a material a second-order elastic material if the stress tensors are functions of the motion, the deformation gradient, and the gradient of the deformation gradient:

(49) T f ( , Grad , Grad Grad ) f ( , F, Grad F)

(50) S F ( , Grad , Grad Grad ) F ( , F, Grad F)

where it is understood that all variables are taken at the same material point at the same instant of time.

These constitutive equations can be further reduced by means of the Euclidean invariance principle (see BERTRAM 2005, therein called PISM), which we assume in the following form.

Principle of Euclidean Invariance. The stress power at the end of a motion (x0 , ) t0 in some time interval

[0,t] equals the stress power after superimposing a rigid body motion upon the original motion

(51) {Q() (x0 , ) + c() t0 }

with arbitrary differentiable time functions Q() Orth and c() V .

The Euclidean transformation (51) has nothing to do with changes of observers. The invariance under such changes of observers has already been used for the objectivity of the stress power (31), which led us to the objectivity of the stress tensors. The invariance under observer changes does not lead to reduced forms (see BERTRAM/ SVENDSEN 2001), in contrast to the Principle of Euclidean Invariance above, as we will show in the sequel.

The action of the group transformation (51) determines the transformation behaviour of all kinematic variables. We will further call a tensor T of arbitrary order invariant if it is not affected by any modification (51), and objective if it is rotated into Q T . For scalars the two properties coincide. As a result, D and grad L turn out to be objective.

Then one can easily show that the above assumption on the stress power is generally fulfilled iff T and S are objective.

In contrast to them, the following tensors are invariant: C, K and their duals S SK , which makes them good candidates for material modelling.

We will further-on denote the binary set of elements like {T, T} consisting of all dyads T and triads T with right subsymmetry by Lin . This space has the dimension 9 + 18 = 27. A subset of this space is formed by all positive-

definite and symmetric second order tensors C and all triads with right subsymmetry K , which we call the space of configurations Conf . This set is imbedded in a space with dimension 6 + 18 = 24. Another subset of Lin is formed by all invertible dyads and all triads with right subsymmetry, which we denote by Inv . We can further restrict this subset to those dyads which are unimodular (determinant equal 1) denoted by Unim .

Reduced forms of the elastic laws (49) and (50) are then

(52) S = k(C, K)

(53) SK = K(C, K)

by two elastic laws which are defined on the space of configurations

k : Conf Lin

K : Conf Lin .


9

This means that every second order elastic material that obeys the Principle of Euclidean Invariance can be brought into these forms.

If, moreover, the elastic material is hyperelastic then there exists a specific elastic energy

w : Conf R

such that the specific stress power after (46) equals

(54) p : = 1/0 [½ S C + SK K ]

= w(C, K) = C w(C, K) CK w(C, K) K

By comparison we obtain the potential relations for (52) and (53)

(55) S = k(C, K) = 20 C w(C, K)

(56) SK = K(C, K) = 0 K w(C, K) .

In the sequel we will distinguish between the elastic and the hyperelastic case, since the latter is a proper subset of the first. So there is a clear mathematical distinction, which of course does not mean that a material which is elastic but not hyperelastic, is physically meaningful.

Change of Reference Placement

The reduced forms (52) and (53) and the hyperelastic energy depend on the choice of the reference placement . If we want to indicate this dependence, we will write for example k( , ) and K( , ) for the elastic laws. Their transformation behaviour under change of the reference placement plays an important role for isomorphisms and symmetry transformations.

While the spatial quantities grad v, grad grad v, T S do not depend on the reference placement, the material ones

like C, K, S, SK do so. We will next investigate their transformation behaviour under change of the reference placement. We will therefore consider a second reference placement indicated by underlining.

For an arbitrary differentiable field we obtain for the two reference placements by the chain rule

(57) Grad = Grad A

where A : = Grad ( 1) is the gradient of the change of reference placement. It is understood that the field is defined on the corresponding reference placement. In particular we find

(58) F = Grad = F A and J = J det(A) and C = AT C A = AT C

and Grad F = Grad (F A)

with (25) = F Grad A + [(Grad F) t A] t

with (57) = F Grad A + [{(Grad F) A}t A] t

with (6) = F Grad A + AT (AT Grad F) .

Thus

(59) K =F –1 Grad F A–1 F –1 {F Grad A + AT (AT Grad F)}

=A–1 Grad A + A–1 F –1 (AT AT Grad F)

=A–1 Grad A + AT (AT A–1 F –1 Grad F)

with (14) =KA + AT ○ K

with

(60) KA =A–1 Grad A .

For the material stresses we obtain

(61) S = J F –1 T F –T


10

with (58) = det(A) J A–1 F –1 T F –T A–T

= det(A) A–1 S A–T

= A–1 JA S with JA : = det(A)

and with (48) for the hyperstresses

(62) SK = F –1 ○ J S

with (58), (7) = (A–1 F –1) ○ (det(A) J S)

with (18) = A–1 ○ [ F –1 ○ (det(A) J S)] = A–1 ○ JA SK

or inversely

(63) SK = A ○ JA–1 SK .

The above formulae hold for arbitrary changes of reference placements. If we particularize these results to rigid rotations of the reference placements A Orth we have

(64) JA = 1 Grad A 0

K AT K SK = AT SK .

Elastic Isomorphy

This concept plays an important role for the formulation of elasticity and elastoplasticity (see BERTRAM 1998, 2005). It is used to precisely define the notion that two elastic points show the same elastic behaviour.

Definition 2. Two elastic material points X and Y are called elastically isomorphic if we can find reference placements X for X and Y for Y such that the following two conditions hold.

In X and Y the mass densities are equal

(65) 0X = 0Y .

With respect to X and Y the elastic laws are identical

(66) kX (X , ) = kY (Y , )

(67) KX (X , ) = KY (Y , ) .

TESTA/ VIANELLO (2005) demand in addition to (65) that also the gradient of the density is equal in the two points, an assumption which makes sense in the context of elastic gradient fluids. In the present context, however, we do not see any reason for such a restriction.

By standard arguments (BERTRAM 2005) one can then show that these conditions are equivalent to the following statement using (58), (59), (61), (63).

Theorem 1. Two elastic material points X and Y with elastic laws kX , KX and kY , KY with respect to arbitrary reference placements are elastically isomorphic if and only if there exist two tensors (P, P) Inv such that

(68) 0X = 0Y det (P)

(69) kY (C, K) = det –1(P) [P kX (PT C, PT ○ K + P )]

(70) KY (C, K) = det –1(P) [P ○ KX (PT C, PT ○ K + P )] (C, K) Conf

hold with 0X and 0Y being the mass densities in the reference placements of X and Y , respectively.


11

The dyad P can be interpreted as the gradient of the change of the reference placement and the triad P as its second gradient. However, in a local theory, these two tensors can be considered as being independent of each other.

The last two conditions are fulfilled iff the specific elastic energy satisfies

(71) wY (C, K) = wX (PT C, PT ○ K + P ) + w0 (C, K) Conf

with some constant w0 .

Material Symmetry

If we particularize the concept of isomorphy to identical points X Y , it defines automorphy or symmetry. In this case we consider only one point so that we can drop the point index, and denote the automorphism by (A, A) Inv to distinguish from the isomorphisms of the previous section. Because of the first isomorphy condition, any

automorphism must be proper unimodular in its first entry: (A, A) Unim . This leads us to the following definition using (69) and (70).

Definition 3. For a gradient elastic material with material laws k and K a symmetry transformation is a pair (A, A) Unim such that

(72) k (C, K ) = A k (AT C, AT ○ K + A)

(73) K (C, K ) = A ○ K (AT C, AT ○ K + A)

holds for all (C, K) Conf .

For the elastic energy the symmetry transformation is

(74) w (C, K) = w (AT C, AT ○ K + A) (C, K) Conf .

The set of all such symmetry transformations represented by such a couple (A, A) Unim represents the symmetry group of the material. In fact, the transformation is a group under composition in the algebraic sense. Its identity is (I, 0) Unim , and the inverse of some (A, A) Unim is (A–1, A) Unim .

This group is used to define isotropy or anisotropy. If the symmetry group is a subgroup of Orth in the first entry

and the zero in the second, (Q, O), these transformations can be interpreted as rigid rotations and we call the respective reference placement an undistorted state. If a material allows for such undistorted states, it is a solid. If it contains all orthogonal dyads in the first entry, then the material is called isotropic. These definitions apply not only to gradient elasticity and hyperelasticity, but also to any inelastic gradient material in an analogous way.

In all of these cases, we obtain after (64) with respect to undistorted states

(75) A k (C, K ) = k (A C, A K)

(76) A K (C, K ) = K (A C, A K)

(77) w (C, K) = w (A C, A K) (C, K) Conf .

Thus, for an isotropic material the elastic laws are isotropic tensor functions.


12

In MURDOCH (1979)5 one finds interesting considerations about symmetry of second gradient materials. MURDOCH uses other configuration variables in his work, namely F and FT Grad F , the latter being a material quantity, in contrast to the first one.

Linear Elasticity

For many applications the elastic deformations are rather small which justifies the linearization of the hyperelastic laws. In this case, one would assume a square form of the configuration tensors for the energy. In order to avoid the introduction of new notations like a generalized Voigt notation, we use a tensor notation.

The following multiplications for a tensor m

D and a higher order tensor

k

E (k > m)

(78)

k

E [m

D ] and [m

D ]

k

E

denote the m-fold contractions from the respective side. Thus

(79)

k

E [m

D ] = E i1 ... ik Dik-m+1

... ik ei1 ... eik-m

(80) [m

D ]

k

E = Dim

... i1 E i1 ... ik e im+1 ... eik .

Note that the in first application the contraction order is reverse to the second one.

The major transposed of an even-order tensor is denoted by

2k

E T and defined by the identity

(81)

2k

E T[

k

D ] : = [k

D ]

2k

E

for arbitrary k

D . Such an even-order tensor is (major) symmetric if

(82) 2k

E =

2k

E T or

2k

E i1 ... i2k =

2k

E i2k ... i1 .

In the physically linear elasticity theory, the elastic energy is assumed to be a square form of the configuration (C, K). Such a form on Conf would have 242 /2 + 24/2 = 300 parameters. In tensor notations it can be represented by

(83) w (C, K) = 0

1

4 [C – Cu]

4

E [C – Cu] + 0

1

2 [C – Cu]

5

E [K – K u] + 0

1

4 [K – K u]

6

E [K – K u]

with higher-order elasticity tensors

4

E ,

5

E ,

6

E and some unloaded configuration (Cu , K u) Conf .

These elasticities can be submitted to the following symmetry conditions:

4

E :

left subsymmetry {ijkl} = {jikl} right subsymmetry {ijkl} = {ijlk} and the major symmetry {ijkl} = {lkji}

with 21 independent constants as customary from classical elasticity

5 see also CROSS (1973), ELZANOWSKI/ EPSTEIN (1992) and DE LEON/ EPSTEIN (1996) for such considerations related to the symmetry group.


13

5

E :

left subsymmetry {ijklm} = {jiklm} right subsymmetry {ijklm} = {ijkml}

with 108 independent parameters

6

E :

subsymmetry in the 1st and 2nd indices {ijklmn} = {jiklmn} right subsymmetry {ijklmn} = {ijklnm} and major symmetry {ijklmn} = {nmlkji}

with 171 independent parameters

This gives in total again 300 constants, which can eventually be reduced by the exploitation of symmetry properties. The isotropic versions of the elastic energy can be found in, e.g., MINDLIN/ ESHEL (1968) and BERTRAM/ FOREST (2013) with only 7 independent parameters including the two Lamé constants from classical elasticity6.

The elastic energy (83) acts as a potential for the stresses with (55) and (56)

(84) k (C, K) =

4

E [C – Cu] +

5

E [K – K u]

(85) K (C, K) = ½ [C – Cu]

5

E + ½

6

E [K – K u] (C, K) Conf .

These laws are straightforward extensions of the St.-Venant-Kirchhoff law to gradient elasticity. They are physically linear, but geometrically nonlinear, and they fulfil the Euclidean invariance requirement. Note that the linear theory depends on the choice of the stress and configuration variables, in contrast to the preceding non-linear theory. However, for small deformations, the differences remain negligible.

In the linear case the isomorphy conditions (69) and (70) become with (P, P) Inv

(86)

4

E Y[C – CuY] +

5

E Y[K – K uY]

= P det –1(P) {4

E X[PT C – CuX] +

5

E X[PT ○ K + P – KuX]}

(87) [C – CuY]

5

E Y +

6

E Y[K – K uY]

= P ○ det –1(P) {[PT C – CuX]

5

E X +

6

E X [PT ○ K + P – KuX]}) .

By a comparison in the independent variables (C, K) Conf one can determine the transformations of the elasticities and the unloaded configuration like

(88)

4

E Y = P det –1(P)

4

E X and CuY = P –T CuX .

The linearity of the elastic laws will not be assumed in what follows, in order to preserve full generality.

Gradient Elastoplasticity

For a gradient theory of elastoplasticity, we consider materials for which both the elastic and the plastic behaviour are assumed to be of gradient type.

6 see SUIKER/ CHANG (2000), DELL´ISOLA/ SCIARRA/ VIDOLI (2009), AUFFRAY/ LE QUANG/ HE (2013)


14

While in an inelastic theory one would expect that the current stresses depend on the entire deformation process, in elastoplasticity the situation is different. Here one assumes that after some deformation process the material is within some elastic range for which elastic laws for the stresses are given. Thus, the stresses can be determined by these current elastic laws. And this holds also for any continuation of the deformation process as long as it does not leave the current elastic range. The latter would mean that the material continuously passes through different elastic ranges, a process which characterizes yielding. We want to make these concepts more precise.

Definition 4. An elastic range is a triple {Ep , kp , Kp} consisting of

1.) a path-connected submanifold with boundary Ep Conf and

2.) elastic laws

(89) S = kp(C, K)

(90) SK = Kp(C, K)

such that after any continuation process {C(), K()} tto

, which remains entirely in Ep

{C(), K()} Ep [to , t]

the stresses are determined by the final values of the process by elastic laws

(91) S(t) = kp(C(t), K(t))

(92) SK(t) = kp(C(t) , K(t)) .

Note that the two elastic laws are physically determined only for configurations within the specific elastic range Ep. However, in the sequel we will extend them to the entire space Conf for simplicity.

Assumption. At each instant the elastoplastic material point is associated with an elastic range.

Isomorphy of the Elastic Ranges

During yielding two effects have to be considered. Firstly, the elastic range has to be changed reflecting the hardening or softening of the material. And secondly, the elastic laws associated to these elastic ranges evolve. We will first address this second effect.

For many materials it is a microphysically and experimentally well-substantiated fact that during yielding the elastic behaviour hardly alters even under very large deformations. This reduces the effort for the identification tremendously, as otherwise one would have to identify the elastic constants at each step of the deformation anew. We now give this assumption a precise form.

Assumption. The elastic laws of all elastic ranges are isomorphic.

As a consequence, if {E1 , k1 , K1} and {E2 , k2 , K2} are two elastic ranges, then according to (68), (69) and (70)

there exist two tensors (P12 , P12) Inv such that

for the mass densities in the reference placements 01 and 02 holds

(93) 01 = 02 det P12

and for the elastic laws we have the equalities

(94) k2(C, K) = det –1(P12) [P12 k1(P12T C, P12

T ○ K + P12)]

(95) K2(C, K) = det –1(P12) [P12 ○ K1(P12T C, P12

T ○ K + P12)] (C, K) Conf .


15

As we have chosen a joint reference placement for all elastic laws of one particular material point (this is, however, not compulsory), we already have 01 02 and therefore P12 must be proper unimodular, so that the first isomorphy condition (93) is always fulfilled.

If all elastic laws belonging to different elastic ranges are mutually isomorphic, then because of the group property of isomorphy transformations, they all are isomorphic to some freely chosen elastic reference laws k0 and K0 . While the current elastic laws kp and Kp vary with time during yielding, these reference laws can always be chosen as constant in time. We thus have the isomorphy condition in the following form.

Theorem 2. Let k0 and K0 be the elastic reference laws for an elasto-plastic material. Then for each elastic range {Ep , kp , Kp} there are two tensors (P, P) Unim such that

(96) S = kp (C, K) = P k0 (PT C, PT ○ K + P)

(97) SK = Kp(C, K) = P ○ K0 (PT C, PT ○ K + P) (C, K) Conf .

Instead of the two elastic reference laws we can also introduce an elastic reference energy w0 after (71) such that

(98) wp (C, K) = w0 (PT C, PT ○ K + P) (C, K) Conf

and the potentials hold in the form of (55) and (56), which again gives (96) and (97)7.

If one linearises these elastic laws, then we obtain after (86) and (87)

(99)

4

E p[C – Cup] +

5

E p[K – K up]

= P {4

E 0[PT C – Cu0] +

5

E 0[PT ○ K + P – Ku0]}

(100) [C – Cup]

5

E p +

6

E p[K – K up]

= P ○ {[PT C – Cu0]

5

E 0 +

6

E 0 [PT ○ K + P – Ku0]})

where the suffix p indicates the (time-dependent) quantities related to the linear forms of kp and Kp , and the

suffix 0 to the (time-independent) ones of the linear forms of the elastic reference laws k0 and K0 . Again one can determine the transformations of the elasticities and the unloaded configuration as in (88)

(101)

4

E p = P

4

E 0 and Cup = P –T Cu0 .

In the present theory, the two variables (P, P) Unim are chosen as the plastic internal variables. One might be tempted to interpret (96) and (97) as both an additive and a multiplicative decomposition of the kinematical variables8. They are, however, not introduced as deformations but rather as a transformation of the current elastic law (not of a placement!) to a time-independent reference law, which results in a natural way from the isomorphy condition. We avoid the introduction of an intermediate configuration or a split of some deformation into elastic and plastic parts since it is misleading in a finite deformation theory9.

Yield Criteria

Let us first consider one particular elastic range {Ep , kp , Kp} . We decompose the Ep topologically into its interior

Epo and its boundary Ep . The latter is called yield surface (in the configuration space). In order to describe it

more easily, we introduce a real-valued tensor-function in the configuration space

7 see the alternative decomposition of FOREST/ SIEVERT (2003) 8 In CHAMBON/ CAILLERIE/ TAMAGNINI (2001) such an interpretation is given. 9 see the comments in BERTRAM (2005) on p. 291.


16

p : Conf R (C , K) p(C , K)

the kernel of which coincides with the yield limit

(102) p(C , K) = 0 (C , K) Ep .

For distinguishing points in the interior and in the exterior of the elastic ranges, we postulate

(103) p(C , K) < 0 (C , K) Ep o

and, consequently,

(104) p(C , K) > 0 (C , K) Conf \ Ep .

We call such an indicator function or level set function a yield criterion, and assume further-on that p is at least piecewise differentiable.

Instants of yielding are characterised by two facts.

1) The configuration is currently on the yield limit and, thus, fulfils its yield condition

(105) p(C , K) = 0 .

2) It is about to leave the current elastic range, i.e., the loading condition

(106) p = Cp C + K p K > 0

is fulfilled.

Such a yield criterion is associated with some particular elastic range. In order to obtain a general yield criterion which holds for all elastic ranges in the same form, we have to introduce additional internal variables Z called hardening variables (although they could also describe softening). These can be tensors of arbitrary order or even a vector of such tensors and, thus, form elements of some finite dimensional linear space, the specification of which depends on the particular hardening model. The general form of the yield criterion is assumed to be like

(107) (P, P, C, K , Z) .

With this extension we obtain for the yield condition (105)

(108) (P, P, C, K , Z) = 0

and for the loading condition (106)

(109) C C + K K > 0

where the hardening variables are kept constant.

Decomposition of the Stress Power

We will next consider the stress power (46) again and specify it for our elastoplastic material. The specific stress power is with (89), (90), (96), (97), (15)

(110) p = 1/0 [½ kp(C, K) C + Kp(C, K) K]

= 1/0 [½ P k0(PT C, PT ○ K + P) C + P ○ K0 (P

T C, PT ○ K + P) K]


17

= 1/0 [½ k0(Ce , Ke) (PT C) + K0 (Ce , Ke) (PT ○ K)]

with the abbreviations

(111) Ce : = PT C P = PT C

(112) Ke : = PT ○ K + P.

This gives for the rates

(113) Ce= (PT C P) = PT C P + 2 sym(P T C P) = PT C + 2 sym(Ce P

–1 P)

where sym stands for the symmetric part, and

(114) Ke = PT ○ K + P

+ ijk P –1 ei) PT ej) PT ek)P –1 ei) PT ej) PT ek)P –1 ei) PT ej) PT ek)}

= PT ○ K + P P –1 P (Ke P) 2 subsym [(Ke P) P –1 P]

the term with subsym being the symmetric part with respect to the right subsymmetry. We substitute this into (110)

p = 1/0 {½ k0(Ce , Ke) [Ce 2 sym(Ce P

–1 P)]

+ K0 (Ce , Ke) [Ke P + P –1 P (Ke P) 2 subsym [(Ke P) P –1 P]}

and because of the symmetries of the stress tensors

= 1/0 {½ k0(Ce , Ke) Ce + K0 (Ce , Ke) Ke

(115) ½ k0(Ce , Ke) (2 Ce P –1 P) K0 (Ce , Ke) [P P –1 P (Ke P) 2 (Ke P) P –1 P]}

with (98) = w0 (Ce , Ke) + Sp P + Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P]

with the plastic stress and plastic superstress tensor defined as

(116) Sp : = – P –T Ce k0(Ce , Ke) = – C S P –T

(117) Sp : = – K0 (Ce , Ke) = – P –1○ SK .

According to (115) the stress power goes into a change of the elastic reference energy and a dissipative part that is only active during yielding and working on the configuration rates P and P .

Flow and Hardening Rules

For the evolution of the internal plastic variables P, P, Z evolution equations are needed, namely two flow rules

(118) P= f (P, P, C, K, Z, C, K)

(119) P = F (P, P, C, K, Z, C, K)

and a hardening rule

(120) Z= h (P, P, C, K, Z, C, K)


18

all assumed to be rate-independent as customary in plasticity. This can be assured in the usual way by the introduction of a plastic consistency parameter 0

(121) P = f (P, P, C, K , Z, C, K)

(122) P = F (P, P, C, K , Z, C, K)

(123) Z = h(P, P, C, K , Z, C, K)

where we normed the increments of the kinematic variables

(124) C : = C/ and K : = K /

by a factor

(125) : = (C2 + L2 K 2)

which is (only) positive during yielding. The positive constant L with the dimension of a length is necessary for dimensional reasons and controls the ratio of yielding due to C and K . We introduced three functions f , F , h, which give the directions of the flow and hardening, while the amount is finally determined by the consistency parameter. The consistency parameter is zero during elastic processes. During yielding it can be calculated by the yield condition (108)

0 = (P, P, C, K , Z)

= P PP P C CK KZ Z

by (121) - (123)

(126) = P f (P, P, C, K , Z , C, K)P F (P, P, C, K , Z , C, K)

C CK KZ h(P, P, C, K , Z , C, K)

which gives

(127) = [C CK K] /

[P f (P, P, C, K , Z , C, K)P F (P, P, C, K , Z , C, K)

Z h(P, P, C, K , Z , C, K)] .

Both, numerator and denominator of this ratio are always negative during yielding as a consequence of the loading condition (109), and, thus, is positive in this case. If we substitute this value of in (121) - (123), we obtain the consistent flow and hardening rules. In all cases (elastic and plastic), the Kuhn-Tucker condition

(128) = 0 with 0 and 0

holds since at any time one of the two factors is zero.


19

II. Thermomechanical Theory

We will next enlarge the concepts of the mechanical theory to a thermodynamic format.

For that purpose we introduce the following additional fields:

the specific internal energy

the heat supply per unit mass and time Q by irradiation and conduction

the material heat flux per unit area in the reference placement and unit time q forming one part of Q

the absolute temperature

the material temperature gradient g : = Grad

the specific entropy .

As usual we can substitute the internal energy by the Helmholtz free energy according to the Legendre transformation

(129) : = – .

The independent variables of this theory are given by the thermo-kinematical processes

{C( ), K( ), ( ), g( ) t0 }

taking values in Conf R + V . In the general case it is assumed that such a process determines the caloro-dynamical state at its end consisting of the stresses, the heat flux, the internal energy, and the entropy

{S(t), SK(t), q(t), (t), (t)}

by constitutive laws.

The Euclidean invariance principle is already identically fulfilled by consequently choosing material variables.

Gradient Thermoelasticity

Definition 5. A thermoelastic gradient material is one for which the current thermo-kinematical state determines the current kaloro-dynamical state by thermoelastic laws

S(t) = k(C(t), K(t), (t), g(t))

SK(t) = K(C(t), K(t), (t), g(t))

(130) q(t) = q(C(t), K(t), (t), g(t))

(t) = (C(t), K(t), (t), g(t))

(t) = (C(t), K(t), (t), g(t)) .

The free energy is then after (129) also a function of the thermo-kinematical state

(131) (t)(C(t), K(t), (t), g(t)): = (C(t), K(t), (t), g(t)) – (C(t), K(t), (t), g(t)) .

These are altogether reduced forms. If the material is heterogeneous then all these functions can depend on the material point as well.

By substituting these laws in the Clausius-Duhem inequality with (46)

(132) 0 – p + 0

1

q g + +

= – 1/0 {½ k(C(t), K(t), (t), g(t)) C + K(C(t), K(t), (t), g(t)) K}


20

+ 0

1

q(C(t), K(t), (t), g(t)) g + C C + K K + + g g

+ (C(t), K(t), (t), g(t))

= [C – 1/0 ½ k(C(t), K(t), (t), g(t))] C + [K – 1/0 K(C(t), K(t), (t), g(t)] K

+ 0

1

q(C(t), K(t), (t), g(t)) g + g g

+ [ + (C(t), K(t), (t), g(t)) ]

we obtain by standard arguments the following necessary and sufficient conditions for the second law to hold:

(133) g = 0 (independence of the free energy of the temperature gradient)

(134) k(C, K, ) = 20 C (potential for the stresses)

(135) K (C, K, ) = 0 K (potential for the hyperstresses)

(136) (C, K, ) = – (potential for the entropy)

(137) 0 q g (heat conduction inequality).

The extension of the concept of material isomorphisms to thermoelasticity is not trivial and needs some explanation. The basic idea behind this concept is the following10. We consider two thermoelastic points as isomorphic if their thermoelastic behavior shows no measurable difference during arbitrary processes. As measurable quantities we consider the stresses (as a result of balance of moments), the heat flux, and the rate of the internal energy (as a result of the energy balance, the first law of thermodynamics) in the local form

(138) Q = p ,

while the entropy or the free energy are certainly not measurable. For thermoelastic materials the mechanical dissipation is zero

(139) 0 = – p + – = Q –

= Q – (C C + K K + ) .

As we consider the heat supply Q as a measurable quantity, then so is the rate of the entropy for thermoelastic materials. So the entropy of two isomorphic thermoelastic materials named X and Y can differ only by a constant

(140) Y (CY , KY , ) = X (CX , KX , ) + c .

By integrating (136) we obtain a split of the free energy

(141) Y (CY , KY , ) = X (CX , KX , ) c + c .

For the internal energy we obtain by (129)

(142) Y (CY , KY , ) = X (CX , KX , ) + c + X (CX , KX , ).

By extending Definition 2 and Theorem 1 we obtain

Definition 6. Two gradient thermoelastic material points X and Y with elastic laws kX , KX , qX , X , X and kY , KY , qY , Y , Y with respect to arbitrary reference placements are elastically isomorphic if there exist two tensors (P, P) Inv and two constants c , c R such that

(143) 0X = 0Y det (P)

(144) kY (C, K, ) = det –1(P) P kX(PT C, PT ○ K + P, )

(145) KY (C, K, ) = det –1(P) P ○ KX

(PT C, PT ○ K + P, )

(146) qY (C, K, , g) = det –1(P) P qX

(PT C, PT ○ K + P, , PT g)

(147) Y (C, K, ) = X

(PT C, PT ○ K + P, ) + c

10 see BERTRAM/ KRAWIETZ (2012)


21

(148) Y(C, K, ) = X (PT C, PT ○ K + P, ) + c

hold for all (C, K, , g) Conf R + V with 0X and 0Y being the mass densities in the reference placements of X and Y , respectively.

The free energy for isomorphic thermoelastic points is then related by

(149) Y (C, K, , g) = X (PT C, PT ○ K + P, ) + c – c .

Gradient Thermoplasticity

The natural extension of the concept of an elastic range to thermodynamics is as follows.

Definition 7. A (thermo)elastic range is a sextuple {Ep , kp , Kp , qp , p , p} consisting of

1) a path-connected submanifold with boundary

Ep Conf R + V

of the space of the thermo-kinematic variables, and

2) a set of thermoelastic laws (as reduced forms) that give for all thermo-kinematical processes out of some initial state

{C( ), K( ), ( ), g( )ttA

} ,

which remain at all times in Ep , the caloro-dynamic state {S(t), SK(t), q(t), (t), (t)} by thermoelastic laws

S(t) = kp(C(t), K(t), (t), g(t))

SK(t) = Kp(C(t), K(t), (t), g(t))

(150) q(t) = qp(C(t), K(t), (t), g(t))

(t) = p(C(t), K(t), (t), g(t))

(t) = p(C(t), K(t), (t), g(t)) .

Instead of the internal energy, we can use the free energy, which allows for a representation after (129) as

(151) (t)=p(C(t), K(t), (t), g(t)) : = p(C(t), K(t), (t), g(t)) – p(C(t), K(t), (t), g(t)) .

We assume further-on that these functions are continuous and continuously differentiable on Ep , and as such extendible on Conf R + V . Two assumptions are needed in analogy to the mechanical part.

Assumption. At each instant the thermoelastoplastic material point is associated with a thermoelastic range.

Assumption. The thermoelastic laws of all elastic ranges are isomorphic.

Using the isomorphy conditions of thermoelasticity (144) - (148), we obtain

kp(C, K, , g) = P k0(PT C, PT ○ K + P, , PT g)

Kp(C, K, , g) = P ○ K0(PT C, PT ○ K + P, , PT g)

(152) qp(C, K, , g) = P q0(PT C, PT ○ K + P, , PT g)

p(C, K, , g) = 0(PT C, PT ○ K + P, , PT g) + c

p(C, K, , g) = 0(PT C, PT ○ K + P, , PT g) + c

or

(153) p(C, K, , g) = 0(PT C, PT ○ K + P, , PT g) + c – c


22

for all (C, K, , g) Conf R + V with (P, P) Unim . Note that the additive constants c and c can not

depend on the current thermo-kinematical variables C , K, , and g after the above assumption.

The boundary Ep of Ep is again called yield limit or yield surface of the thermo-elastic range.

However, there is no material known for which the yield limit depends on the temperature gradient, so that Ep is trivial in its last component V . We will in the sequel suppress this last component of Ep , so that Ep is considered as a subset of only Conf R +.

The yield criterion associated with some thermo-elastic range is then a mapping

(154) p : Conf R + R {C, K, } p(C, K, )

the kernel of which forms the yield surface

(155) p(C, K, ) = 0 {C, K, } Ep .

For the distinction of states in the interior Epo and beyond the thermo-elastic range, we demand

(156) p(C, K, ) < 0 (C, K, ) Epo.

The loading condition (106) now becomes

(157) p(C, K, ) = Cp C + K p K + p > 0 .

For the general yield criterion of all elastic ranges we use the ansatz with the vector of hardening variables Z

(158) p(C, K, ) = (P, P, C, K, , Z)

assumed to be differentiable in all arguments. The yield condition is then

(159) (P, P, C, K, , Z) = 0

and the loading condition

(160) C C + K K + > 0 ,

which is not the complete time-derivative of .

By analogy to the mechanical ansatz (121) - (123) for the rate-independent yield and hardening rules we set

(161) P = p (P, P, C, K, , g, Z, C, K, )

(162) P = P (P, P, C, K, , g, Z, C, K, )

(163) Z= h(P, P, C, K, , g, Z, C, K, )

with the normed increments

(164) C : = C

/ K : = K / : = /

by a positive factor defined as a generalization of (125)

(165) : = (C2+ L2K 2+ 2/02)

with respect to an arbitrarily chosen reference temperature 0 . The consistency parameter is assumed to have a switcher, which sets the values to zero if both the yield criterion and the loading condition are not simultaneously fulfilled. We introduce the abbreviations for the yield directions and for the hardening direction

P : = P / = p(P, P, C, K, , g, Z, C, K, )

(166) P : = P / = P(P, P, C, K, , g, Z, C, K, )

Z : = Z / = h(P, P, C, K, , g, Z, C, K, ) .

As the yield condition must permanently hold during yielding, we obtain the consistency condition

0 = (P, P, C, K, , Z)

(167) = P P + P P + C C + K K + + Z Z


23

= P P + P P + C C + K K + + Z Z

which allows for determining the plastic multiplier as

(168) (P, P, C, K, , g0 , Z, C, K, )

= (C C + K K + )(P P + P P + Z Z)1.

Both factors are negative for all loadings, and, hence, is positive during yielding. If the yield condition and the loading condition are not simultaneously fulfilled, then is set to zero. If we substitute from (168) into the rules (161) - (163), we obtain the consistent yield and hardening rules as rate forms for the internal variables. Because of the switcher in these are incrementally nonlinear, which is typical for elastoplasticity. However, for cases of yielding, is linear in the increments C, K, . Again, the Kuhn-Tucker condition holds in the form (128).

The additive constants in the free energy and in the entropy c and c must remain constant during elastic

processes because of the assumption of isomorphic thermo-elastic ranges, and thus can not depend on C, K, , or

g. They can only depend on those variables which are constant during elastic processes which are P, P, and Z, i.e.,

c(P, P, Z) , c(P, P, Z) . We will assume that these dependencies are differentiable. Consequently, after (152) and (153) we obtain

p(C, K, , g0) = 0(PT C, PT ○ K + P, , PT g) + c(P, P, Z)

(169) p(C, K, , g0) = 0(PT C, PT ○ K + P, , PT g) + c(P, P, Z)

p(C, K, , g0) = 0(PT C, PT ○ K + P, , PT g) + c(P, P, Z) – c(P, P, Z).

In the literature an additive split of the free energy into elastic and plastic parts is often assumed. In the present context it is a consequence of the isomorphy assumption. Note that the plastic parts of the internal energy and the entropy can not depend on the temperature, while the plastic part of the free energy

(170) c(P, P, , Z) : = c(P, P, Z) – c(P, P, Z)

is linear in the temperature.

The material time-derivative of the free energy is with ge : = PT g

= 0(Ce , Ke , , ge) + c(P, P, Z) – c(P, P, Z) – c(P, P, Z)

(171) = Ce 0 (PT C P) + Ke

0 (PT ○ K + P) + 0 + ge0 (PT g)

+ (P c – P c) P + (P c –

P c) P + (Z c – Z c) Z – c .

Without physical effect in the present setting, we assume the symmetry of Ce0 and the right subsymmetry of

Ke0 . All these derivatives have to be taken at {Ce , Ke , , ge} = {PT C, PT ○ K + P, , PT g}. By use of

(113) and (114) we continue with (166)

= Ce0 (PT C + 2 Ce P

–1 P) + Ke0 [PT ○ K + P P –1 P (Ke P)

2 (Ke P) P –1 P] + 0 + ge 0 (PT g + PT g) + (P c –

P c) P

+ (P c – P c) P + (Z c –

Z c) Z – c

(172) = (P Ce 0) C + (P ○ Ke

0) K + ( 0 – c) + (P ge

0) g

+ Ce 0 (2 Ce P

–1 P) + Ke 0 [P P –1 P (Ke P) 2 (Ke P) P –1 P]

+ (P c – P c) P + (P c –

P c) P + ge0 (PT g)

+ (Z c – Z c) Z.

We substitute this and the stress power density after (110) into the Clausius-Duhem inequality in the same form as (132)

0 – p + 0

1

q g + +

= – 1/0 [½ P k0(PT C, PT ○ K + P) C + P ○ K0 (P

T C, PT ○ K + P) K]


24

+ 1/(0 ) q g + (P Ce0) C + (P ○ Ke

0) K + ( 0 – c) + (P ge

0) g

+ Ce0 (2 Ce P

–1 P) + Ke0 [P P –1 P (Ke P) 2 (Ke P) P –1 P]

+ (P c – P c) P + (P c –

P c) P + (Pge 0) g

(173) + (Z c – Z c) Z +

= 1/0 [ ½ P k0(PT C, PT ○ K + P) + P Ce

0] C

+ [ 1/0 P ○ K0 (PT C, PT ○ K + P)+ P ○ Ke

0] K

+ [ 0 +0] + (P ge

0) g + 1/(0 ) q g

+ Ce0 (2 Ce P

–1 P) + Ke0 [ P –1 P (Ke P) 2 (Ke P) P –1 P] + ge

0 (PT g)

+ (P c – P c) P + (P c –

P c + Ke 0) P + (Z c –

Z c) Z.

We will exploit this inequality first for cases without yielding, where only remain

0 1/0 [ ½ P k0(PT C, PT ○ K + P) + P Ce

0] C

(174) + [ 1/0 P ○ K0 (PT C, PT ○ K + P)+ P ○ Ke

0] K

+ [ 0 +0] + (P ge

0) g + 1/(0 ) q g

which leads by standard arguments to the thermoelastic relations:

(175) ge 0 = 0 (independence of the free energy of the temperature gradient)

(176) k0(Ce , Ke , ) = 20 Ce 0 (potential for the stresses)

(177) K0 (Ce , Ke , ) = 0 Ke 0 (potential for the hyperstresses)

(178) 0 = – 0 (potential for the elastic part of the entropy)

(179) 0 q g (heat conduction inequality).

In the case of yielding, the above findings must still hold because of continuity. Additionally, we obtain the residual inequality

(180) 0 Ce0 (2 Ce P

–1 P) + Ke 0 [ P –1 P (Ke P) 2 (Ke P) P –1 P]

+ (P c – P c) P + (P c –

P c + Ke 0) P + (Z c –

Z c) Z

or with (166) and a positive and with (170)

(181) 0 (2 P –T Ce Ce0 + P c) P + Ke

0 [2 (Ke P) P –1 P P –1 P (Ke P)]

+ (Ke 0 + P c) P + Z c Z

posing a restriction on the flow rules and the hardening rule (166). Note that not each of these terms has to be negative, but only the sum of them. Thus, yield against the stresses is not excluded by the second law11.

We will state these findings in the following

Theorem 3. The Clausius-Duhem inequality (173) is fulfilled for every thermo-kinematical process if and only if the following conditions hold.

The free energy does not depend on the temperature gradient.

The free energy is a potential for the stresses and the elastic part of the entropy after (176) - (178).

The heat conduction inequality (179) holds at any instant.

The residual inequality (181) holds during yielding. 11 see BERTRAM/ KRAWIETZ (2012).


25

The first three conditions are familiar from thermo-elasticity. They must hold for the thermo-elastic reference laws, and are then automatically valid for all isomorphic laws, including the current ones. The stress power (115) becomes with the potentials (176) to (178)

p = Ce0(Ce, Ke , ) Ce

+ Ke0(Ce, Ke , ) Ke

(182) + Sp P + Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P]

= 0(Ce , Ke , ) + 0(Ce , Ke , )

+ Sp P + Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P] .

Temperature Changes

In order to determine the change of the temperature of the material point under consideration, we use the local form of the first law of thermodynamics (138) with the heat supply per unit mass and time Q , which results from irradiation and conduction. We substitute the stress power (182) and the internal energy (169) into the first law (138)

(183) Q = 0(Ce, Ke , ) + c(P, P, Z) 0(Ce, Ke , ) 0(Ce, Ke , )

Sp P Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P] .

y using (129) for the elastic parts we get

(184) Q = 0(Ce , Ke , ) + c(P, P, Z) Sp P

Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P] .

This can be split into an elastic heat generation using (176) - (178)

(185) Qe : = 0(Ce , Ke , )

= Ce0 Ce

Ke0 Ke

0

= 0 + (20)

1 k0(Ce, Ke , ) Ce + 0

1 K0(Ce, Ke , ) Ke

and a plastic heat generation

(186) Qp : = c(P, P, Z) + Sp P + Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P] .

This can be solved for the temperature change

(187) = {Q + R Ce + R Ke

c(P, P, Z)

+ Sp P + Sp [P P –1 P (Ke P) 2 (Ke P) P –1 P]} / c

with the abbreviations for

the specific heat c (Ce , Ke , ) : = 0

the 2nd order stress-temperature tensor R(Ce , Ke , ) : = – Ce 0

the 3rd order stress-temperature tensor R(Ce , Ke , ) : = – Ke 0 .

By equation (187), we can integrate the temperature along the process and so determine the final temperature after some arbitrary elasto-plastic process. After (187), temperature changes are caused by

the heat supply Q from the outside

thermoelastic transformations due to the second term in Eq. (187)


26

the heat Qp generated by plastic yielding and hardening, which can be determined by use of the flow and hardening rules (161) - (163).

Conclusions

The present text can be considered as a continuation of the work of FOREST/ SIEVERT (2003). A finite gradient plasticity theory is there envisaged only in its basics. Unfortunately, Rainer Sievert passed away too early to finish his promising concepts. So we tried to continue with this task in the spirit of Rainer´s ideas.

The starting point there and here is a higher order expression of the stress power12. The additional variables for extending the theory of simple materials are the third-order curvature tensor and a work-conjugate hyperstress tensor. By giving both of them a material form which is invariant under Euclidean transformations, we obtain reduced forms for the elastic law and the elastoplastic model. The choice of these variables, however, is immaterial for the results. Any other choice would describe exactly the same material behaviour.

While the introduction of finite gradient elasticity appears to be rather straightforward, finite plasticity does certainly not. Here we are again confronted with conceptual questions which already have been discussed during the last 5 decades for finite plasticity of simple materials. We tried to solve these problems by the same concept of isomorphic elastic ranges which gave good results already in the much simpler context of non-gradient plasticity. It turns out that also in the case of gradient plasticity this concept leads to a natural introduction of additional plastic variables with clear mathematical properties, thus avoiding all the problems and controversies associated with the concept of unloaded intermediate configurations.

By enlarging these concepts to the thermomechanical format, we are able to describe materials with thermoelastic ranges, where the gradient terms appear both in the elastic laws and in the plastic parts of the model.

The intention of this paper is to give a general framework for material models of gradient elasticity and plasticity which is based on clear physical assumptions like

Euclidean invariance

isomorphy of the elastic ranges

Clausius-Duhem inequality

trying to avoid hidden or tacit assumptions and those lacking a clear physical substantiation. Naturally, the span of this theory turns out to be rather large. This makes the theory, at least in some parts, also more complicated. The latter, however, can be reduced in many cases by restricting to isotropy, linearity, etc.

Acknowledgment. The author wants to thank Thomas Böhlke, Samuel Forest, Rainer Glüge, Christian Reiher, and, in particular, Arnold Krawietz for stimulating discussions and helpful comments.

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