Comparison of hyperelastic micromorphic, micropolar and microstrain continua

International Journal of Non-Linear Mechanics 77 (2015) 115–127

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm

Comparison of hyperelastic micromorphic, micropolar and microstrain continua T. Leismann, R. Mahnken n Chair of Engineering Mechanics (LTM), University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 20 February 2015 Received in revised form 3 June 2015 Accepted 4 August 2015 Available online 12 August 2015

Micromorphic continua are equipped with additional degrees of freedom in comparison to the classical continuum, representing microdeformations of the material points of a body. Secondary they are provided with a higher order gradient. Therefore, they are able to account for material size-effects and to regularize the boundary value problem, when localization phenomena arise. Arbitrary microdeformations are allowed for in the micromorphic continuum, while the special cases micropolar continuum and microstrain continuum merely allow for microrotation and microstrain, respectively. Amongst these cases, the micropolar case has been covered most extensively in the literature. One goal of this paper is to make the transition from a full micromorphic continuum to a micropolar or microstrain continuum, by varying the constitutive equations. To this end two different possibilities are presented for hyperelasticity with large deformations. This leads to four different material models, which are compared and illustrated by numerical examples. Another goal is to present a constitutive model encompassing the micromorphic, micropolar and microstrain continua as special cases and enabling arbitrary mixtures of micropolar and microstrain parts, allowing the representation of versatile material behaviour. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Micromorphic continuum Micropolar continuum Microstrain continuum Higher order gradient Enhanced continuum theory Hyperelasticity

1. Introduction Many materials show size-dependent behaviour, e.g. for metals and ceramics the indentation hardness increases with decreasing indenter size for micron-size indents, see [27,21,24,1]. Additionally, there are localization phenomena, such as shear bands, which occur under softening. The classical continuum theory is not able to account for these phenomena, which is why different extensions of classical continuum theory have been developed. Supplementary quantities are introduced, that lead to non-local behaviour, meaning that the stresses of a material point are dependent on a finite neighbourhood. To govern the non-locality an internal length scale is introduced. The non-local behaviour leads to a regularization of the boundary value problem, when localization phenomena arise, and the internal length scale can be used to represent size dependence. When higher order gradients of strain or internal variables are used to account for non-local behaviour, the theories are called gradient theories [23,18]. The so-called micromorphic theories,

n

Corresponding author. E-mail addresses: [email protected] (T. Leismann), [email protected] (R. Mahnken). http://dx.doi.org/10.1016/j.ijnonlinmec.2015.08.004 0020-7462/& 2015 Elsevier Ltd. All rights reserved.

which incorporate micropolar and microstrain theories as special cases, introduce additional degrees of freedom for that purpose. To every material point an additional micro-continuum representing a deformation of this point is attached. Micromorphic models intend to capture the microstructure of a material, by introducing additional degrees of freedom, which are labelled as micro degrees of freedom, but they are still entirely macroscale models. Micromorphic materials were introduced by Eringen in [6]. The subsequent publications, e.g. [7], focussed on micropolar continua, which are a special case of micromorphic continua. Micropolar continua have a rigid micro-continuum, which can only rotate, whereas a micromorphic continuum has fully deformable microcontinua. The micropolar continuum is covered extensively in the literature. Concerning plasticity and parameter identification, see e.g. [26,4,5,22]. Another special case is the microstrain continuum. It is a micromorphic continuum without rigid body rotation of the micro-continua and was introduced in [11]. Plasticity for micromorphic continua has been the subject of various publications in recent years, see e.g. [9,10,12–14,17]. The papers [9,13,14] are also concerned with damage. A comparison of elastic micropolar and microstretch continua for small strains has been conducted in [19]. Micromorphic, micropolar and microstrain continua can be applied to model different materials. For example a hybrid composite of ceramics and polymers consists of deformable

116

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

polymer particles, which are embedded in a ceramic foam. This microstructure can be represented by a micromorphic continuum. Sand on the other hand consists of nearly rigid particles, which are able to rotate against each other. This behaviour can be captured by a micropolar continuum. The microstrain continuum could be useful to model materials with a deformable microstructure with few rigid body rotation, e.g. metal foam. The material motivating this paper is a cold box sand, as described in [3], which is used for cold box casting. It consists of sand and a binder, which creates a matrix for the sand particles. For the composites of this material different material models would be suitable. For sand, as mentioned before, micropolar models are often considered, because of the nearly rigid particles. In contrast to that, the binder is able to experience strain, and rotation is not an important factor. A micromorphic model should be well suited for a homogenization of these two phases. For this purpose, it seems to be advantageous to have independent influence on the micropolar and microstrain parts of the model. In this work new material models for the micropolar and microstrain continuum are presented, based on a hyperelastic micromorphic model introduced in [15]. Furthermore, an additive micromorphic continuum model is proposed, which is able to represent the micromorphic, micropolar and microstrain case and combinations of the three by varying the material parameters. In this way it is possible to investigate the influence of the micropolar and microstrain parts with one model in a consistent way. The aforementioned micropolar, microstrain and micromorphic models are used to validate the results of the additive micromorphic model. For this purpose and to illustrate the differences between the material models two numerical examples are presented. In this paper, Section 2 presents the framework for micromorphic elasticity, with kinematic relations in Section 2.1, followed by balance laws and weak forms in Section 2.2. Hyperelastic constitutive models are introduced in Section 2.3 and the discretization for a finite element implementation is shown in Section 2.4. Section 3 presents numerical examples to compare and illustrate the material models. Finally Section 4 gives a conclusion and an outlook.

micromorphic continuum, as presented in [15] and utilized in this paper, has ðndim Þ2 additional degrees of freedom, which are called micro degrees of freedom. The components of the so-called microdeformation map, which represents the deformation of a point in the micro-continuum (as does the deformation gradient for the macro-continuum), are introduced as micro degrees of freedom. This is done instead of introducing micro-displacements, because otherwise C1-continuity would be necessary for the elements in a finite element implementation, see [20]. 2.1. Kinematic relations A material point has coordinates x in the spatial configuration and coordinates X in the material configuration. The motion φ maps the latter to the former 1:

x ¼ φðXÞ;

2:

FðXÞ ¼ ∇φðXÞ:

ð1Þ

F is the deformation gradient for the macroscale, with the material gradient represented by ∇. Additionally, there are micro-continua attached to each macromaterial point. To simplify the theory these can only experience homogeneous deformations. Consequently, the deformation gradient of the micro-continua is not dependent on placements x and X of the micro-material points in the spatial and material configuration, respectively. It only depends on the placement of the macro-material points. This is why we skip to introduce a micromotion and directly introduce a micro-deformation gradient FðXÞ, which is able to represent the deformation of the microcontinuum assigned to X. We also introduce the macro-material gradient of the micro-deformation gradient as GðXÞ ¼ ∇FðXÞ:

ð2Þ

The material gradient over the macro-domain is represented by ∇ðÞ. The kinematic relations are illustrated in Fig. 1 with a body in the material configuration B0 and the spatial configuration Bt . Additionally, there is a micro-continuum B in both configurations. 2.2. Balance laws and weak form

2. Micromorphic continuum model In a micromorphic continuum, the classical macro-continuum is enhanced by adding micro-continua to each material point. These micro-continua may experience affine, kinematically independent deformation, consisting of stretch and rotation, see e.g. [16]. When the deformation is limited to rotation, the continuum is called micropolar, also known as Cosserat continuum. A full

The local balance of linear momentum in a material form, see e.g. [2, p. 144], for a quasistatic problem is Div P þb ¼ 0;

ð3Þ

which describes the macroscale. DivðÞ is the material divergence. For a micromorphic continuum we obtain an additional balance of moment of momentum for the microscale, see [8, p. 45]: Div Q  P ¼ 0:

ð4Þ

The stresses are of Piola type and denoted as macro-stress P, micro-stress P and micro double stress Q , following [15], where the latter is a tensor of third order. b is the body force. The following boundary conditions, acting in B0 , apply

Fig. 1. Kinematic variables.

1:

φ ¼ φpre on ∂Bφ0 ;

3:

P  N ¼ t on ∂BP0 ;

2: 4:

F ¼F

pre

on ∂BF0

Q  N ¼ 0 on ∂BQ 0 :

ð5Þ

where N is the outward normal vector on the material surface and t is the macro-traction on the Neumann boundary ∂BP0 . A possible micro-traction on the boundary BQ 0 is neglected, because to the authors’ knowledge it cannot be applied in real experiments. A product with one dot denotes a simple contraction. So called strong solutions for every material point with the equations above are generally not available. This is why variational methods are used to gain weak forms. By multiplying Eqs. (3) and (4) with test functions  δφ and δF, respectively, and integrating over B0 , we

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

obtain the global equation Z Z    ðDiv P þbÞ  δφ dV ¼ Div Q  P : δF dV; B0

B0

8 δφ; δF:

117

ð6Þ

wherein : denotes a double contraction. By transforming the divergence with the product rule and Gauss's theorem and using Eq (1.2) and (2.2), we can write the Principal of Virtual Power for a micromorphic continuum in the following form: Z   P : δF þP : δF þ Q : δG  b  δφ dV B0 Z ¼ P  N  δφ dA; 8 δφ; δF; ð7Þ ∂B0

  wherein : denotes a triple contraction, δF ¼ ∇ δφ δG ¼ ∇ δF . With the definitions Z 1: aðφ; F; δφ; δFÞ≔ P : δF þ P : δF þQ : δG dV; B0 Z Z P  N  δφ dA: 2: f ðδφÞ≔ b  δφdV þ B0

and

ð8Þ

∂B 0

Fig. 2. Polar decomposition of the deformation gradient F.

Eq. (7) rewrites as aðφ; F; δφ; δFÞ  f ðδφÞ ¼ 0;

8 δφ; δF

ð9Þ

with the solution ðφ; FÞ A H for all ðδφ; δFÞ A H . The space H is a Hilbert space of order one. We use the convention that for semilinear forms, such as að; Þ, the form is linear with respect to arguments on the right side of the semicolon. 1

1

The Piola-type stresses are derived from Eqs. (10) as

1

1:

P ¼ DF W ¼ ðλ ln J  μÞF  T þ μF pðF  FÞ

2:

P ¼ DF W scale ¼ pðF  FÞ

3:

Q ¼ DG W mic ¼ μl G: 2

ð13Þ

For the subsequent investigation the following derivatives of stresses will be needed:

2.3. Hyperelastic constitutive model Let us define U ¼ Uðφ; F; F; GÞ as the total potential energy density per unit volume dV in the material domain B0 . Then, for a hyperelastic material the stresses are obtained from the derivatives of U with respect to their energetically conjugate deformation variables at fixed material placements:

1:

DF P ¼ λF  T  F  T ðλ ln J  μÞF  T  F  1 þ ðμ þ pÞI  I

2:

DF P ¼  p I  I

3:

DF P ¼  p I  I

P≔DF U;

ð10Þ

4:

DF P ¼ p I  I

The total potential energy density U ¼ W þ V consists of internal and external contributions W and V, respectively.

5:

DF Q ¼ 0

6:

DG Q ¼ μl I  ðI  IÞ;

P≔DF U;

Q ≔DG U:

2.3.1. Micromorphic case Following [15], a hyperelastic constitutive model for the internal stored energy density W is introduced as WðF; F; GÞ ¼ W mac þ W mic þ W scale :

ð11Þ mac

With a Neo-Hooke type law for the macroscale W , a quadratic term for the microscale W mic and a difference in the scale transition term for the coupling of both scales W scale , we specialize 1:

W mac ðFÞ ¼ 12 λ ln J þ 12 μðF : F n dim  2 ln J Þ 2

mic

ðGÞ ¼ 12 μl G : G 2

2:

W

3:

W scale ðF; FÞ ¼ 12 pðF  FÞ : ðF  FÞ:

ð12Þ

In the above equations λ and μ are Lamé constants, l is an internal length scale, p is a penalty parameter, ndim is the number of dimensions in space and J is the determinant of F, J ¼ det F. This model is chosen to simulate the elastic part of the mechanical behaviour of a cold box sand, as described in [3]. One-dimensional tests have shown that the macro-behaviour can be captured by a hyperelastic law, such as the Neo-Hooke type law in Eq. (12.1). The microscale term in Eq. (12.2) is chosen, so that it yields simple terms for the stresses. The scale transition term in Eq. (12.3) is advantageous for the transition from micromorphic to micropolar and microstrain continua, as will be seen in Sections 2.3.2 and 2.3.3.

ð2;3Þ

ð2;4Þ

ð2;4Þ ð2;4Þ

ð2;4Þ

2 ð2;3Þ

ð2;4Þ

ð14Þ

where I is the second order identity tensor. The standard dyadic ðÞ

product is represented by  . The products  are modified dyadic products between tensors of arbitrary order, where the numbers above the product sign indicate which number the indices of the right hand tensor get among the indices of the resulting tensor, e.g. with a tensor of 3rd order A and a tensor of 2nd order B, the product ð2;4Þ

is A  B ¼ Aikm Bjl ei  ej  ek  el  em . 2.3.2. Micropolar case The micropolar continuum is a special case of the full micromorphic continuum. The micro degrees of freedom are restricted to rotation, so there is no microstrain. To achieve this, we make use of the polar decomposition of the deformation gradient F ¼ R  U into a rotation tensor R and a stretch tensor U, see Fig. 2. Analogously, the micro-deformation gradient F is decomposed into a micro-rotation tensor R and a micro-stretch tensor U. For a micropolar continuum the micro-deformation gradient is equal to the micro-rotation tensor F ¼ R and the micro-stretch tensor is equal to the second order identity tensor U ¼ I. R is a proper orthogonal tensor. With respect to an orthonormal basis ei its Euler–Rodrigues representation is R ¼ ðe1  e1 þ e2  e2 Þ cos φ þ ðe2  e1  e1  e2 Þ sin φ þ e3  e3 ;

ð15Þ

118

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

if the axis of rotation coincides with e3 . φ represents the microrotation angle. In the two-dimensional case the matrix of coefficients of R with respect to the basis e1 , e2 is " # cos φ  sin φ R¼ : ð16Þ sin φ cos φ This equation shows that R only has one degree of freedom φ , while F in the micromorphic case has four. Analogous to Eq. (11) we introduce the internal stored energy density WðF; R ; ∇R Þ ¼ W mac þ W mic þ W scale ;

ð17Þ

where W mac is the Neo-Hooke term in Eq. (12.1) and where mic

W

W scale ðR; R Þ ¼ 12 pðR  RÞ : ðR  RÞ:

ð18Þ

The rotation tensor is computed from the polar decomposition of the macro-deformation gradient R ¼ F  U  1 . Furthermore, the stretch tensor for the two-dimensional case is computed from    1=2   1=2 1=2 U ¼ trðCÞ þ 2ðdetðCÞÞ ðdetðCÞÞ IþC ; ð19Þ with the right Cauchy-Green deformation tensor C ¼ FT  F. The trace of a tensor is abbreviated by trðÞ. In Eq. (19) the square root of a tensor (in this case the two-dimensional tensor C) is computed. It is taken from [28] and is based on Cayley–Hamilton's theorem. The scale transition term W scale in Eq. (18.2) is changed compared to Eq. (12.3), such that it includes the difference of the micro-rotation tensor and the macro-rotation tensor R  R. With the parameter p this difference is penalized and so the micro-deformation is affine to the macro-rotation. According to Eq. (10), the equations for the stresses are 1:

P ¼ DF W ¼ ðλ ln J  μÞF  T þ μF pðR  RÞ

2:

P ¼ DF W scale ¼ pðR  RÞ

3:

Q ¼ DG W mic ¼ μl ∇R :

ð1;2j 1;2Þ

:

DF R

2

ð20Þ

ð1;2j 1;2Þ

The product : is a double contraction over the first and the second index of the tensors on the left and the right side of the product sign, e.g. for a tensor of third order A and a tensor of ð1;2j 1;2Þ fourth order B, the product is A : B ¼ Aijk Bijlm ek  el  em . For the subsequent investigation the following derivatives will be needed: 1:

DF P ¼ λF

F

T

 ðλ ln J  μÞF

T

ð2;3Þ

 F

þ p DF R : DF R pðR  RÞ : DF ðDF R Þ 2:

ð2;4Þ

4:

DR P ¼ p I  I

5:

DF Q ¼ 0

6:

D∇R Q ¼ μl I  ðI  IÞ

ð2;4Þ

2

ð2;3Þ

ð2;4Þ

ð2;4Þ

with

ð2;3Þ

1 ðI  I  R  R T Þ trðUÞ

7:

DF R ¼

8:

Dφ R ¼  ðe1  e1 þ e2  e2 Þ sin φ þ ðe2  e1  e1  e2 Þ cos φ

9:

  Dφ Dφ R ¼  ðe1  e1 þ e2  e2 Þ cos

φ  ðe2  e1  e1  e2 Þ sin φ :

ð21Þ

The result for DF R in (21.7) is taken from [25].

ð∇RÞ ¼ μl ∇R : ∇R

1:

T

DF P ¼  p I  I : DF R

2

2:

1 2

ð2;4Þ

3:

DR P ¼  p I  I

1

ð2;4Þ

þμ I  I

2.3.3. Microstrain case The microstrain continuum is another special case of the micromorphic continuum. The micro degrees of freedom are in this case restricted to strain, so there is no micro-rotation. For a microstrain continuum the micro-deformation gradient is equal to the micro-stretch tensor F ¼ U and the micro-rotation tensor is equal to the second order identity tensor R ¼ I. U is a symmetric, positive definite tensor, so it has only three independent components in the two-dimensional case. Analogous to Eq. (11) we introduce the internal stored energy density WðF; U; ∇UÞ ¼ W mac þW mic þ W scale ;

ð22Þ

where W mac is the Neo-Hooke term in Eq. (12.1) and where 1: 2:

W mic ð∇UÞ ¼ 12 μl ∇U : ∇U 2

W

scale

ðU; UÞ ¼

1 2 pðU  UÞ

: ðU  UÞ;

ð23Þ

wherein U can be computed according to Eq. (19). The scale transition term W scale in Eq. (23.2) includes the difference of the micro-stretch tensor and the macro-stretch tensor U  U. With the parameter p this difference is penalized and so the microdeformation is affine to the macro-stretch. According to Eq. (10), the equations for the stresses are 1:

P ¼ DF W ¼ ðλ ln J  μÞF  T þ μF  pðU  UÞ : DF U

2:

P ¼ DF W scale ¼ pðU  UÞ

3:

Q ¼ DG W mic ¼ μl ∇U: 2

ð24Þ

For the subsequent investigation the following derivatives will be needed: 1:

ð2;3Þ

ð2;4Þ

DF P ¼ λF  T  F  T ðλ ln J  μÞF  T  F  1 þ μ I  I þ p DF U : DF U  pðU  UÞ : DF ðDF UÞ

Fig. 3. Function plots of pRU and pUR (a) pRU ðpR ; pU Þ if 0 r pU o pR and (b) pUR ðpR ; pU Þ if 0r pR o pU .

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

continuum. Analogous to Eq. (11) we introduce the internal stored energy density

Table 1 Overview of material models for the numerical examples. Material model

Micro-quantity Material parameters

For all models Micromorphic

/ F

E ¼ 1000 N=mm , ν ¼ 0:4 / 2.3.1 p ¼ 20E, l ¼ 20 mm

Micropolar

R

p ¼ 20E, l ¼ 20 mm

2.3.2

Microstrain

U

2.3.3

Additive micromorphic

R and U

p ¼ 20E, l ¼ 20 mm pR ¼ pU ¼ 20E,

R

lR ¼ lU ¼ 20 mm pR ¼ 20E, lR ¼ 20 mm,

Additive micropolar Additive microstrain

U

Additive microstrain 25% R and U

119

Section

WðF; R; U; ∇R; ∇UÞ ¼ W mac þ W mic þ W scale ;

ð26Þ

mac

2

2.3.4

is the Neo-Hooke term in Eq. (12.1). The additive where W structure of micropolar and microstrain parts is incorporated in the micro and scale transition part of the internal stored energy density as 1:

2.3.4

W mic ð∇R ; ∇UÞ ¼ 12 μl 2R ∇R : ∇R þ 12 μl 2U ∇U : ∇U |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} W

pU ¼ lU ¼ 0 pU ¼ 20E, lU ¼ 20 mm,

2.3.4

pR ¼ lR ¼ 0 pU ¼ 20E, lU ¼ 20 mm,

2.3.4

2:

mic R

W

and

mic U

W scale ðR; U; R; UÞ ¼ 12 p R ðR  RÞ : ðR  RÞ þ 12 p U ðU  UÞ : ðU  UÞ þ 12 pRU ðU IÞ : ðU IÞ þ 12 pUR ðR  IÞ : ðR  IÞ;

pR ¼ 5E, lR ¼ 5 mm

ð27Þ with separate material parameters for the micropolar part (lR , pR ) and the microstrain part (lU , pU ). The additional parameters pRU and pUR are computed from pR and pU as ( pR exp  αpU =pR if 0 r pU opR 1: pRU ¼ 0 else ( pU exp  βpR =pU if 0 r pR o pU ð28Þ 2: pUR ¼ 0 else: with positive constants α and β. An illustration of both function plots for α ¼ 40 and β ¼ 2000 is given in Fig. 3. The advantage of this model is that we can weight the micropolar and the microstrain parts differently and thus permit all kinds of combinations of both parts. The model offers three special cases: 1. Additive micromorphic:Micropolar material constants are equal to the microstrain ones pU and lU lR ¼ lU ) pRU ¼ pUR ¼ 0). 2. Additive micropolar:Microstrain material constants zero (pU ¼ lU ¼ 0 ) pRU ¼ pR , pUR ¼ 0). 3. Additive microstrain:Micropolar material constants zero (pR ¼ lR ¼ 0 ) pUR ¼ pU , pRU ¼ 0).

Fig. 4. Plate with hole under tension. (a) Geometry, constraints and load, and (b) FE mesh.

ð2;4Þ

2:

DU P ¼  p I  I

3:

DF P ¼ p I  I : DF U

4:

DU P ¼ p I  I

5:

DF Q ¼ 0

6:

D∇U Q ¼ μl I  ðI  IÞ   ð2;4Þ ð2∣1Þ DF U ¼ R T  I  I  DF R  U :

7:

ð2;4Þ

ð2;4Þ

2

ð2;3Þ

ð2;4Þ

ð25Þ

1:

P ¼ DF W ¼ ðλ ln J  μÞF  T þ μF

2:

P ¼ DF W ¼ p R ðR RÞ : DF R þ p U ðU UÞ : DF U

 pR ðR  RÞ : DF R  pU ðU  UÞ : DF U 2.3.4. Additive micromorphic continuum The proposed model consists of an additive composition of a micropolar and a microstrain part and therefore allows for microrotation and microstrain, which produces a micromorphic

approach approach

In the second line of Eq. (27.2), there are additional terms, which are only relevant for the two special cases additive micropolar and additive microstrain. In the additive micropolar case without these additional terms, R is affine to R, because of the scale transition term in the first line of Eq. (27.2), but this does not concern U. In the micropolar model this is sufficient, because there is only R and no U, but in the additive micropolar case both R and U are present, so information about U is missing in the scale transition term. This is why the additional term with ðU  IÞ is added, which causes U to be equal to the second order identity. This is done analogously for the additive microstrain case, so that there are two additional terms. These are weighted with the parameters pRU and pUR , which are only different from zero in the additive micropolar and microstrain cases, respectively. A different approach for the scale transition term W scale in Eq. (27), which avoids the additional terms in the second line of Eq. (27.2), but only works for the additive micropolar and microstrain and not the mixed cases, is discussed in Appendix A. According to Eqs. (10), (12.1) and (27) the stresses result as

ð2∣1Þ

The product  denotes a simple contraction over the second index of the left and the first index of the right tensor.

pR and lR (pR ¼ pU ,

þ pUR ðR IÞ : DF R þ pRU ðU  IÞ : DF U |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} P

scale

120

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

Fig. 5. Plate with hole under tension: contour plot of Green Lagrange strain Eyy. (a) Classical, (b) micromorphic, (c) additive micromorphic, (d) micropolar, (e) additive micropolar, (f) microstrain, and (g) additive microstrain.

þ μlR ∇R : DF ð∇RÞ þ μlU ∇U : DF ð∇UÞ : |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

2

P

mic R

P

ð29Þ

mic U

There is no double stress Q , conjugate to G, because the corresponding part of W is incorporated in P, conjugate to F. Otherwise, derivatives of ∇R and ∇U with respect to G would become necessary. Nevertheless, the derivatives are more involved in this case and rather lengthy and can be found in Appendix Appendix B. 2.4. Discretization For a finite element implementation the body B is discretized in space with nel finite elements. The quantities at each material point are interpolated utilizing the values of this ^ and the element quantity at the element nodes, denoted by ðÞ, shape functions N. We distinguish between macro-shape functions N φ and micro-shape functions N F . The scale transition term in Eq. (12.3) contains the difference F  F. The

coefficients of F are incorporated in the finite element model as additional degrees of freedom at the element nodes, so the values at the Gaussian points are computed with the microshape functions, while the values of F are computed with the derivatives of the macro-shape functions. It is desirable for the macro-shape functions to be one order higher than the microshape functions, e.g. quadratic macro- and linear micro-shape functions, though there is no formal need since the method is still consistent. If both shape functions were of the same order, the approximation of F would be one order lower than that of F. Different orders of approximation result in different numbers of nodes nnφ and nn for the macro degrees of freedom and F the micro degrees of freedom, respectively. 2.4.1. Micromorphic case The discretized version of the macro-deformation of Eq. (1) is n

1:

φh ¼

nφ X φ^ Ni φ ;

i¼1

i

n

2:

δφ h ¼

nφ X φ ^ N j δφ ;

j¼1

j

ð30Þ

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

with

The discretized micro-deformation map of Eq. (2) is nn

1:

h

F ¼

F X

k¼1

N Fk F^ k ;

nn

2:

h

δF ¼

F X

l¼1

NFl

δF^ l

ext

ð31Þ

n

1:

F ¼

nφ X i¼1

n

φ^  ∇Nφi ; i

2:

3:

k¼1

nφ X j¼1

nn

F X G ¼ F^ k  ∇N Fk ;

h

δF ¼ h

δφ^ j  ∇N φj

4:

e¼1

Be0

and

ext

fJ

¼ 0;

ð34Þ

ð33Þ

with increments of the degrees of freedom Δu. We can rewrite the linearized coupled problem of Eq. (35) as 2 3 3 # 2 ext int φφ φF " f I f I Δφ^ L K K IM 6 IL 7 5 ¼ 4 ext ð36Þ 4 Fφ 5 int FF ΔF M f J fJ K JL K JM |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

el

2:

N I b dV

∂u R  Δu ¼  R

Be0

Z   n ext h h R FJ ¼ A Q  ∇N FJ þ P N FJ dV  f J

φ h

Be0

ð32Þ

l¼1

The residual vector R consists of the two components: Z nel φ φ ext 1: R I ¼ A P h  ∇N I dV f I and e¼1

e¼1

Z

to vanish for the system to be in equilibrium. A non-linear problem like this cannot be solved directly, instead it is commonly solved with the Newton–Raphson method. Therefore it needs to be linearized:

nn

F X δG ¼ δF^ l  ∇N Fl :

h

nel

¼ A

if only external bulk potential is considered. The residual vector is h iT ^ F^ and needs dependent on the nodal degrees of freedom u≔ φ

and the discretized gradient variables of Eqs. (1) and (2) are h

fI

121

∂u R≔K

ð35Þ

Δu

R

Fig. 6. Plate with hole under tension: contour plot of Green Lagrange strain Exy. (a) Classical, (b) micromorphic, (c) additive micromorphic, (d) micropolar, (e) additive micropolar, (f) microstrain, and (g) additive microstrain.

122

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

Fig. 7. Plate with hole under tension: contour plot of micro-rotation angle φ and macro-rotation angle φ. (a) Micromorphic φ, (b) micromorphic φ, and (c) microstrain φ.

3:

nel

Fφ

K JL ¼ ∂φL R FJ ¼ A

e¼1

Z

nel

¼ A

e¼1

4:

Z   h h φ φ DF ðQ  ∇N FJ Þ  ∇N L þ DF ðP N FJ Þ  ∇N L dV Be0

φ

B e0

φF

DF P h N FJ  ∇N L dV ¼ K LJ nel

Z 

e¼1

Be0

K FJMF ¼ ∂F M R FJ ¼ A

 h h DG ðQ  ∇N FJ Þ  ∇N FM þ DF ðP N FJ ÞN FM dV:

ð37Þ 2.4.2. Micropolar and microstrain case The micropolar model of Section 2.3.2 has the micro-quantity R, which has only one degree of freedom φ^ , according to Eq. (16). Derivatives for the residual and stiffness matrix have to be taken with respect to this degree of freedom, which is done using the chain rule: h

∂φ^ ðÞ ¼ ∂R ðÞ : ∂φ^ R ;

ð38Þ

h

wherein R is computed for the two-dimensional case as nn

h

R ¼

F X

k¼1

N Fk ðe1  e1 þe2  e2 Þ cos φ^ k þ ðe2  e1  e1  e2 Þ sin φ^ k : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ≔R k

ð39Þ F

This leads to the following micro-residual vector R : Z   nel h h ext R FJ ¼ A Q  ∇N FJ þP NFJ : ∂φ^ R K dV  f J e¼1

Fig. 8. Plate under bending. (a) Geometry, constraints and load, and (b) FE mesh.

with the component matrices of the global tangential stiffness matrix K 1:

2:

φφ

φ

nel

K IL ¼ ∂φL R I ¼ A

e¼1

φF

φ

nel

K IM ¼ ∂F M R I ¼ A

e¼1

Z

φ

B e0

h

DF ðP 

φ

∇N I ÞN FM

dV

ð40Þ

J

and the component matrices of K Z   nel φF φ φ DF ðP h  ∇N I ÞNFM : ∂φ^ R K dV 1: K IM ¼ ∂φ^ R I ¼ A e¼1

M

2: 3:

Z B e0

φ

DF ðP h  ∇NI Þ  ∇N L dV

Be0

Be

M

Z 0  h Fφ φ ð1;2j 1;2Þ F : ∂φ^ R K dV K JL ¼ ∂φ^ R J ¼ A DF ðP N FJ Þ  ∇N L J L e ¼ 1 Be 0 Z  nel h FF K JM ¼ ∂φ^ R FJ ¼ A DG ðQ  ∇N FJ : ∂φ^ R K Þ  ∇NFM : ∂φ^ R K : nel

e¼1

M

h

Be0



þ Q  ∇NFJ : ∂φ^ ∂φ^ R K Þ M J

J

M

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

123

Fig. 9. Plate under bending: contour plot of Green Lagrange strain Eyy. (a) Classical, (b) micromorphic, (c) additive micromorphic, (d) micropolar, (e) additive micropolar, (f) microstrain and (g) additive microstrain.

Fig. 10. Plate under bending: contour plot of Green Lagrange strain Exy. (a) Classical, (b) micromorphic, (c) additive micromorphic, (d) micropolar, (e) additive micropolar, (f) microstrain and (g) additive microstrain.

124

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127 h

þDF ðP N FJ : ∂φ^ R K ÞN FM : ∂φ^ R K J M   h þP N FJ : ∂φ^ ∂φ^ R K Þ dV: M

J

ð41Þ

For a numerical implementation of the micropolar model into a finite-element-program two possibilities can be distinguished in the two-dimensional case. The first is to use one degree of freedom φ , which leads to the above modifications of the residual and stiffness matrix in comparison to Eqs. (37) and (33) for the micromorphic continuum. The second possibility is to use four degrees of freedom for the coefficients of R , which avoids these modifications. However, this approach results in higher computational costs. If we compare the equations for W mac , W mic and W scale of the micromorphic and micropolar continua (Eqs. (12) and (18)), we see that they only differ insofar that F is replaced with R and in the scale transition term F is substituted with R for the micropolar continuum. If four micro degrees of freedom are used for the components of R in the micropolar continuum, the only difference to the micromorphic case in a numerical implementation is subtracting R instead of F in the scale transition term, which is therefore the essential point in the transition from micromorphic to micropolar. A microstrain continuum can be implemented numerically with three or four micro degrees of freedom, analogous to the micropolar case, with the same reasoning. The reduction of computational costs is of course not as high in this case. 3. Numerical examples This section considers two numerical examples to compare the basic characteristics of micromorphic, micropolar and microstrain continua and to illustrate the differences. To this end, it is important to have non-uniform deformations, because otherwise no gradient G of the micro-deformation gradient would be activated. Table 1 shows an overview of the utilized material models and the corresponding material parameters with the elastic modulus E and Poisson's ratio ν. The Lamé constants λ and μ are computed from E and ν as 1: λ ¼

Eν ; ð1 þ νÞð1  2νÞ

2: μ ¼

E : 2ð1 þ νÞ

ð42Þ

3.1. Plate with hole under tension The first example is a plate with a circular hole under tension, as shown in Fig. 4a. Because of symmetry, only a quarter of the plate is simulated. Plane strain conditions are prescribed. The load is q ¼ 366 N=mm and the length is

r ¼ 10 mm. The finite element mesh consists of quadratic triangular elements with linear approximations for the micro degrees of freedom and is depicted in Fig. 4b. In Fig. 5 we see the macro-deformation of the classical, micromorphic, micropolar and microstrain continua and a contour plot of the Green–Lagrange normal strain in the direction of the force Eyy . The corresponding cases of the additive micromorphic model are shown as well. The classical continuum shows the highest and the micromorphic continuum the smallest deformation. In the case of the micropolar continuum, the deformation is similar to that of the classical continuum. The behaviour of the microstrain continuum seems, at a first glance, to be exactly the same as for the micromorphic continuum. However, there is a difference in the bottom right part of the structure. The same holds, if we compare the classical and micropolar continua. The bottom part of the right edge of the classical and microstrain continua is slightly curved, while the edges of the micromorphic and micropolar continua are nearly straight. This difference is more obvious for the shear strain Exy in Fig. 6. Only for the micromorphic and micropolar models, the maximum strain Exy arises at the bottom edge. This effect can be explained with micro- and macro-rotations in Fig. 7, which shows contour plots of the micro-rotation angle φ for the micromorphic continuum and of the macro-rotation angle φ for the micromorphic and microstrain continua. In Fig. 7a there is microrotation for the micromorphic continuum all over the body, with maximum values on the edge of the hole. For the microstrain continuum there is by definition no micro-rotation. A comparison of Fig. 7b and c shows that the macro-rotation for the micromorphic continuum is smaller than for the microstrain continuum, which causes the aforementioned differences in the strain Exy . The macrodeformation becomes smaller, when a corresponding microdeformation occurs. This is a consequence of the balance law in Eq. (7), in which the sum of the internal macro- and micro-contributions has to be equal to the external macro-energy. There is no external micro-energy. In this example the deformation is dominated by normal strains, so a micropolar continuum, in contrast to the microstrain continuum, does not show much difference to a classical continuum, because it only allows micro-rotation. For the micropolar continuum the strain is distributed in the same way as for the classical continuum, but more evenly so, due to gradient effects. The results of the additive model are nearly identical to those of the corresponding micromorphic, micropolar or microstrain model. For the depicted accuracy there is no difference at all, except slight differences in the maximum strain values for the additive micromorphic case. The observations made for Eyy also hold for Exy .

Fig. 11. Plate under bending: Contour plot of Green Lagrange strain Eyy and Exy. (a) Eyy: additive micropolar 25%, (b) Exy: additive micropolar 25%, (c) Eyy: additive microstrain 25%, and (d) Exy: additive microstrain 25%

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

3.2. Plate under bending The second example is a plate under bending, see Fig. 8a. The load is f ¼ 20 N=mm and the length is L ¼ 5 mm. Again, plane strain conditions are prescribed. In this example rotation is an important part of the deformation. The same material models, as for the plate under tension, are used. Fig. 9 shows that the classical continuum and the microstrain continuum have qualitatively the same strain distribution, but the strain gradient is not as high for the enhanced continuum and so are the maximum strain values. There are high strains on the lower part of the left edge, getting lower from bottom to top and the same on the opposite edge with negative strains. This leads to curved side edges. For the micromorphic and micropolar continua high strain values only occur at the bottom corners and the side edges are nearly straight. Fig. 10 shows the Green–Lagrange strain Exy . Again, the strain distributions are qualitatively equal for the micromorphic and micropolar continua and for the classical and microstrain continua. This is in contrast to the first example with tension, where strain distributions are qualitatively equal for the micromorphic and microstrain continua and for the classical and micropolar continua. This can be explained by the fact that rotation is not significant in the first example, but very relevant in the second example. So it is to be expected that in the first example the microstrain part of a micromorphic continuum plays the more important role in comparison to the micropolar part and vice versa for the second example. Another important observation in both examples is that if the problem leads to the activation of micro-deformations, which are represented in the considered continuum, the corresponding parts of the macrodeformation are lessened, which is a consequence of the energy balance. The same external energy has to be equal to the macro part of the internal energy plus an additional micro part. Additionally, Fig. 11 shows both strain components for an additive micropolar and microstrain continuum, where the values of the material constants of the respective other part are 25% instead of 0%. The results for this additive micropolar and microstrain model are between those of the respective additive model with 0% and the micromorphic continuum, but closer to the respective additive model, which is the expected and desired result. The magnitude of the described differences of the enhanced models to the classical model depends on the additional material parameters. In this paper the material parameters are chosen so that they lead to clearly observable differences. A lower internal length l leads to a decreasing effect of the higher order gradient. For smaller parameters p the influence of the micro-quantities is smaller as well. Higher values of l and p lead to a higher stiffness of the material. Additionally for higher values of p the difference between macro- and micro-deformation becomes smaller, e.g. the values of the macro- and micro-rotation angles φ and φ , as shown in Fig. 7a and b, would be closer together for a higher value of p.

4. Conclusion and outlook Micropolar and microstrain continua are special cases of the micromorphic continuum. This paper is concerned with the transition from a micromorphic to a micropolar or a microstrain continuum. Additionally, the role of the micropolar and microstrain part in a micromorphic continuum is demonstrated. Three simple constitutive models are shown, which represent the micromorphic, micropolar and microstrain continua. On this basis an additive constitutive model is presented, which incorporates all three as special cases and furthermore allows for arbitrary mixtures of micropolar and microstrain parts. The model consists of an additive composition of a micropolar and

125

a microstrain part. Both parts can be weighted differently. This makes the model very versatile, but also leads to additional material parameters. The numerical examples show nearly the same results for the separate models of the micromorphic, micropolar and microstrain continua and for the respective special cases of the additive model. Concerning the influence of the micropolar and microstrain parts, the two numerical examples show that the macro-deformation is lessened, when the corresponding micro-deformation is active. The deformation in the first example is dominated by normal strain. This strain is significantly lower for the micromorphic and microstrain continua than for the classical and micropolar continua. In the second example rigid body rotation is an important part of the deformation. In this case the macro-deformation of the micropolar continuum is also much lower. The reason for this behaviour is the energy balance of the enhanced continua, which has an additional internal micro part. Furthermore, the deformation of the enhanced continua is distributed more evenly, because they are equipped with a higher order gradient. The scale of these effects is influenced by the additional material parameters, one of which is the internal length scale l. A goal for future research is to add plasticity with damage to the presented micromorphic model to simulate localization phenomena, such as shear bands. These simulations in combination with corresponding experiments can be used to solve an inverse problem for parameter identification.

Acknowledgements This paper is based on investigations supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant MA 1979/17-1, which is gratefully acknowledged. We also thank the anonymous reviewers for their helpful comments. Appendix A. Different approach for the scale transition term in the additive micromorphic model An alternative to create an additive model, as shown in Section 2.3.4, is to use the differences F  R and F  U in the scale transition term W scale of Eq. (27). With this approach it would be possible to create the additive micropolar and microstrain cases without the additional terms, because this leads to the same structure as in the micropolar or microstrain models in Sections 2.3.2 and 2.3.3, respectively. In the micropolar case the micro-deformation gradient F is equal to the micro-rotation tensor R , so the difference F R effectively becomes R  R. This applies analogous to the microstrain case. However, in the additive micromorphic case or other mixed cases both micropolar and microstrain parts are non-zero and both differences of F to R and U are penalized, but F cannot resemble R and U simultaneously, so we get differences between two different geometric objects. With this approach it is not possible to represent the micromorphic or other mixed cases, only the micropolar and microstrain cases. Therefore, this approach is not discussed any further.

Appendix B. Derivatives of the stresses for the additive micromorphic model The derivatives of the stresses in Eq. (29) in Section 2.3.4 are 1:

ð2;3Þ

ð2;4Þ

DF P ¼ λF  T  F  T ðλ ln J  μÞF  T  F  1 þ μ I  I   ð1;2j 1;2Þ þ pR DF R : DF R þðR  R Þ : DF ðDF RÞ

126

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

  ð1;2∣1;2Þ þ pU DF U : DF U þðU  UÞ : DF ðDF UÞ ð1;2∣1;2Þ

:

DF R  pU DF U

with

DF P ¼ pR DF R

3:

DF P ¼ pR DF R

4:

þ DF P þ DF P DF P ¼ DF P   ð1;2j 1;2Þ scale DF P ¼ pR þ pUR DF R : DF R      þ pR R R þ pUR R  I : DF ðDF R Þ   ð1;2j 1;2Þ þ pU þ pRU DF U : DF U      þ pU U  U þ pRU U  I : DF ðDF UÞ   mic R 2 DF ðP Þ ¼ μlR DF ð∇RÞ : DF ð∇RÞ þ ∇R : DF DF ð∇RÞ   mic U 2 Þ ¼ μlU DF ð∇UÞ : DF ð∇UÞ þ∇U : DF DF ð∇R Þ : DF ðP

ð1;2∣1;2Þ

:

scale

5:

6: 7:

DF R  pU DF U

micR

:

DF U

ð1;2∣1;2Þ

:

DF U

micU

ðB:1Þ

A≔trðCÞ þ 2ðdetðCÞÞ

1:

;

B≔ðdetðCÞÞ

2:

1=2

IþC

ðB:2Þ

0

n

and use the abbreviations ðÞ ≔∇ðÞ and ðÞ ≔DF ðÞ to make expressions clearer. This results in the following equations for U and its derivatives: 1:

U ¼ A  1=2 B

2:

U ¼  12 A  3=2 B An þ A  1=2 Bn

n

3:

  ð3Þ 0 0 n0 U ¼ 34 A  5=2 B  An  A0  12 A  3=2 B  An þ B0  An þ Bn  A0 þ A  1=2 Bn

4:

U

n″

h  7=2 ¼  15 B  An  A0  A0 þ 34 A  5=2 B  An  A″ 8 A   0 ð4;5Þ þ B  An  A0     ð3Þ ð4;5Þ ð3Þ 0 þ B0  An  A0 þ B  An þ B0  An þBn  A0  A0  ð3;4;5Þ 0 ð3;6;7Þ 0 ð3Þ ″  12 A  3=2 B  An þ B0  An þ B0  An þ B″  An  0 ð4;5Þ 0 ″ ðB:3Þ þ Bn  A″ þ Bn  A0 þ B þ  A0 þ A  1=2 Bn :

a

A0 ¼ a : C n0

A ¼C

3:

C ¼F

n

4:

C ¼F

5:

C

n″

n0

:





a þa : C  1  1 1 0 1 a ¼ ð det CÞÞ2 12 C C : C þ ðC Þ0 0 ð1;2j 1;2Þ 0

4:

A″ ¼ C

5:

An ¼ C

″

:

a þa : C

n ð1;2j 1;2Þ ″

:

a″ ¼ 12 C

1

þ ðC

a þC

C

7:

DC C



8:

ðDC C

9:

ðC

n0 ð1;2j 1;2Þð2;3Þ 0

:

0 ð1;2j 1;2Þ 0

:



a þC

n0 ð1;2j 1;2Þ 0

1

:

″

a þa : C

n″

  1 0 ð3;4Þ

n

6:

Bn ¼ I  12 ðdetCÞ2 C : C þC |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

7:

B0 ¼ I  b : C þ C

8:

Bn ¼ I  C

8:1

b ¼ 12 a0

9:

B″ ¼ I  C

10:

Bn ¼ 12

0

Þ  b:C

n

0 n0

n ð1;2j 1;2Þ 0

:

b þI  b : C þC

n0

0

″

0 ð1;2j 1;2Þ 0



n0 ð1j 1Þð2;4;5Þ 0



T

F þF  F

F þF

n0

n0 ð1j 1Þð1;4;5Þ 0

¼ C





F

1

F

:C

0

 1 ð2;4Þ

1 0

C

Þ ¼  ðC

1 ″

Þ ¼ ðDC C

T

 1 0 ð2;4Þ

Þ C

1

 ðC

 1 0 ð1;3Þ

Þ C

 1 0 ð3;4j 1;2Þð3;4Þ 0

Þ

:

C þDC C

1

1 ″

:C :

ðB:5Þ

Modified contractions are used in the equations above. The first parenthesis above the contraction sign indicates over which indices to sum. The second parenthesis indicates which number the remaining indices of the tensor on the right side of the product sign get among the indices of the resulting tensor, e.g. with a third order tensor A and a second order tensor B the contraction is ð1j 2Þð2Þ A  B ¼ Alik Bjl ei  ej  ek . For the derivatives of R, we use the results for the derivatives of U, which results in   n n ð2j 1Þð2Þ  1  U 1: R ¼ G  R  U  0  n0 0 ð2j 1Þð2;3Þ n n0 ð2j 1Þð2Þ  1 2: R ¼ G  R  U R  U  U   n ð2j 1Þð2;4;5Þ  1  þ G R  U U  ″  n″ n0 ð1j 2Þð1;4;5Þ 0 n0 ð1j 2Þð1;6;7Þ 0 n″ ð2j 1Þð2;3Þ n 3: R ¼  R  U þU  R þU  R þR  U  0  1 0 ð2j 1Þð2;3Þ n n0 ð2j 1Þð2;6;7Þ 1   U R  U ðU Þ0  U þ G R  0  0 ð2j 1Þð2;3Þ n n0 ð2j 1Þð2;4;5Þ 1  þ G R  U R  U ðU Þ0   n ð2j 1Þð2;4;5;6;7Þ 1  þ G R  U ðU Þ″ ðB:6Þ

ð2j 1Þð2Þ

″

1:

G ¼0

2:

ðU

3:

ðU

   1 ð2;4Þ  1 0 Þ ¼ U U :U

1 0

1 ″

Þ ¼  ðU

 1 0 ð2;4Þ

 1 ð3;4j 1;2Þð3;4Þ

0

Þ U  U  1  1 ″ ð2j 1Þð2Þ  1 0 ð1j 2Þð1;3;4Þ 0 : U  ðU Þ  U þU  U

ðB:7Þ



b 0

0 ð1j 1Þð2;3Þ n

References 0

0

1

1

F þF

F þF

ð1j 1Þð1;6;7Þ 0

Þ ¼ DC C

1



n0 ð1j 1Þð2;6;7Þ 0 n0

n

0 ð1j 1Þð1;3Þ n

″

a þC  b : C þðC  1 1 Þ  b : C þ ð det CÞ2 ðC Þ″

1 0

F þF

1 0

ðC

T

F þF  F

n0 ð1j 1Þð2Þ

¼F

6:

0

n ð1;2j 1;2Þ 0

n ð1j 1Þð2Þ

n0

0

3:1:

5:1:

3:

with

For Eq. (B.3.2)–(B.3.4) we need the derivatives   1 n :C 1: An ¼ I þðdet CÞ1=2 C |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2:

T ð2;4Þ

2:

þF

For the derivatives of U we define 1=2

ð2;3Þ T

0

C ¼ I  F þF  I   ð2;4Þ ð2;3Þ ð3;5Þ ″ C ¼ I  IþI  I  I

1:

ð1;2∣1;2Þ

2:

″

: b þI  b : C þ C  ″  n″ n″ þC I  An  I : C

″

ðB:4Þ

[1] M.R. Begley, J.W. Hutchinson, The mechanics of size-dependent indentation, J. Mech. Phys. Solids 46 (1998) 2049–2068. [2] A. Bertram, Elasticity and Plasticity of Large Deformations—An Introduction, Springer, Berlin, 2005. [3] I. Caylak, R. Mahnken, Thermomechanical characterisation of cold box sand including optical measurements, Int. J. Cast Met. Res. 23 (3) (2010) 176–184. [4] W. Ehlers, W. Volk, On theoretical and numerical methods in the theory of porous media based on polar and non-polar elasto-plastic solid materials, Int. J. Solids Struct. 35 (1998) 4597–4617. [5] W. Ehlers, W. Volk, An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material, Arch. Appl. Mech. 77 (2007) 911–931. [6] A.C. Eringen, Mechanics of micromorphic materials, in: H. Görtler (Ed.), Proceedings of the 11th International Congress of Applied Mechanics, 1964. [7] A.C. Eringen, Theory of micropolar continua, in: Proceedings of the Ninth Midwestern Mechanics Conference, Wisconsin, August 16–18, 1965. [8] A.C. Eringen, Microcontinuum Field Theories, Springer, New York, 1999.

T. Leismann, R. Mahnken / International Journal of Non-Linear Mechanics 77 (2015) 115–127

[9] S. Forest, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech. 135 (2009) 117–131. [10] S. Forest, E.C. Aifantis, Some links between recent gradient thermo-elastoplasticity theories and the thermomechanics of generalized continua, Int. J. Solids Struct. 47 (2010) 3367–3376. [11] S. Forest, R. Sievert, Nonlinear microstrain theories, Int. J. Solids Struct. 43 (2006) 7224–7245. [12] P. Grammenoudis, Ch. Tsakmakis, Micromorphic continuum. Part I: Strain and stress tensors and their associated rates, Int. J. Non-Linear Mech. 44 (2009) 943–956. [13] P. Grammenoudis, Ch. Tsakmakis, D. Hofer, Micromorphic continuum. Part II: Finite deformation plasticity coupled with damage, Int. J. Non-Linear Mech. 44 (2009) 957–974. [14] P. Grammenoudis, Ch. Tsakmakis, D. Hofer, Micromorphic continuum. Part III: Small deformation plasticity coupled with damage, Int. J. Non-Linear Mech. 45 (2010) 140–148. [15] C.B. Hirschberger. A treatise on micromorphic continua. Theory, homogenization, computation (Ph.D. thesis), Technische Universität Kaiserslautern, 2008. [16] C.B. Hirschberger, E. Kuhl, P. Steinmann, On deformational and configurational mechanics of micromorphic hyperelasticity—theory and computation, Comput. Methods Appl. Mech. Eng. 196 (2007) 4027–4044. [17] C.B. Hirschberger, P. Steinmann, Classification of concepts in thermodynamically consistent generalized plasticity, J. Eng. Mech. 135 (2009) 156–170. [18] M. Jirásek, S. Rolshoven, Localization properties of strain-softening gradient plasticity models. Part I: strain-gradient theories, Int. J. Solids Struct. 46 (2009) 2225–2238.

127

[19] N. Kirchner, P. Steinmann, Mechanics of extended continua: modeling and simulation of elastic microstretch materials, Comput Mech. 40 (2007) 651–666. [20] V.G. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans, Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy, Comput. Methods Appl. Mech. Eng. 193 (2004) 5525–5550. [21] Q. Ma, D.R. Clarke, Size dependent hardness of silver single crystals, J. Mater. Res. 10 (1995) 853–863. [22] P. Neff, J. Jeong, A. Fischle, Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature, Acta Mech. 211 (2010) 237–249. [23] R.H.J. Peerlings, M.G.D. Geers, R. de Borst, W.A.M. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua, Int. J. Solids Struct. 38 (2001) 7723–7746. [24] W.J. Poole, M.F. Ashby, N.A. Fleck, Micro-hardness of annealed and workhardened copper polycrystals, Scr. Mater. 34 (1996) 559–564. [25] L. Rosati, Derivatives and rates of the stretch and rotation tensors, J. Elast. 56 (1999) 213–230. [26] P. Steinmann, A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity, Int. J. Solids Struct. 31 (1994) 1063–1084. [27] N.A. Stelmashenko, M.G. Walls, L.M. Brown, Y.U.V. Milman, Microindentations on W and Mo oriented single crystals: an STM study, Acta Metall. Mater. 41 (1993) 2855–2865. [28] J. Stickforth, The square root of a three-dimensional tensor, Acta Mech. 67 (1987) 233–235.