Mechanics of Cosserat media An introduction - CiteSeerX

Mechanics of Cosserat media An introduction Samuel Forest1 Ecole des Mines de Paris / CNRS Centre des Mat´eriaux / UMR 7633 BP 87, 91003 Evry, France

1

Introduction

The idea of a material body endowed with both translational and rotational degrees of freedom stems from the seminal work of the Cosserat brothers (Cosserat and Cosserat, 1909). A triad of orthonormal directors (d i )i=1,3 is associated to the microstructure of each material point. The material transformation describes the displacement u of the material point and the rotation R ∼ with respect to the initial position X and orientations d i0 : u =x −X,

di = R (X ).d i0 ∼

(1)

A Timoshenko beam (resp. Mindlin shell) is an example of one–dimensional (resp. two– dimensional) Cosserat continuum, for which the directors are attached to the beam cross–section. The theory presented in this course deals with the full 3D case for which the volume element of mechanics actually has a finite extension and a microstructure. The intuitive view of the Cosserat volume element is given in figure 1. In contrast to the classical infinitesimal volume element of continuum mechanics which can be subjected only to volume and surface forces, there is “enough room” on each edge to apply a gradient of forces, i.e. a surface couple... The rigorous derivation of balance equations for such a continuum is given in section 2. The Cosserat continuum belongs to the larger class of generalized continua which introduce intrinsic length scales into continuum mechanics via higher order gradients, additional degrees of freedom of fully non local constitutive equations (Eringen, 1999; Eringen, 2002; Forest, 2005). m zy

σ yy

σ xy

σ yx m zx

y

+

=

σ xx

x

σ xx

L

Figure 1: The material point of a Cosserat continuum 1

Tel.: +33-1-60-76-30-51; Fax: +33-1-60-76-31-50, [email protected]

1

mzx

2

The method of virtual power

The method of virtual power, based on d’Alembert’s principle of virtual work, provides a systematic and straightforward way of deriving balance equations and boundary conditions in various mechanical situations (Germain, 1973a; Germain, 1973b; Maugin, 1980). Forces and stresses are not introduced directly but by the value of the virtual power they produce for a given class of virtual motions. If one extends the class of considered virtual motions, one refines the description of forces and stresses. 2.1

The virtual motions of a Cosserat continuum

For a given current configuration Ω of a simply connected body, a virtual motion ϑ∗ is a vector field on Ω, called the field of virtual (generalized) velocities. Let V be the topological vector space of all virtual velocities. The set V is not empty since it always contains the subspace of rigid body motions. Indeed, virtual velocities like real velocities are defined with respect to a given frame or observer. The velocity fields of the same virtual motion followed by two different observers differ only by a rigid body velocity field. In a Cosserat medium, a material point can translate with the velocity u˙ . On the other hand, a triad of rigid directors is assumed to be attached to each material point and can rotate with the c microrotation rate W represented by a skew–symmetric second rank tensor. An axial (pseudo-) ∼ c ˙ by the relations vector Φ is uniquely associated with the skew–symmetric second rank tensor W ∼ ∀x ,

c ˙ ∧ x =  : (Φ ˙ ⊗ x) W .x = Φ ∼ ∼

(2)

where ∧ and ∼ respectively denote the vector product and the third rank Levi–Civita permutation tensor. The following relations hold between a skew–symmetric tensor and its axial vector2 : ˙ = − 1  : W c, Φ 2∼ ∼

c ˙ W = −∼.Φ ∼

(3)

As a result, the set of virtual velocities of the Cosserat continuum is V := {u˙ ,

˙ } Φ

(4)

The field of virtual microrotation rate is generally not a compatible field, meaning that it is generally not the gradient of a micromotion field3 . The actual and virtual microrotation rate fields ˙ are assumed to be a priori independent of the velocity field u˙ . In particular the microrotation Φ is not bound to follow the material rotation rate associated with the rotation part of the velocity gradient. Such an internal constraint can be enforced but then the Cosserat theory degenerates into the so–called couple–stress theory or Koiter theory (Koiter, 1963; Fleck and Hutchinson, 1997). The (generalized) virtual velocities will be assumed to be at least continuous and piecewise differentiable fields on Ω. In fact, the virtual fields play the role of the test functions in distribution theory (Schwartz, 1984). It would be therefore sufficient to consider virtual velocities that are differentiable at any order, with a compact support. However, we will not adopt the language of distributions in the following even though it would be the most appropriate one. Accordingly, the 2

The components of the skew–symmetric tensors and axial vectors can be sorted out in the following matrix form:   c c  ˙3 ˙2  0 −Φ Φ 0 W12 −W31 c c c ˙ ˙1  0 W23  =  Φ3 [W ] =  −W12 0 −Φ ∼ c c ˙ ˙ W31 −W23 0 − Φ2 Φ1 0

3 It is known from classical continuum mechanics that a rotation field on a body Ω is compatible iff it is homogeneous (Forest and Amestoy, 2004).

2

actual velocity and microrotation fields, i.e. the real motion of the body Ω under the considered system of forces, are generally examples of virtual velocities but not always. Indeed, actual fields are only required to be differentiable almost everywhere. This allows for instance for the existence of shock waves. Such discontinuities can be investigated using regular test functions. However, for the sake of brevity, the jump conditions that can be derived for such an analysis are not reported in the present analysis (cf. (Nowacki, 1986)). Within the context of a first gradient theory, only the first gradient4 of the vector fields of virtual and actual velocity and microrotation is considered and put into the set V ∇ := {∇u˙ ,

˙ } ∇Φ

The case of second gradient theory can be found in (Germain, 1973a; Forest et al., 2000b) and references quoted therein. 2.2

The principle of virtual power

The system of forces one wants to consider is defined by a linear continuous application P:

V −→ IR

ϑ∗ 7→ P(ϑ∗ ) The real number P(ϑ∗ ) is the virtual power produced by the system of forces in the virtual motion ϑ∗ . The principle of virtual power, or d’Alembert’s statement, stipulates that: The virtual power of the system of all forces acting on a body with respect to a Galilean frame, vanishes for any considered virtual motion. The statement holds also in the dynamical case provided that inertial forces are added to the set of forces. The various forces acting on a mechanical system are usually classified into two classes: external forces represent the dynamical effects on Ω due to the interaction with other systems that share no common part with Ω, the internal forces represent the mutual dynamical effects of subsystems of Ω. It is crucial to notice that the definition of the virtual power of internal forces is subject to a limitation expressed as the following axiom. The axiom of virtual power of internal forces stipulates that5 : The virtual power of internal forces acting on any subdomain D ⊂ Ω for a given virtual motion is invariant with respect to any change of observer. The considered changes of observer are described by Euclidean transformations, i.e. any time– dependent homogeneous translations and rotations: x0 = Q (t).x + b (t) ∼

(5)

where Q is a proper orthogonal tensor, i.e. a rotation. In other words, the virtual power of internal ∼ forces takes the same value whatever the frame of observation is. It is equivalent to the following statement: The power of internal forces vanishes for all rigid body motions. The reason is that the equation (5) formally can be associated with a rigid body motion. 4

In a Cartesian orthonormal coordinate system, the gradient of a vector field is the second rank tensor ∇u with the components ui,j where the comma denotes partial derivation with respect to the spatial coordinate xj . 5 This axiom represents in fact no real restriction on the development of continuum theory. It is in fact concomitant of the chosen representation of forces. It is universally accepted in contrast to the principle of form invariance used in material theory (see section 3). The reason is that violating the principle of Euclidean Frame Invariance probably amounts to questioning Newtonian equations of motions themselves...

3

The axiom of power of internal forces is the counterpart of the principle of Euclidean Frame Indifference or, simply, principle of objectivity (Liu, 2002; Bertram, 2005). External forces are not subjected to a similar limitation: the power of inertial forces for instance are obviously not invariant with respect to changes of observer. 2.3

The virtual power of internal forces of the Cosserat continuum

The method of virtual power applied to a specific continuum theory always starts with the expression of the virtual power of internal forces for three reasons. Firstly, the representation of internal forces in the fundamentally new concept brought by the continuum theory compared to existing frameworks. Secondly, this expression is limited by the axiom of power of internal forces introduced in the previous section. Lastly, the analysis of the power of internal forces will dictate the possible forms that the power of external forces can take. The virtual power P (i) of internal forces is assumed to admit6 a power density p(i) : Z (i) ∗ P (ϑ ∈ V) = − p(i) (ϑ∗ ) dv (6) D

The power density p(i) is assumed to be a linear form on V and V ∇ . The principle of objectivity excludes in fact the direct dependence of p(i) on u˙ and limits its dependence on the rotation and microrotation rates: c ˙ p(i) (ϑ∗ ) = σ : (∇u˙ − W )+m : ∇Φ (7) ∼ ∼ ∼ c ˙ are objective quantities meaning that they transform In this expression, σ ,m , ∇u˙ −W and ∇Φ ∼ ∼ ∼ like objective tensors under change of observer (Liu, 2002). This ensures the invariance of p(i) c with respect to Euclidean transformations. The difference (∇u˙ − W ) represent the relative ∼ ˙ is deformation rate with respect to a frame attached to the microstructure. The gradient ∇Φ the tensor of curvature rate. The dual quantities, defining the linear form of power of internal forces, are called the force stress tensor σ and the couple stress tensor m . They are generally ∼ ∼ not symmetric. The stress tensors are assumed to be (almost everywhere) continuously differentiable. The application of the divergence theorem to the relation (6) and taking (7) into account, leads to the following expression: Z  Z  c (i) ∗ ˙ ˙ .m).n da P (ϑ ) = (σ .∇).u˙ + (m .∇).Φ + σ :W dv − (u˙ .σ +Φ (8) ∼ ∼ ∼ ∼ ∼ ∼ D

∂D

where ∂D is the boundary of the subdomain D. The field n denotes the unit normal vector at any point of the boundary of the domain D. The notations σ .∇ stands for the divergence of the ∼ 7 force stress tensor . This expression dictates the form of 2.4

The virtual power of external forces

It can be split into a virtual power density of volumic forces representing long range external actions: Z (d) ∗ ˙ ) dv (9) P (ϑ ) = (f .u˙ + c .Φ D 6

It is assumed that the power density of internal forces is defined at every point x ∈ Ω independently of the subdomain D ⊂ Ω. This means that the considered internal forces are short range forces. Long range forces are bound to a non local theory and are excluded from the Cosserat model. The Cosserat theory can be used to model size effects but it remains a local theory in the sense of (Truesdell and Noll, 1965). 7 In a Cartesian orthonormal coordinate system, σ .∇ denotes the vectors of components σij,j . ∼

4

and a virtual power density of contact forces: Z (c) ∗ ˙ ) da P (ϑ ) = (t .u˙ + M .Φ

(10)

∂D

These virtual powers are linear forms on the space V of the virtual motions and of their first ˙ were not introduced in (9) and (10) gradient V ∇ . In fact, the terms linear in ∇u˙ and ∇Φ because they have no counterparts in the power of internal forces (8), thus anticipating on the consequences of the principle of virtual power. The traction vector t represents a surface density of forces. The couple stress vector M represents a surface density of couples. A possible meaning for this couple stress vector is given in figure 1. The power of inertial forces is defined as the opposite of the derivative of the kinetic energy. It takes the form: Z (a) ∗ ˙ ˙ ) dv P (ϑ ) := −K = − (ρa .u˙ + ρIΓ .Φ (11) D

The vector fields a and Γ denote the actual acceleration and microgyration fields, computed as the time derivatives of the actual velocity and microrotation fields. The mass density field is called ρ. An isotropic microrotational inertia I is introduced. A more detailed derivation of Cosserat inertia terms can be found in (Germain, 1973b; Eringen, 1999). 2.5

Balance of momentum and balance of moment of momentum: field equations

According to the principle of virtual power, the total virtual power of all forces vanishes on any ˙ ): subdomain D ⊂ Ω and any virtual motion ϑ∗ = (u˙ , Φ ∀D ⊂ Ω, ∀ϑ∗ ∈ V,

P (i) (ϑ∗ ) + P (d) (ϑ∗ ) + P (c) (ϑ∗ ) + P (a) (ϑ∗ ) = 0

(12)

The substitution of (8), (9), (10) and (11) into (12) leads to the following variational equation: Z  Z 
˙ dv σ .∇ + f − ρa dv + m .∇ + c −  : σ − ρIΓ .Φ . u ˙ ∼ ∼ ∼ ∼ D

D

Z −

Z (σ .n − t ) .u˙ da − ∼

∂D

˙ da = 0 (m .n − M ) .Φ ∼

(13)

∂D

˙ are chosen in such a way that they vanish outside First assume that the virtual fields u˙ and Φ a compact subset interior to D. In the previous sum, only the volume integral remains. It must be zero for this large class of virtual motions. As a result, the integrand must vanish at any point of the interior of D where it is continuous. In the process, the velocity and microrotation rate can be varied independently. This leads therefore to two field equations known as the balance of momentum and balance of moment of momentum equations: σ .∇ + f = ρ¨ u, ∼ ¨, m .∇ − ∼ : σ + c = ρI Φ ∼ ∼

◦

∀x ∈ Ω ◦

∀x ∈ Ω

(14) (15)

where the expressions of acceleration and microgyration have been substituted for the actual fields. After taking the previous field equations into account, the equation (13) reduces to the surface integral terms which must vanish for all virtual motions. Accordingly, the traction vector and couple stress vectors are8 linear functions of the unit normal vector n : t =σ .n , ∼

M =m .n , ∼

∀x ∈ ∂D

(16)

These relations hold in particular at the boundary ∂Ω of the considered body where t and M may be prescribed. 8

at least at points where the traction vector and the couple stress vector are continuous, which may not be the case at the front of shock waves for instance.

5

2.6

Energy balance

According to the first principle of thermodynamics, the material time derivative of the total energy in a subdomain D ⊂ Ω is the sum of the power of external forces acting on it and of the rate Q˙ of heat supply into it. The total energy is the sum of the kinetic energy K and of internal energy E having specific internal energy e, the energy principle can be written: E˙ + K˙ = P (d) + P (c) + Q˙

(17)

The principle of virtual power (12) applied to the real motion9 is nothing but the kinetic energy theorem: K˙ = P (i) + P (d) + P (c) (18) By substitution into (17), one obtains another expression of the first principle: E˙ = −P (i) + Q˙

(19)

The rate of heat supply is assumed to take the general form involving the rate of volumic heat r and the heat flux vector q : Z Z Z Q˙ = r dv − q .n da = (r − ∇.q ) dv (20) D

D

∂D

The heat flux vector is assumed to be objective. It results from that and from the axiom of virtual power of internal forces, that the rate of internal energy is invariant with respect to Euclidean transformations. The local form of the first principle is obtained by applying the global form (17) to any subdomain V ⊂ Ω. It reads c ˙ + r − ∇.q ρ˙ = σ : (∇u˙ − W )+m : ∇Φ ∼ ∼ ∼

2.7

(21)

Strain measures and field equations in the context of small perturbations

In the context of small perturbations, strain measures are deduced from the relative deformation rate and curvature rate by time integration: e = ∇u + ∼.Φ , ∼

κ = ∇Φ ∼

(22)

These strain measures are called the relative deformation e and the curvature tensor κ . The ∼ ∼ gradient operators can be applied with respect to the initial configuration. One can replace also the Eulerian nabla operator by the Lagrangian one in the field equations (14) and (15). Since microrotations are small, the Cosserat rotation R takes the simple form ∼ R =1 − ∼.Φ ∼ ∼

(23)

For a complete description of the strain measures at finite deformation, the reader is referred to (Kafadar and Eringen, 1971; Forest and Sievert, 2003).

3

Cosserat material theory

The 6 field equations (14) and (15) are not sufficient to determine a total of 24 unknown fields, namely the six translational and micrororational degrees of freedom and the 18 components of the stress tensors. It is the purpose of material theory to link stresses and deformations via constitutive 9

in the absence of shock waves.

6

equations. The constitutive theory is restricted here to the context of small perturbations for the sake of simplicity. Material non linearity is envisaged within the context of continuum thermodynamics with internal variables thus generalizing the concepts introduced for the classical continuum in (Germain et al., 1983; Lemaitre and Chaboche, 1994; Besson et al., 2001). Non–linear evolution rules are usually proposed for the internal variables. Questions of existence and uniqueness for such systems are tackled in (Alber, 1998). 3.1

Entropy principle

The specific internal energy , entropy η and Helmholtz free energy Ψ =  − T η are introduced as functions of state and internal variables. The global form of the second principle reads : S˙ ≥ Qs where S is the global entropy of the system and Qs is the total flux of entropy. Z Z q S = ρη dv, Qs = − J η .n ds and J η = T D ∂D

(24)

where J η is the entropy flux vector. No extra–entropy flux is assumed in the present theory, although this can be considered for internal variables influencing the heat conduction as in (Maugin, 1990). The following local form of the entropy inequality is adopted: ρη˙ + J η .∇ ≥ 0

(25)

Combining (21) and (25) leads to the Clausius–Duhem inequality is obtained: ˙ + η T˙ ) + p(i) − −ρ(Ψ

1 q .∇T ≥ 0 T

(26)

This inequality is used to derive the state laws and the remaining intrinsic dissipation D. 3.2

State laws

The strain measures are decomposed into elastic and plastic parts: e p e =e +e , ∼ ∼ ∼

e p κ =κ +κ ∼ ∼ ∼

(27)

The free energy is a function of the elastic contributions and, possibly, of internal variables q. Taking the expression (7) of the work of internal forces, the Clausius–Duhem inequality (26) becomes: (σ −ρ ∼

∂Ψ ∂Ψ ∂Ψ ˙ ∂Ψ 1 ):e ˙ e + (m −ρ e) : κ ˙ e − (η + )T + σ :e ˙p + m :κ ˙p−ρ q˙ − q .∇T ≥ 0 (28) ∼ ∼ ∼ ∼ ∼ ∼ ∼ e ∂e ∂κ ∂T ∂q T ∼ ∼

e e For any given values of {e ,κ , T, e ˙ e, κ ˙ e , T˙ }, there is a thermodynamic process having these values ∼ ∼ ∼ ∼ at point (x , t). Suitable external volume forces and couples or external heat supply may be e e required for this to hold (Liu, 2002). In other words, for given {e ,κ , T }, the inequality (28) ∼ ∼ e e must hold for arbitrary values of e ˙ ,κ ˙ and T˙ , in which the inequality is linear (Coleman and ∼ ∼ Noll, 1963; Coleman and Gurtin, 1967). Consequently, their coefficients must vanish:

σ =ρ ∼

∂Ψ , e ∂e ∼

m =ρ ∼

∂Ψ , e ∂κ ∼

η=−

∂Ψ , ∂T

R := −ρ

∂Ψ ∂q

(29)

These relations are the state laws. The thermodynamic force associated with the internal variable q was called R. 7

3.3

Residual dissipation and dissipation potential

In the isothermal case, the residual dissipation after enforcing the previous state laws reduces to the intrinsic dissipation: :κ ˙ p + Rq˙ (30) D := σ :e ˙p + m ∼ ∼ ∼ ∼ An efficient way of ensuring the positivity of the dissipation for any thermodynamic processes is to assume the existence of a dissipation potential Ω(σ ,m , R) which is a convex function of its ∼ ∼ arguments: ∂Ω ∂Ω ∂Ω e ˙p = κ ˙p = q˙ = (31) ∼ ∼ ∂σ ∂m ∂R ∼ ∼ These equations are the plastic flow rules and the evolution equation for the internal variable. Materials possessing such a potential are called standard generalized materials by (Halphen and Nguyen, 1975) who extended the pioneering work of J.J. Moreau to elastoviscoplasticity. The Legendre–Fenchel transform of the convex potential Ω can be used to define the dual potential Ω∗ (e ˙ p, κ ˙ p , q): ˙ ∼ ∼ Ω∗ (e ˙ p, κ ˙ p , q) ˙ = sup (σ :e ˙p + m :κ ˙ p − Ω(σ ,m , R)) ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ σ ,m,R ∼ ∼

(32)

This dual potential is such that σ = ∼

4 4.1

∂Ω? ∂e ˙p ∼

m = ∼

∂Ω? ∂κ ˙p ∼

R=

∂Ω? ∂ q˙

(33)

Examples Single vs. multi–criterion Cosserat plasticity

Two main classes of potentials have been used in the past. In the first class, the potential is a coupled function of force and couple–stresses, whereas in the second class the potential is a sum of two independent functions of force stress and couple stress respectively : Ωtot = Ω(σ , R) + Ωc (m , Rc ) ∼ ∼

(34)

in the spirit of (Koiter, 1960) and (Mandel, 1965). Both situations can be illustrated for the rate–independent material behaviour. The first class of models involves a single yield function f (σ ,m , R) and a single plastic multiplier p˙ : ∼ ∼ e ˙ p = p˙ ∼

∂f , ∂σ ∼

κ ˙ p = p˙ ∼

∂f , ∂m ∼

q˙ = p˙

∂f ∂R

(35)

The second class of models requires two yield functions f (σ , R, Rc ) and fc (m , R, Rc ) and two ∼ ∼ plastic multipliers : e ˙ p = p˙ ∼

∂f , ∂σ ∼

κ ˙ p = κ˙ ∼

∂fc , ∂m ∼

q˙ = p˙

∂f , ∂R

q˙c = κ˙

∂fc ∂Rc

(36)

In the latter case, coupling between deformation and curvature comes from the balance equations and possibly coupled hardening laws. This type of coupling between several hardening variables has been investigated within the framework of multi–mechanism based plasticity theory in (Cailletaud and Sai, 1995) for the classical continuum. The treatment of the Cosserat continuum is very similar. The first trials for an extension of classical von Mises elastoplasticity to the Cosserat continuum are due to (Sawczuk, 1967), (Lippmann, 1969), (Besdo, 1974), (M¨ uhlhaus and Vardoulakis, 1987) 8

and (Borst, 1991; Borst, 1993). They belong to the class of single criterion plasticity models. The following form of the extended von Mises criterion encompasses these previous models : f (σ ,m , R) = J2 (σ ,m ) − R(p) ∼ ∼ ∼ ∼ J2 (σ ,m )= ∼ ∼

q

(37)

0 : σ 0 + a σ 0 : σ 0 T + b m : m + b m : mT a1 σ 2∼ 1 ∼ 2 ∼ ∼ ∼ ∼ ∼ ∼

(38)

0 where σ is the deviatoric part of σ , ai , bi are material parameters. The flow rules and plastic ∼ ∼ multiplier then read : 0 0T a1 σ + a2 σ ∼ ∼ e ˙ = p˙ , ∼ J2 (σ ,m ) ∼ ∼

T b1 m + b2 m ∼ ∼ κ ˙ = p˙ ∼ J2 (σ ,m ) ∼ ∼

p

s p˙ =

a21

p

(39)

a1 a2 b1 b2 e˙ p : e ˙p + 2 e˙ p : e ˙ pT + 2 κ˙ p : κ ˙p + 2 κ˙ p : κ ˙ pT ∼ ∼ ∼ ∼ 2∼ 2∼ 2∼ 2∼ − a2 a2 − a1 b1 − b2 b2 − b1

(40)

The use of the consistency condition f˙ = 0 under plastic loading yields the following expression of the plastic multiplier : N :E :e ˙+N :C :κ ˙ ∼ ∼ ∼ c ∼ ∼ ∼ ∼ ∼ p˙ = (41) H +N :E :N +N :C :N ∼ ∼ ∼ c ∼ c ∼ ∼ ∼

∼

This expression involves the normal tensors N and N to the yield surface, the hardening modulus ∼ ∼ c H and the tensors of elastic moduli E and C for linear elasticity (for a material admitting at least ∼ ∼ ∼ ∼ point symmetry) : N = ∼

∂f , ∂σ ∼

N = ∼ c

∂f , ∂m ∼

H=

∂R , ∂p

E = ∼ ∼

∂2Ψ , e ∂ee ∂e ∼ ∼

C = ∼ ∼

∂2Ψ e ∂κe ∂κ ∼ ∼

(42)

The condition of plastic loading for the material point is that the numerator of equation (41) is positive, provided that the denominator remains positive, which still allows softening behaviours (H < 0). This is however not the only possible extension of von Mises plasticity since a multi–criterion framework can also be adopted : f (σ , R) = J2 (σ ) − R(p, κ) ∼ ∼

fc (m , Rc ) = J2 (m ) − Rc (p, κ) ∼ ∼

q 0 : σ 0 + a σ 0 : σ 0T , J2 (σ ) = a1 σ 2∼ ∼ ∼ ∼ ∼

J2 (m )= ∼

(43)

q

T b1 m :m + b2 m :m ∼ ∼ ∼ ∼

(44)

There are then two distinct plastic multipliers r p˙ =

s

a1 a2 e˙ p : e ˙p + 2 e˙ p : e˙ pT , ∼ 2 2∼ a1 − a2 a2 − a21 ∼ ∼

κ˙ =

b21

b1 b2 κ˙ p : κ ˙p + 2 κ˙ p : κ˙ pT ∼ 2∼ − b2 b2 − b21 ∼ ∼

(45)

The exploitation of two consistency conditions f˙ = 0 and f˙c = 0 under plastic loading leads to a system of two equations for the unknowns p, ˙ κ˙ : (H + N :E :N )p˙ + H pc = N :E :e ˙, ∼ ∼ ∼ ∼ ∼ ∼ ∼

∼

H pc p˙ + (H c + N :C :N )κ˙ = N :C :κ ˙ ∼ c ∼ c ∼ c ∼ ∼ ∼ ∼

(46)

∼

where a coupling hardening modulus appears : H pc = ∂R/∂κ = ∂Rc /∂p. Whether both plastic mechanisms are active or not, is determined by the sign of the solutions (p, ˙ κ) ˙ of the previous 9

plastic α

h

2 elastic

1











































































 





















































 































































 Figure 2: Simple glide test for a Cosserat infinite layer : elastic and elastoplastic domains, boundary conditions.

x2 plastic

h

elastic

x1

M

plastic

Figure 3: Simple bending test for a Cosserat material : elastic and elastoplastic domains, boundary conditions.

system. If the determinant of the system vanishes, the value of the plastic multipliers can remain indeterminate (Mandel, 1965). The choice of one viscoplastic potential for deformation or curvature can be used as a regularization procedure to settle the indeterminacy : p˙ = −

∂Ω ∂R

or κ˙ = −

∂Ωc ∂Rc

(47)

Such mixed plastic–viscoplastic potentials are already recommended in the classical case (Mandel, 1971; Cailletaud and Sai, 1995). An example of multi–mechanism elastoviscoplastic Cosserat material is the case of Cosserat crystal plasticity described in (Forest et al., 1997; Forest et al., 2000a). Single and multi–criterion plasticity including generalized kinematic hardening variables can be found in (Forest, 1999). Non–associative flow rules are necessary in the case of geomaterials for which the yield function appearing in equations (35) and (36) must be replaced by a different function of the same arguments. Some extensions of classical compressible plasticity models are reported in (Chambon et al., 2001).

10

4.2

Application to simple glide and bending

It is important to see the respective role of Cosserat characteristic lengths appearing in the elastic and plastic constitutive equations in some simple situations. The difference between the use of single or multi–mechanism Cosserat elastoplasticity can also be shown. Analytical solutions for an isotropic elastic-ideally plastic Cosserat material involving one or two yield functions can be worked out in the case of the Cosserat glide and bending tests. The considered boundary value problems are depicted on figures 2 and 3 respectively. The detailed solutions are provided in appendices A and B. Two characteristic lengths can be defined : s r a β (48) , lp = le = b µ in the simple case a1 = a, a2 = 0, b1 = b, b2 = 0 (see also equation (A50) for the definition of isotropic Cosserat elastic bending modulus β). In the glide and bend tests, the material can be divided into elastic and plastic zones (figures 2 and 3). Characteristic length le explicitly appears in the solution in the elastic zone, whereas the solution in the plastic zone is driven by length lp . Classical solutions are retrieved for vanishing le and lp . The use of a single coupled yield criterion (38) leads to non–homogeneous distribution of force and couple stress in the plastic zone for both glide and bending, as can be seen from figures 4 and 5. In contrast, if no hardening is introduced, the use of two uncoupled criteria (44) gives rise to constant values of the force and couple–stress components in the plastic zone of the bent beam.

11

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 Figure 4: Simple glide test for a single criterion von Mises elastoplastic Cosserat infinite layer : force stress and couple stress profiles along a vertical line. A micro–rotation φ = 0.001 is prescribed at the top h = 5lu . The material parameters are : E = 200000 MPa, ν = 0.3, µc =100000 MPa, β=76923 MPa.lu2 , R0 =100MPa, a1 = 1.5, a2 = 0, b1 = 1.5lu−2 , b2 = 0. The micro–couple prescribed at the top is µ032 = 80MPa.lu . lu is a length unit.

12

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Figure 5: Simple bending test of an elastoplastic Cosserat material : influence of the characteristic length lp on the profiles of stress components σ11 and m31 , obtained for a fully plastic beam. The parameters are the same as in figure 4 except that b = 15 when lc = 0.26lu . The beam thickness is h = 5lu .

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Appendices A

Simple glide in Cosserat elastoplasticity

A two–dimensional layer of Cosserat material with infinite extension in direction 1 and height h is considered on figure 2. The unknowns of the problem are u = [u(x2 ), 0, 0]T and φ = [0, 0, φ(x2 )]T . Various types of boundary conditions are possible. For example, we consider : u(0) = 0, φ(0) = 0, t = σ12 e 1 = 0, m = m32 e 3 = m032 e 3

(A49)

Note that the solution of this problem for the classical Cauchy continuum would be a vanishing u. The material exhibits an elastoplastic behaviour with a generalized von Mises yield function (38) or (44). Let us recall the elasticity relations in the isotropic case : e e} e{ σ = λ(trace e )1 + 2µ{ e + 2µ}c e , ∼ ∼ ∼ ∼ ∼

e e} e{ m = α(trace κ )1 + 2β { κ + 2γ } κ ∼ ∼ ∼ ∼ ∼

(A50)

where λ, µ are the Lam´e constants and µc , α, β, γ are additional moduli. The brackets {} (resp. }{) denote the symmetric (resp. skew–symmetric) part of the tensor. One usually takes β = γ atpleast in the two–dimensional case (Borst, 1991). An elastic Cosserat characteristic length le = β/µ can be defined. Under the prescribed boundary conditions, a plastic zone develops starting from the top. Elastic zone, 0 ≤ x2 ≤ α The evaluation of elasticity law and balance equations leads to the following equations : σ12 = (µ + µc )u,2 + 2µc φ, σ12,2 = 0,

σ21 = (µ − µc )u,2 − 2µc φ,

m32 = 2βφ,2

m32,2 + σ21 − σ12 = 0

(A51) (A52)

from which two differential equations are deduced : φ,22 = ωe2 φ,

2µc φ, u,2 = − µ + µc

s ωe =

2µµc β(µ + µc )

(A53)

Taking the boundary conditions at the bottom into account, the solutions follow, including an integration constant B to be determined : φ(x2 ) = B sinh(ωe x2 ),

u(x) =

m32 = 2Bβωe cosh(ωe x2 ),

2µc B (1 − cosh(ωe x2 )) ωe (µ + µc )

σ21 = −

4µµc B sinh(ωe x2 ) µ + µc

(A54)

(A55)

Plastic zone, α ≤ x2 ≤ h In the generalized von Mises criteria (38) or (44), the simplifying assumption a1 = a, a2 = 0, b1 = b, b2 = 0 is adopted, together with a constant threshold R = R0 . The yield criterion (38) requires : 2 aσ21 + bm232 = R02 (A56)

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Combining this condition with balance equations (A52), the solution takes the following form including integration constants C and D : m32 = C cos(ωp x2 ) + D sin(ωp x2 ),

σ21 = ωp (C sin(ωp x2 ) − D cos(ωp x2 ))

1 ωp = = lp

r

b a

(A57)

(A58)

where a charateristic length lp comes into play. The constants C and D are solutions of the following system of equations : C 2 + D2 =

R02 , b

C cos(ωp h) + D sin(ωp h) = m032

(A59)

The continuity of surface couple vector and yield condition at x2 = α provides the system of equations for the unknowns B and α : 2βωe B cosh(ωe α) = C cos(ωp α) + D sin(ωp α)  16a

µµc µ + µc

2

B 2 sinh2 (ωe α) + 4bβ 2 ωe2 B 2 cosh2 (ωe α) = R02

(A60)

(A61)

The numerical resolution of both systems of equations leads to a semi–analytical solution of the simple glide test, that can be used as test for the implementation of Cosserat elastoplasticity in a Finite Element code. This has been checked for the simulation presented on figure 4. In contrast, the use of two separate criteria (44) without hardening and assuming plastic loading for both deformation and curvature, leads to a plastic zone with no extension with constant force and couple stress σ21 and m32 at the upper boundary.

B

Simple bending in Cosserat elastoplasticity

Simple bending is a well–suited test to investigate the effect of curvature on the overall response of the material. The bending of metal sheets has been studied experimentally in the elastic regime (Schivje, 1966) and in the plastic regime (St¨olken and Evans, 1998) : size effect have been observed only in the latter case. The solution of the simple bending problem is given here for the elastic and elastoplastic cases. Elastic solution The beam of thickness h and width W of figure 3 is considered for simple bending under plane stress conditions and for an elastic isotropic Cosserat material. Two types of boundary conditions are possible : imposed couple M on the beam, or rotation of left and right sides of the beam. In the latter case, on can prescribe a micro–rotation equal to the rotation of the section, but it does not matter in the sense of Saint–Venant. The solution takes the form : u1 = Ax1 x2 ,

A D u2 = − x21 + (x22 − x23 ), 2 2

φ1 = Dx3 ,

φ2 = 0,

u3 = Dx2 x3

φ3 = −Ax1

(B62) (B63)

in the coordinate frame defined in figure 3. Under these conditions the non–vanishing components of the deformation and curvature tensors are : e11 = Ax2 ,

e22 = e33 = Dx2 15

κ31 = −A,

κ13 = D

(B64)

which shows that the solution is in principle fully three–dimensional. The fact that the deformation tensor is found to be symmetric means that there is no relative rotation between material lines and the Cosserat directors. The plane stress condition implies that the constant A and D are related by D = −νA. The non–vanishing stress components are then : σ11 = EAx2 ,

m13 = −A(β(1 + ν) − γ(1 − ν))

m31 = −A(β(1 + ν) + γ(1 − ν)) = −β ? A

(B65) (B66)

The couple stress component m13 is an out–of–plane component that should vanish under plane couple stress condition. This can be regarded as a reaction stress that will not be considered here in order to keep the simple form of the solution. Note also that it vanishes for the choice γ = (1 + ν)/(1 − ν). The resulting moment M with respect to axis 3 is computed as : Z Eh3 M = (σ11 x2 + m31 )dx2 dx3 = W A( + β ? h) (B67) 12 which gives A for a given couple M . The additional resistance due to the Cosserat character of the material can be readily seen in the term β ? . Formula (B67) reduces to the classical solution when the Cosserat characteristic length goes to zero or when le is much smaller than h. Plastic case Some elements of the solution of the bending problem for Cosserat elastoplasticity with a single coupled plastic potential (38) are provided here, that can be compared to finite element simulations. We still look for a solution of the form : u1 = Ax1 x2 ,

φ3 = −Ax1

where A is the loading parameter. The non–vanishing stress components are σ11 et m31 . The component m13 may exist but is not taken into account in the present two–dimensional solution. The yield condition reads : 2 2 aσ + bm231 = R02 (B68) 3 11 As in the classical solution, a plastic zone starts from the top and the bottom up to x2 = ±α. In the plastic zone, the plastic deformation and curvature are deduced from the flow rules (39) : ep11 = p

2 a σ11 , 3 R0

ep22 = ep33 = −p

1 a σ11 , 3 R0

κp31 = p

b m31 R0

(B69)

Combining the elastic and plastic parts of deformation and curvature, we get : 1 2 a + p)σ11 E 3 R0 1 b =( + p)m31 2β R0

e11 = Ax2 = ee11 + ep11 = (

(B70)

κ31 = −A = κe31 + κp31

(B71) (B72)

Eliminating p from (B70) using (B71), we get : x2 2a 1 1 1 1 a + = ( − ) σ11 3 b m31 A E 3β b

(B73)

This equation combined with (B68) leads to a system of two equations in σ11 and m31 . It can be shown that σ11 then is a root of an algebraic equation of degree 3, the coefficients depending on 16

the material parameters and on x2 . An simple solution can be given however if the right–hand side of (B73) is neglected, which is possible for sufficiently large values of prescribed A and of βb/a which is proportional to le2 /lp2 . In this case, we get, for a fully plastic beam (α = 0) : R0 lp

m31 = r

3 a + bx22 2

(B74)

and the profile of σ11 is deduced from the yield condition (B68). The role played by the plastic characteristic length p lp defined by (A58) appears clearly. When lp tends toward zero, the classical solution σ11 = ±R0 3/2a is retrieved. This dependence on lp is illustrated in figure 5. The approximate solution is found to be a good one when compared to a finite element simulation. In contrast, the use of two potentials f and fc according to equation (44) leads to a linear √ plastic zone of the profile of σ11 in the elastic zone and a constant value σ11 = ±R0 / a in the √ beam. The component m31 remains constant in the beam with the value Rc0 / b.

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