# Nonlinear Pseudo-plastic Orthotropic Solids (LAWS28, 50 and 68)

## Conventional Nonlinear Pseudo-plastic Orthotropic Solids (LAW28 and LAW50)

These laws are generally used to model honeycomb material structures as crushable foams. The microscopic behavior of this kind of materials can be considered as a system of three independent orthogonal springs. The nonlinear behavior in orthogonal directions can then be determined by experimental tests. The behavior curves are injected directly in the definition of law. Therefore, the physical behavior of the material can be obtained by a simple law. However, the microscopic elasto-plastic behavior of a material point cannot be represented by decoupled unidirectional curves. This is the major drawback of the constitutive laws based on this approach. The cell direction is defined for each element by a local frame in the orthotropic solid property. If no property set is given, the global frame is used.
The Hooke matrix defining the relation between the stress and strain tensors is diagonal, as there is no Poisson's effect:(1)
$\left[\begin{array}{l}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{23}\\ {\sigma }_{31}\end{array}\right]=\left[\begin{array}{cccccc}{E}_{11}& 0& 0& 0& 0& 0\\ 0& {E}_{12}& 0& 0& 0& 0\\ 0& 0& {E}_{33}& 0& 0& 0\\ 0& 0& 0& {G}_{12}& 0& 0\\ 0& 0& 0& 0& {G}_{23}& 0\\ 0& 0& 0& 0& 0& {G}_{31}\end{array}\right]\left[\begin{array}{l}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\epsilon }_{12}\\ {\epsilon }_{23}\\ {\epsilon }_{31}\end{array}\right]$
An isotropic material may be obtained if:(2)
${E}_{11}={E}_{22}={E}_{33}\text{ }\text{and}\text{ }{G}_{12}={G}_{23}={G}_{31}=\frac{{E}_{11}}{2}$
Plasticity may be defined by a volumic strain or strain dependent yield curve (Figure 2). The input yield stress function is always positive. If the material undergoes plastic deformation, its behavior is always orthotropic, as all curves are independent to each other.

The failure plastic strain may be input for each direction. If the failure plastic strain is reached in one direction, the element is deleted. The material law may include strain rate effects (LAW50) or may not (LAW28).

## Cosserat Medium for Nonlinear Pseudo-plastic Orthotropic Solids (LAW68)

Conventional continuum mechanics approaches cannot incorporate any material component length scale. However, a number of important length scales as grains, particles, fibers, and cellular structures must be taken into account in a realistic model of some kinds of materials. To this end, the study of a microstructure material having translational and rotational degrees-of-freedom is underlying. The idea of introducing couple stresses in the continuum modelling of solids is known as Cosserat theory which returns back to the works of brothers Cosserat in the beginning of 20th century. 1 A recent renewal of Cosserat mechanics is presented in several works of Forest 2 3 4 5 A short summary of these publications is presented in this section.

Cosserat effects can arise only if the material is subjected to non-homogeneous straining conditions. A Cosserat medium is a continuous collection of particles that behave like rigid bodies. It is assumed that the transfer of the interaction between two volume elements through surface element dS occurs not only by means of a traction and shear forces, but also by moment vector as shown in Figure 3.
Surface forces and couples are then represented by the generally non-symmetrical force-stress and couple-stress tensors ${\sigma }_{ij}$ and ${\mu }_{ij}$ (units MPA and MPa-m):(3)
${t}_{i}={\sigma }_{ij}{n}_{j}\text{ }:\text{ }{m}_{i}={\mu }_{ij}{n}_{j}$

The force and couple stress tensors must satisfy the equilibrium of momentums:

${\sigma }_{ij,j}+{f}_{i}=\rho {\stackrel{¨}{u}}_{i}$
(4)
${\mu }_{ij,j}-{\epsilon }_{ikl}{\sigma }_{kl}+{c}_{i}=I{\stackrel{¨}{\phi }}_{i}$
Where,
${f}_{i}$
Volume forces
${c}_{i}$
Volume couples
$\rho$
Mass density
$I$
Isotropic rotational inertia
${\epsilon }_{ikl}$
Signature of the perturbation (i,k,l)
In the often used couple-stress, the Cosserat micro-rotation is constrained to follow the material rotation given by the skew-symmetric part of the deformation gradient:(5)
${\phi }_{i}=-\frac{1}{2}{\epsilon }_{ijk}{u}_{j,k}$

The associated torsion-curvature and couple stress tensors are then traceless. If a Timoshenko beam is regarded as a one-dimensional Cosserat medium, constraint Equation 5 is then the counterpart of the Euler-Bernoulli conditions.

The resolution of the previous boundary value problem requires constitutive relations linking the deformation and torsion-curvature tensors to the force- and couple-stresses. In the case of linear isotropic elasticity, you have:(6)
$\begin{array}{l}{\sigma }_{ij}=\lambda {e}_{kk}{\delta }_{ij}+2\mu \text{\hspace{0.17em}}{e}_{ij}^{Symm.}+2{\mu }_{c}{e}_{ij}^{Skew\text{\hspace{0.17em}}Symm.}\\ {\mu }_{ij}=\alpha \text{\hspace{0.17em}}{k}_{kk}{\delta }_{ij}+2\beta \text{\hspace{0.17em}}{\kappa }_{ij}^{Symm.}+2\gamma {\kappa }_{ij}^{Skew\text{\hspace{0.17em}}Symm.}\end{array}$

Where, ${e}_{ij}^{Symm.}$ and ${e}_{ij}^{Skew\text{\hspace{0.17em}}Symm.}$ are respectively the symmetric and skew-symmetric part of the Cosserat deformation tensor. Four additional elasticity moduli appear in addition to the classical Lamé constants.

Cosserat elastoplasticity theory is also well-established. von Mises classical plasticity can be extended to micropolar continua in a straightforward manner. The yield criterion depends on both force- and couple-stresses:(7)
$f\left(\sigma ,\mu \right)=\sqrt{\frac{3}{2}\left({a}_{1}{s}_{ij}{s}_{ij}+{a}_{2}{s}_{ij}{s}_{ji}+{b}_{1}{\mu }_{ij}{\mu }_{ij}+{b}_{2}{\mu }_{ij}{\mu }_{ji}\right)}-R$
Where,
$s$
Stress deviator
${a}_{i}$ and ${b}_{i}$
Material constants

Cosserat continuum theory can be applied to several classes of materials with microstructures as honeycombs, liquid crystals, rocks and granular media, cellular solids and dislocated crystals.

1 Cosserat E. and Cosserat F., “Theory of Deformable Bodies”, Hermann, Paris, 1909.
2 Forest S. and Sab K., “Cosserat overall modeling of heterogeneous materials”, Mechanics Research Communications, Vol. 25, No. 4, pp. 449-454, 1998.
3 Forest S., Cailletaud G. and Sievert R., “A Cosserat Theory for Elastoviscoplastic Single Crystals at Finite Deformation”, Archives of Mechanics, Vol. 49, pp. 705-736, 1997.
4 Besson J., Bultel F., and Forest S., “Plasticity Cosserat media. Application to particular composites particles”, 4th Symposium Calculation of Structures, CSMA/Teksea, Toulouse, pp. 759-764, 1999.
5 Forest S., “Cosserat Media”, ed. by K.H.J. Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer and S. Mahajan, Encyclopedia of Materials, Science and Technology, Elsevier, 2001.