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

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

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 20^{th} 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.

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

- ${f}_{i}$
- Volume forces
- ${c}_{i}$
- Volume couples
- $\rho $
- Mass density
- $I$
- Isotropic rotational inertia
- ${\epsilon}_{ikl}$
- Signature of the perturbation (i,k,l)

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.

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.

- $s$
- 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.