1. Introduction#

1.1. Behavioral relationships described in this document#

In the STAT_NON_LINE [U4.51.03] operator (or DYNA_NON_LINE [U4.53.01]), two types of behaviors can be processed:

incremental behavior: keyword factor COMPORTEMENT,
nonlinear elasticity behavior: keyword factor COMPORTEMENT.

For each behavior you can choose:

the behavioral relationship: keyword RELATION,
the method of calculating the deformations: keyword DEFORMATION.

For more details, refer to document [U4.51.03] in the user manual, as the behaviors described here only fall under the keyword factor COMPORTEMENT.

The relationships covered in this document are:

VMIS_ISOT_LINE:	Von Mises with linear isotropic work hardening,
VMIS_ISOT_TRAC:	Von Mises with isotropic work hardening given by a traction curve,
VMIS_ISOT_PUIS:	Von Mises with isotropic work hardening given by an analytical curve,
VMIS_JOHN_COOK:	Von Mises with Johnson-Cook isotropic work hardening,
VMIS_CINE_LINE:	Von Mises with linear kinematic work hardening.

1.2. Purpose of integration#

To solve the global nonlinear structure problem, document [R5.03.01] describes the algorithm used in code_aster for nonlinear statics and document [R5.05.05] describes the method used for nonlinear dynamics.

These two algorithms are based on the calculation of local quantities (at each point of integration of each finite element) that result from the integration of behavioral relationships.

At each iteration \(n\) of the Newton method [R5.03.01] we must calculate the nodal forces \(R({{u}_{i}}^{n})={Q}^{T}{\sigma }_{i}^{n}\) (options RAPH_MECA and FULL_MECA), the constraints \({\sigma }_{i}^{n}\) being calculated at each integration point of each element from the displacements \({u}_{i}^{n}\) through the behavioral relationship. We must also build the tangent operator to calculate \({K}_{i}^{n}\) (option FULL_MECA).

Before the first iteration, for the prediction phase, \({K}_{i-1}\) is calculated (option RIGI_MECA_TANG). The calculation of \({K}_{i-1}\), which is necessary for the initialization phase [R5.03.01], corresponds to the calculation of the tangent operator deduced from the speed problem.

This operator is not identical to the one used to calculate \({K}_{i}^{n}\) by option FULL_MECA, during Newton iterations. Indeed, this last operator is tangential to the discretized problem in an implicit manner.

For the behavioral relationships VMIS_ISOT_LINE, VMIS_ISOT_TRAC, VMIS_ISOT_PUIS, VMIS_JOHN_COOK and VMIS_CINE_LINE, we describe here the calculation of the tangent matrix of the prediction phase, \({K}_{i-1}\), then the calculation of the stress field from a deformation increment, the calculation of the nodale forces \(R\) and the tangent matrix \({K}_{i}^{n}\).