r5.03.02 Integration of Von Mises elasto-plastic behavior relationships

Summary:

This document describes the quantities calculated by the operator STAT_NON_LINE necessary for the implementation of the quasistatic nonlinear algorithm described in [R5.03.01] in the case of elasto-plastic behaviors. These quantities are calculated by the same subprograms in the operator DYNA_NON_LINE in the case of a dynamic load [R5.05.05].

This description is presented according to the various keywords that allow the user to choose the desired behavioral relationship. The behavioral relationships covered here are:

• the Von Mises behavior under isotropic work hardening (linear or non-linear)

• the Von Mises behavior with linear kinematic work hardening (Prager model)

The integration method used is based on a direct implicit formulation. From the initial state, or from the previous calculation instant, the stress field resulting from a deformation increment is calculated. The tangent operator is also calculated.

• 1. Introduction
  • 1.1. Behavioral relationships described in this document
  • 1.2. Purpose of integration
• 2. General notations and assumptions about deformations
  • 2.1. Partition of deformations (small deformations)
  • 2.2. Update
  • 2.3. Initial conditions
• 3. Von Mises relationship to isotropic work hardening
  • 3.1. Expression of behavioral relationships
  • 3.2. Tangent operator. Option RIGI_MECA_TANG
  • 3.3. Calculation of constraints and internal variables
  • 3.4. Tangent operator. Option FULL_MECA
  • 3.5. Internal behavior variables VMIS_ISOT_LINE, VMIS_ISOT_PUIS,, VMIS_ISOT_TRAC, and VMIS_JOHN_COOK
• 4. Von Mises relationship to linear kinematic work hardening
  • 4.1. Expression of the behavioral relationship, general case
  • 4.2. Expression of the behavioral relationship in 1D
  • 4.3. Tangent operator. Option RIGI_MECA_TANG
  • 4.4. Calculation of constraints and internal variables
  • 4.5. Tangent operator. Option FULL_MECA
  • 4.6. Model VMIS_CINE_LINE Internal Variables
• 5. Bibliography