r7.02.22 Calculation of modified J-integral in presence of initial state

Summary:

This document aims at describing the theoretical formulations and implementation to compute the modified J -integral in*3D and 2D* via the macro_command POST_JMOD.

The theoretical background of*J* -integral is presented in Section 1. It is first introduced by the continuous mathematical description of the standard*J* in*2D* [1]. This standard formulation without initial state is extended successively to the full*3D* formulation [2] and the modified formulations to account for the presence of an initial strain field [3] and the non-proportional loading conditions [4]. In Section 2, the domain integral formulations are detailed in discretized form adapted for the finite element method. The overall strategy for the implementation of the macro_command POST_JMOD in*code_aster*, and general information regarding the supported element types, loads, and required mechanics fields are presented in Section 3. Specific details of the procedures used in the identification of key terms such as the virtual crack extension field, the cracked surface advancement and the gradient of the deformation and strain energy terms are presented in Section 2. Finally, an overview list of the test cases used to validate POST_JMOD is presented in Section 3.

The current development version of POST_JMOD is applicable for:

• elastic and plastic material behaviors;

• proportional/non-proportional and mixed mode loads;

• initial strain (associated with an equilibrium residual stress);

• Standard FEM with a meshed crack

Table of Contents

Table of Figures

• 1. Theoretical background
  • 1.1. Standard J-integral formulation in 2D
  • 1.2. Extension of standard J-integral 3D
  • 1.3. Modified J-integral for initial strain under proportional loads
  • 1.4. Modified J-integral for initial strain under non-proportional loads
  • 1.5. Evaluation of initial strain
• 2. Discretized formulation implemented within POST_JMOD
• 3. Algorithm of POST_JMOD J-integral in code_aster
  • 3.1. Approach to Compute Fracture Mechanics Parameters in Presence of Initial Strain
  • 3.2. Steps in strategy implementation
  • 3.3. Restrictions on Element Types and Loads
  • 3.4. Required existing code_aster concepts
• 4. Developments specific to POST_JMOD formulation
  • 4.1. Computation of Virtual Crack Extension Field Q
  • 4.2. Extension of contour domain identification from 2D to 3D
  • 4.3. Identifying node and element sets within the Integral domain
  • 4.4. Computation of the area increase of the cracked surface
  • 4.5. Computation of gradients of deformation terms in code_aster
  • 4.6. Computation of Strain Energy Density Term and Its Gradient
• 5. Test cases
• 6. References