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**r7.02.22** Calculation of modified J-integral in presence of initial state
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**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 :ref:`1 <Ref14766102>`. It is first introduced by the continuous mathematical description of the standard*J* in*2D* [:ref:`1 <1>`]. This standard formulation without initial state is extended successively to the full*3D* formulation [:ref:`2 <2>`] and the modified formulations to account for the presence of an initial strain field [:ref:`3 <3>`] and the non-proportional loading conditions [:ref:`4 <4>`]. In Section :ref:`2 <Ref14766126>`, 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 :ref:`3 <Ref14766135>`. 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 :ref:`2 <Ref14766142>`. Finally, an overview list of the test cases used to validate POST_JMOD is presented in Section :ref:`3 <Ref14766157>`.

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


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.. toctree::
    :hidden:

    self

.. toctree::
    :maxdepth: 2
    :numbered:

    Theoretical_background
    Discretized_formulation_implemented_within_POST_JMOD
    Algorithm_of_POST_JMOD_J-integral_in_code_aster
    Developments_specific_to_POST_JMOD_formulation
    Test_cases
    References