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Isotropic Plastic Hardening Axial Response

The axial response of the beam is described below using uni-dimensional equations. The following quantities are defined, in the context of the J2 flow theory.

The total axial strain is used as the main input to determine, if yielding has occurred at the current time-step. The algorithm updates the force and the axial strains, but also the material Jacobian and zero axial strain intercept which are needed in the assembly of the global system of finite element equations. The algorithm used is based on a “predictor” force, which is calculated purely on elastic behaviour. Therefore, the material Jacobian is defined as:

is defined in elastic terms only, thus linking the increment in force, , to the increment in axial strain, . It follows that “predictor” force is:

The effective "predictor" force is:

The elastic axial strain at the beginning of the time-step, , is updated with the increment in axial strain:

Using equation (3), the effective “predictor” force, , is calculated and compared against the yield force at the beginning of the time-step in order to determine if the material has yielded. If yielding does not occur, , then the deformation is entirely elastic. Hence, the new values for axial force and elastic axial strain are those given by equations (2) and (4), respectively. However, if the effective “predictor” force is larger than the calculated yield axial force at the beginning of the time-step, , then the elastic and plastic axial strains as well as the force need to be corrected. In this case, the following equation is solved iteratively for the increment in the equivalent plastic axial strain, .

Once is known, and hence the equivalent plastic axial strain at end of the time-step is known, the solution is fully defined and is known. The force can be updated to the end of the time-step as:

The increment in plastic axial strain is:

Therefore, the elastic axial at the end of the time-step is corrected to:

The plastic and equivalent plastic axial strains at the end of the time-step are found to be:

The material Jacobian needs to be updated to reflect the new hardening moduli:

where

and

Finally, the zero curvature intercept if found as