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Isotropic Plastic Hardening Bending Response

The bending response of the beam is determined using the coupled local-y and local-z bending. The following quantities are defined, in the context of the J2 flow theory.

The total local-y and local-z curvatures are used as the main input to determine, if yielding has occurred at the current time-step. The algorithm updates the moments and curvatures, but also the material Jacobian and zero curvature intercept which are needed in the assembly of the global system of finite element equations. The algorithm used is based on a “predictor” moment, 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 the moment, , to the increment in curvature, . It follows that “predictor” moment is:

The effective "predictor" moment is:

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

Using equation (3), the effective “predictor” moment, , is calculated and compared against the yield moment 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 moment and elastic curvature are those given by equations (2) and (4), respectively. However, if the effective “predictor” moment is larger than the calculated yield moment at the beginning of the time-step, , then the components of the elastic and plastic curvatures tensors as well as the moment tensor need to be corrected. In this case, the following equation is solved iteratively for the increment in the equivalent plastic curvature, .

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

The increment in plastic curvature is:

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

The plastic and equivalent plastic curvatures 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 i and j are placeholders for y and z,

and

Finally, the zero curvature intercept if found as