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SUMMARYTwo-dimensional (2-D) and three-dimensional (3-D) mesh-grading finite elements for problems with localized phenomena are presented. These quadrilaterals and hexahedrons permit mesh grading without element distortion and, if desired, can be used in a convenient recursive form. They are particularly well suited to adaptive finite element methods. Constraints associated with mesh grading are embedded in the basis functions, making the elements computationally efficient and easy to implement in standard finite element programs. Dramatic computational savings have been achieved. The computational complcxity for an implicit analysis of a point load on a 3-D uniform mesh is O(n7). For the equivalent graded mesh it is O(logzn). The corresponding 2-D analyses have computational complexities of O(n") and O(log,n).

Prediction of the residual stresses caused by welding is important when post-weld stress relief is not feasible. Phase changes and transformation plasticity have a significant effect on the residual stresses generated by welding and heat-treatment of some alloys. Transformation plasticity occurs when the stresses generated by the transformation of individual grains interact with the macroscopic stress state to produce plastic strains. Heuristic methods requiring empirical constants have been used in the past. A method based on the fundamental laws of plasticity and basic material properties is proposed to incorporate transformation plasticity in a finite element program. The transformation plasticity which occurs depends on the stress state. During any increment the stress state can change substantially. If the step size is too large, the analysis may become unstable. A method which allows larger steps while eliminating the instability and improving the convergence is presented. A three-dimensional (3D) analysis of a short longitudinal pipe weld in a typical pressure vessel steel is shown. The significance of this phenomenon in welds is demonstrated by comparing the residual stress states predicted with and without transformation plasticity.

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