Abstract
In this paper, we present a new one-point integration method generalizing Flanagan and Belytschko's method (Internat. J. Numer. Methods Engrg. 17 (1981) 679–706) and a modification of Belytschko and Bindeman's method (Comput. Methods Appl. Mech. Engrg. 88 (1991) 311–340), both in the frame of large deformation elastoplastic analysis. These stabilization methods are combined with the radial return method used to integrate the constitutive law. Plane strain problems are first considered, and the method is then generalized to axisymmetrical situations. The explicit time integration scheme with its critical timestep is also considered. A few examples are presented that slow the great time savings that can be obtained with reduced integration without any loss of accuracy, and even with a gain in the solution quality, since the underintegrated elements prove to be ‘flexurally superconvergent’.
Highlights
In this paper, we present a new one-point integration method generalizing Flanagan and Belytschko's method
The use of one-point integration rules in modelling large inelastic strains via finite elements provides two advantages: (i) it prevents locking from appearing in nearly incompressible media (NIM), (ii) it reduces the number of operations in the integration of the constitutive laws for each element
Belytschko and Bachrach [9] presented a quintessential bending element (QBE), with no locking for NIM. This element is based on a first order development of the strain field and on the use of the Hu-Washizu variational principle for determining the stress field
Summary
In a non-linear situation, the hourglass stresses in (3.1) are incrementally computed, but there is no problem of objectivity, since the expression (3.1) is frame independent. In the case of deviatoric plasticity (Von Mises yield surface for example), a simple and efficient method for the integration of the constitutive law over a load increment is the radial return of Wilkins [14]. This method consists of finding an approximate plastic stress along the radius issued from an elastic predictor (see Fig. 2 for the case of the Von Mises yield criterion with isotropic hardening). This radial return method is nearly equivalent to using an 'effective shear modulus' μ throughout the current step. This suggests to compute the hourglass stresses as follows:.
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