Analysis of strain localization in strain-softening hyperelastic materials, using assumed stress hybrid elements

Newly developed assumed stress finite elements, based on a mixed variational principle which includes unsymmetric stress, rotation (drilling degrees of freedom), pressure, and displacement as variables, are presented. The elements are capable of handling geometrically nonlinear as well as materially nonlinear two dimensional problems, with and without volume constraints. As an application of the elements, strain localization problems are investigated in incompressible materials which have strain softening elastic constitutive relations. It is found that the arclength method, in conjunction with the Newton Raphson procedure, plays a crucial role in dealing with problems of this kind to pass through the limit load and bifurcation points in the solution paths 9 The numerical examples demonstrate that the present numerical procedures capture the formation of shear bands successfully and the results are in good agreement with analytical solutions. List of Symbols u displacement R rotation U right stretch tensor r* Biot stresss tensor t first Piola Kirchhoff stress tensor r Cauchy stress tensor I identity tensor F deformation gradient := I + (Vu) r t J__ ab := a, bjg g dyad a.b := a,b '= dot product A.b := A .bJg ' ; A-B := Aik B~ g' gJ A:B :=A B 'j tJ v velocity := 6 W spin tensor := 1~ D rate of stretch := (d t* UL rate of r* [ UL rate of t 6 v Kronecker's delta V := g i ~ ] := detF = det{I + (Vu) r} 1 symm (A) := ~ (A + A r) 1 skew (A) := (A A T) 2 trace (A) "= A" 9 ,t Communicated by S. N. Atluri, 30 March 1994 Computational Mechanics Center, Georgia Institute of Technology, Atlanta, Georgia 30332-0356 This research is supported by the Office of Naval Research. The first author wishes to express his appreciation to Dr. H. Murakawa, Dr. E. F, Punch, Mr. A. Cazzani, and Dr. H. Okada for fruitful discussions on the subject 549 Computational Mechanics 14 (1994) lal 2 := a.a = a,a' [AI 2 :=A:A =AjA 'J where a, b; vectors, A, B; tensors (second order), At; transpose of A, (g,); base vectors. 550