A new 3D finite element for the finite deformation of nearly incompressible hyperelastic solids

Abstract

Volumetric locking is exhibited by nearly incompressible solids such as rubber, resulting in over-stiffening response of the finite element mesh. In this work, we developed the displacement-based computationally efficient volumetric locking-free 3D finite element using smoothening of determinant of deformation gradient (J-Bar method) within the framework of isotropic hyperelasticity. The developed methodology is employed to analyse a rubber block undergoing finite stretch and bending deformations. The convergence study for finite stretch and bending of rubber block is presented. Results of the analysis show that J-Bar method efficiently removes the volumetric locking.

A linearized consistent mixed displacement-pressure formulation for hyperelasticity

“…The proposed approach avoids ambiguities in dealing with the derivatives of symmetric and unsymmetric second-order tensors. Substituting Eqns ( 23)- (27) in Eqn (20), and invoking the vanishing condition for all the test functions, we obtain the following nonlinear equations…”

“…To overcome the numerical issues in solving saddle-point problems, the widely followed approach in computational solid mechanics is to impose the incompressibility constraint weakly using quasi-incompressible approximations. Some popular approaches for simulating quasiincompressible hyperelastic materials are F -bar formulation [6,7], enhanced-strain method [8], average nodal strain formulation [9,10], reduced integration method with hourglass control [11,12], enhanced assumed strain (EAS) methods [13,14], the two-field displacement-pressure formulation [3,4,15], the three-field formulation [16], mixed stabilised formulations [17][18][19][20][21][22][23][24][25], energy-sampling stabilisation [26,27], least-squares formulations [3,[28][29][30][31], F -bar projection methods [32]. Some noteworthy recent contributions to incompressible and quasi-incompressible computational solid mechanics are in [27,[33][34][35][36][37][38][39][40][41][42][43]…”