A stabilised large-strain elasto-plastic Q1-P0 method

The mean dilatation method e!ectively involves bi-linear displacements and a constant pressure and is often known as the Q1-P0 formulation. Its non-linear implementation was originally derived as a three-"eld formulation which included the volume ratio via the Jacobian, J, of the deformation gradient as an additional separate variable. However, the latter term was not directly required in the numerical implementation once J was assumed constant along with the pressure. This formulation will here be termed the non-linear Q1-P0 method. It is known to give good solutions for many practical large-strain elasto-plastic problems. However, for some problems, it has been shown to be prone to severe &hour-glassingʹ. With a view to remedying this situation, we here re-visit the three-"eld formulation and derive a modi"ed form, which although variationally valid, is over-sti! in comparison to the original procedure (here simply called the Q1-P0 method). However, the concepts lead to a natural method for stabilising the Q1-P0 technique. The associated tangent sti!ness matrix is symmetric.