|M.Sc Thesis||Department of Aerospace Engineering|
|Supervisor:||Prof. Durban David|
|Full Thesis text - in Hebrew|
The effect of initial uniform tension (compression) of elastoplastic response of compressible solids, under superposed simple shear, is examined within the framework of finite strain J2 theories. Exact continuum plasticity equations are derived and solved analytically for the flow theory, deformation theory and hypoelastic deformation-type theory. Isotropic hardening is considered with arbitrary strain hardening (softening) characteristics. Elastic compressibility is accounted for in all models, yet practical relations are given for the deep plastic range with vanishing compressibility. An elegant by-product is that with elastic-perfectly plastic response we recover Prager's classical hardening (tanh) rule for superposed shear.
The solution method centers on the effective Mises stress as the independent variable and on quadratues for field variables in flow and hypoelastic models. For the deformation theory (which is essentially a Cauchy elastic model) we obtain algebraic expressions. Initial strain is shown to have an appreciable effect on superposed simple shear behavior, particularly with the deformation theory and its hypoelastic version, due to weakening of instantaneous shear modulus. We provide a detailed comparison among the shear stress - strain curves (based on identical axial stress - strain curves) of the three competing models, with special emphasis on points of instability (maximum shear stress). Also discussed, are the normal stress components needed to sustain simple shear and thickness changes during shear.
Key results are supported by asymptotic relations along with a sample of numerical illustrations. A few analytical approximations are shown to agree quite well with exact results and contact is made with earlier studies on restricted versions of the problem (e.g. assuming complete incompressibility, or neglecting initial strain).
Finally, we show how the present analysis can be used to solve the large strain elastoplastic problem of simple torsion applied to circular (compressible) cylinders with axial pre-stress.