T-criterion
The T-failure criterion is a set of material failure criteria that can be used to predict both brittle and ductile failure. These criteria were designed as a replacement for the von Mises yield criterion which predicts the unphysical result that pure hydrostatic tensile loading of metals never leads to failure. The T-criteria use the volumetric stress in addition to the deviatoric stress used by the von Mises criterion and are similar to the Drucker Prager yield criterion. T-criteria have been designed on the basis of energy considerations and the observation that the reversible elastic energy density storage process has a limit which can be used to determine when a material has failed. Description Only in the case of pure shear does the strain energy density stored in the material and calculated by the area under the - curve, represent the total amount of energy stored. In all other cases, there is a divergence between the actual and calculated stored energy in the material, which is maximum in the case of pure hydrostatic loading, where, according to the von Mises criterion, no energy is stored. This paradox is resolved if a second constitutive equation is introduced, that relates hydrostatic pressure p with the volume change . These two curves, namely and (p-) are essential for a complete description of material behaviour up to failure. Thus, two criteria must be accounted for when considering failure and two constitutive equations that describe material response. According to this criterion, an upper limit to allowable strains is set either by a critical value ΤV,0 of the elastic energy density due to volume change (dilatational energy) or by a critical value ΤD,0 of the elastic energy density due to change in shape (distortional energy). The volume of material is considered to have failed by extensive plastic flow when the distortional energy Τd reaches the critical value ΤD,0 or by extensive dilatation when the dilatational energy Τv reaches a critical value ΤV,0. The two critical values ΤD,0 and ΤV,0 are considered material constants independent of the shape of the volume of material considered and the induced loading, but dependent on the strain rate and temperature. Deployment for Isotropic Metals For the development of the criterion, a continuum mechanics approach is adopted. The material volume is considered to be a continuous medium with no particular form or manufacturing defect. It is also considered to behave as a linear elastic isotropically hardening material, where stresses and strains are related by the generalized Hook’s law and by the incremental theory of plasticity with the von Mises flow rule. For such materials, the following assumptions are considered to hold: (a) The total increment of a strain component is decomposed into the elastic and the plastic increment and respectively: (1) (b) The elastic strain increment is given by Hooke’s law: (2) where the shear modulus, the Poisson’s ratio and the Krönecker delta. (c) The plastic strain increment is proportional to the respective deviatoric stress: (3) where and an infinitesimal scalar. (3) implies