T-criterion - Deployment For Isotropic Metals

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 generalised 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 that the plastic strain increment:

• depends on the value of stresses, not on their variation
• is independent of the hydrostatic component of the Cauchy stress tensor
• is collinear with the deviatoric stresses (isotropic material)

(d) The increment in plastic work per unit volume using (4.16) is:
(4)
and the increment in strain energy, equals to the total differential of the potential :
(5)
where, and for metals following the von Mises yield law, by definition
(6)
(7)
are the equivalent stress and strain respectively. In (5) the first term of the right hand side, is the increment in elastic energy for unit volume change due to hydrostatic pressure. Its integral over a load path is the total amount of dilatational strain energy density stored in the material. The second term is the energy required for an infinitesimal distortion of the material. The integral of this quantity is the distortional strain energy density. The theory of plastic flow permits the evaluation of stresses, strains and strain energy densities along a path provided that in (3) is known. In elasticity, linear or nonlinear, . In the case of strain hardening materials, can be evaluated by recording the curve in a pure shear experiment. The hardening function after point “y” in Figure 1 is then:
(8)
and the infinitesimal scalar is: (9)
where is the infinitesimal increase in plastic work (see Figure 1). The elastic part of the total distortional strain energy density is:
(10)
where is the elastic part of the equivalent strain. When there is no nonlinear elastic behaviour, by integrating (4.22) the elastic distortional strain energy density is:
(11)
Similarly, by integrating the increment in elastic energy for unit volume change due to hydrostatic pressure, the dilatational strain energy density is:
(12)
assuming that the unit volume change is the elastic straining, proportional to the hydrostatic pressure, p (Figure 2):
or (13)
where, and the bulk modulus of the material.
In summary, in order to use (12) and (13) to determine the failure of a material volume, the following assumptions hold:

• The material is isotropic and follows the von Mises yield condition
• The elastic part of the stress-strain curve is linear
• The relationship between hydrostatic pressure and unit volume change is linear
• The derivative (hardening slope) must be positive or zero