FLAC3D Theory and Background • Constitutive Models

# Von-Mises Model

The yield envelope for this model involves a von-Mises criterion. The position of a stress point on this envelope is controlled by an associated flow rule for shear failure.

Formulations

The yield function of the von-Mises model with kinematic hardening is defined as

(1)$f^s = \left[ 1.5 (s_{ij}-\alpha_{ij}) (s_{ij}-\alpha_{ij}) \right]^{0.5} - \sigma_Y = 0$

where the Einstein summation convention applies, $$s_{ij}$$ is the deviatoric stress tensor, $$\alpha_{ij}$$ is the back-stress tensor under the condition $$\alpha_{ii}=0$$), and $$\sigma_Y$$ is a positive material constant denoting yield strength under uniaxial conditions. Note that if there is no hardening, or $$\alpha_{ij} = 0$$, the yield function becomes

(2)$f^s = q - \sigma_Y = 0$

where $$q = ( 1.5\ s_{ij} s_{ij} )^{0.5}$$.

Incremental Elastic Law

The incremental expression of Hooke’s law, in terms of the generalized stress and stress increments, has the form

(3)$\Delta s_{ij} = 2G \Delta e^e_{ij}$

and

(4)$\Delta p = -3K \Delta \varepsilon^e_v$

where $$p=-\sigma_{ii}/3$$, $$G$$ is the elastic shear modulus, $$K$$ is the elastic bulk modulus, $$e^e_{ij}$$ is the elastic part of deviatoric strain tensor, and $$\varepsilon^e_v$$ is the elastic part of volumetric strain.

Composite Failure Criterion and Flow Rule

The potential function $$g^s$$ corresponds in general to an associated law, and has the form

(5)$g^s = f^s$

Hardening Rule

The hardening rule is assumed

(6)$\Delta \alpha_{ij} = H \Delta \epsilon^p_{ij}$

where $$H$$ is a material constant called kinematic plastics modulus. See the schematic in Figure 1 for the physical meaning of the plastic modulus in uniaxial loading.

Plastic Corrections

First, considering shear failure, partial differentiation of Equation (1) yields plastic strain

(7)$\epsilon^p_{ij} = {\lambda}^s {{\partial g^s}\over{\partial \sigma_{ij}}} = {\lambda}^s {{1.5(s_{ij}-\alpha_{ij})}\over{[ 1.5(s_{ij}-\alpha_{ij}) (s_{ij}-\alpha_{ij})]^{0.5}}}$

apparently, $$\varepsilon^p_v = 0$$, and $$e^p_{ij} = \epsilon^p_{ij}$$.

The consistency condition of yield function

(8)$\Delta f^s(s^I_{ij} + \Delta s_{ij}, \alpha^I_{ij} + \Delta \alpha_{ij}) = 0$

yields

(9)${\lambda}^s = {{f^s(s^I_{ij}, \alpha^I_{ij})} \over {3G+H}}$

von-mises Model Properties

Use the following keywords with the zone property (FLAC3D) or block zone property (3DEC) command and with the structure shell property (or liner/geogrid) command to set these properties of the von Mises model.

von-mises
bulk f

bulk modulus, $$K$$

modulus-plastic f

kinematic plastic (hardening) modulus, $$H$$ (see Figure 2). The default value is zero. Be aware that this is not the elastoplastic tangent modulus, $$E_t$$. The relationship between $$H$$ and $$E_t$$ is shown in Figure 2.

poisson f

Poisson’s ratio, $$\nu$$

shear f

shear modulus, $$G$$

strength-yield f

yield strength, $$\sigma_Y$$

young f

Young’s modulus, $$E$$

Notes

• Only one of the two options is required to define the elasticity: bulk modulus $$K$$ and shear modulus $$G$$, or Young’s modulus $$E$$ and Poisson’s ratio $$\nu$$. When choosing the latter, Young’s modulus $$E$$ must be assigned in advance of Poisson’s ratio $$\nu$$.