Constitutive model for jointed materials

We consider a particular joint a oriented by the normal to the joint surface ${\mathbf{n}}_a$. We define ${\mathbf{t}}_{a\alpha },\alpha =1,2$ as two unit, orthogonal vectors in the joint surface. The local stress components are the pressure stress across the joint

and the shear stresses across the joint

where $\mathit{\sigma }$ is the stress tensor. We define the shear stress magnitude as

The local strain components are the normal strain across the joint

and the engineering shear strain in the $\alpha$-direction in the joint surface

where $\mathit{\epsilon }$ is the strain tensor.

A linear strain rate decomposition is assumed, so that

where $d\mathit{\epsilon }$ is the total strain rate, $d{\mathit{\epsilon }}^{el}$ is the elastic strain rate, and $d{\mathit{\epsilon }}^{pl}$ is the inelastic (plastic) strain rate. Supposing that several systems are active (we designate an active system by i, where $i=b$ indicates the bulk material system and $i=a$ is a joint system a), we write

When all joints at a point are closed, the elastic behavior of the material is assumed to be isotropic and linear. The material cannot be elastically incompressible (Poisson's ratio must be less than 0.5).

We use a stress-based joint opening criterion whereas joint closing is monitored based on strain. Joint system a opens when the estimated pressure stress across the joint (normal to the joint surface) is no longer positive:

In this case the material is assumed to have no elastic stiffness with respect to direct strain across the joint system. Open joints, thus, create anisotropic elastic response at a point. The joint system remains open as long as

where ${\epsilon }_{an}^{el}$ is the component of direct elastic strain across the joint and ${\epsilon }_{an\left(ps\right)}^{el}$ is the component of direct elastic strain across the joint calculated in plane stress as

where E is the Young's modulus of the material, $\nu$ is the Poisson's ratio, and

are the direct stresses in the plane of the joint.

The shear response of open joints is governed by the shear retention parameter, $f_{sr}$, which represents the fraction of the elastic shear modulus retained when the joints are open ($f_{sr}$=0 means no shear stiffness associated with open joints, while $f_{sr}$=1 corresponds to elastic shear stiffness in open joints; any value between these two extremes can be used).

The failure surface for sliding on joint system a is defined by

where ${\beta }_a$ is the friction angle for system a, and $d_a$ is the cohesion for system a (see Figure 1).

Figure 1. Joint system material model.

As long as $f_a<0$, joint system a does not slip. When $f_a=0$ joint system a slips. The inelastic (“plastic”) strain on the system is then given by

where $d{\gamma }_{a\alpha }^{pl}$ is the rate of inelastic shear strain in direction $\alpha$ on the joint surface, $d{\overline{\epsilon }}_a^{pl}$ is the magnitude of the inelastic strain rate, ${\psi }_a$ is the dilation angle for this joint system (choosing ${\psi }_a=0$ provides pure shear flow on the joint, while ${\psi }_a>0$ causes dilation of the joint as it slips), and $d{\epsilon }_{an}^{pl}$ is the inelastic strain normal to the joint surface. In order to add the plastic flow contributions from different systems we write the tensorial plastic strain rate for joint a as

The sliding of the different joint systems at a point is independent, in the sense that sliding on one system does not change the failure criterion or the dilation angle for any other joint system at the same point. The model provides for up to three joint systems at a point.

In addition to the joint systems, the model includes a bulk material failure mechanism. This is based on the Drucker-Prager failure criterion,

where $q\stackrel{\mathrm{def}}{=}\sqrt{\frac{3}{2}\mathbf{S}:\mathbf{S}}$ is the Mises equivalent deviatoric stress (here $\mathbf{S}$ is the deviatoric stress $\mathbf{S}\stackrel{\mathrm{def}}{=}\mathit{\sigma }+p\mathbf{I}$), $p\stackrel{\mathrm{def}}{=}-\frac{1}{3}\mathbf{I}:\mathit{\sigma }$ is the equivalent pressure stress, ${\beta }_b$ is the friction angle for the bulk material, and $d_b$ is the cohesion for the bulk material (see Figure 2).

Figure 2. Bulk material model.

If this failure criterion is reached, the bulk inelastic flow is defined by

where

is the flow potential. Here, $d{\overline{\epsilon }}_b^{pl}$ is the magnitude of the inelastic flow rate (chosen so that $d{\overline{\epsilon }}_b^{pl}=\left|{\left(d{\epsilon }_b^{pl}\right)}_{11}\right|$ in uniaxial compression in the 1-direction) and ${\psi }_b$ is the dilation angle for the bulk material. This bulk failure model is a simplified version of the extended Drucker-Prager model described in Models for granular or polymer behavior. As with the joint systems, this bulk failure system is independent of the joint systems, in that bulk inelastic flow does not change the behavior of any joint system.

If $\psi \ne \beta$ in any system the flow in that system is “nonassociated.” This implies that the material stiffness matrix is not symmetric, so that the unsymmetric matrix solution scheme should be invoked by the user. If the difference between $\psi$ and $\beta$ is not large, a symmetric approximation to the matrix can provide an acceptable rate of convergence of the equilibrium equations, and hence a lower overall solution cost. For this reason the unsymmetric solver is not automatically invoked for this material behavior. However, it is recommended for all cases where $\psi$ and $\beta$ are very different on any joint system.

The constitutive equations described above are integrated using the backward Euler method generally used with the plasticity models in Abaqus. A material Jacobian consistent with this integration operator is used for the overall equilibrium iterations.