FRIC_COEF

Abaqus provides user subroutine FRIC_COEF, in which complex dependencies of a friction coefficient can be defined on slip rate, pressure, temperature, and field variables. This example verifies the capability by considering the contact response for a Coulomb friction law in which the friction coefficient is of the form

where ${\dot{\gamma }}_{eq}$ is the slip rate, $d_c$ is the decay coefficient, and ${\mu }_s$ and ${\mu }_k$ are the static and dynamic coefficients of friction, respectively. Both the static and dynamic coefficients are functions of contact pressure, $p$ and the average temperature between the two contacting surfaces, $\overline{\theta }$

where $C_s$, $D_s$, $C_k$, and $D_k$ are constants.

The verification test consists of a rod perpendicular to a fixed rigid surface forced into contact with the rigid surface by a concentrated load applied in the axial direction at the top of the rod. Subsequently, prescribed temperatures and displacements are applied to the rod, forcing the rod to slide along the surface. The contact between the bottom end of the rod and the rigid surface is modeled by specifying a main-secondary contact pair. A node-based secondary surface is defined on the bottom end of the rod. This secondary surface has a contact area of unity; hence, the normal force applied on the rod is equal to the contact pressure.

A second identical rod, subjected to the same loading sequence, serves as the reference solution. The friction behavior for this reference model is entered as tabulated data.

In this test the friction coefficient is assumed to be a function of a solution-dependent state variable that is a linear function of equivalent contact slip. A saturation type dependence of the friction coefficient on the solution-dependent variable is assumed:

where the friction coefficient $\mu$ varies between an initial value of ${\mu }_I$ and an asymptotic value of ${\mu }_f$ . In the simplest case the solution-dependent variable $s$ is chosen as a linear homogeneous function of equivalent contact slip ${\gamma }_{eq}$ ; that is, $s=\alpha {\gamma }_{eq}+\beta$ with parameter $\beta =0$ . The parameter value $\alpha =-0.2$ is chosen.

The test consists of a block meshed with a single element sliding on another block with a single row of elements. The slider block is chosen for the secondary surface, while the main surface is chosen on the other block. Contact is established in the first step. In the second step the slider block is moved with specified displacements. The value of equivalent contact slip is accessed by a call to the utility routine GETVRC, which returns the converged value at the previous increment. The friction coefficient is updated as the slider moves, but its dependence on the solution-dependent variable and, thereby, the equivalent slip is explicit since it corresponds to the converged value for the previous increment. In the third step, which is a dummy step, the slider is held in place; but we allow the solution-dependent variable to evolve to its final value based on the final values of equivalent contact slip. The first three steps use the static procedure. Additional linear steps for steady-state dynamics and frequency extraction are added to verify the user subroutine with other procedure types.

This test incorporates anisotropic friction by making the friction coefficient in each slip direction depend on the slip component in that direction. A bilinear dependence is assumed where the friction coefficient varies linearly with the slip component magnitude until a slip tolerance is reached and remains constant thereafter:

where $I=1,2$ are indices for the two local tangent directions. For simplicity, identical values of slip tolerance ${\gamma }_I^{Tol}=0.1$ , coefficients $a_I=0.3$ , and peak friction coefficients ${\mu }_I^{\max }=0.3$ are chosen for both directions. The friction behavior is still anisotropic since the slip components might not be equal as the solution evolves. The derivative of the friction coefficient is a step function at ${\gamma }_I=0$ , jumping between values $+a_I/{\gamma }_I^{tol}$ and $-a_I/{\gamma }_I^{tol}$ . In this example the step function for the derivative is replaced by a linear variation with a finite slope and with a zero value at ${\gamma }_I=0$ to help convergence.

The test consists of a block meshed with a single element sliding on another block. The slider block is chosen for the secondary surface, while the main surface is chosen on the fixed block. Contact is established in the first step. In the second step the slider block is moved with specified displacements to trace a slip path along a line and returning along the same path. The current values of the slip components are passed via an argument into the user subroutine.

A variant of the user subroutine with utility routine GETVRC to retrieve the slip components is also provided. In this case the slip components denote converged values from the previous increment. Since friction coefficient values do not depend implicitly on the slip component values at the current time, the derivatives of the friction coefficient with respect to the slip values are zero.