Frictional Behavior

It is possible to remove, to modify, or to add a friction model that does not involve a user subroutine to a contact property definition in any particular step of an Abaqus/Standard simulation. In some models, such as shrink-fit contact interference problems, friction should not be added until after the first steps have been completed. In other models friction might be removed or lowered to represent the introduction of a lubricant between the bodies.

You must identify which contact property definition or contact element set is being changed.

In Abaqus/Explicit the friction definition is specified as part of the model definition for a general contact analysis and as part of the history definition for a contact pair analysis. See Assigning Contact Properties for General Contact in Abaqus/Explicit and Assigning Contact Properties for Contact Pairs in Abaqus/Explicit for information on changing aspects of any contact property definition during an Abaqus/Explicit analysis.

In the default model you define the coefficient of friction directly as

where ${\dot{\gamma }}_{eq}$ is the equivalent slip rate, p is the contact pressure, $\overline{\theta }=\frac{1}{2}\left({\theta }_A+{\theta }_B\right)$ is the average temperature at the contact point, and ${\overline{f}}^{\alpha }=\frac{1}{2}\left(f_A^{\alpha }+f_B^{\alpha }\right)$ is the average predefined field variable $\alpha$ at the contact point. ${\theta }_A$ , ${\theta }_B$ , $f_A^{\alpha }$ , and $f_B^{\alpha }$ are the temperature and predefined field variables at points A and B on the surfaces. Point A is a node on the secondary surface, and point B corresponds to the nearest point on the opposing main surface. The temperature and field variables are interpolated along the surface at location B. If the main surface consists of a rigid body, the temperature and field variable at the reference node are used.

The friction coefficient can depend on slip rate, contact pressure, temperature, and field variables. By default, it is assumed that the friction coefficients do not depend on field variables.

The coefficient of friction can be set to any nonnegative value. A zero friction coefficient means that no shear forces will develop and the contact surfaces are free to slide. You do not need to define a friction model for such a case.

Experimental data show that the friction coefficient that opposes the initiation of slipping from a sticking condition is different from the friction coefficient that opposes established slipping. The former is typically referred to as the “static” friction coefficient, and the latter is referred to as the “kinetic” friction coefficient. Typically, the static friction coefficient is higher than the kinetic friction coefficient.

In the default model the static friction coefficient corresponds to the value given at zero slip rate, and the kinetic friction coefficient corresponds to the value given at the highest slip rate. The transition between static and kinetic friction is defined by the values given at intermediate slip rates. In this model the static and kinetic friction coefficients can be functions of contact pressure, temperature, and field variables.

Abaqus also provides a model to specify a static and a kinetic friction coefficient directly. In this model it is assumed that the friction coefficient decays exponentially from the static value to the kinetic value according to the formula:

where ${\mu }_k$ is the kinetic friction coefficient, ${\mu }_s$ is the static friction coefficient, $d_c$ is a user-defined decay coefficient, and ${\dot{\gamma }}_{eq}$ is the slip rate (see Oden, J. T. and J. A. C. Martins, 1985). This model can be used only with isotropic friction and does not allow dependence on contact pressure, temperature, or field variables. There are two ways of defining this model.

Providing the Static, Kinetic, and Decay Coefficients Directly

You can provide the static friction coefficient, the kinetic friction coefficient, and the decay coefficient directly (see Figure 2).

Exponential decay friction model.

Using Test Data to Fit the Exponential Model

Alternatively, you can provide test data points to fit the exponential model. At least two data points must be provided. The first point represents the static coefficient of friction specified at ${\dot{\gamma }}_{eq}=0.0$, and the second point, (${\dot{\gamma }}_2$, ${\mu }_2$) (shown in Figure 3), corresponds to an experimental measurement taken at a reference slip rate ${\dot{\gamma }}_2$. An additional data point can be specified to characterize the exponential decay. If this additional data point is omitted, Abaqus will automatically provide a third data point, (${\dot{\gamma }}_{\mathrm{\infty }}$, ${\mu }_{\mathrm{\infty }}$), to model the assumed asymptotic value of the friction coefficient at infinite velocity. In such a case ${\mu }_{\mathrm{\infty }}$ is chosen such that $\left({\mu }_2-{\mu }_{\mathrm{\infty }}\right)/\left({\mu }_1-{\mu }_{\mathrm{\infty }}\right)=0.05$.

Exponential decay friction model specified with test data points.

In Abaqus/Explicit a shear stress limit cannot be used when a contact pair uses a node-based surface as one of the surfaces.

Abaqus/Explicit provides three options for computing weight factors for determination of $s_{combined}\left(\theta \right)=w_1{s}_{surf1}\left(\theta \right)+w_2{s}_{surf2}\left(\theta \right)$. $w_1=w_2=\frac{1}{2}$ for all cases with ${ϵ}_{surf1}={ϵ}_{surf2}$. For cases with ${ϵ}_{surf1}\ne {ϵ}_{surf2}$, $w_1$ is established according to one of the following three weighting methods (and $w_2=1-w_1$):

1. Balanced weighting, in which $w_1=\frac{1}{2}$.
2. ${ϵ}_s$-proportional weighting, in which $w_1=\frac{{ϵ}_{surf1}}{{ϵ}_{surf1}+{ϵ}_{surf2}}$.
3. Maximum-${ϵ}_s$-dominant weighting, in which $w_1=1$ if ${ϵ}_{surf1}>{ϵ}_{surf2}$ and $w_1=0$ if ${ϵ}_{surf1}<{ϵ}_{surf2}$.

Consider a contact interaction in which only one surface introduces directional preference, with ${ϵ}_{surf1}=\frac{1}{3}$ and ${ϵ}_{surf2}=0$. In this case balanced weighting leads to ${ϵ}_{combined}=\frac{1}{6}$, as shown in Figure 12. The other two weighing methods lead to $w_1=1$ and $w_2=0$ for this example, in which case the combined scaling ellipse is identical to the scaling ellipse for the surface with directional preference and ${ϵ}_{combined}=\frac{1}{3}$.

Balanced weighting for anisotropic friction with one surface contributing directional preference.

The effect of balanced weighting for a situation with ${ϵ}_{surf1}=\frac{1}{3}$ and ${ϵ}_{surf2}=\frac{1}{5}$ with a $45°$ angle between major axes of the surface scaling ellipses is represented in Figure 13. Weight factors for this combination of surface scaling ellipses according to the different weighting methods are:

• With balanced weighting: $w_1=\frac{1}{2}$ and $w_2=\frac{1}{2}$.
• With ${ϵ}_s$-proportional weighting: $w_1=\frac{5}{8}$ and $w_2=\frac{3}{8}$. With this weighting, the contact scaling ellipse would be closer to the scaling ellipse for surface 1 than the contact scaling ellipse for balanced weighting.
• With maximum-${ϵ}_s$-dominant weighting: $w_1=1$ and $w_2=0$. With this weighting, the contact scaling ellipse would be identical to the scaling ellipse for surface 1.

Balanced weighting for anisotropic friction with both surfaces contributing directional preference.

As explained, the critical shear stress surface is proportional to the average, or nominal, friction coefficient. For convenience, you can also specify the minimum or maximum friction coefficient rather than the average one. Internally the value will be converted to the average friction coefficient using the following formulas:

The measure of eccentricity of the scaling ellipse for each surface is defined using the frictional directional preference factor $ϵ$. Sometimes it is useful to specify the scaling ellipse using the ratio of the friction coefficients:

The relationship between $ϵ$ and $r$ is given by the following formula:

The preferred local directions for each surface in contact can be output by requesting the generic output variable CORIENT (with the respective vector components CORIENT1 and CORIENT2).

To activate softened tangential behavior in Abaqus/Explicit, specify the slope of the shear stress versus elastic slip relationship ($\kappa$ in Figure 15). User subroutine VFRIC cannot be used in conjunction with softened tangential behavior.

The stiffness method used for friction in Abaqus/Standard is a penalty method that permits some relative motion of the surfaces (an “elastic slip”) when they should be sticking (similar to the allowable elastic slip defined with softened tangential behavior in Abaqus/Explicit). While the surfaces are sticking (i.e., ${\tau }_{eq}<{\tau }_{crit}$ for the basic isotropic Coulomb friction), the magnitude of sliding is limited to this elastic slip. Abaqus continually adjusts the magnitude of the penalty constraint to enforce this condition.

The stiffness method in Abaqus/Standard requires the selection of an allowable elastic slip, ${\gamma }_{crit}$. Using a large ${\gamma }_{crit}$ in the simulation makes convergence of the solution more rapid at the expense of solution accuracy (there is greater relative motion of the surfaces when they should be sticking). Behavior in which no slip is permitted in the sticking state is approximated more accurately by allowing only a small ${\gamma }_{crit}$. If ${\gamma }_{crit}$ is chosen very small, convergence problems may occur; in that case, it may be better to use the Lagrange multiplier method to apply the sticking constraint (see Lagrange Multiplier Method for Imposing Frictional Constraints in Abaqus/Standard later in this section).

The default value of allowable elastic slip used by Abaqus/Standard generally works very well, providing a conservative balance between efficiency and accuracy. Abaqus/Standard calculates ${\gamma }_{crit}$ as a small fraction of the “characteristic contact surface length,” ${\overline{l}}_{char}$ , and scans all of the facets of all the secondary surfaces when calculating ${\overline{l}}_{char}$ . Abaqus/Standard reports the value of ${\overline{l}}_{char}$ used for each contact pair in the data (.dat) file if you request detailed printout of contact constraint information (see Controlling the Amount of analysis input file processor Information Written to the Data File). The allowable elastic slip is given as ${\gamma }_{crit}=F_f{\overline{l}}_{char}$ , where $F_f$ is the slip tolerance; the default value of $F_f$ is 0.005.

This method of calculating the allowable elastic slip is used for all analysis procedures in Abaqus/Standard except steady-state transport analysis (Steady-State Transport Analysis), in which the penalty constraint is based on a maximum allowable slip rate, ${\dot{\gamma }}_{crit}$. The maximum slip rate is calculated as

where $\omega$ is the angular spinning rate and R is the radius of the rolling structure.

If the stiffness method is used for an anisotropic friction model, ${\gamma }_{crit}$ is a nominal allowable elastic slip (or slip rate). If ${\gamma }_1^e$ and ${\gamma }_2^e$ represent components of elastic slip in the $t_1$ and $t_2$ tangent directions, respectively, the transition from stick to slip will occur when ${\gamma }_{eff}={\gamma }_{crit}$, where ${\gamma }_{eff}$ is computed as

For example, if ${\gamma }_2^e=0$, the stick/slip transition will occur at ${\gamma }_1^e=\sqrt{\frac{1}{2}\left(1+\frac{{\mu }_2^2}{{\mu }_1^2}\right)}{\gamma }_{crit}$. Depending on values of ${\mu }_1$ and ${\mu }_2$, this ${\gamma }_1^e$ can be greater or smaller than ${\gamma }_{crit}$. As another example, if the “1” and “2” components of elastic slip are equal, the stick/slip transition will occur at ${\gamma }_1={\gamma }_2=\frac{{\gamma }_{crit}}{\sqrt{2}}$, such that the magnitude of the elastic slip, ${\gamma }_{mag}=\sqrt{{\gamma }_1^2+{\gamma }_2^2}={\gamma }_{crit}$.

The stiffness method used for friction with the general contact algorithm in Abaqus/Explicit and, optionally, with the contact pair method in Abaqus/Explicit is a penalty method that permits some relative motion of the surfaces (an “elastic slip”) when they should be sticking (similar to the allowable elastic slip defined with softened tangential behavior in Abaqus/Explicit). While the surfaces are sticking (i.e., ${\tau }_{eq}<{\tau }_{crit}$), the magnitude of sliding is limited to this elastic slip. Abaqus continually adjusts the magnitude of the penalty constraint to enforce this condition.

In Abaqus/Explicit you can choose to have contact constraints for the contact pair algorithm enforced with the penalty method; the general contact algorithm always uses a penalty method (see Contact Constraint Enforcement Methods in Abaqus/Explicit).

The default penalty stiffness for frictional constraints is chosen automatically by Abaqus/Explicit and is the same as would be used for normal hard contact constraints. Softening in the normal direction does not affect the penalty stiffness used to enforce stick conditions. If tangential softening is specified (see Defining Tangential Softening in Abaqus/Explicit above), the penalty stiffness will be equal to the value specified for the slope of the shear stress versus elastic slip relationship. You can specify a scale factor to adjust the penalty stiffness, as discussed in Contact Controls for General Contact in Abaqus/Explicit and Contact Controls for Contact Pairs in Abaqus/Explicit.

In Abaqus/Standard the sticking constraints at an interface between two surfaces can be enforced exactly by using the Lagrange multiplier implementation. With this method there is no relative motion between two closed surfaces until ${\tau }_{eq}={\tau }_{crit}$. However, the Lagrange multipliers increase the computational cost of the analysis by adding more degrees of freedom to the model and often by increasing the number of iterations required to obtain a converged solution. The Lagrange multiplier formulation may even prevent convergence of the solution, especially if many points are iterating between sticking and slipping conditions. This effect can occur particularly if locally there is a strong interaction between slipping/sticking conditions and contact stresses.

Because of the added cost of using the Lagrange friction formulation, it should be used only in problems where the resolution of the stick/slip behavior is of utmost importance, such as modeling fretting between two bodies. In typical metal forming applications or for contact of rubber components, accurate resolution of the stick/slip behavior is not important enough to justify the added costs of the Lagrange multiplier formulation.

By default, the contact pair algorithm in Abaqus/Explicit uses a kinematic method for imposing frictional constraints (see Contact Constraint Enforcement Methods in Abaqus/Explicit). The kinematic method applies sticking constraints in a way similar to the optional Lagrange multiplier method in Abaqus/Standard; however, the algorithm is quite different. The value of the force required to enforce sticking at a node is first calculated using the mass associated with the node; the distance the node has slipped; the time increment; and additionally for softened contact, the current value of the elastic slip and the elastic slip versus shear stress slope. For hard contact this sticking force is that which is required to maintain the node's position on the opposite surface in the predicted configuration. For softened contact this force is consistent with the user-specified value for the slope of the shear stress versus elastic slip relationship. The sticking force for each node is calculated using the mass associated with the node, the distance the node has slipped, the shear traction-elastic slip slope (if softened contact is specified in the tangential direction), and the time increment. If the shear stress at the node calculated using this force is less than ${\tau }_{crit}$ , the node is considered to be sticking and this force is applied to each surface in opposing directions. If the shear stress exceeds ${\tau }_{crit}$ , the surfaces are slipping and the force corresponding to ${\tau }_{crit}$ is applied. In either case the forces result in acceleration corrections tangential to the surface at the secondary node and either the nodes of the main surface facet or the points on the analytical rigid surface that it contacts.

You can define a generic frictional behavior between contacting surfaces using user subroutine FRIC in Abaqus/Standard. In Abaqus/Explicit the generic frictional behavior for contact pairs is defined in user subroutine VFRIC, while the generic frictional behavior for general contact is defined in user subroutine VFRICTION.

Abaqus provides a simple way to specify complex dependence of friction coefficients with user subroutines FRIC_COEF (Abaqus/Standard) and VFRIC_COEF (Abaqus/Explicit). VFRIC_COEF can be used only with general contact. These user subroutines have a much narrower scope and are much simpler to create than user subroutines that control all aspects of a friction model (FRIC and VFRIC, discussed in Defining Generic Frictional Behavior). In addition, user subroutine FRIC_COEF preserves heuristics built into Abaqus/Standard friction algorithms to assist convergence behavior; many of these built-in heuristics are bypassed with user subroutine FRIC.

FRIC_COEF can be used for isotropic or anisotropic friction behavior. Friction coefficients can depend on contact pressure, temperature, and a number of contact slip–related variables at the current time. VFRIC_COEF is limited to controlling a single friction coefficient per contact constraint for isotropic friction or for the anisotropic friction model discussed in Anisotropic Friction with Directional Preference as a Surface Property. In addition to temperature and contact pressure dependence, the friction coefficient can also depend on equivalent contact slip and contact slip rates.

Friction constraints produce unsymmetric terms when the surfaces are sliding relative to each other. These terms have a strong effect on the convergence rate if frictional stresses have a substantial influence on the overall displacement field and the magnitude of the frictional stresses is highly solution dependent. Abaqus/Standard will automatically use the unsymmetric solution scheme if $\mu >0.2$ or if $\mu$ is pressure-dependent. If desired, you can turn off the unsymmetric solution scheme; see Matrix Storage and Solution Scheme in Abaqus/Standard.

No slip occurs with rough friction; the contribution to the stiffness will be fully symmetric, and Abaqus/Standard will use the symmetric solution scheme by default.