Frictional behavior

Abaqus assumes by default that the interaction between contacting bodies is frictionless. You can include a friction model in a contact property definition for both surface-based contact and element-based contact.

SURFACE INTERACTION, NAME=interaction_property_name
FRICTION

INTERFACE or GAP, ELSET=name
FRICTION

Interaction module: contact property editor: MechanicalTangential Behavior

The methods used to change friction properties during an analysis differ between Abaqus/Standard and Abaqus/Explicit.

The basic concept of the Coulomb friction model is to relate the maximum allowable frictional (shear) stress across an interface to the contact pressure between the contacting bodies. In the basic form of the Coulomb friction model, two contacting surfaces can carry shear stresses up to a certain magnitude across their interface before they start sliding relative to one another; this state is known as sticking. The Coulomb friction model defines this critical shear stress, ${\tau }_{crit}$, at which sliding of the surfaces starts as a fraction of the contact pressure, p, between the surfaces (${\tau }_{crit}=\mu p$). The stick/slip calculations determine when a point transitions from sticking to slipping or from slipping to sticking. The fraction, $\mu$, is known as the coefficient of friction.

For the case when the slave surface consists of a node-based surface, the contact pressure is equal to the normal contact force divided by the cross-sectional area at the contact node. In Abaqus/Standard the default cross-sectional area is 1.0; you can specify a cross-sectional area associated with every node in the node-based surface when the surface is defined or, alternatively, assign the same area to every node through the contact property definition. In Abaqus/Explicit the cross-sectional area is always 1.0, and you cannot change it.

The basic friction model assumes that $\mu$ is the same in all directions (isotropic friction). For a three-dimensional simulation there are two orthogonal components of shear stress, ${\tau }_1$ and ${\tau }_2$, along the interface between the two bodies. These components act in the local tangent directions for the contact surfaces or contact elements. The local tangent directions for contact surfaces are defined in Contact formulations in Abaqus/Standard, and those for contact elements are defined in the sections describing contact modeling with those elements.

Abaqus combines the two shear stress components into an “equivalent shear stress,” ${\tau }_{eq}$, for the stick/slip calculations, where ${\tau }_{eq}=\sqrt{{\tau }_1^2+{\tau }_2^2}$. In addition, Abaqus combines the two slip velocity components into an equivalent slip rate, ${\dot{\gamma }}_{eq}=\sqrt{{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2}$. The stick/slip calculations define a surface (see Figure 1 for a two-dimensional representation) in the contact pressure–shear stress space along which a point transitions from sticking to slipping.

Figure 1. Slip regions for the basic Coulomb friction model.

There are two ways to define the basic Coulomb friction model in Abaqus. In the default model the friction coefficient is defined as a function of the equivalent slip rate and contact pressure. Alternatively, you can specify the static and kinetic friction coefficients directly.

In Abaqus/Explicit you can establish friction coefficients as mathematical combinations of coefficients specified as surface properties. For example, you can assign a particular friction coefficient to a surface associated with all steel parts, you can assign a second friction coefficient to a surface associated with all rubber parts, and likewise for other materials. These surface-based friction coefficients apply to interactions between the same material. A combinatorial rule is used to determine friction coefficients for interactions between different materials. You can override approximate friction coefficients computed in this manner by the traditional approach of assigning friction coefficients as contact property assignments based on combinations of surfaces.

The combinatorial approach reduces the required user input. For example, a simulation involving six materials could involve contact interactions with 21 unique material combinations, as shown in Figure 4. For simulations involving many materials, it may suffice to

• determine approximate friction coefficients for contact between like surfaces (corresponding to entries 1 through 6 of the table of Figure 4) and a subset of other surface combinations from experiments or available references; and
• allow a combinatorial rule to determine friction coefficients for the remaining surface combinations, at least for early stages of a design.

Figure 4. Example with six materials and 21 unique combinations of two materials in contact.

Abaqus/Explicit uses the equation

where ${\mu }_A\le {\mu }_B$ to compute the friction coefficient, ${\mu }_{AB}$, for an interaction between surfaces A and B if ${\mu }_A$ and ${\mu }_B$ are assigned as surface properties to the respective surfaces. With this combinatorial rule, ${\mu }_{AB}$ is at most $1+\alpha$ times greater than the smaller of the two surface-based friction coefficients. The default value of $\alpha$ is 0.3.

Consider an example with ${\mu }_1=0.05$, ${\mu }_2=0.1$, and ${\mu }_3=0.3$. The default combinatorial rule gives ${\mu }_{12}=0.0575$, ${\mu }_{13}=0.0625$, and ${\mu }_{23}=0.12$. For example, you may choose to override the value of ${\mu }_{23}$ using the traditional contact property assignment approach.

The user interface for assigning surface-based friction coefficients is discussed in Surface-based friction coefficients.

COMBINATORIAL RULE, ALPHA=$\alpha$

FRICTION, COEFFICIENT PRECEDENCE=CONTACT PROPERTY

FRICTION, COEFFICIENT PRECEDENCE=SURFACE PROPERTY COMBINATION

You can specify an optional equivalent shear stress limit, ${\tau }_{max}$, so that, regardless of the magnitude of the contact pressure stress, sliding will occur if the magnitude of the equivalent shear stress, ${\tau }_{eq}$, reaches this value (see Figure 5). A value of zero is not allowed.

Figure 5. Slip regions for the friction model with a limit on the critical shear stress.

This shear stress limit is typically introduced in cases when the contact pressure stress may become very large (as can happen in some manufacturing processes), causing the Coulomb theory to provide a critical shear stress at the interface that exceeds the yield stress in the material beneath the contact surface. A reasonable upper bound estimate for ${\tau }_{max}$ is ${\sigma }_y/\sqrt{3}$, where ${\sigma }_y$ is the Mises yield stress of the material adjacent to the surface; however, empirical data are the best source for ${\tau }_{max}$.

FRICTION, TAUMAX=${\tau }_{max}$

Interaction module: contact property editor: MechanicalTangential Behavior: Friction formulation: Penalty or Lagrange Multiplier: Shear Stress, Shear stress limit: Specify: ${\tau }_{max}$

You can specify an anisotropic friction model in Abaqus/Explicit for which directional preferences are specified as surface properties, while the nominal, or average, friction coefficient is specified as a contact interaction property in the same manner as for isotropic friction. The resulting critical contact shear stress surface is elliptical in the ${\tau }_1$–${\tau }_2$ plane, as shown in Figure 6. Points on the critical shear stress surface satisfy the equation:

where $s_{combined}$ represents the combined effects of surface-based directional preferences (and these combined effects evolve as the relative surface orientations change), ${\mu }_{nom}$ is the specified nominal (average) friction coefficient, and $P$ is the contact pressure. Maximum and minimum values of $s_{combined}$, corresponding to directions along the major and minor axes of the critical shear stress surface, are $s_{combined}^{\max }=1+{ϵ}_{combined}$ and $s_{combined}^{\min }=1-{ϵ}_{combined}$.

Figure 6. Critical shear stress surface for anisotropic friction.

$ϵ$ is the frictional directional preference factor. It is a unitless parameter that can range from –1.0 to 1.0 and is a measure of the eccentricity of the scaling ellipse. The most commonly used eccentricity measure for ellipses is $e=\frac{c}{a}$ (see Figure 7). The relationship between $ϵ$ and $e$ is:

Figure 7. Geometrical quantities for an ellipse.

The critical contact shear stress surface influences the friction algorithm as follows:

1. Abaqus/Explicit computes a candidate contact shear stress necessary to enforce stick conditions:
   $${\tau }^{cand}={\tau }^{stick}=({\tau }_1^{stick},{\tau }_2^{stick})\mathrm{.}$$
2. If ${\tau }^{cand}$ lies on or within the critical shear stress surface, as shown in Figure 8, Abaqus/Explicit accepts the candidate contact shear stress as the current contact shear stress, such that stick conditions are in effect.
   Figure 8. Example of candidate contact shear stress within the sticking region for anisotropic friction.
   
   
3. Otherwise, if ${\tau }^{cand}$ lies outside the critical shear stress surface, Abaqus/Explicit sets the contact shear stress equal to ${\tau }^{crit}$ on the critical shear stress surface where the normal to the critical shear stress surface passes through ${\tau }^{cand}$, as shown in Figure 9, and sets the direction of incremental slip to be normal to the critical shear stress surface.
   Figure 9. Example of candidate contact shear stress outside of the sticking region for anisotropic friction.
   
   

$s_{combined}$ can be thought of as a scaling ellipse calculated as a weighted average of surface-based scaling ellipses:

where the weight factors sum to unity ($w_1+w_2=1$). $s_{surf1}$ and $s_{surf2}$ are scaling ellipses representing directional preferences of respective surfaces at a contact location. The maximum and minimum principal values of each scaling ellipse are of the form $s_i=1+{ϵ}_i$ and $s_i=1-{ϵ}_i$, respectively. A lack of directional preference corresponds to ${ϵ}_i=0$.

If both surfaces of a contact interaction contribute directional preference to the frictional behavior, the shape of the $s_{combined}$ scaling ellipse evolves as the (relative) orientations of the contacting surfaces change. For example, for contact between like surfaces with equal weighting factors ($w_1=w_2=\frac{1}{2}$):

where ${ϵ}_s{=ϵ}_{surf1}={ϵ}_{surf2}$ and $\gamma$ is the angle between major axes of the surface scaling ellipses across the contact interface. Consider the following specific cases for contact between like surfaces:

• $\gamma =0°$: This corresponds to aligned directional preferences of contacting surfaces, as shown in Case 1 of Figure 10 and Figure 11. In this situation ${ϵ}_{combined}={ϵ}_s$ and $s_{combined}=s_{surf1}=s_{surf2}$.
• $\gamma =90°$: This corresponds to orthogonal directional preferences of contacting surfaces, as shown in Case 2 of Figure 10 and Figure 11. In this situation opposing direction preferences of the surfaces cancel each other, such that ${ϵ}_{combined}=0$ and $s_{combined}$ corresponds to a unit circle.
• $\gamma =45°$: ${ϵ}_{combined}={\frac{\sqrt{2}}{2}ϵ}_s$ and the major axis of $s_{combined}$ bisects the major axes of the respective surface scaling ellipses).

Figure 10. Examples of contact between like surfaces with directional preferences aligned (Case 1) and orthogonal (Case 2).

Figure 11. Scaling ellipse combinations when directional preferences of like surfaces are aligned (Case 1) and orthogonal (Case 2).

FRICTION, ANISOTROPIC BEHAVIOR=SURFACE PROPERTY

SURFACE PROPERTY ASSIGNMENT, PROPERTY=ORIENTATION
FRICTION, DEPENDENCIES=n 
$\mu$, ${\dot{\gamma }}_{eq}$, p, $\overline{\theta }$, ${\overline{f}}^{\alpha }$

Directional preference for this anisotropic friction model is specified as a contact property and is implicitly associated with one surface of a contact interaction. This friction model is available in Abaqus/Standard and is less general than the anisotropic friction model discussed in Anisotropic friction with directional preference as a surface property, which allows both surfaces of a contact interaction to contribute directional preference characteristics.

If you indicate that this anisotropic friction model should be used, you must specify two friction coefficients, where ${\mu }_1$ is the coefficient of friction in the first local tangent direction and ${\mu }_2$ is the coefficient of friction in the second local tangent direction. The critical contact shear stress surface for this friction model (see Figure 14) is elliptical in the ${\tau }_1$–${\tau }_2$ plane, like the anisotropic friction model discussed in Anisotropic friction with directional preference as a surface property. The shape of the critical contact shear stress surface remains constant in this case, with the extreme points being ${\tau }_1^{crit}={\mu }_{1}p$ and ${\tau }_2^{crit}={\mu }_{2}p$.

The orientation of the critical contact shear stress for this friction model evolves with the local tangent directions of the contact interaction, which are discussed in Local tangent directions on a surface. For example, local tangent directions for the finite-sliding, surface-to-surface contact formulation are established from, and evolve with, the slave surface of the contact interaction, so, in this example, directional preference for this anisotropic friction model would evolve with the orientation of the slave surface (and would be independent of the master surface).

The size of the critical-shear-stress ellipse will change with the change in contact pressure between the surfaces. The direction of slip, $d{\gamma }_{\alpha }$, is orthogonal to the critical shear stress surface.

The optional equivalent shear stress limit, ${\tau }_{max},$ is applied to the scaled equivalent shear stress, ${\overline{\tau }}_{eq}$, for anisotropic friction. See Anisotropic friction for the definition and discussion of ${\overline{\tau }}_{eq}$.

Figure 14. Critical shear stress surface for the anisotropic friction model.

The friction coefficients 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.

FRICTION, ANISOTROPIC BEHAVIOR=LEGACY, DEPENDENCIES=n 
${\mu }_1$, ${\mu }_2$, ${\dot{\gamma }}_{eq}$, p, $\overline{\theta }$, ${\overline{f}}^{\alpha }$

Interaction module: contact property editor: MechanicalTangential Behavior: Friction formulation: Penalty: Friction, Directionality: Anisotropic

Abaqus offers the option of specifying an infinite coefficient of friction ($\mu =\infty$). This type of surface interaction is called “rough” friction, and with it all relative sliding motion between two contacting surfaces is prevented (except for the possibility of “elastic slip” associated with penalty enforcement) as long as the corresponding normal-direction contact constraints are active. In most cases Abaqus/Standard uses a penalty method to enforce these tangential constraints; however, a Lagrange multiplier method is used during general (non-perturbation) analysis steps if the corresponding normal-direction constraints have directly enforced “hard contact” or exponential pressure-overclosure behavior. Abaqus/Explicit uses either a kinematic or penalty method, depending on the contact formulation chosen.

Rough friction is intended for nonintermittent contact; once surfaces close and undergo rough friction, they should remain closed. Convergence difficulties may arise in Abaqus/Standard if a closed contact interface with rough friction opens, especially if large shear stresses have developed. The rough friction model is typically used in conjunction with the no separation contact pressure-overclosure relationship for motions normal to the surfaces (see Using the no separation relationship), which prohibits separation of the surfaces once they are closed.

When rough friction is used with the no separation relationship for hard contact in Abaqus/Explicit specified with the kinematic contact method, no relative motions of the surfaces will occur. For hard contact in Abaqus/Explicit specified with the penalty contact method, relative motions will be limited to the elastic slip and penetration corresponding to the inexact satisfaction of the contact constraints by the applied penalty forces. When softened tangential behavior is specified in Abaqus/Explicit (see Defining tangential softening in Abaqus/Explicit below), the relative surface motions will be governed by the specified softening behavior.

FRICTION, ROUGH

Interaction module: contact property editor: MechanicalTangential Behavior: Friction formulation: Rough

In some cases some incremental slip may occur even though the friction model determines that the current frictional state is “sticking.” In other words, the slope of the shear (frictional) stress versus total slip relationship may be finite while in the “sticking” state, as shown in Figure 15.

Figure 15. Elastic slip versus shear traction relationship for sticking and slipping friction.

The relationship shown in this figure is analogous to elastic-plastic material behavior without hardening: $\kappa$ corresponds to Young's modulus, and ${\tau }_{crit}$ corresponds to yield stress; sticking friction corresponds to the elastic regime, and slipping friction corresponds to the plastic regime. A finite value of the sticking stiffness may reflect a user-specified physical behavior or may be characteristic of the constraint enforcement method.

Frictional constraints are enforced with a stiffness (penalty method) by default in Abaqus/Standard and for the general contact algorithm in Abaqus/Explicit; in this case the sticking stiffness will have a finite value. An infinite sticking stiffness, in which case the elastic slip is always zero, can be achieved with the optional Lagrange multiplier method for imposing frictional constraints in Abaqus/Standard or with the kinematic constraint method (available only for contact pairs) in Abaqus/Explicit. In Abaqus/Explicit some tangential contact damping acts on the elastic slip rate by default, as discussed in Contact damping. Tangential softening to reflect a physical behavior is available only in Abaqus/Explicit.

You can define the shear stress between contacting surfaces through a user subroutine when the friction behavior provided by Abaqus is not sufficient. The shear stress can be defined as a function of a number of variables such as slip, slip rate, temperature, and field variables. You can also introduce a number of solution-dependent state variables that you can update and use within the friction user subroutines. You can declare a number of properties or constants associated with your friction model and use these values in the user subroutine.

In addition to the friction user subroutines, subroutines are available for defining the complete mechanical interaction between surfaces, including the interaction in the normal direction as well as the frictional behavior in the tangential direction; see User-defined interfacial constitutive behavior for more information.

By default, in Abaqus/Explicit slip increment calculations for friction do not account for the incremental rotation of shell and beam thickness offsets, and frictional constraints do not apply a moment to nodes offset from the contact interface due to shell or beam thicknesses. This behavior can be modified for general contact; for details see Consideration of incremental rotation of shell and beam thickness offsets for frictional contact.

Several features of the frictional interaction of surfaces can have a strong influence on the rate of convergence in an Abaqus/Standard simulation.

In fully coupled temperature-displacement analysis and fully coupled thermal-electrical-structural analysis, all dissipated mechanical (frictional) energy is converted to heat and distributed equally between the two surfaces by default. This behavior can be modified; for details about this and other thermal surface interactions, see Thermal contact properties.

Temperature and field-variable distributions in beam and shell elements can generally include gradients through the cross-section of the element. Contact between these elements occurs at the reference surface; therefore, temperature and field-variable gradients in the element are not considered when determining friction properties that depend on these variables.

Abaqus provides output of the shear stresses at points on the slave surface that use a surface interaction model containing frictional properties. The shear stresses, CSHEAR1 and CSHEAR2, are given in the two orthogonal local tangent directions, which are constructed on the master surface (see Contact formulations in Abaqus/Standard). There is only one local tangent direction in two-dimensional problems. Details about how to request contact surface variable output are given in About contact pairs in Abaqus/Standard and About contact pairs in Abaqus/Explicit.

Contour plots of these variables can also be plotted in Abaqus/CAE.