# Coulomb friction

 Abaqus provides an extended version of the classical isotropic Coulomb friction model for use with all contact analysis cababilities. The extensions include an additional limit on the allowable shear stress, anisotropy, and the definition of a “secant” friction coefficient. The following topics are discussed:
 Related Topics In Other Guides Frictional behavior

ProductsAbaqus/StandardAbaqus/Explicit

The standard Coulomb friction model assumes that no relative motion occurs if the equivalent frictional stress

$τe⁢q=τ12+τ22$

is less than the critical stress, $τc⁢r⁢i⁢t$, which is proportional to the contact pressure, $p$, in the form

$τc⁢r⁢i⁢t=μ⁢p,$

where $μ$ is the friction coefficient that can be defined as a function of the contact pressure, $p$; the slip rate, $γ˙e⁢q$; the average surface temperature at the contact point; and the average field variables at the contact point. Rate-dependent friction cannot be used in a static Riks analysis since velocity is not defined. In Abaqus it is possible to put a limit on the critical stress:

$τc⁢r⁢i⁢t=m⁢i⁢n⁢(μ⁢p,τm⁢a⁢x),$

where $τm⁢a⁢x$ is user-specified. If the equivalent stress is at the critical stress $(τe⁢q=τc⁢r⁢i⁢t)$, slip can occur. If the friction is isotropic, the direction of the slip and the frictional stress coincide, which is expressed in the form

$τiτe⁢q=γ˙iγ˙e⁢q,$

where $γ˙i$ is the slip rate in direction $i$ and $γ˙e⁢q$ is the magnitude of the slip velocity,

$γ˙e⁢q=γ˙12+γ˙22.$

As will be shown later, the same laws can be used for anisotropic friction after some simple transformations.

The above behavior can be modeled in Abaqus/Standard in two different ways. By default, the condition of no relative motion is approximated by stiff elastic behavior. The stiffness is chosen such that the relative motion from the position of zero shear stress is bounded by a value $γc⁢r⁢i⁢t$. (In Frictional behavior $γc⁢r⁢i⁢t$ is referred to as the allowable maximum elastic slip.) The critical slip value, $γc⁢r⁢i⁢t$, can be specified by the user. If it is not specified by the user, $γc⁢r⁢i⁢t$ is, by default, set to 0.5% of the average length of all contact elements in the model. It is worth noting that this approximate implementation method can also be considered an implementation of a nonlocal friction model; that is, a friction model for which the Coulomb condition is not applied pointwise but weighted over a small area with a so-called mollifying function (Zhong, 1989). See Oden and Pires (1983) for further discussion of nonlocal friction models.

Optionally, the relative motion in the absence of slip can be made exactly zero with the use of a Lagrange multiplier formulation. Although this procedure appears attractive because of the exact sticking constraint, it has two disadvantages:

1. The additional Lagrange multipliers increase the cost of the analysis.

2. The presence of rigid constraints tends to slow or sometimes prevent convergence of the Newton solution scheme used in Abaqus/Standard. This is likely to occur in areas where contact conditions change.

A special case of friction in Abaqus/Standard is so-called rough friction, where it is assumed that there is no bound on the shear stress; that is, no relative motion can occur as long as the surfaces are in contact. Rough friction is implemented with the Lagrange multiplier method.

In Abaqus/Explicit the relative motion in the absence of slip is always equal to zero if the kinematic contact algorithm is used with hard tangential surface behavior; at the end of each increment the positions of the nodes on the contact surfaces are adjusted so that the relative motion is zero. With the penalty contact algorithm in Abaqus/Explicit the relative motion in the absence of slip is equal to the friction force divided by the penalty stiffness.