Products: ABAQUS/Standard ABAQUS/Explicit
An extended version of the classical isotropic Coulomb friction model is provided in ABAQUS 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 standard Coulomb friction model assumes that no relative motion occurs if the equivalent frictional stress
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 . (In the ABAQUS Analysis User's Manual is referred to as the allowable maximum elastic slip.) The critical slip value, , can be specified by the user. If it is not specified by the user, 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:
The additional Lagrange multipliers increase the cost of the analysis.
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.
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.
In the elastic stick formulation in ABAQUS/Standard, the “elastic” tangential slip is defined as the reversible relative tangential motion from the point of zero shear stress. The elastic slip is related to the interface shear stress with the relation
The above expressions hold if the equivalent shear stress remains less than the critical stress. If the equivalent stress exceeds the critical stress, slip must be taken into consideration so that the condition is satisfied. Let the starting situation be characterized by the elastic slip . The critical stress at the end of the increment follows from the contact pressure, , and the slip rate, .
Let the (as yet unknown) elastic slip at the end of the increment be and the slip increment be . Consistency requires that
For the iterative solution scheme this equation must be linearized. Some straightforward algebra yields
If the user specifies different velocities for two bodies in contact as a predefined field, it is assumed that the slip velocity follows from the prescribed motion and is independent of the displacement values. The frictional stresses then follow from
At the contact nodes at which the velocity differential is imposed, the linearized shear stress can be expressed in the following form:
For steady-state transport a viscous stick formulation is used. In this case the “viscous” tangential slip rate, , is related to the interface shear stress by
In ABAQUS/Standard Lagrange multipliers are used to enforce exact sticking conditions. A constraint term enforced with Lagrange multipliers is added to the virtual work statement:
where is a Lagrange multiplier. By taking the rate of change of Equation 5.2.3–4, the rate of virtual work is obtained: where the last term results from a nonlinear relation between nodal displacements and relative motion . This term is nonzero only for contact with finite sliding (see Finite-sliding interaction between deformable bodies, Section 5.1.2, and Finite-sliding interaction between a deformable and a rigid body, Section 5.1.3).To obtain a complete formulation, a relationship between , , and must be defined. For sticking conditions a suitable relation is
If the element is slipping, can take an arbitrary value. Consequently the terms involving in Equation 5.2.3–4 and Equation 5.2.3–5 vanish. With the backward difference method, the frictional stress is obtained as
and the linearized form is The rate of virtual work can, hence, be written in the form Observe that for two-dimensional problems the term involving vanishes. In the case of dynamics is scaled by the Hilber-Hughes-Taylor time integration operator parameters.To determine whether an element sticks or slips, the following tests are used. If the element is currently sticking, it is checked whether the magnitude of the frictional stress exceeds the critical value. Hence, if
If the element is slipping, the direction of slip is compared with the frictional stress applied at the start of the iteration. If at the end of the iteration
In anisotropic friction the friction coefficients in the two (principal) directions are different; hence, the critical shear stresses are different:
If slip occurs, it is assumed that the direction of slip is governed by an associated slip law:
In the anisotropic elastic stick formulation, it is assumed that the stiffness in stick in terms of the scaled stress and slip is constant:
In terms of the actual stress and strain in the - and -directions, this implies
In addition to the friction models described above, ABAQUS allows for the definition of a “viscous” shear stress that is proportional to the relative tangential velocity . This viscous damping in the tangential direction occurs if
damping in the contact direction is included and the tangent fraction is nonzero; or
contact controls are used to stabilize rigid body modes automatically in multi-body contact analysis.
The virtual work contribution associated with the viscous shear stress is
In static analysis the velocity is defined as the displacement increment divided by the time increment. Therefore, , and the stiffness contribution reduces to
In the case of dynamics is defined by the dynamic time integration operator and the stiffness contribution can be written as