Products: ABAQUS/Standard ABAQUS/Explicit
The contact modeling capabilities in ABAQUS allow access to a library of “surface constitutive models.” Part of this library in ABAQUS/Standard is the definition of the contact pressure between two surfaces at a point, 
, as a function of the “overclosure,” 
, of the surfaces (the interpenetration of the surfaces). Two models for 
are provided as described below.
In this case
The contact constraint is enforced with a Lagrange multiplier representing the contact pressure in a mixed formulation. The virtual work contribution is
This model provides an exponential – relationship, as shown in Figure 5.2.1–1.
The user defines an initial contact distance, 
, and a typical pressure value, 
, which is the pressure value at zero overclosure (
). Then, we define
), a linearized pressure-overclosure relation with continuous slope is used.The pressure-overclosure (-) relationship can be entered directly in tabular form as shown in Figure 5.2.1–2.
The linear pressure-overclosure relationship is similar to the tabular relationship except that the linear form requires only a single value to be input to define the slope and the curve always passes through the origin.
A mixed formulation is used because the exponential stiffness associated with softened contact tends to slow down convergence or, due to the development of excessive contact stresses, may cause divergence. For the mixed formulation the virtual work contribution is
where is the contact pressure, is the actual overclosure, and 
is the overclosure associated with the contact pressure, . A local Newton loop is used to calculate 
for the current value of . The linearized form of this contribution is where 
is evaluated for the overclosure . Since there is no term involving 
, there is a zero on the diagonal of the Jacobian. A zero on the diagonal is not desirable because it may lead to equation solver problems if a rigid body mode is constrained only by contact elements. Hence, a small reference stiffness 
is introduced by splitting the contact pressure as follows: 
is a Lagrange multiplier and is a small reference stiffness (see Figure 5.2.1–1): 
Substituting for the pressure in Equation 5.2.1–1 and Equation 5.2.1–2, we obtain
in Equation 5.2.1–4, 
In the mixed formulation the difference between the actual and the calculated overclosure 
will go to zero as part of the iterative solution process. The difference must be sufficiently small to obtain an accurate solution. The admissible error in is set to 
for 
. For 
the admissible error is interpolated linearly between and 
, where represents the tolerance level at 
; alternatively, the tolerances can be specified by the user as part of the solution controls.
In addition to the surface constitutive models described above, where the contact pressure is a function of the surface overclosure, ABAQUS/Standard allows for the definition of a “viscous” pressure that is proportional to the relative velocity, 
, at which the surfaces approach or separate from each other. This option is intended for the regularization of snap-through problems involving contact where convergence difficulties arise due to the sudden violation of contact constraints.
The damping pressure, 
, is given by
is the damping coefficient. This coefficient is specified as a function of the overclosure, , as follows: 
is the value of the damping coefficient at zero overclosure and 
is the fraction of the overclosure interval 
over which the damping coefficient is equal to .The virtual work contribution associated with the damping pressure 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
and 
are the Hilber-Hughes-Taylor time integration operator parameters. The viscous damping option cannot be used in a Riks analysis since velocity is not defined.
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