Product: ABAQUS/Standard
ABAQUS/Standard provides two formulations for modeling the interaction between two deformable bodies. The first is a small-sliding formulation in which the contacting surfaces can undergo only relatively small sliding relative to each other but arbitrary rotation of the surfaces is permitted. This formulation is discussed in “Small-sliding interaction between bodies,” Section 5.1.1. The second is a finite-sliding formulation where separation and sliding of finite amplitude and arbitrary rotation of the surfaces may arise. The formulation for two-dimensional and axisymmetric analysis, as well as for tube-in-tube analysis, is discussed in this section.
Depending on the type of contact problem, two approaches are available to the user for specifying the finite-sliding capability: (1) defining possible contact conditions by identifying and pairing potential contact surfaces and (2) using contact elements. With the first approach ABAQUS automatically generates the appropriate contact elements.
In axisymmetric problems with asymmetric deformations, ISL21A and ISL22A elements can be used to model contact with CAXA or SAXA elements. Sliding of tubes inside each other can be modeled with ITT21 and ITT31 elements.
To define a sliding interface between two surfaces, one of the surfaces (the “slave” surface) is covered with ISL or ITT elements. The other surface (the “master” surface) is defined as a slide line surface composed of a series of nodes ordered in sequence. The slide line itself can consist of linear or quadratic segments. If smoothing is used, these segments are connected with quadratic or cubic segments such that full slope continuity is achieved. The smoothing procedure is described later in this section.
Consider contact of a node on the slave surface 
with a segment of the master surface described by nodes 
, 
, 
, where the number of nodes depends on the order of the segment. For a linear segment the number of nodes is 2, whereas for a quadratic segment the number of nodes is 3. For a smoothed section of a linear slide line, the number of nodes is also 3; and for a smoothed section of a quadratic slide line, the number of nodes is 5. If the contact occurs at the (convex) vertex of two segments, only a single node will enter the equations. A typical linear segment is shown in Figure 5.1.2–1, and a quadratic segment is displayed in Figure 5.1.2–2. Smoothed segments are shown later in this section.
To derive the equations governing these elements, we consider the coordinates in the plane of the slide line. For the axisymmetric elements, this plane coincides with the two-dimensional space. First, we determine the point 
on the segment closest to the point 
on the slave surface. We also determine the normal 
and tangent 
to the segment at that point. The point and the normal can be related to the overclosure 
with the relation
is on the segment, its position is defined completely by the interpolation function 
for the segment, the position 
, and the position 
of the nodes 
that are part of the segment. That allows us to write for Equation 5.1.2–1 where 
and 
, 
, are functions of . For instance, for a linear segment you obtain 
, 
. For a quadratic segment you use 
, 
, 
. Similar expressions are obtained for smoothed segments of the slide line. The tangent to the slide line at point follows with where 
is determined from the condition that the normal and tangent must be orthogonal, which leads to the following equation for : 
To obtain the contact/slip equation, the position equation (Equation 5.1.2–2) is linearized. This linearization yields
is the slip. In the direction of contact, 
this yields and in the direction of slip, 
one finds 
is on the slide line; hence, it will be assumed that 
. From this follows To obtain the initial stress stiffness terms, the second variations of and 
must be calculated. From Equation 5.1.2–4 follows The first term is expressed readily in terms of 
with the help of Equation 5.1.2–5: The rate of change of the normal can be re-expressed as In this equation 
can be obtained from Equation 5.1.2–3: where use was made of Equation 5.1.2–5 and 
(the segment curvature) is defined as For straight segments obviously vanishes. Substitution of Equation 5.1.2–7 to Equation 5.1.2–9 in Equation 5.1.2–6 yields the final result: This expression is symmetric, as should be expected. The second variation in can be derived along similar lines: Note that Substitution of Equation 5.1.2–7, Equation 5.1.2–9, and Equation 5.1.2–13 in Equation 5.1.2–12 yields This expression is nonsymmetric. The second variations of and vanish if no slip in the slide plane occurs 
In the case of tube-tube interface elements it is assumed that the inner tube can be considered the slave surface and the outer tube the master surface. The tube-tube interface elements differ from the axisymmetric slide line elements in two ways. In the first place there is assumed to be a finite clearance between the tubes, which has the effect that the separation distance is finite even when contact occurs. In the second place for the three-dimensional element ITT31 there is a second possible slip direction in the plane of the cross-section of the tubes. The derivations for the ITT elements follow much along the same line as the derivations for the ISL elements. The contact equation can be written in the form
is a point on the outer tube that potentially contacts point on the inner tube, as shown in Figure 5.1.2–3.In this equation 
is the (positive) radial clearance between the tubes. In a similar way as was done for the ISL elements, the contact equation can be written in the form
. In the contact direction this yieldsand in the direction along the pipe,With use of 
, it readily follows which transforms Equation 5.1.2–18 into where is the segment curvature as defined by Equation 5.1.2–10. For the three-dimensional tube-tube interface element, one can define the transverse slip direction 
, which yields The initial stress stiffness terms are again obtained by taking the second variations in , , and 
. From Equation 5.1.2–17 follows
Substitution of Equation 5.1.2–20 and Equation 5.1.2–23 in Equation 5.1.2–22 yields
This expression is symmetric. In the-direction the virtual work term has the form 
is defined by In this expression the terms involving 
vanish for element type ITT21. Substitution of Equation 5.1.2–20 and Equation 5.1.2–27 in Equation 5.1.2–26 yields In the -direction one simply looks at the first variation in 
: Note that 
At the junction of two segments along a slide line, a discontinuity in slope may occur. This discontinuity can cause convergence problems since during iteration a contact point might move back and forth between two segments. Hence, it is useful to smooth the transition between segments. Consider first the transition between two linear segments (Figure 5.1.2–4).
The junction of the two segments is connected by a Hermitian polynomial between the points 
and 
, which are located on the segments:
The positions of and are, hence, determined by the smoothing factor 
, which is specified directly by the user in the range 
.
We choose as the parameter coordinate on the smoothed segment, with 
. At the extreme values for the coordinates are and , and for the coordinate derivatives we choose
of a point on the segment as a function of : 
Note that the smoothing has in fact been done with a second-order polynomial. In the formulation of the interface contact and friction equations, the treatment of a smoothing segment is identical to the treatment of a regular segment. A transition between quadratic segments is shown in Figure 5.1.2–5.
It is readily established that in this case
is specified directly by the user. Substitution of these equations in the Hermite shape functions yields If 
and 
, the equations reduce to the same equations as were used for smoothing between linear segments.Self contact is available in the context of a surface folding and touching itself. Appropriate contact elements are generated internally for each node on the surface. A node is allowed to contact all of the surface segments, with the exception of the segments that are adjacent to the node. Since a node is simultaneously a master and a slave (producing symmetric master-slave relationships), overconstraints would occur, for example, if the meshes on both sides of the interface matched exactly. If only a pair of nodes matches in two dimensions, no problem occurs due to the smoothing carried out on the master side of the interface. To avoid solver problems due to overconstraining, three-dimensional self-contact is activated only if penalty-type contact is used.
The mathematical treatment of contact elements is the same as described above, with a few modifications in two dimensions. When two adjacent segments of the surface fold forming a sharp crack, the contact algorithm becomes pure master-slave instead of symmetric to prevent redundant contact constraints. It has been arbitrarily chosen that of these two segments the shortest is the slave and the longest is the master.
