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Self contact
Self contact is available in the context of a two-dimensional surface folding and touching itself. Appropriate slide line elements are generated internally for each node of the surface. At the same time the surface itself describes a slide line. Each node in a slide line element is allowed to contact all of the segments of the slide line, with the exception of the segments that are adjacent to the node. The mathematical treatment of each of these slide line elements is the same as described above, with a few modifications. Since a node is simultaneously a master and a slave (producing symmetric master-slave relationships), potentially undefined constraints would occur if the meshes on both sides of the interface matched exactly. If only a pair of nodes matches, no problem occurs due to the smoothing carried out on the master side of the interface. When two adjacent segments of the surface fold forming a sharp crack, the contact algorithm becomes pure master-slave instead of symmetric to prevent undefined contact constraints. It has been arbitrarily chosen that of these two segments the shortest is the slave and the longest is the master.
5.1.3 Finite-sliding interaction between a deformable and a rigid body
ABAQUS/Standard provides two formulations for modeling the interaction between a deformable body and an arbitrarily shaped rigid body that may move during the history being modeled. The first is a small-sliding formulation in which the contacting surfaces can only undergo 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. This formulation is discussed in this section.
The finite-sliding rigid contact capability is implemented by means of a family of contact elements that ABAQUS automatically generates based on the data associated with the *CONTACT PAIR option. At each integration point these elements construct a measure of overclosure (penetration of the point on the surface of the deforming body into the rigid surface) and measures of relative shear sliding. These kinematic measures are then used, together with appropriate Lagrange multiplier techniques, to introduce surface interaction theories (contact and friction). A library of interaction theories is provided in ABAQUS--these may be thought of as a library of "surface constitutive models." In this section we discuss only the kinematics of the interacting surfaces. The surface constitutive models are described in Chapter 4, "Mechanical Constitutive Theories."
Let A be a point on the deforming mesh, with current coordinates \mathbf { x } _ { A } . Let C be the "rigid body reference node"--the node that defines the position of the rigid body--with current coordinates \mathbf { x } _ { C } . Let A ^ { \prime } be the closest point on the surface of the rigid body to A at which the normal to the surface of the rigid body, n, passes through A. Define r as the vector from C to A ^ { \prime } . The geometry described by these quantities is shown in Figure 5.1.3-1.
Figure 5.1.3-1 Rigid surface interface geometry.
Interface Modeling
text_image
A A' n r C
Let h be the distance from A ^ { \prime } to A along ¡n: the "overclosure" of the surfaces. From the definitions introduced above,
\mathbf {n} h = - \mathbf {x} _ {A} + \mathbf {x} _ {C} + \mathbf {r}.
Then if h < - c there is no contact between the surfaces at A, and no further surface interaction calculations need be done at this point. Here c is the clearance below which contact occurs. For a "hard" surface c = 0 , but ABAQUS/Standard also allows a "softened" surface to be introduced in which c may be nonzero (although c is usually very small compared to other dimensions). If h \geq - c the surfaces are in contact. To enforce the contact constraint we will need the first variation of h , \delta h , , and its second variation, d±h. These quantities are now derived.
Let S ^ { \alpha } , \alpha = 1 ; 2 be locally orthogonal, distance measuring surface coordinates on the surface at A ^ { \prime } . . The S ^ { \alpha } measure distance along the tangents \mathbf { t } _ { \alpha } to the surface at A ^ { \prime } { : } : these tangents are constructed according to the standard ABAQUS convention for such tangents to a surface in space. As the point A and the rigid body move, the projected point A0 will move along. The movement consists of two parts: movement due to motion of the rigid body and motion relative to the body
\delta \mathbf {x} _ {A ^ {\prime}} = \delta \mathbf {x} _ {C} + \delta \mathbf {r} | _ {\gamma_ {\alpha}} + \delta \mathbf {r} | _ {\phi_ {C}} = \delta \mathbf {x} _ {C} + \delta \pmb {\phi} _ {C} \times \mathbf {r} + \mathbf {t} _ {\alpha} \delta \gamma_ {\alpha},
where \delta \gamma _ { \alpha } is the "slip" of point A0 . The normal n will also change due to rotation of the rigid surface and due to slip along the surface
\delta \mathbf {n} = \delta \mathbf {n} | _ {\gamma_ {\alpha}} + \delta \mathbf {n} | _ {\phi_ {C}} = \delta \boldsymbol {\phi} _ {C} \times \mathbf {n} + \frac {\partial \mathbf {n}}{\partial S _ {\alpha}} \delta \gamma_ {\alpha}.
The linearized form of the contact equation, thus, becomes
\mathbf {n} \delta h + h (\delta \pmb {\phi} _ {C} \times \mathbf {n} + \frac {\partial \mathbf {n}}{\partial S _ {\alpha}} \delta \gamma_ {\alpha}) = - \delta \mathbf {x} _ {A} + \delta \mathbf {x} _ {C} + \delta \pmb {\phi} _ {C} \times \mathbf {r} + \mathbf {t} _ {\alpha} \delta \gamma_ {\alpha}.
For "hard" contact h = 0 exactly, and for soft contact we will assume h = 0 as well. The linearized kinematic equation, thus, becomes
Interface Modeling
\mathbf {n} \delta h = - \delta \mathbf {x} _ {A} + \delta \mathbf {x} _ {C} + \delta \pmb {\phi} _ {C} \times \mathbf {r} + \mathbf {t} _ {\alpha} \delta \gamma_ {\alpha}.
This equation can be split into normal and tangential components, which yields the contact equation,
\delta h = - \mathbf {n} \cdot (\delta \mathbf {x} _ {A} - \delta \mathbf {x} _ {C}) + (\mathbf {r} \times \mathbf {n}) \cdot \delta \boldsymbol {\phi} _ {C},
and the slip equations,
\delta \gamma_ {\alpha} = \mathbf {t} _ {\alpha} \cdot (\delta \mathbf {x} _ {A} - \delta \mathbf {x} _ {C}) - (\mathbf {r} \times \mathbf {t} _ {\alpha}) \cdot \delta \pmb {\phi} _ {C}.
To obtain the second variation of h , it will again be assumed that h = 0 . In addition, it will be assumed that d h = \delta h = 0 , which is accurate for relatively "hard" contact. It then directly follows that
\mathbf {n} d \delta h = d \delta \mathbf {r},
and from the linearized kinematic equation follows
\mathbf {n} d \delta h = d (\delta \pmb {\phi} _ {C} \times \mathbf {r}) | _ {\gamma_ {\beta}} + \delta \pmb {\phi} _ {C} \times d \mathbf {r} | _ {\phi_ {C}} + d \mathbf {t} | _ {\gamma_ {\beta}} \delta \gamma_ {\alpha} + d \mathbf {t} | _ {\phi_ {C}} \delta \gamma_ {\alpha} + \mathbf {t} _ {\alpha} d \delta \gamma_ {\alpha},
where we have used d \delta { \bf x } _ { A } = d \delta { \bf x } _ { C } = d \delta \phi _ { C } = 0 . The first term corresponds to a second-order variation on the vector r for rigid body rotations around point C and is given by (see ``Rotation variables,'' Section 1.3.1):
d (\delta \pmb {\phi} _ {C} \times \mathbf {r}) | _ {\gamma_ {\beta}} = \delta \pmb {\phi} _ {C} \cdot d \pmb {\phi} _ {C} \mathbf {r} - \frac {1}{2} \delta \pmb {\phi} _ {C} d \pmb {\phi} _ {C} \cdot \mathbf {r} - \frac {1}{2} \mathbf {r} \cdot \delta \pmb {\phi} _ {C} d \pmb {\phi} _ {C}.
The second term in the expression for the second variation is obtained with the previously used expression for the "slip" along the surface:
\delta \boldsymbol {\phi} _ {C} \times d \mathbf {r} | _ {\phi_ {C}} = \delta \boldsymbol {\phi} _ {C} \times \mathbf {t} _ {\alpha} d \gamma_ {\alpha}.
The third term follows from the expression for the rigid body rotation:
d \mathbf {t} | _ {\gamma_ {\beta}} \delta \gamma_ {\alpha} = d \pmb {\phi} _ {C} \times \mathbf {t} _ {\alpha} \delta \gamma_ {\alpha}.
Finally, the fourth term is obtained by differentiation along the surface coordinates:
d \mathbf {t} | _ {\phi_ {C}} \delta \gamma_ {\alpha} = \frac {\partial \mathbf {t} _ {\alpha}}{\partial S _ {\beta}} d \gamma_ {\beta} \delta \gamma_ {\alpha} = \delta \gamma_ {\alpha} \pmb {\kappa} _ {\alpha \beta} d \gamma_ {\beta},
where
Interface Modeling
\pmb {\kappa} _ {\alpha \beta} = \frac {\partial \mathbf {t} _ {\alpha}}{\partial S _ {\beta}} = \frac {\partial \mathbf {t} _ {\beta}}{\partial S _ {\alpha}} = \frac {\partial^ {2} \mathbf {r}}{\partial S _ {\alpha} \partial S _ {\beta}}
is the surface curvature matrix.
Substitution of the last four expressions in the expression for the second variation yields
\mathbf {n} d \delta h = \delta \pmb {\phi} _ {C} \cdot d \pmb {\phi} _ {C} \mathbf {r} - \frac {1}{2} \delta \pmb {\phi} _ {C} d \pmb {\phi} _ {C} \cdot \mathbf {r} - \frac {1}{2} \mathbf {r} \cdot \delta \pmb {\phi} _ {C} d \pmb {\phi} _ {C}
+ \delta \pmb {\phi} _ {C} \times \mathbf {t} _ {\alpha} d \gamma_ {\alpha} + d \pmb {\phi} _ {C} \times \mathbf {t} _ {\alpha} \delta \gamma_ {\alpha} + \delta \gamma_ {\alpha} \pmb {\kappa} _ {\alpha \beta} d \gamma_ {\beta} + \mathbf {t} _ {\alpha} d \delta \gamma_ {\alpha}.
As in the first variation, one can split the second variation into a normal and tangential components. For the normal component one finds
d \delta h = (\mathbf {n} \cdot \mathbf {r}) \delta \pmb {\phi} _ {C} \cdot d \pmb {\phi} _ {C} - \frac {1}{2} \delta \pmb {\phi} _ {C} \cdot (\mathbf {n r} + \mathbf {r n}) \cdot d \pmb {\phi} _ {C}
+ \delta \pmb {\phi} _ {C} \cdot (\mathbf {t} _ {\alpha} \times \mathbf {n}) d \gamma_ {\alpha} + d \pmb {\phi} _ {C} \cdot (\mathbf {t} _ {\alpha} \times \mathbf {n}) \delta \gamma_ {\alpha} + \delta \gamma_ {\alpha} \mathbf {n} \cdot \pmb {\kappa} _ {\alpha \beta} d \gamma_ {\beta}
and for the tangential components,
d \delta \gamma_ {\alpha} = - (\mathbf {t} _ {\gamma} \cdot \mathbf {r}) \delta \pmb {\phi} _ {C} \cdot d \pmb {\phi} _ {C} + \frac {1}{2} \delta \pmb {\phi} _ {C} \cdot (\mathbf {t} _ {\gamma} \mathbf {r} + \mathbf {r t} _ {\gamma}) \cdot d \pmb {\phi} _ {C} + \delta \gamma_ {\alpha} \mathbf {t} _ {\gamma} \cdot \pmb {\kappa} _ {\alpha \beta} d \gamma_ {\beta}.
The expression involving \kappa _ { \alpha \beta } can be simplified somewhat. Observe that \begin{array} { r } { \mathbf { n } \cdot \mathbf { t } _ { \alpha } = 0 ; } \end{array} ; hence,
\mathbf {n} \cdot \pmb {\kappa} _ {\alpha \beta} = \mathbf {n} \cdot \frac {\partial \mathbf {t} _ {\alpha}}{\partial S _ {\beta}} = - \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {n}}{\partial S _ {\beta}} = \mathbf {t} _ {\beta} \cdot \frac {\partial \mathbf {n}}{\partial S _ {\alpha}}.
Similarly
\mathbf {t} _ {\gamma} \cdot \pmb {\kappa} _ {\alpha \beta} = \mathbf {t} _ {\gamma} \cdot \frac {\partial \mathbf {t} _ {\alpha}}{\partial S _ {\beta}} = - \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {t} _ {\gamma}}{\partial S _ {\beta}}.
If the local surface coordinate system is created by projection of a tangential Cartesian X-Y system onto the surface, it is readily established that the last terms vanish. Hence, we will assume that the last term in the second variation is zero. The final result is obtained by substitution of the expressions for the first-order variation of the slip in the expressions for the second variation. After some reordering and with \kappa _ { \alpha \beta } \equiv \mathbf { n } \cdot \pmb { \kappa } _ { \alpha \beta } this furnishes
d \delta h = \left(\delta \mathbf {x} _ {A} - \delta \mathbf {x} _ {C}\right) \cdot \mathbf {t} _ {\alpha} \kappa_ {\alpha \beta} \mathbf {t} _ {\beta} \cdot \left(d \mathbf {x} _ {A} - d \mathbf {x} _ {C}\right)
Interface Modeling
+ (\delta \mathbf {x} _ {A} - \delta \mathbf {x} _ {C}) \cdot (\mathbf {t} _ {\alpha} \kappa_ {\alpha \beta} (\mathbf {t} _ {\beta} \times \mathbf {r}) + \mathbf {t} _ {\alpha} (\mathbf {t} _ {\beta} \times \mathbf {n})) \cdot d \pmb {\phi} _ {C}
+ \delta \pmb {\phi} _ {C} \cdot \left((\mathbf {t} _ {\alpha} \times \mathbf {r}) \kappa_ {\alpha \beta} \mathbf {t} _ {\beta} + (\mathbf {t} _ {\alpha} \times \mathbf {n}) \mathbf {t} _ {\beta}\right) \cdot (d \mathbf {x} _ {A} - d \mathbf {x} _ {C})
+ \delta \pmb {\phi} _ {C} \cdot (\mathbf {t} _ {\alpha} \times \mathbf {r}) \kappa_ {\alpha \beta} (\mathbf {t} _ {\beta} \times \mathbf {r}) \cdot d \pmb {\phi} _ {C}
+ (\mathbf {n} \cdot \mathbf {r}) \delta \pmb {\phi} _ {C} \cdot d \pmb {\phi} _ {C} - \frac {1}{2} \delta \pmb {\phi} _ {C} \cdot (\mathbf {n r} + \mathbf {r n}) \cdot d \pmb {\phi} _ {C}
d \delta \gamma_ {\alpha} = - (\mathbf {t} _ {\alpha} \cdot \mathbf {r}) \delta \pmb {\phi} _ {C} \cdot d \pmb {\phi} _ {C} + \frac {1}{2} \delta \pmb {\phi} _ {C} \cdot (\mathbf {t} _ {\alpha} \mathbf {r} + \mathbf {r t} _ {\alpha}) \cdot d \pmb {\phi} _ {C}.
The first two terms of the expression for d±h will only need to be included if slip occurs, whereas the expression for d \delta \gamma _ { \alpha } only needs to be taken into account if frictional forces are transmitted.
For dynamic applications we need the velocity and acceleration terms \dot { h } and \ddot { h } to calculate impact forces and impulses correctly. These terms are
\dot {h} = - \mathbf {n} \cdot (\dot {\mathbf {x}} _ {A} - \dot {\mathbf {x}} _ {C} - \dot {\pmb {\phi}} _ {C} \times \mathbf {r})
(this is the same form as the first variation of h ) ; and
\begin{array}{l} \ddot {h} = - \mathbf {n} \cdot (\ddot {\mathbf {x}} _ {A} - \ddot {\mathbf {x}} _ {C} - \ddot {\pmb {\phi}} _ {C} \times \mathbf {r}) + \mathbf {n} \cdot \dot {\pmb {\phi}} _ {C} \mathbf {r} \cdot \dot {\pmb {\phi}} _ {C} - \mathbf {n} \cdot \mathbf {r} \dot {\pmb {\phi}} _ {C} \cdot \dot {\pmb {\phi}} _ {C} \\ - \left(\dot {\mathbf {x}} _ {A} - \dot {\mathbf {x}} _ {C} - \dot {\pmb {\phi}} _ {C} \times \mathbf {r}\right) \cdot \left(\dot {\pmb {\phi}} _ {C} \times \mathbf {n} + \frac {\partial \mathbf {n}}{\partial S ^ {\alpha}} \mathbf {t} _ {\alpha} \cdot \left(\dot {\mathbf {x}} _ {A} - \dot {\mathbf {x}} _ {C} - \dot {\pmb {\phi}} _ {C} \times \mathbf {r}\right)\right). \\ \end{array}
5.2 Surface interactions
5.2.1 Contact pressure definition
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, p , as a function of the "overclosure," h, of the surfaces (the interpenetration of the surfaces). Two models for p = p ( h ) are provided as described below.
Hard contact
In this case
p = 0 \quad \text { for } h < 0 \quad \text { and }
p = \overline {{k}} h, \quad \text { with } \overline {{k}} \text { infinite for } h \geq 0.
Softened contact defined with an exponential pressure-overclosure relationship
This model provides an exponential p-h relationship, as shown in Figure 5.2.1-1.
Figure 5.2.1-1 "Softened" pressure-overclosure relationship using an exponential law.
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exponential p-h relationship p p° -h -c -k₀ = dp/dh | h = -0.9999c -0.9999c -0.9999c
The user defines an initial contact distance, c, and a typical pressure value, p ^ { o } , which is the pressure value at zero overclosure (h = 0). Then, we define
\begin{array}{l} \boxed {p = 0 \quad \text {for} h \leq - c,} \\ p = \frac {p ^ {o}}{(\exp (1) - 1)} \left[ \left(\frac {h}{c} + 1\right) \left(\exp \left(\frac {h}{c} + 1\right) - 1\right) \right] \quad \text {for} h > - c, \\ \end{array}
and
\begin{array}{l} \frac {d p}{d h} = 0 \quad \text {for} h \leq - c, \\ \frac {d p}{d h} = \frac {p ^ {o}}{(\exp (1) - 1)} \left[ \frac {1}{c} \left(\frac {h}{c} + 2\right) \exp \left(\frac {h}{c} + 1\right) - \frac {1}{c} \right] \quad \text {for} h > - c, \\ \end{array}
Softened contact defined with a tabular pressure-overclosure relationship
The pressure-overclosure (p-h) relationship can be entered directly in tabular form as shown in Figure 5.2.1-2.
Figure 5.2.1-2 "Softened" pressure-overclosure relationship defined in tabular form.
line
| Overclosure h | Pressure p |
|---|---|
| 0 | (p₂,h₂) |
| 0 | (p₃,h₃) |
| 0 | (pₙ,hₙ) |
Softened contact defined with a linear pressure-overclosure relationship
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.
Softened contact implementation
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
Equation 5.2.1-1
\delta \Pi = \delta p (h - \overline {{h}}) + p \delta h,
where p is the contact pressure, h is the actual overclosure, and \overline { { h } } ( p ) is the overclosure associated with the contact pressure, p . A local Newton loop is used to calculate \overline { { h } } for the current value of p . The linearized form of this contribution is
Equation 5.2.1-2
\mathrm{d} \delta \Pi = \delta p \mathrm{d} h + \mathrm{d} p \delta h - \frac {d \overline {{h}}}{d p} \delta p \mathrm{d} p,
where d \overline { { h } } / d p = ( d p / d \overline { { h } } ) ^ { - 1 } is evaluated for the overclosure \overline { { h } } . Since there is no term involving ±hdh, 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 k _ { 0 } is introduced by splitting the contact pressure as follows:
p = q + k _ {0} (h + c) \quad \text { and } \quad \delta p = \delta q + k _ {0} \delta h,
where q is a Lagrange multiplier and k _ { 0 } is a small reference stiffness (see Figure 5.2.1-1):
Interface Modeling
k _ {0} = \left. \frac {d p}{d h} \right| _ {h = - 0. 9 9 9 9 c}.
Substituting for the pressure p in Equation 5.2.1-1 and Equation 5.2.1-2, we obtain
Equation 5.2.1-3
\delta \Pi = \delta q (h - \overline {{h}}) + \delta h \left[ k _ {0} (h - \overline {{h}}) + q + k _ {0} (h + c) \right]
and
Equation 5.2.1-4
d \delta \Pi = \delta q (d h - d \overline {{{h}}}) + \delta h \left[ k _ {0} (d h - d \overline {{{h}}}) + d q + k _ {0} d h \right].
Further,
d \overline {{h}} = \frac {d \overline {{h}}}{d p} d p = \frac {d \overline {{h}}}{d p} (d q + k _ {0} d h).
Therefore, substituting for d \overline { { h } } in Equation 5.2.1-4,
d \delta \Pi = \delta q \left[ (1 - \frac {d \overline {{h}}}{d p} k _ {0}) d h - \frac {d \overline {{h}}}{d p} d q \right] + \delta h \left[ (2 - \frac {d \overline {{h}}}{d p} k _ {0}) k _ {0} d h + (1 - \frac {d \overline {{h}}}{d p} k _ {0}) d q \right].
Thus, the corresponding system of equations in matrix form is
\left[ \begin{array}{cc}(2 - \frac{d\overline{h}}{dp} k_{0})k_{0} & 1 - \frac{d\overline{h}}{dp} k_{0}\\ 1 - \frac{d\overline{h}}{dp} k_{0} & -\frac{d\overline{h}}{dp} \end{array} \right]\left\{ \begin{array}{c}d h\\ dq \end{array} \right\} = -\left\{ \begin{array}{c}q + k_{0}(h + c) + k_{0}(h - \overline{h})\\ h - \overline{h} \end{array} \right\} .
In the mixed formulation the difference between the actual and the calculated overclosure h - \overline { { h } } 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 h - \overline { { h } } is set to 0:005c for p \geq p ^ { 0 } . For 0 \le p \le p ^ { 0 } the admissible error is interpolated linearly between 0:005c and 0:1c, where 0:1c represents the tolerance level at p = 0 . 0 ; alternatively, the tolerances can be specified by the user with the *CONTROLS option.
Viscous damping option
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, \dot { h } , 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, f , is given by
Interface Modeling
f = f (h, \dot {h}) = \mu (h) \dot {h},
where \mu is the damping coefficient. This coefficient is specified as a function of the overclosure, h , as follows:
\mu (h) = \left\{ \begin{array}{l l} 0 & \text {for} h < - c _ {0} \\ \mu_ {0} (h + c _ {0}) / (1 - \eta) c _ {0} & \text {for} - c _ {0} \leq h < - \eta c _ {0} \\ \mu_ {0} & \text {for} - \eta c _ {0} \leq h \leq 0 \end{array} \right.
where \mu _ { 0 } is the value of the damping coefficient at zero overclosure and \eta is the fraction of the overclosure interval [ - c _ { 0 } , 0 ] over which the damping coefficient is equal to \mu _ { 0 } .
The virtual work contribution associated with the damping pressure is
\delta \Pi = f \delta h.
The contribution to the stiffness matrix for the Newton solution is given by the linearized form of the virtual work contribution:
\begin{array}{l} \mathrm{d} \delta \Pi = \delta h \mathrm{d} f = \delta h \left(\frac {\partial f}{\partial h} + \frac {\partial f}{\partial \dot {h}} \frac {\partial \dot {h}}{\partial h}\right) \mathrm{d} h \\ = \delta h \left(\frac {d \mu}{d h} \dot {h} + \mu \frac {\partial \dot {h}}{\partial h}\right) \mathrm{d} h, \\ \end{array}
where
\frac {d \mu}{d h} = \left\{ \begin{array}{l l} 0 & \mathrm{for} h < - c _ {0} \\ \mu_ {0} / (1 - \eta) c _ {0} & \mathrm{for} - c _ {0} \leq h < - \eta c _ {0} \\ 0 & \mathrm{for} - \eta c _ {0} \leq h \leq 0 \end{array} \right.
In static analysis the velocity is defined as the displacement increment divided by the time increment. Therefore, \dot { h } = \Delta h / \Delta t , and the stiffness contribution reduces to
\mathrm{d} \delta \Pi = \delta h \left(\frac {d \mu}{d h} \frac {\Delta h}{\Delta t} + \frac {\mu}{\Delta t}\right) \mathrm{d} h.
In the case of dynamics \partial \dot { h } / \partial h is defined by the dynamic time integration operator, and the stiffness contribution can be written as
\mathrm{d} \delta \Pi = \delta h \left(\frac {d \mu}{d h} \dot {h} + \frac {\gamma \mu}{\beta \Delta t}\right) \mathrm{d} h,
where \gamma and \beta 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.
5.2.2 Pressure and fluid flow in pore pressure contact
ABAQUS/Standard provides a surface-based capability, which uses the *CONTACT PAIR option, to model fully saturated porous media. The surface-based capability can be used for small or finite sliding; however, tangential flow cannot be modeled.
The pore fluid constraints on the contact interface
Let p _ { 1 } and p _ { 2 } be the pore pressures at the two sides of the interface. It is at all times required that the pore pressures on opposite sides be equal:
p _ {2} - p _ {1} = 0.
Similarly, let q _ { 1 } and q _ { 2 } be the volume flow rate densities normal to the interface at the two sides, and let { \dot { h } } = v _ { 2 } - v _ { 1 } = \mathbf { n } \cdot \left( \mathbf { v _ { 2 } } - \mathbf { v _ { 1 } } \right) be the relative velocity of the two sides in the direction of the interface normal.
It is assumed that the interface is filled with fluid at all times. Hence, continuity requires that
q _ {1} + q _ {2} = - \dot {h},
whereas the difference
q _ {2} - q _ {1} = \hat {q}
is undetermined and is to be treated as an independent variable. Inversion of these equations yields
q _ {1} = - \frac {1}{2} (\dot {h} + \hat {q}),
q _ {2} = - \frac {1}{2} (\dot {h} - \hat {q}).
The transient equations
The contribution of the interfacial virtual work equation and its linearized form are first obtained in the general form including finite sliding. The equations are then specialized to the various formulations implemented in ABAQUS.
Since we want to achieve force and volume flow rate equilibrium at each side of the interface, as well as obtain continuity in the pore pressures, we add the following integral to the virtual work equation:
\delta \Pi^ {i p p} = \int_ {A} [ q _ {1} \delta p _ {1} + q _ {2} \delta p _ {2} - p _ {2} \delta v _ {2} + p _ {1} \delta v _ {1} - (p _ {1} - p _ {2}) \delta \lambda ] d A,
where \delta \lambda is an arbitrary Lagrange multiplier and A is the interface area. Eliminating q _ { 1 } and q _ { 2 } and