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Interface Modeling
The three-dimensional small-sliding rigid contact formulation can also be derived from its deformable counterpart by generalizing some of the expressions that were introduced in the previous section. In particular, in this formulation the rigid reference node can undergo an arbitrary finite rotation described by the rotation vector \phi _ { r s } . Consequently, Equation 5.1.1-13 for the variations of the anchor point coordinates and the contact plane's tangent generalize to
\delta \mathbf {x} _ {0} = \delta \mathbf {u} _ {r s} + \widehat {\delta \pmb {\theta}} _ {r s} \cdot \mathbf {r},
\delta \mathbf {t} _ {i} = \widehat {\delta \pmb {\theta}} _ {r s} \cdot \mathbf {t} _ {i}, \qquad i = 1, 2,
where \widehat { \delta \pmb { \theta } } _ { r s } is the skew-symmetric matrix associated with the linearized rotation \delta \pmb { \theta } _ { r s } , , as explained in ``Rotation variables,'' Section 1.3.1.
Replacing \mathbf { v } _ { i } by \mathbf { t } _ { i } in Equation 5.1.1-8 through Equation 5.1.1-10 and substituting for \delta \mathbf { x } _ { 0 } and \delta \mathbf { t } _ { i } from above results in the following expressions for \delta h , \delta s _ { 1 } , and \delta s _ { 2 } :
Equation 5.1.1-16
\delta h = - \mathbf {n} \cdot \left[ \delta \mathbf {u} _ {N + 1} - \delta \mathbf {u} _ {r s} - \widehat {\delta \pmb {\theta}} _ {r s} \cdot (\mathbf {r} + \xi_ {1} \mathbf {t} _ {1} + \xi_ {2} \mathbf {t} _ {2}) \right],
Equation 5.1.1-17
\delta s _ {1} \stackrel {\mathrm{def}} {=} \delta \xi_ {1} = \mathbf {t} _ {1} \cdot \left[ \delta \mathbf {u} _ {N + 1} - \delta \mathbf {u} _ {r s} - \widehat {\delta \pmb {\theta}} _ {r s} \cdot (\mathbf {r} + \xi_ {2} \mathbf {t} _ {2}) \right],
Equation 5.1.1-18
\delta s _ {2} \stackrel {\mathrm{def}} {=} \delta \xi_ {2} = \mathbf {t} _ {2} \cdot \left[ \delta \mathbf {u} _ {N + 1} - \delta \mathbf {u} _ {r s} - \widehat {\delta \pmb {\theta}} _ {r s} \cdot (\mathbf {r} + \xi_ {1} \mathbf {t} _ {1}) \right].
5.1.2 Finite-sliding interaction between deformable bodies
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. For ISL and ITT elements, the other surface (the "master" surface) is defined by a series of nodes ordered in sequence in the *SLIDE LINE option. 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.
Two-dimensional and axisymmetric slide line elements
Consider contact of a node on the slave surface n _ { 1 } with a segment of the master surface described by nodes n _ { 2 } , n _ { 3 } , . . . , 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 x on the segment closest to the point \mathbf { x } _ { 1 } on the slave surface. We also determine the normal n and tangent t to the segment at that point. The point x and the normal n can be related to the overclosure h with the relation
Equation 5.1.2-1
\mathbf {n} h = \mathbf {x} - \mathbf {x} _ {1}.
Figure 5.1.2-1 Linear slide line segment.
text_image
X₁ n t X₂ X X₃
Figure 5.1.2-2 Quadratic slide line segment.
Interface Modeling
text_image
X₁ n X₃ X t X₂ X₄
Since x is on the segment, its position is defined completely by the interpolation function N _ { i } for the segment, the position g, and the position \mathbf { x } _ { i } of the nodes n _ { i } that are part of the segment. That allows us to write for Equation 5.1.2-1
Equation 5.1.2-2
\mathbf {n} h = N _ {i} (g) \mathbf {x} _ {i},
where N _ { 1 } = - 1 and N _ { 2 } , N _ { 3 } , \ldots are functions of g. For instance, for a linear segment you obtain N _ { 2 } = { \textstyle { \frac { 1 } { 2 } } } ( 1 - g ) , N _ { 3 } = { \textstyle { \frac { 1 } { 2 } } } ( 1 + g ) . For a quadratic segment you use \begin{array} { r } { N _ { 2 } = \frac { 1 } { 2 } g ( g - 1 ) , N _ { 3 } = 1 - g ^ { 2 } } \end{array} , \begin{array} { r } { N _ { 4 } = \frac 1 2 g ( g + 1 ) } \end{array} . Similar expressions are obtained for smoothed segments of the slide line. The tangent t to the slide line at point x follows with
Equation 5.1.2-3
\mathbf {t} \stackrel {\mathrm{def}} {=} \frac {\mathrm{d} \mathbf {x}}{\mathrm{d} s} = \frac {\mathrm{d} \mathbf {x}}{\mathrm{d} g} \bigg / \left| \frac {\mathrm{d} \mathbf {x}}{\mathrm{d} g} \right|,
where
\frac {\mathrm{d} \mathbf {x}}{\mathrm{d} g} = \frac {\mathrm{d} N _ {i}}{\mathrm{d} g} \mathbf {x} _ {i}.
The position of point x is determined from the condition that the normal and tangent must be orthogonal, which leads to the following equation for g \dot { . }
\mathbf {n} \cdot \mathbf {t} = N _ {i} (g) \frac {\mathrm{d} N _ {j} (g)}{\mathrm{d} g} \mathbf {x} _ {i} \cdot \mathbf {x} _ {j} = 0.
For linear segments this yields a linear equation, which can be solved directly. For quadratic and cubic segments it leads to third- or fifth-order equations, which must be solved iteratively. The equation is solved using Newton's method preceded by a number of bisections to find the true minimum distance solution.
To obtain the contact/slip equation, the position equation ( Equation 5.1.2-2) is linearized. This linearization yields
\delta \mathbf {n} h + \mathbf {n} \delta h = \frac {\mathrm{d} N _ {i}}{\mathrm{d} g} \mathbf {x} _ {i} \delta g + N _ {i} \delta \mathbf {x} _ {i} = \mathbf {t} \delta s + N _ {i} \delta \mathbf {x} _ {i},
Interface Modeling
where ±s is the slip. In the direction of contact, n; this yields
Equation 5.1.2-4
\delta h = N _ {i} \mathbf {n} \cdot \delta \mathbf {x} _ {i};
and in the direction of slip, t; one finds
\delta s = - N _ {i} \mathbf {t} \cdot \delta \mathbf {x} _ {i} - h \mathbf {t} \cdot \delta \mathbf {n}.
It is assumed that slip is relevant only if the node \mathbf { x } _ { 1 } is on the slide line; hence, it will be assumed that h = 0. From this follows
Equation 5.1.2-5
\delta s = - N _ {i} \textbf {t} \cdot \delta \mathbf {x} _ {i}.
To obtain the initial stress stiffness terms, the second variations of h and s must be calculated. From Equation 5.1.2-4 follows
Equation 5.1.2-6
\mathrm{d} \delta h = \delta \mathbf {x} _ {i} \cdot \mathbf {n} \frac {\mathrm{d} N _ {i}}{\mathrm{d} g} \mathrm{d} g + N _ {i} \delta \mathbf {x} _ {i} \cdot \mathrm{d} \mathbf {n}.
The first term is expressed readily in terms of dxi with the help of Equation 5.1.2-5:
Equation 5.1.2-7
\frac {\mathrm{d} N _ {i}}{\mathrm{d} g} \mathrm{d} g = \frac {\mathrm{d} N _ {i}}{\mathrm{d} g} \frac {\mathrm{d} g}{\mathrm{d} s} \frac {\mathrm{d} s}{\mathrm{d} g} \mathrm{d} g = \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathrm{d} s = - \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} N _ {j} \mathbf {t} \cdot \mathrm{d} \mathbf {x} _ {j}.
The rate of change of the normal can be re-expressed as
Equation 5.1.2-8
\mathrm{dn} = (\textbf {n n} + \textbf {t t}) \cdot \mathrm{dn} = \textbf {t t} \cdot \mathrm{dn} = - \textbf {t n} \cdot \mathrm{dt}.
In this equation n dt can be obtained from Equation 5.1.2-3:
Equation 5.1.2-9
\mathbf {n} \cdot \mathrm{dt} = \mathbf {n} \cdot \mathrm{d} \left(\frac {\mathrm{dx}}{\mathrm{dg}} \Bigg / \left| \frac {\mathrm{dx}}{\mathrm{dg}}\right) \right| = \mathbf {n} \cdot \left(\mathbf {x} _ {i} \frac {\mathrm{d} ^ {2} N _ {i}}{\mathrm{dg} ^ {2}} \mathrm{dg} + \frac {\mathrm{d} N _ {i}}{\mathrm{dg}} \mathrm{dx} _ {i}\right) \Bigg / \left| \frac {\mathrm{dx}}{\mathrm{dg}} \right|
= - \rho_ {n} \mathrm{d} s + \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathbf {n} \cdot \mathrm{d} \mathbf {x} _ {i} = \rho_ {n} N _ {i} \mathbf {t} \cdot \mathrm{d} \mathbf {x} _ {i} + \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathbf {n} \cdot \mathrm{d} \mathbf {x} _ {i},
where use was made of Equation 5.1.2-5 and \rho _ { n } (the segment curvature) is defined as
Equation 5.1.2-10
\rho_ {n} \stackrel {\mathrm{def}} {=} - \mathbf {n} \cdot \frac {\mathrm{d} ^ {2} \mathbf {x}}{\mathrm{d} g ^ {2}} \bigg / \left| \frac {\mathrm{d} \mathbf {x}}{\mathrm{d} g} \right| ^ {2}.
Interface Modeling
For straight segments \rho _ { n } 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:
Equation 5.1.2-11
d \delta h = - \delta \mathbf {x} _ {i} \cdot \left(\mathbf {n} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} N _ {j} \mathbf {t} + \mathbf {t} N _ {i} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n} + \mathbf {t} N _ {i} \rho_ {n} N _ {j} \mathbf {t}\right) \cdot \mathrm{d} \mathbf {x} _ {j}.
This expression is symmetric, as should be expected. The second variation in s can be derived along similar lines:
Equation 5.1.2-12
\mathrm{d} \delta s = - \delta \mathbf {x} _ {i} \cdot \mathbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} g} \mathrm{d} g - N _ {i} \delta \mathbf {x} _ {i} \cdot \mathrm{d} \mathbf {t}.
Note that
Equation 5.1.2-13
d \mathbf {t} = (\mathbf {n n} + \mathbf {t t}) \cdot d \mathbf {t} = \mathbf {n n} \cdot d \mathbf {t}.
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
Equation 5.1.2-14
\mathrm{d} \delta s = \delta \mathbf {x} _ {i} \cdot \left(\mathbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} N _ {j} \mathbf {t} - \mathbf {n} N _ {i} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n} - \mathbf {n} N _ {i} \rho_ {n} N _ {j} \mathbf {t}\right) \cdot \mathrm{d} \mathbf {x} _ {j}.
This expression is nonsymmetric. The second variations of h and s vanish if no slip in the slide plane occurs ( \mathrm { d } s = \delta s = 0 ) :
Tube-tube interface elements
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 h 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
Equation 5.1.2-15
- \mathbf {n} (h + h _ {o}) = \mathbf {x} _ {1} - \mathbf {x}.
Here x is a point on the outer tube that potentially contacts point \mathbf { x } _ { 1 } on the inner tube, as shown in Figure 5.1.2-3.
Figure 5.1.2-3 Tube-tube contact.
text_image
Outer tube Inner tube x t n x₁
In this equation h _ { o } 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
Equation 5.1.2-16
\mathbf {n} (h + h _ {o}) = - N _ {i} (g) \mathbf {x} _ {i}.
The sign reversal as compared to Equation 5.1.2-2 is related to the internal nature of the contact as compared to the external nature for the regular slide line elements. The linearized form of Equation 5.1.2-16 is
\delta \mathbf {n} (h + h _ {o}) + \mathbf {n} \delta h = - \mathbf {t} \delta s - N _ {i} \delta \mathbf {x} _ {i}.
As before, we assume that during contact h = 0. In the contact direction this yields
\delta h = - N _ {i} \mathbf {n} \cdot \delta \mathbf {x} _ {i},
Equation 5.1.2-17
and in the direction along the pipe,
\delta s = - h _ {o} \mathbf {t} \cdot \delta \mathbf {n} - N _ {i} \mathbf {t} \cdot \delta \mathbf {x} _ {i}.
Equation 5.1.2-18
With use of n ¢ t = 0, it readily follows
\mathbf {t} \cdot \delta \mathbf {n} = - \mathbf {n} \cdot \delta \mathbf {t} = - \rho_ {n} N _ {i} \mathbf {t} \cdot \delta \mathbf {x} _ {i} - \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathbf {n} \cdot \delta \mathbf {x} _ {i},
Equation 5.1.2-19
which transforms Equation 5.1.2-18 into
\delta s = - (1 - \rho_ {n} h _ {o}) N _ {i} \mathbf {t} \cdot \delta \mathbf {x} _ {i} + h _ {o} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathbf {n} \cdot \delta \mathbf {x} _ {i},
Equation 5.1.2-20
where \rho _ { n } is the segment curvature as defined by Equation 5.1.2-10. For practical applications
Interface Modeling
\rho _ { n } h _ { o } < < 1 and will be neglected. For the three-dimensional tube-tube interface element, one can define the transverse slip direction \mathbf { s } \ { \stackrel { \mathrm { d e f } } { = } } \ \mathbf { t } \times \mathbf { n } , which yields
Equation 5.1.2-21
\delta s _ {2} = - N _ {i} \mathbf {s} \cdot \delta \mathbf {x} _ {i} (= h _ {o} \mathbf {s} \cdot \delta \mathbf {n}).
The initial stress stiffness terms are again obtained by taking the second variations in h _ { o } , s , , and s _ { 2 } . From Equation 5.1.2-17 follows
Equation 5.1.2-22
\mathrm{d} \delta h = - \delta \mathbf {x} _ {i} \cdot \left(\mathbf {n} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathrm{d} s + N _ {i} \mathrm{d} \mathbf {n}\right),
and with Equation 5.1.2-19 and Equation 5.1.2-21 follows
Equation 5.1.2-23
\mathrm{dn} = \textbf {t t} \cdot \mathrm{dn} + \textbf {s s} \cdot \mathrm{dn} = \left(- \textbf {t} \rho_ {n} N _ {i} \textbf {t} - \textbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \textbf {n} - \textbf {s} h _ {o} ^ {- 1} N _ {i} \textbf {s}\right) \cdot \mathrm{dx} _ {i}.
Substitution of Equation 5.1.2-20 and Equation 5.1.2-23 in Equation 5.1.2-22 yields
Equation 5.1.2-24
\begin{array}{l} \mathrm{d} \delta h = \delta \mathbf {x} _ {i} \cdot \left(\mathbf {n} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} N _ {j} \mathbf {t} - \mathbf {n} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} h _ {o} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n} \right. \\ \left. + \mathbf {t} N _ {i} \rho_ {n} N _ {j} \mathbf {t} + \mathbf {t} N _ {i} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n} + \mathbf {s} N _ {i} h _ {o} ^ {- 1} N _ {j} \mathbf {s}\right) \cdot \mathrm{d} \mathbf {x} _ {j}. \\ \end{array}
This expression is symmetric. In the t-direction the virtual work term has the form
F _ {t} (\delta s + h _ {o} \mathbf {t} \cdot \delta \mathbf {n}) \stackrel {\mathrm{def}} {=} F _ {t} \delta \overline {{s}},
which yields with Equation 5.1.2-18
Equation 5.1.2-25
\delta \overline {{s}} = - N _ {i} \mathbf {t} \cdot \delta \mathbf {x} _ {i}.
For the second variation follows
Equation 5.1.2-26
\mathrm{d} \delta \overline {{s}} = \delta \mathbf {x} _ {i} \cdot \left(- \mathbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathrm{d} s - \mathrm{d} \mathbf {t} N _ {i}\right).
Note that
Equation 5.1.2-27
\begin{array}{l} \mathrm{dt} = \mathrm{d} \left(\frac {\mathrm{dx}}{\mathrm{ds}}\right) = \frac {\mathrm{d} ^ {2} \mathbf {x}}{\mathrm{dg} ^ {2}} \mathrm{dg} / \left| \frac {\mathrm{dx}}{\mathrm{dg}} \right| - \frac {\mathrm{dx}}{\mathrm{dg}} \frac {\mathrm{dx}}{\mathrm{dg}} \cdot \frac {\mathrm{d} ^ {2} \mathbf {x}}{\mathrm{dg} ^ {2}} \mathrm{dg} / \left| \frac {\mathrm{dx}}{\mathrm{dg}} \right| ^ {3} \\ = \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} (\mathbf {I} - \textbf {t t}) \cdot \mathrm{d} \mathbf {x} _ {i} + \frac {\mathrm{d} ^ {2} \mathbf {x}}{\mathrm{d} g ^ {2}} \bigg / \left| \frac {\mathrm{d} \mathbf {x}}{\mathrm{d} g} \right| ^ {2} \cdot (\mathbf {I} - \textbf {t t}) \mathrm{d} s \\ = \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} (\mathbf {n n} + \mathbf {s s}) \cdot \mathrm{d} \mathbf {x} _ {i} - (\rho_ {n} \mathbf {n} + \rho_ {s} \mathbf {s}) \mathrm{d} s, \\ \end{array}
where the transverse segment curvature \rho _ { s } is defined by
Equation 5.1.2-28
\rho_ {s} \stackrel {\mathrm{def}} {=} - \mathbf {s} \cdot \frac {\mathrm{d} ^ {2} \mathbf {x}}{\mathrm{d} g ^ {2}} \Bigg / \left| \frac {\mathrm{d} \mathbf {x}}{\mathrm{d} g} \right| ^ {2}.
In this expression the terms involving s 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
Equation 5.1.2-29
\begin{array}{l} \mathrm{d} \delta \overline {{s}} = \delta \mathbf {x} _ {i} \cdot \left(\mathbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} N _ {j} \mathbf {t} - \mathbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} h _ {o} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n} - \mathbf {s} N _ {i} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {s} \right. \\ \left. - \mathbf {s} N _ {i} \rho_ {s} N _ {j} \mathbf {t} + \mathbf {s} N _ {i} \rho_ {s} h _ {o} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n}\right) \cdot \mathrm{d} \mathbf {x} _ {j}. \\ \end{array}
In the last term \rho _ { s } h _ { o } is very small and, therefore, can be neglected. Also, observe that d±s does not vanish for zero slip, except when h _ { o } = 0 . In the s-direction one simply looks at the first variation in \delta s _ { 2 } :
Equation 5.1.2-30
\mathrm{d} \delta s _ {2} = \delta \mathbf {x} _ {i} \cdot \left(- \mathbf {s} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathrm{d} s - N _ {i} \mathrm{d} \mathbf {s}\right).
Note that
\mathrm{ds} = \mathbf {n} \mathbf {n} \cdot \mathrm{ds} + \mathbf {t} \mathbf {t} \cdot \mathrm{ds} = - \mathbf {n} \mathbf {s} \cdot \mathrm{dn} - \mathbf {t} \mathbf {s} \cdot \mathrm{dt}.
With Equation 5.1.2-20, Equation 5.1.2-23, and Equation 5.1.2-27 follows
\mathrm{d} \mathbf {s} = \left(\mathbf {n} h _ {o} ^ {- 1} N _ {i} \mathbf {s} - \mathbf {t} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathbf {s} - \mathbf {t} \rho_ {s} N _ {i} \mathbf {t} + \mathbf {t} \rho_ {s} h _ {o} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} \mathbf {n}\right) \cdot \mathrm{d} \mathbf {x} _ {i}.
Substituted in Equation 5.1.2-30 this yields
Equation 5.1.2-31
\begin{array}{l} \mathrm{d} \delta s _ {2} = \delta \mathbf {x} _ {i} \cdot \left(\mathbf {s} \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} N _ {j} \mathbf {t} - s \frac {\mathrm{d} N _ {i}}{\mathrm{d} s} h _ {o} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n} + \mathbf {t} N _ {i} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {s} \right. \\ \left. + \mathbf {t} N _ {i} \rho_ {s} N _ {j} \mathbf {t} - \mathbf {t} N _ {i} \rho_ {s} h _ {o} \frac {\mathrm{d} N _ {j}}{\mathrm{d} s} \mathbf {n}\right) \cdot \mathrm{d} \mathbf {x} _ {i}, \\ \end{array}
where use was made of the relation \delta h = 0 . Again, the term involving \rho _ { s } h _ { o } can be neglected. The expression does not vanish for zero slip unless h _ { o } = 0 .
Slide line smoothing
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).
Figure 5.1.2-4 Transition between linear segments.
text_image
X₂ X₃ Xₐ X X₁ X_b X₄
The junction of the two segments is connected by a Hermitian polynomial between the points \mathbf { x } _ { a } and \mathbf { x } _ { b } , which are located on the segments:
\mathbf {x} _ {a} = \alpha \mathbf {x} _ {2} + (1 - \alpha) \mathbf {x} _ {3}, \quad \mathbf {x} _ {b} = \alpha \mathbf {x} _ {4} + (1 - \alpha) \mathbf {x} _ {3}.
The positions of \mathbf { x } _ { a } and \mathbf { x } _ { b } are, hence, determined by the smoothing factor \alpha , which is specified directly by the user in the range 0 \le \alpha \le 0 . 5 .
We choose g as the parameter coordinate on the smoothed segment, with - 1 \leq g \leq 1 . At the extreme values for g the coordinates are \mathbf { x } _ { a } and \mathbf { x } _ { b } , and for the coordinate derivatives we choose
\frac {\mathrm{d} \mathbf {x} _ {a}}{\mathrm{d} g} = \mathbf {x} _ {3} - \mathbf {x} _ {a} = \alpha (\mathbf {x} _ {3} - \mathbf {x} _ {2}), \quad \frac {\mathrm{d} \mathbf {x} _ {a}}{\mathrm{d} g} = \mathbf {x} _ {b} - \mathbf {x} _ {3} = \alpha (\mathbf {x} _ {4} - \mathbf {x} _ {3}).
With the Hermitian interpolation functions we can calculate the position x of a point on the segment as a function of g \dot { . }
\begin{array}{l} \mathbf {x} = \frac {1}{4} (g ^ {3} - 3 g + 2) \mathbf {x} _ {a} + \frac {1}{4} (- g ^ {3} + 3 g + 2) \mathbf {x} _ {b} \\ + \frac {1}{4} (g ^ {3} - g ^ {2} - g + 1) \frac {\mathrm{d} \mathbf {x} _ {a}}{\mathrm{d} g} + \frac {1}{4} (g ^ {3} + g ^ {2} - g - 1) \frac {\mathrm{d} \mathbf {x} _ {b}}{\mathrm{d} g}. \\ \end{array}
Combining the above equations yields
Interface Modeling
Equation 5.1.2-32
\mathbf {x} = \frac {\alpha}{4} (g ^ {2} - 2 g + 1) \mathbf {x} _ {2} + \frac {1}{4} (4 - 2 \alpha (g ^ {2} + 1)) \mathbf {x} _ {3} + \frac {\alpha}{4} (g ^ {2} + 2 g + 1) \mathbf {x} _ {4}.
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.
Figure 5.1.2-5 Transition between quadratic segments.
text_image
X₂ X₃ Xₐ X₁ X X₄ X_b X₅ X₆
It is readily established that in this case
\mathbf {x} _ {a} = (2 \alpha^ {2} - \alpha) \mathbf {x} _ {2} + (- 4 \alpha^ {2} + 4 \alpha) \mathbf {x} _ {3} + (1 - 3 \alpha + 2 \alpha^ {2}) \mathbf {x} _ {4}
\mathbf {x} _ {b} = (2 \alpha^ {2} - \alpha) \mathbf {x} _ {6} + (- 4 \alpha^ {2} + 4 \alpha) \mathbf {x} _ {5} + (1 - 3 \alpha + 2 \alpha^ {2}) \mathbf {x} _ {4}.
For the coordinate derivatives we choose
\frac {\mathrm{d} \mathbf {x} _ {a}}{\mathrm{d} g} = - \alpha \frac {\mathrm{d} \mathbf {x} _ {a}}{\mathrm{d} \alpha} = (\alpha - 4 \alpha^ {2}) \mathbf {x} _ {2} + (- 4 \alpha + 8 \alpha^ {2}) \mathbf {x} _ {3} + (3 \alpha - 4 \alpha^ {2}) \mathbf {x} _ {4}
\frac {\mathrm{d} \mathbf {x} _ {b}}{\mathrm{d} g} = \alpha \frac {\mathrm{d} \mathbf {x} _ {b}}{\mathrm{d} \alpha} = (- \alpha + 4 \alpha^ {2}) \mathbf {x} _ {6} + (4 \alpha - 8 \alpha^ {2}) \mathbf {x} _ {5} + (- 3 \alpha + 4 \alpha^ {2}) \mathbf {x} _ {4}.
As before, ® is specified directly by the user. Substitution of these equations in the Hermite shape functions yields
Equation 5.1.2-33
\mathbf {x} = \frac {1}{4} \big (\alpha [ - 2 \alpha g ^ {3} + (4 \alpha - 1) g ^ {2} + (- 2 \alpha + 2) g - 1 ] \mathbf {x} _ {2}
+ \alpha [ 4 \alpha g ^ {3} + (- 8 \alpha + 4) g ^ {2} + (4 \alpha - 8) g + 4 ] \mathbf {x} _ {3}
+ [ (8 \alpha^ {2} - 6 \alpha) g ^ {2} + (- 6 \alpha + 4) ] \mathbf {x} _ {4}
+ \alpha [ - 4 \alpha g ^ {3} + (- 8 \alpha + 4) g ^ {2} + (- 4 \alpha + 8) g + 4 ] \mathbf {x} _ {5}
+ \alpha [ 2 \alpha g ^ {3} + (+ 4 \alpha - 1) g ^ {2} + (2 \alpha - 2) g - 1 ] \mathbf {x} _ {6}).
If \begin{array} { r } { { \bf x } _ { 3 } = \frac { 1 } { 2 } ( { \bf x } _ { 2 } + { \bf x } _ { 4 } ) } \end{array} and \begin{array} { r } { { \bf x } _ { 5 } = \frac { 1 } { 2 } ( { \bf x } _ { 4 } + { \bf x } _ { 6 } ) } \end{array} , the equations reduce to the same equations as were used for smoothing between linear segments.