김경종
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23 KiB
Markdown
307 lines
23 KiB
Markdown
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which small-sliding or infinitesimal-sliding assumptions would be preferred, the contact pair algorithm should be used (see “Contact formulations for contact pairs in Abaqus/Explicit,” Section 38.2.2).
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Abaqus/Explicit is designed to simulate highly nonlinear events or processes. Because it is possible for a node on one surface to contact any of the facets on the opposite surface, Abaqus/Explicit must use sophisticated search algorithms for tracking the motions of the surfaces. The finite-sliding contact search algorithm is designed to be robust, yet computationally efficient. This algorithm assumes that the incremental relative tangential motion between surfaces does not significantly exceed the dimensions of the master surface facets, but there is no limit to the overall relative motion between surfaces. It is rare for the incremental motion to exceed the facet size because of the small time increment used in explicit dynamic analyses. In cases involving relative surface velocities that exceed material wave speeds it may be necessary to reduce the time increment.
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The contact search algorithm uses a global search when a contact interaction is first introduced, and a hierarchical global/local search algorithm is used thereafter. No user control of the search algorithm is needed.
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# Local tangent directions for contact
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Local tangent directions for contact provide a reference frame for select general contact output variables in Abaqus/Explicit (see “Defining general contact interactions in Abaqus/Explicit,” Section 36.4.1). These local tangent directions are separate from local coordinate systems associated with user subroutines VFRICTION and VUINTERACTION. Abaqus/Explicit establishes and updates the orientation of the first local contact tangent direction, $\mathbf { t } _ { 1 }$ , at slave and edge nodes according to the logic described below for different contact formulation types within general contact. The orientation of the second local tangent direction, $\mathbf { t } _ { 2 } ,$ is found as the cross product of the contact normal direction, with $\mathbf { t } _ { 1 }$ . A change in the predominant contact formulation type that is active at a node may lead to a sudden change in the local tangent directions.
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• Finite-sliding, node-to-surface formulation for non-analytical surfaces: The $\mathbf { t } _ { 1 }$ -direction is initialized at a slave node upon first contact using the standard convention for calculating a first local surface tangent direction (see “Conventions,” Section 1.2.2). In subsequent increments, if the slave node belongs to an element-based surface, the $\mathbf { t } _ { 1 }$ -direction rotates with the slave surface for geometrically nonlinear analyses; otherwise, the standard convention is used.
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• Finite-sliding, node-to-surface formulation with an analytical surface: The $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n } }$ for contact is initialized at a slave node upon first contact to be aligned with the convention for the $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n } }$ of the analytical rigid surface discussed in “Analytical rigid surface definition,” Section 2.3.4 at the point of contact. In subsequent increments, the $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n } }$ for contact at a slave node will evolve such that it continues to be aligned with the $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n } }$ of the analytical rigid surface at the current point of contact.
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• Finite-sliding, edge-to-edge formulation: The $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n } }$ for an edge-to-edge contact constraint is initialized upon first contact to be in the axial direction of one of the edges involved in the contact and will evolve to remain aligned with the axial direction of this edge until a local transition to another edge occurs, and then the axial direction of that edge will be adopted as the $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n } }$ .
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Local tangent directions associated with slave nodes will often differ across a contact interface. For example, respective local $\mathbf { t } _ { 1 } { \mathrm { - d i r e c t i o n s } }$ (CTANDIR1) on opposite sides of an interface will evolve differently if surface rotations across the interface are not the same. The respective local $\mathbf { t } _ { 2 } { \mathrm { - d i r e c t i o n s } }$ (CTANDIR2) on opposite sides of an interface are typically in opposing directions initially, due to slave nodes on opposite sides of an interface having opposing contact normal directions.
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# 38.2.2 CONTACT FORMULATIONS FOR CONTACT PAIRS IN Abaqus/Explicit
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Products: Abaqus/Explicit Abaqus/CAE
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# References
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• “Surfaces: overview,” Section 2.3.1
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• “Defining contact pairs in Abaqus/Explicit,” Section 36.5.1
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• \*CONTACT PAIR
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• “Defining surface-to-surface contact,” Section 15.13.7 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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The contact formulation for the contact pair algorithm in Abaqus/Explicit includes:
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• the contact surface weighting (balanced or pure master-slave); and
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• the sliding formulation (finite, small, or infinitesimal).
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You can also specify the method that is used to enforce contact constraints in the contact pair; these methods are discussed in “Contact constraint enforcement methods in Abaqus/Explicit,” Section 38.2.3.
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# Contact surface weighting
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Both the pure master-slave and the balanced master-slave contact algorithms are available in Abaqus/Explicit. By default, Abaqus/Explicit will decide which algorithm to use for any given contact pair based on the nature of the two surfaces forming the contact pair and whether kinematic or penalty enforcement of contact constraints is used. You can override the defaults in some cases.
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# Default choices for the contact pair weighting
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Abaqus/Explicit uses the pure master-slave, kinematic contact algorithm, by default, in the following situations (the first surface in each situation listed is designated the master surface):
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• when a rigid surface contacts a deformable surface;
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• when an element-based surface contacts a node-based surface; or
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• when a surface based on continuum elements contacts a surface based on shell or membrane elements.
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By default, Abaqus/Explicit uses the balanced master-slave, kinematic contact algorithm in the following situations:
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• when a single surface contacts itself (referred to as self-contact or single-surface contact); or
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• when two deformable surfaces that are meshed with similar elements (i.e., either both surfaces have shells or membranes or both have continuum elements) contact each other.
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<!-- source-page: 804 -->
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If the penalty contact algorithm is specified, Abaqus/Explicit uses pure master-slave weighting, by default, in the following situations (the first surface in each situation listed is designated the master surface):
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• when an analytical rigid surface contacts a deformable surface; or
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• when an analytical rigid surface or an element-based surface contacts a node-based surface.
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If the penalty contact algorithm is specified, Abaqus/Explicit chooses balanced master-slave weighting, by default, in the following situations:
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• when a single surface contacts itself (referred to as self-contact or single-surface contact); or
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• when two element-based surfaces contact each other.
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Balanced master-slave weighting means that the corrections produced by both sets of contact calculations are weighted equally.
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# Modifying the default choices for the contact pair weighting
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When the kinematic contact method is chosen, you can override the default contact pair weighting only when two separate deformable element-based surfaces are contacting each other, which corresponds to the last situation in each list for kinematic contact given in the previous section.
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The following aspects should be considered when deciding whether or not to override the default choice. First, the balanced master-slave contact algorithm requires more computational time, but it is typically more accurate. Second, when the densities differ by orders of magnitude, the less dense body should be a pure slave surface. Contact-induced noise can occur if a surface on a much denser body is at all weighted as a slave surface. Finally, to avoid significant penetration for hard contact, the surface with the finer mesh should not be the master surface in the pure master-slave contact pair.
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When the penalty contact method is chosen, you can choose to specify a pure master-slave weighting to reduce computational time. When two originally flat surfaces contact one another, a more uniform penetration distance distribution (and consequently pressure distribution) may result with pure masterslave weighting as compared to balanced master-slave weighting. This can be particularly evident if the mesh densities of the contacting surfaces differ significantly—with balanced weighting the contact penetrations will be smaller near the nodes of the coarsely meshed surface. However, balanced masterslave weighting provides better enforcement of contact constraints in most cases.
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You define a weighting factor, f, to specify the master-slave weighting. Set f=1.0 to designate the first surface in the contact pair as the master surface and the second surface as the slave surface. Set $\scriptstyle { f = 0 . 0 }$ to designate the first surface in the contact pair as the slave surface and the second surface as the master surface. Specifying any value of f between 0 and 1.0 invokes the balanced master-slave contact algorithm. When $\scriptstyle f = 0 . 5$ , which is the default for balanced master-slave contact pairs, Abaqus/Explicit weights each set of corrections equally. In contrast, Abaqus/Standard uses a pure master-slave contact algorithm; the slave surface must always be given first, as in the $\scriptstyle { f = 0 . 0 }$ case above.
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Input File Usage: \*CONTACT PAIR, WEIGHT=f
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Abaqus/CAE Usage: Interaction module: interaction editor: Weighting factor Specify f
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<!-- source-page: 805 -->
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In Abaqus/Explicit there are three approaches to account for the relative motion of the two surfaces forming a contact pair:
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• finite sliding, which is the most general and allows any arbitrary motion of the surfaces;
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• small sliding, which assumes that although two bodies may undergo large motions, there will be relatively little sliding of one surface along the other; or
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• infinitesimal sliding and rotation, which assumes that both the relative motion of the surfaces and the absolute motion of the contacting bodies are small.
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The small-sliding and infinitesimal-sliding formulations cannot be used for contact pairs using the penalty contact algorithm or involving self-contact or analytical rigid surfaces.
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# Using the finite-sliding formulation
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The finite-sliding formulation allows for arbitrary separation, sliding, and rotation of the surfaces. Abaqus/Explicit uses this formulation by default. Only the finite-sliding approach is available for self-contact or contact involving analytical rigid surfaces.
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Input File Usage: \*CONTACT PAIR
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Abaqus/CAE Usage: Interaction module: interaction editor: Sliding formulation: Finite sliding
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# Example
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The following input defines finite-sliding contact between the surfaces ASURF and BSURF, shown in Figure 38.2.2–1, with ASURF acting as the slave surface:
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```txt
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* SURFACE, NAME=ASURF
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ESETA,
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* SURFACE, NAME=BSURF
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ESETB,
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* CONTACT PAIR, INTERACTION=PAIR1, WEIGHT=0.0
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ASURF, BSURF
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* SURFACE INTERACTION, NAME=PAIR1
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```
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In the example shown in Figure 38.2.2–1 slave node 101 may come into contact anywhere along the master surface BSURF. While in contact, it is constrained to slide along BSURF, irrespective of the orientation and deformation of this surface. This behavior is possible because Abaqus/Explicit tracks the position of node 101 relative to the master surface BSURF as the bodies deform. Figure 38.2.2–2 shows the possible evolution of the contact between node 101 and its master surface BSURF. Node 101 is in contact with the element face with end nodes 201 and 202 at time $t _ { 1 }$ . The load transfer at this time occurs between node 101 and nodes 201 and 202 only. Later on, at time $t _ { 2 }$ , node 101 may find itself in contact with the element face with end nodes 501 and 502. Then the load transfer will occur between node 101 and nodes 501 and 502.
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<!-- source-page: 806 -->
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<details>
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<summary>text_image</summary>
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ESETB
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201
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202
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501
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502
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BSURF
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</details>
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<details>
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<summary>text_image</summary>
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ESETA
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101 102 103
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ASURF
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</details>
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Figure 38.2.2–1 Contacting bodies.
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<details>
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<summary>line</summary>
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| Point | Value |
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|-------|-------|
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| t = t₁ | 201 |
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| t = t₂ | 502 |
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| t = 0 | 101 |
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| t = t₁ | 202 |
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| t = t₂ | 501 |
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</details>
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Figure 38.2.2–2 Trajectory of node 101 in finite-sliding contact.
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Finite sliding in a geometrically linear analysis
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Finite-sliding simulations usually include nonlinear geometric effects because such simulations generally involve large deformations and large rotations. However, it is also possible to use the finite-sliding formulation in a geometrically linear analysis (see “Geometric nonlinearity” in “General and linear perturbation procedures,” Section 6.1.3). The load transfer paths between the surfaces and the contact direction are updated in finite-sliding, geometrically linear analysis. This capability is useful for analyzing finite sliding between two stiff bodies that do not undergo large rotations.
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# Using the small-sliding formulation
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For a large class of contact problems the general tracking of the finite-sliding formulation is unnecessary, even though geometric nonlinearity must be considered. Abaqus/Explicit provides a small-sliding
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<!-- source-page: 807 -->
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contact formulation for such problems. This formulation assumes that the surfaces may undergo arbitrarily large rotations but that a slave node will interact with the same local area of the master surface throughout the analysis. Contact pairs that use the small-sliding formulation must be defined in the first step of the simulation, although they may remain active after the first step.
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A large-displacement formulation (the default) should be used for the step in which the small-sliding contact formulation should be used.
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In a small-sliding analysis every slave node interacts with its own local tangent plane on the master surface (see Figure 38.2.2–3). The slave node is constrained not to penetrate this local tangent plane. Each local tangent plane, which is a line in two dimensions, is defined by an anchor point, $\mathbf { X } _ { 0 }$ , on the master surface and an orientation vector at the anchor point (see Figure 38.2.2–3).
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["1"] --> B["2"]
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B --> C["3"]
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C --> D["4"]
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D --> E["5"]
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F["102"] --> G["103"]
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H["104"] --> I["103"]
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J["1"] --> K["N2"]
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K --> L["X0"]
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M["N3"] --> N["N(X0)"]
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O["N4"] --> P["local tangent plane"]
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Q["slave surface"] --> R["102"]
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S["master surface"] --> T["X0"]
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```
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</details>
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Figure 38.2.2–3 Definition of the anchor point and local tangent plane for node 103.
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Having a local tangent plane for each slave node means that for the small-sliding formulation Abaqus/Explicit does not have to monitor slave nodes for possible contact along the entire master surface. Therefore, small-sliding contact is less expensive computationally than finite-sliding contact. The cost savings are most dramatic in three-dimensional contact problems.
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When the balanced master-slave contact algorithm is invoked with the small-sliding formulation, anchor points and tangent planes will be computed for both surfaces.
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Input File Usage: Use both of the following options:
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\*STEP, NLGEOM=YES
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\*CONTACT PAIR, SMALL SLIDING
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<!-- source-page: 808 -->
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For example, the following options define small-sliding contact between the two bodies shown in Figure 38.2.2–1:
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```prolog
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*STEP, NLGEOM=YES
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...
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*SURFACE, NAME=ASURF
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ESETA,
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*SURFACE, NAME=BSURF
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ESETB,
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*CONTACT PAIR, SMALL SLIDING, WEIGHT=0.0
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ASURF, BSURF
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```
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Abaqus/CAE Usage: Interaction module: interaction editor: Sliding formulation: Small sliding Step module: step editor: Nlgeom: On
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# Anchor point and tangent plane definition
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The anchor point and the tangent plane orientation are chosen before the analysis starts using the initial configuration of the model. The anchor point and the tangent plane orientation remain fixed with respect to the master surface facet for all steps in which the contact pair is active. No contact constraints are enforced for slave nodes whose nearest point lies on the free perimeter of the master surface in the original configuration and that do not project onto any master surface facet.
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Abaqus/Explicit chooses the anchor point as the nearest point on the master surface. The orientation of the tangent plane is calculated by default from the normals at the master surface nodes, or you can specify it directly.
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• Master surface normals: The first step in defining the tangent plane orientation is to construct the unit normal vectors at each node of the master surface. Abaqus/Explicit forms these nodal normals by averaging the normals of the element faces making up the master surface; only the element faces in the surface definition will contribute to the nodal normals. The tangent plane orientation is then calculated from the master surface nodal normals and the element shape functions at the anchor point.
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Figure 38.2.2–3 shows the nodal unit normals for a master surface, the anchor point , and the local tangent plane associated with slave node 103. Abaqus/Explicit uses the closest point on the master surface as the anchor point. is the contact direction for slave node 103 and defines the orientation of the local tangent plane. In this example, as in many cases, the local tangent plane is only an approximation of the actual mesh geometry.
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• Master surface normals at symmetry planes: Sometimes the master surface normal and the local tangent plane that Abaqus/Explicit calculates are not suitable for the desired analysis. The most common situation where unsuitable surface normals are calculated occurs when a curved master surface ends at a symmetry plane and the boundary conditions have been specified in direct format rather than in symmetry “type” format (XSYMM, YSYMM, or ZSYMM—see “Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1). In this case the correct normals should be in the symmetry plane; however, because the surface facets that abut the symmetry plane usually form an angle with the plane, the normal will project away from the
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<!-- source-page: 809 -->
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symmetry plane. The effect of this behavior can be that a slave node does not project onto any master surface facet (the slave node is said not to “intersect” the master surface). No contact constraints will be enforced for such slave nodes. However, if symmetry “type” format boundary conditions are specified, contact constraints will be enforced as described below. The finite-sliding formulations use no special treatment for master surfaces ending at a symmetry plane.
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Figure 38.2.2–4 shows two concentric cylinders that contact each other; the inner cylinder is chosen as the master surface CSURF, and a half-symmetry model is used. Since Abaqus/Explicit calculates the nodal normals from the approximate, finite element model, the nodal normal $\mathbf { N _ { 1 } }$ does not point along the symmetry plane, which means that slave node 100 has no anchor point within the perimeter of the master surface. Whether or not contact is enforced for node 100 depends on how the symmetry boundary condition is specified. If the individual components are specified rather than a symmetry “type” boundary condition, slave node 100 will be free to penetrate the master surface. If the symmetry “type” format is used, the master normal at the node on the symmetry plane will be corrected to lie along the symmetry plane and contact will be enforced on the tangent plane as shown in Figure 38.2.2–5. Defining a YSYMM “type” boundary condition at node 1 to specify the symmetry plane will allow slave node 100 to see the master surface CSURF.
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<details>
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<summary>text_image</summary>
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master surface CSURF
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slave surface DSURF
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symmetry plane
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N₁
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1
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100
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y
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x
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</details>
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Figure 38.2.2–4 Master surface normal at node 1 in a small-sliding model of concentric cylinders. With the default $\mathbf { N _ { 1 } }$ slave node 100 will never contact CSURF.
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• Modifying the local tangent plane orientation: In some cases the contact direction, $\bf N ( X _ { 0 } )$ , defined from the master surface averaged normals will not define the contact surface accurately. The most common example of this is a circular surface meshed with nonuniform length facets. Figure 38.2.2–6 shows how the averaged master normals will not be oriented correctly in the radial direction. In this case you should specify the contact direction directly for each slave node by defining spatially varying initial clearances (see “Specifying initial clearance values precisely” in “Adjusting initial surface positions and specifying initial clearances for contact pairs in Abaqus/Explicit,” Section 36.5.4). The location of the anchor point is not affected by reorienting the tangent plane using an initial clearance definition.
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<!-- source-page: 810 -->
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<details>
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<summary>text_image</summary>
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master surface CSURF
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slave surface DSURF
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N₁
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1
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100
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y
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x
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tangent plane
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</details>
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Figure 38.2.2–5 The modified master surface normal at node 1 of CSURF now allows slave node 100 to contact CSURF.
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<details>
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<summary>text_image</summary>
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averaged master normal
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actual surface
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2
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3
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4
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master surface
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1
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5
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</details>
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Figure 38.2.2–6 Poorly oriented averaged master surface normals for an irregularly meshed circular surface.
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# Local tangent plane rotation
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The local tangent plane is always orthogonal to the contact direction. The contact direction is taken as the interpolated normal of the master surface at the anchor point, $\bf N ( X _ { 0 } )$ , or as the direction specified with a spatially varying clearance definition (see “Specifying initial clearance values precisely” in “Adjusting initial surface positions and specifying initial clearances for contact pairs in Abaqus/Explicit,” Section 36.5.4). Once the contact direction has been defined, the orientation of the
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