김경종
271 lines
21 KiB
Markdown
271 lines
21 KiB
Markdown
<!-- source-page: 771 -->
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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["initial configuration"] --> B["master surface"]
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B --> C["slave surface"]
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C --> D["local tangent plane"]
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D --> E["large deformation"]
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style A fill:#f9f,stroke:#333
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style B fill:#ccf,stroke:#333
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style C fill:#cfc,stroke:#333
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style D fill:#fcc,stroke:#333
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style E fill:#ffc,stroke:#333
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```
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</details>
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Figure 38.1.1–15 Master surface deformation in a small-sliding contact analysis can cause problems with the local tangent planes.
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# Calculating the initial local tangent directions for a two-dimensional surface
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Two-dimensional and standard axisymmetric models have only one local tangent direction, $\mathbf { t } _ { 1 }$ . Abaqus/Standard defines the orientation of this direction by the cross product of the vector into the plane of the model (0., 0., 1.0) and the contact normal vector.
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Models consisting of generalized axisymmetric bodies have a second local tangent direction, $\mathbf { t } _ { 2 } ,$ , to account for the component of slip associated with relative differences in circumferential twist between contacting bodies. The first local tangent direction at any point on the surface is always tangent to the master surface in the local $\pmb { r } - z$ plane. The second local tangent direction is orthogonal to this plane in the local circumferential direction. For more information about generalized axisymmetric models, see “Generalized axisymmetric stress/displacement elements with twist” in “Choosing the element’s dimensionality,” Section 27.1.2.
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# Calculating the initial local tangent directions for a three-dimensional surface
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By default, Abaqus/Standard determines the initial orientation of the two local tangent directions, $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 } .$ , using the following conventions:
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<!-- source-page: 772 -->
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• Finite-sliding, surface-to-surface formulation: The default initial orientations of the two local tangent directions are based on the slave surface normal, using the standard convention for calculating surface tangents (see “Conventions,” Section 1.2.2) with the assumption that the contact normal corresponds to the negative normal to the slave surface.
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• Finite-sliding, node-to-surface formulation: For contact involving a slave surface based on three-dimensional beam-type elements, the first and second local tangent directions are defined along the length of the beam and transverse to the beam, respectively. For contact involving an analytical rigid surface and a slave surface that is not based on three-dimensional beam-type elements, the first local tangent direction is tangential to the cross-section used to generate the analytical rigid surface, and the second local tangent direction is orthogonal to the plane of the cross-section in which the contact occurs.
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In other cases, default initial orientations of the two local tangent directions are calculated by first computing tentative $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ directions. For element-based slave surfaces the tentative directions are based on the slave surface using the standard convention for calculating surface tangents. For node-based slave surfaces the tentative $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ directions are set at each node to coincide with the global $\pmb { x } -$ and y-axes, respectively. Abaqus constructs an orthogonal triad of $\mathbf { t } _ { 1 }$ , $\mathbf { t } _ { 2 } .$ , and (where $\mathbf { n } = \mathbf { t } _ { 1 } \times \mathbf { t } _ { 2 } )$ , then rotates this triad such that becomes aligned with the master surface normal at the tracked point on the master surface.
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• Small-sliding, surface-to-surface formulation: The default initial orientations of the two local tangent directions are based on the slave surface normal, using the standard convention for calculating surface tangents, except for contact involving analytical rigid surfaces, in which case the local tangent directions are based on the master surface normal.
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• Small-sliding, node-to-surface formulation: The default initial orientations of the two local tangent directions are calculated at each point on the master surface based on the master surface normal, using the standard convention for calculating surface tangents.
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# Defining alternative initial local tangent directions for contact pair surfaces
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If the default local tangent directions are not convenient to prescribe an anisotropic friction model or to view contact output, you can define the local tangent directions for three-dimensional contact pair surfaces. You cannot redefine the local tangent directions for the following types of surfaces:
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• Surfaces in a general contact domain
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• Analytical rigid surfaces
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• Two-dimensional surfaces
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You define the local tangent directions by associating an orientation definition (see “Orientations,” Section 2.2.5) with a contact pair surface. You can assign an orientation only to one surface of a contact pair. The surface on which an orientation can be defined is the same surface on which the default orientation would be calculated (see the conventions given previously). For example, an orientation can be defined only on the slave surface in deformable versus deformable finite-sliding contact. If a second orientation is also given, an error message is issued. Therefore, it is not possible to redefine the local tangent directions for finite-sliding contact between a deformable slave surface and an analytical rigid surface.
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<!-- source-page: 773 -->
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An orientation that is defined on a slave surface of a contact pair that is generated from threedimensional truss-type elements or from a list of nodes without rotational degree of freedoms will not be rotated if the slave surface undergoes finite motion. In this case a warning message is issued during input processing.
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# Input File Usage:
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\*CONTACT PAIR, INTERACTION=interaction\_property\_name slave surface name, master surface name, orientation for slave surface slave surface name, master surface name, , orientation for master surface
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# Abaqus/CAE Usage:
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You cannot define alternative local tangent directions for contact pairs in Abaqus/CAE.
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# Evolution of the local tangent directions
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For geometrically nonlinear analyses the local tangent directions rotate with the surface on which these directions were initially calculated or redefined using an orientation definition as described above with the exception that the local tangent direction rotates with the master surface for the small-sliding, surfaceto-surface formulation. These rotated local tangent directions are further rotated to ensure that the normal vector, computed using the cross product of the rotated local tangent directions, corresponds to the normal vector on the master surface when the slave node comes into contact.
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<!-- source-page: 774 -->
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<!-- source-page: 775 -->
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# 38.1.2 CONTACT CONSTRAINT ENFORCEMENT METHODS IN Abaqus/Standard
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Products: Abaqus/Standard Abaqus/CAE
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# References
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• “Defining general contact interactions in Abaqus/Standard,” Section 36.2.1
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• “Defining contact pairs in Abaqus/Standard,” Section 36.3.1
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• “Mechanical contact properties: overview,” Section 37.1.1
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• “Contact pressure-overclosure relationships,” Section 37.1.2
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• \*SURFACE BEHAVIOR
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• \*CONTACT CONTROLS
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• “Defining general contact,” Section 15.13.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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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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• “Defining a contact interaction property,” Section 15.14.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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Contact constraint enforcement methods in Abaqus/Standard:
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• are specified as part of the surface interaction definition;
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• determine how contact constraints imposed by a physical pressure-overclosure relationship (see “Contact pressure-overclosure relationships,” Section 37.1.2) are resolved numerically in an analysis;
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• can either strictly enforce or approximate the physical pressure-overclosure relationships;
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• can be modified to resolve convergence difficulties due to overconstraints; and
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• sometimes utilize Lagrange multiplier degrees of freedom.
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The available constraint enforcement methods for normal contact in Abaqus/Standard are discussed in detail in this section. The frictional constraint enforcement methods in Abaqus/Standard are assigned independently of those for the normal contact constraints and are discussed in “Frictional behavior,”
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Section 37.1.5. The use of Lagrange multipliers in contact calculations is also covered in this section.
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# Available constraint enforcement methods in Abaqus/Standard
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There are three contact constraint enforcement methods available in Abaqus/Standard:
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• The direct method attempts to strictly enforce a given pressure-overclosure behavior per constraint, without approximation or use of augmentation iterations.
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<!-- source-page: 776 -->
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• The penalty method is a stiff approximation of hard contact.
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• The augmented Lagrange method uses the same kind of stiff approximation as the penalty method, but also uses augmentation iterations to improve the accuracy of the approximation.
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The default constraint enforcement method depends on interaction characteristics, as follows:
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• The penalty method is used by default for finite-sliding, surface-to-surface contact (including general contact) if a “hard” pressure-overclosure relationship is in effect.
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• The augmented Lagrange method is used by default for three-dimensional self-contact with nodeto-surface discretization if a “hard” pressure-overclosure relationship is in effect.
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• The direct method is the default in all other cases.
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You should consider the following factors when choosing the contact enforcement method:
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• The direct method must be used for contact pairs with a “softened” pressure-overclosure relationship (see “Contact pressure-overclosure relationships,” Section 37.1.2).
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• The direct method strictly enforces the specified pressure-overclosure behavior consistent with the constraint formulation
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• The penalty or augmented Lagrange constraint enforcement methods sometimes provide more efficient solutions (generally due to reduced calculation costs per iteration and a lower number of overall iterations per analysis) at some (typically small) sacrifice in solution accuracy. See the discussions of the penalty and augmented Lagrange methods below.
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• Overconstraints due to overlapping contact definitions or the combination of contact and other constraint types (see “Overconstraint checks,” Section 35.6.1) should be avoided for directly enforced hard contact.
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# Direct method
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The direct method strictly enforces a given pressure-overclosure behavior for each constraint, without approximation or use of augmentation iterations.
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Input File Usage: Use both of the following options: \*SURFACE INTERACTION, NAME=interaction\_property\_name \*SURFACE BEHAVIOR, DIRECT
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Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Normal Behavior: Constraint enforcement method: Direct (Standard)
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# Direct method for hard pressure-overclosure behavior
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The direct method can be used to strictly enforce a “hard” pressure-overclosure relationship. Lagrange multipliers are always used in this case.
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# Direct method for softened pressure-overclosure relationships
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The direct method is the only method that can be used to enforce “softened” pressure-overclosure relationships. The direct method can be used to model softened contact behavior regardless of the
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<!-- source-page: 777 -->
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type of contact formulation; however, modeling stiff interface behavior with a contact formulation that is prone to overconstraints can be difficult. Lagrange multipliers are used if the slope of the pressure-overclosure curve exceeds 1000 times the underlying element stiffness (as computed by Abaqus/Standard); otherwise, the constraints are enforced without Lagrange multipliers. The usage of Lagrange multipliers, thus, depends on the contact pressure. Softened pressure-overclosure relationships are discussed in more detail in “Contact pressure-overclosure relationships,” Section 37.1.2.
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# Limitations of the direct method
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Because of its strict interpretation of contact constraints, hard contact simulations utilizing the direct enforcement method are susceptible to overconstraint issues. As a result, directly enforced hard contact is not available for contact pairs defined using three-dimensional self-contact with node-to-surface discretization. In this instance you can use an alternate enforcement method or the direct method with a softened pressure-overclosure relationship.
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You may experience similar overconstraint problems with symmetric master-slave contact pairs (see “Using symmetric master-slave contact pairs to improve contact modeling” in “Defining contact pairs in Abaqus/Standard,” Section 36.3.1). Although directly enforced hard contact is the default for these contact pairs, it is recommended that you use an alternate enforcement method or a softened contact relationship.
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Certain second-order element faces do not perform well in directly enforced hard contact relationships. See “Three-dimensional surfaces with second-order faces and a node-to-surface formulation” in “Common difficulties associated with contact modeling in Abaqus/Standard,” Section 39.1.2, for details on this issue.
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# Penalty method
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The penalty method approximates hard pressure-overclosure behavior. With this method the contact force is proportional to the penetration distance, so some degree of penetration will occur. Advantages of the penalty method include:
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• Numerical softening associated with the penalty method can mitigate overconstraint issues and reduce the number of iterations required in an analysis.
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• The penalty method can be implemented such that no Lagrange multipliers are used, which allows for improved solver efficiency.
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# Choosing a penalty method
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Abaqus/Standard offers linear and nonlinear variations of the penalty method. With the linear penalty method the so-called penalty stiffness is constant, so the pressure-overclosure relationship is linear. With the nonlinear penalty method the penalty stiffness increases linearly between regions of constant low initial stiffness and constant high final stiffness, resulting in a nonlinear pressure-overclosure relationship. The default penalty method is linear.
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A comparison of the linear and nonlinear pressure-overclosure relationships with the default settings is shown in Figure 38.1.2–1.
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<!-- source-page: 778 -->
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<details>
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<summary>line</summary>
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| Overclosure | Contact pressure (Nonlinear) | Contact pressure (Linear) |
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| ----------- | ----------------------------- | ------------------------- |
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| 0 | 0 | 0 |
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| e | ~0.1K_lin | ~0.05 |
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| d | ~10K_lin | ~0.25 |
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</details>
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Figure 38.1.2–1 Comparison of linear and nonlinear pressure-overclosure relationships with default settings.
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# Linear penalty method
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When the linear penalty method is used, Abaqus/Standard will, by default, set the penalty stiffness to 10 times a representative underlying element stiffness. You can scale or reassign the penalty stiffness, as discussed in “Modifying a linear penalty stiffness” below. Contact penetrations resulting from the default penalty stiffness will not significantly affect the results in most cases; however, these penetrations can sometimes contribute to some degree of stress inaccuracy (for example, with displacement-controlled loading and a coarse mesh). The linear penalty method is used by default for the finite-sliding, surfaceto-surface contact formulation.
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Input File Usage: Use both of the following options to specify the linear penalty method:
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\*SURFACE INTERACTION, NAME=interaction\_property\_name
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\*SURFACE BEHAVIOR, PENALTY=LINEAR
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Abaqus/CAE Usage: Interaction module: contact property editor: Mechanical→Normal Behavior:
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Constraint enforcement method: Penalty (Standard), Behavior: Linear
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# Nonlinear penalty method
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With the nonlinear penalty method, the pressure-overclosure curve has four distinct regions shown in Figure 38.1.2–2.
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• Inactive contact regime: The contact pressure remains zero for clearances greater than $c _ { 0 }$ . The default setting of $c _ { 0 }$ is zero.
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• Constant initial penalty stiffness regime: The contact pressure varies linearly, with a slope equal to $K _ { i }$ for penetrations (overclosures) in the range $- c _ { 0 }$ to $e .$ The default initial penalty stiffness, $K _ { i } ,$ is equal to the representative underlying element stiffness. The default value of is 1% of a characteristic length computed by Abaqus/Standard to represent a typical facet size.
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<!-- source-page: 779 -->
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<details>
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<summary>line</summary>
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| Point Label | Overclosure | Contact pressure |
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| ----------------- | ----------- | ---------------- |
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| Initial stiffness Kᵢ | C₀ | Kᵢ |
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| Final stiffness K_f | d | K_f |
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</details>
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<details>
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<summary>line</summary>
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| Overclosure | Penalty stiffness |
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| ----------- | ----------------- |
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| C₀ | Kᵢ |
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| e | Kᵢ |
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| d | Kᵢ |
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</details>
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Figure 38.1.2–2 Nonlinear penalty pressure-overclosure relationship.
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• Stiffening regime: The contact pressure varies quadratically for penetrations in the range to $d ,$ while the penalty stiffness increases linearly from $K _ { i }$ to $K _ { f }$ . The default final penalty stiffness, $K _ { f } { \mathrm { : } }$ , is equal to 100 times the representative underlying element stiffness. The default value of is 3% of the same characteristic length used to compute (discussed above).
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• Constant final penalty stiffness regime: The contact pressure varies linearly, with a slope equal to $K _ { f }$ for penetrations greater than $d .$
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The low initial penalty stiffness typically results in better convergence of the Newton iterations and better robustness, while the higher final stiffness keeps the overclosure at an acceptable level as the contact pressure builds up.
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<!-- source-page: 780 -->
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<table><tr><td>Input File Usage:</td><td>Use both of the following options to specify the nonlinear penalty method:*SURFACE INTERACTION, NAME=interaction_property_name*SURFACE BEHAVIOR, PENALTY=NONLINEAR</td></tr></table>
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<table><tr><td>Abaqus/CAE Usage:</td><td>Interaction module: contact property editor: Mechanical→Normal Behavior: Constraint enforcement method: Penalty (Standard), Behavior: Nonlinear</td></tr></table>
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# Modifying the penalty stiffness
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If you are interested in investigating the effects of modifying the penalty stiffness, it is generally recommended that you consider order-of-magnitude changes. Increasing the penalty stiffness above the threshold value discussed above will, by default, introduce Lagrange multipliers.
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# Modifying a linear penalty stiffness
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As part of the surface behavior definition, you can specify the linear penalty stiffness, shift the pressureoverclosure relationship by specifying the clearance at which the contact pressure is zero, or scale the default or specified penalty stiffness by a factor.
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<table><tr><td>Input File Usage:</td><td>To modify the linear penalty behavior in the surface behavior definition:*SURFACE BEHAVIOR, PENALTY=LINEARpenalty stiffness, clearance at zero pressure, factor</td></tr></table>
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<table><tr><td>Abaqus/CAE Usage:</td><td>To modify the linear penalty behavior in the surface behavior definition:Interaction module: contact property editor: Mechanical→Normal Behavior: Constraint enforcement method: Penalty (Standard), Behavior: Linear, Stiffness value: Specify: penalty stiffness, Stiffness scale factor: factor, Clearance at which contact pressure is zero: clearance at zero pressure</td></tr></table>
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# Modifying a nonlinear penalty stiffness
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As part of the surface behavior definition, you can specify the final nonlinear penalty stiffness, shift the pressure-overclosure relationship by specifying the clearance at which the contact pressure is zero, or scale the default or specified penalty stiffness by a factor. In addition, you can control directly the ratio of the initial to the final penalty stiffness, the scale factor, and the ratio that determines and .
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<table><tr><td>Input File Usage:</td><td>To modify the nonlinear penalty behavior in the surface behavior definition:*SURFACE BEHAVIOR, PENALTY=NONLINEARfinal penalty stiffness, clearance at zero pressure, factor, upperquadratic limit scale factor, ratio of initial penalty stiffness over finalpenalty stiffness, lower quadratic limit ratio</td></tr></table>
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<table><tr><td>Abaqus/CAE Usage:</td><td>To modify the nonlinear penalty behavior in the surface behavior definition:Interaction module: contact property editor: Mechanical→NormalBehavior: Constraint enforcement method: Penalty (Standard),Behavior: Nonlinear, Maximum stiffness value: Specify: final</td></tr></table>
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