Computing pointwise contact between bodies: a class of formulations based on master–master approach

Abstract

In the context of pointwise contact interaction between bodies, a formulation based on surface-to-surface description (master–master) is employed. This leads to a four-variable local contact problem, which solution is associated with general material points on contact surfaces, where contact mechanical action-reaction are represented. We propose here a methodology that permits, according to necessity, a selective dimension reduction of this local contact problem. Thus, the formulation includes curve-to-curve, point-to-surface, curve-to-surface or other contact descriptions as particular degenerations of the surface-to-surface approach. This is done by assuming convective coordinates in the original local contact problem. An operator for performing the so-called “local contact problem degeneration” is presented. It modifies automatically the dimension of the local contact problem and related requirements for its solution. The proposed method is particularly useful for handling singularity scenarios. It also creates a possibility for representing conformal contact by pointwise actions on a non-uniqueness scenario. We present applications and examples that demonstrate benefits for beam-to-beam contact. Ideas and developments, however, are general and may be applied to other geometries of contacting bodies.

Appendix: Surface parameterization of a beam with super elliptical cross section

The here presented parameterization was taken from [26], where the reader finds more detailed explanations. Here it is presented for self-containing of present work.

Let ${\mathbf{x}}$ represent the beam axis position. Vector ${\mathbf{a}}$ is orthogonal to ${\mathbf{x}}$ at each beam cross section and connects the beam axis to the cross-section boundary. It is proposed the following parameterization for a surface ${\Gamma }$:

$$ {\Gamma }\left( {{\upzeta },{\uptheta }} \right) = {\mathbf{x}}\left( {\upzeta } \right) + {\mathbf{a}}\left( {{\upzeta },{\uptheta }} \right), $$

(59)

where ${\upzeta }$ describes a convective coordinate along the beam axis direction and ${\uptheta }$ describes a convective coordinate along circumferential direction in the plane of the cross section. The vector ${\mathbf{a}}\left( {{\upzeta },{\uptheta }} \right)$ describes the cross-section shape in its local orientation xy, as seen in Fig. 21. We employed a linear interpolation for beam axis, between two extreme nodes located at ${\mathbf{x}}_{\text{A}}$ and ${\mathbf{x}}_{\text{B}}$, such that:

$$ {\mathbf{x}}\left( {\upzeta } \right) = \frac{{\left( {1 - {\upzeta }} \right) }}{2}{\mathbf{x}}_{\text{A}} + \frac{{\left( {1 + {\upzeta }} \right) }}{2}{\mathbf{x}}_{\text{B}} , $$

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Fig. 21

Parameterization of the beam a axis, b cross section (taken from [26, 27])

Cross-sections are chosen to have super elliptical shapes, described by

$$ \left| {\frac{{\text{x}}}{{\text{a}}}} \right|^{\text{n}} + \left| {\frac{{\text{y}}}{{\text{b}}}} \right|^{\text{n}} = 1, $$

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where ${\text{a}}$ and ${\text{b}}$ are semi-axis. The choice for ${\text{a}}$, ${\text{b}}$ and the exponent ${\text{n}}$ makes this a versatile choice to recover circular, elliptical or almost-rectangular cross-sections. Figure 22 shows some examples.

Fig. 22

Examples for super elliptical cross sections with a = 2b and a n = 2, b n = 4, c n = 15 (taken from [26])

At reference configuration, one may describe each cross section by:

$$ {\mathbf{a}}^{{\mathbf{r}}} \left( {\uptheta } \right) = \left( {{\text{a}}\phi \left( {\uptheta } \right)\cos {\uptheta },{\text{b}}\phi \left( {\uptheta } \right){\text{sin}}\uptheta} \right), $$

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where

$$ \phi \left( {\uptheta } \right) = \frac{1}{{\sqrt[{\text{n}}]{{\left| {\sin {\uptheta }} \right|^{\text{n}} + \left| {\cos {\uptheta }} \right|^{\text{n}} }}}}. $$

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With that, the final form for ${\Gamma }\left( {{\upzeta },{\uptheta }} \right)$ is expressed including ${\mathbf{Q}}_{\text{A}}$ and ${\mathbf{Q}}_{\text{B}}$ as the rotation tensors for each beam extreme node:

$$ {\Gamma }\left( {{\upzeta },{\uptheta }} \right) = {\text{N}}_{\text{A}} \left( {\upzeta } \right)\left[ {{\mathbf{x}}_{\text{A}} + {\mathbf{Q}}_{\text{A}} {\mathbf{a}}^{{\mathbf{r}}} \left( {\uptheta } \right)} \right] + {\text{N}}_{\text{B}} \left( {\upzeta } \right)\left[ {{\mathbf{x}}_{\text{B}} + {\mathbf{Q}}_{\text{B}} {\mathbf{a}}^{{\mathbf{r}}} \left( {\uptheta } \right)} \right]. $$

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