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A bi-potential contact formulation of orthotropic adhesion between soft bodies

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Abstract

An orthotropic adhesion model is proposed based on the bi-potential method to solve adhesive contact problems with orthotropic interface properties between hyperelastic bodies. The model proposes a straightforward description of interface adhesion with orthotropic adhesion stiffness, whose components are conveniently expressed according to the local coordinate system. Based on this description, a set of extended unilateral and tangential contact laws has been formulated. Furthermore, we use an element-wise scalar parameter \(\beta \) to characterize the strength of interface adhesive bonds, and the effects of damage. Therefore, complete cycles of bonding and de-bonding of adhesive links with the account for orthotropic interface effects can be modelled. The proposed model has been tested on cases involving both tangential and unilateral contact kinematics. The test cases allowed emergence of orthotropic interface effects between elastomer bodies involving hyperelasticity. Meanwhile, the model can be implemented with minimum effort, and provides inspiration for the modelling of adhesive interface effects in areas of applications such as biomechanics.

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Authors and Affiliations

  1. Université Paris-Saclay, University of Evry, LMEE, 91020, Evry, France

    L. B. Hu, Y. Cong, C. Renaud & Z.-Q. Feng

  2. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, China

    Z.-Q. Feng

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  1. L. B. Hu
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  2. Y. Cong
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  3. C. Renaud
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  4. Z.-Q. Feng
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Correspondence to Z.-Q. Feng.

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Appendix

Appendix

To solve the orthotropic adhesive interface law between hyperelastic bodies, a contact algorithm based on bi-potential theory is used. This algorithm, according to its description of contact kinematics, can be attributed to the category of “node-to-segment” approaches and, with regard to the resolution technique that enforces the contact geometry, belongs to the class of augmented Lagrangian methods. Let us refer to the present contact algorithm with “NTS-AL” (meaning “node-to-segment” contact using augmented Lagrangian resolution), and compare it with other established contact algorithms using alternative schemes of contact kinematics and resolution. In this regard, we consider the widely adopted contact patch test introduced by Taylor and Papodopoulos [55] and compare our results with those reported in [40]. The contact patch test investigates the capacity of a contact algorithm to correctly evaluate the normal contact stresses on contact interface, regardless of its discretization.

As depicted in Fig. 15a, the test case under consideration consists of two surfaces discretized with non-conforming meshes put into normal contact. A homogeneous pressure is prescribed on the upper side of elements that define the slave surface. We investigate both the geometrical configuration of the contact surfaces (see Fig. 15b–f), and the normal pressure distribution on the contact interface (see Fig. 16).

Fig. 15
figure 15

Magnified contact interface configuration with and without surface penetration: comparison of the present contact algorithm (“NTS-AL”) to other algorithms based on results reported in [40]. Here, “NTS” refers to “node-to-segment” contact; “AR” to the technique of area regularization; “ME” to moment equilibrium; “AL” to augmented Lagrangian and “VTS” to the “ Virtual-slave-node-To-Segment” approach

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Fig. 16
figure 16

Contact patch test: comparison of several contact algorithms regarding the interface normal stresses. “NTS” refers to “node-to-segment” contact; “AR” to the technique of area regularization; “ME” to moment equilibrium; “AL” to augmented Lagrangian and “VTS” to the “ Virtual-slave-node-To-Segment” approach. The comparison highlights our result (“NTS-AL”) among existing established methods, based on results reported in [40]

Full size image

As has been extensively studied by Zavarise et al. [40] and recalled in Fig. 16, classical NTS contact algorithms, especially those using one-pass approaches introduce significant errors to contact stresses evaluation on non-conforming meshes. To obtain acceptable behaviours using classical NTS description, it is necessary to implement two-pass sequential schemes in conjunction with Lagrangian multiplier method, or, develop improved one-pass schemes, for example the VTS (“virtual-node-to-segment”) method. VTS method extends the classical NTS approach by considering additional virtual slave nodes on the slave surface, leading to augmented slave segments.

In Figs. 15b–f and 16, we confront the presented NTS-AL approach to existing methods, which include one- or two-pass classical NTS approaches with or without contact area regularization (“AR”), and the improved VTS method proposed by the work of Zavarise et al. We observed satisfactory contact geometry in Fig. 15f and the same level of accuracy as VTS method in Fig. 16 which confirm the capacity of augmented Lagrangian methods in enforcing geometrical relations of contact surfaces and improving the computational accuracy.

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Hu, L.B., Cong, Y., Renaud, C. et al. A bi-potential contact formulation of orthotropic adhesion between soft bodies. Comput Mech 69, 931–945 (2022). https://doi.org/10.1007/s00466-021-02122-1

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