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A New Contact Formulation for Large Frictional Sliding and Its Implement in the Explicit Numerical Manifold Method

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Abstract

Modelling discontinuous systems involving large frictional sliding is one of the key requirements for numerical methods in geotechnical engineering. The contact algorithms for most numerical methods in geotechnical engineering is based on the judgement of contact types and the satisfaction of contact conditions by the open–close iteration, in which penalty springs between contacting bodies are added or removed repeatedly. However, the simulations involving large frictional sliding contact are not always convergent, particularly in the cases that contain a large number of contacts. To avoid the judgement of contact types and the open–close iteration, a new contact algorithm, in which the contact force is calculated directly based on the overlapped area of bodies in contact and the contact states, is proposed and implemented in the explicit numerical manifold method (NMM). Stemming from the discretization of Kuhn–Tucker conditions for contact, the equations for calculating contact force are derived and the contributions of contact force to the global iteration equation of explicit NMM are obtained. The new contact algorithm can also be implemented in other numerical methods (FEM, DEM, DDA, etc.) as well. Finally, five numerical examples are investigated to verify the proposed method and illustrate its capability.

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Abbreviations

\({\varvec{\upsigma}}\text{, }\widehat{{\varvec{\upsigma}}}\) :

Cauchy stress and its corotational form

\({\dot{\mathbf{\varepsilon }}},\;\widehat{{{\dot{\mathbf{\varepsilon }}}}}\) :

Rate of deformation and its corotational form

b :

Body force vector

\({\mathbf{u}},{\dot{\mathbf{u}}},{\mathbf{\ddot{u}}}\) :

Displacement, velocity and acceleration vectors

D :

Constitutive equation

R :

Rotation matrix

\({\mathbf{t}}_{\text{c}}^{\left( i \right)}\) :

Contact traction between two bodies

\({\mathbf{p}}_{k}^{\left( i \right)} ,{\mathbf{p}}_{k,j}^{\left( i \right)}\) :

Normal contact force acted on the segment \(c_{k}^{\left( i \right)}\) or sub-segment \(c_{k,j}^{\left( i \right)}\)

\({\mathbf{P}}_{k}^{\left( i \right)} ,{\mathbf{P}}_{k,j}^{\left( i \right)} ,{\mathbf{x}}_{k}^{\left( i \right)} ,{\mathbf{x}}_{k,j}^{\left( i \right)}\) :

Equivalent normal point load of \({\mathbf{p}}_{k}^{\left( i \right)} ,{\mathbf{p}}_{k,j}^{\left( i \right)}\) and their point of force application

\({\mathbf{t}}_{k}^{\left( i \right)} ,{\mathbf{t}}_{k,j}^{\left( i \right)}\) :

Tangential contact force acted on the segment \(c_{k}^{\left( i \right)}\) or sub-segment \(c_{k,j}^{\left( i \right)}\)

\({\mathbf{n}}_{\text{c}}\) :

Equivalent normal direction \({\mathbf{n}}_{\text{c}}\) of the overlapped contact area

M :

Mass matrix

\({\mathbf{f}}_{\text{int}} ,{\mathbf{f}}_{\text{ext}} ,{\mathbf{f}}_{\lambda } ,{\mathbf{f}}_{c}\) :

Internal nodal force, external nodal force, boundary constraint nodal force and contact nodal forces

\(\rho\) :

Density of material

\(g_{\text{N}}\) :

Penetration of two contact bodies

\(p_{N} ,t_{T}\) :

Normal and tangential components of \({\mathbf{t}}_{\text{c}}^{\left( i \right)}\)

\(\alpha_{\text{N}} ,\alpha_{\text{T}}\) :

Normal and tangential penalty coefficient

M I :

Mathematical cover (MC) patch I

P i :

Physical cover (PC) patch i

\(w_{i}\) :

Weight function of PC patch Pi

\(c_{k}^{\left( i \right)}\) :

Contact segment k of body i

\(c_{k,j}^{\left( i \right)}\) :

Sub segment j of \(c_{k}^{\left( i \right)}\)

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Acknowledgements

The work reported in this paper has received financial support from National Key R&D Program of China (No. 2018YFC1505005), Fundamental Research Funds for the Central Universities (No. 2042018kf0028), and Natural Science Foundation of Hubei Province (No. 2016CFA083). These supports are gratefully acknowledged.

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Authors and Affiliations

  1. Hubei Provincial Key Laboratory of Safety for Geotechnical and Structural Engineering, School of Civil Engineering, Wuhan University, Wuhan, 430072, People’s Republic of China

    Wei Wei & Qinghui Jiang

  2. Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan, 430072, People’s Republic of China

    Wei Wei & Qinghui Jiang

  3. Department of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, M5S 1A4, Canada

    Qi Zhao & Giovanni Grasselli

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  1. Wei Wei
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  2. Qi Zhao
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  3. Qinghui Jiang
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Correspondence to Qinghui Jiang.

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Wei, W., Zhao, Q., Jiang, Q. et al. A New Contact Formulation for Large Frictional Sliding and Its Implement in the Explicit Numerical Manifold Method. Rock Mech Rock Eng 53, 435–451 (2020). https://doi.org/10.1007/s00603-019-01914-5

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