A concise frictional contact formulation based on surface potentials and isogeometric discretization

Abstract

This work presents a concise theoretical and computational framework for the finite element formulation of frictional contact problems with arbitrarily large deformation and sliding. The aim of this work is to extend the contact theory based on surface potentials (Sauer and De Lorenzis in Comput Methods Appl Mech Eng 253:369–395, 2013) to account for friction. Coulomb friction under isothermal conditions is considered here. For a consistent friction formulation, we start with the first and second laws of thermodynamics and derive the governing equations at the contact interface. A so-called interacting gap can then be defined as a kinematic variable unifying both sliding/sticking and normal/tangential contact. A variational principle for the frictional system can then be formulated based on a purely kinematical constraint. The direct elimination approach applied to the tangential part of this constraint leads to the so-called moving friction cone approach of Wriggers and Haraldsson (Commun Numer Methods Eng 19:285–295, 2003). Compared with existing friction formulations, our approach reduces the theoretical and computational complexity. Several numerical examples are presented to demonstrate the accuracy and robustness of the proposed friction formulation.

Appendices

A Linearization of the kinematical constraint

Given the current position of $\varvec{x}_k$ and $\varvec{x}_{\ell }({\hat{\varvec{\xi }}}^n)$, the sliding point $\varvec{\xi }_\mathrm {m}$ is determined by solving nonlinear Eq. (43) with the Newton–Raphson method. Accordingly, the Taylor series of $f_\alpha (\xi ^\beta )$ about the point $\xi ^\beta + \Delta \xi ^\beta$ is given by

$$\begin{aligned} f_\alpha (\xi ^\beta + \Delta \xi ^\beta ) \approx f_\alpha (\xi ^\beta ) + \displaystyle \frac{\partial {f_\alpha }}{\partial {\xi ^\beta }}\,\Delta \xi ^\beta ~. \end{aligned}$$

(65)

With this, the increment $\Delta \xi ^\beta$ for the iterative procedure is determined from setting $f_\alpha (\xi ^\beta + \Delta \xi ^\beta )=0$, giving

$$\begin{aligned} \Delta \xi ^\beta = c^{\alpha \beta }\,f_\alpha (\xi ^\beta ), \end{aligned}$$

(66)

where $c^{\alpha \beta }$ are the components of the matrix

$$\begin{aligned}{}[c^{\alpha \beta }] = \displaystyle \left[ \frac{\partial {f_\alpha }}{\partial {\xi ^\beta }}\right] ^{-1}~. \end{aligned}$$

(67)

Here, following from Eq. (43), we have

$$\begin{aligned} \displaystyle \frac{\partial {f_\alpha }}{\partial {\xi ^\beta }} = -\varvec{c}_\alpha \cdot \varvec{a}_\beta + (\varvec{g}- \varvec{g}^\mathrm {max}_{\tau })\cdot \varvec{a}_{\alpha ,\beta } - \varvec{d}_\alpha ^\gamma \cdot \varvec{a}_{\gamma ,\beta }, \end{aligned}$$

(68)

where we have denoted

$$\begin{aligned} \varvec{c}_\alpha:= & {} \varvec{a}_\alpha - \mu \,\mathrm {sign}\,(g_\mathrm {n})\tau _\alpha \,\varvec{n},\nonumber \\ \varvec{d}_\alpha ^\beta:= & {} \displaystyle \mu \,\frac{\Vert \varvec{g}_{\mathrm {n}}\Vert }{\Vert \hat{\varvec{g}}^n_{\tau }\Vert }(\delta _\alpha ^\beta - \tau _\alpha ^\beta )\,\hat{\varvec{g}}_{\mathrm {n}}^n - \mu \,\mathrm {sign}\,(g_\mathrm {n})\,\tau _\alpha \,\varvec{n}\,g^\beta , \end{aligned}$$

(69)

with $g_\mathrm {n}:=\varvec{g}_\mathrm {e}\cdot \varvec{n}$, $g^\alpha :=\varvec{g}_\mathrm {e}\cdot \varvec{a}^\alpha$, $\tau _\alpha :=\varvec{\tau }\cdot \varvec{a}_\alpha$, $\tau _{\alpha \beta }:=\tau _\alpha \,\tau _\beta$, and $\tau _{\alpha }^{\beta }:=\tau _{\alpha \gamma }\,a^{\gamma \beta }$.

B Tangent matrices

The tangent matrices for the full-pass algorithm follow from the linearization of Eq. (49). In general, we have

$$\begin{aligned} \Delta \delta \Pi _\mathrm {c}= \displaystyle \int _{\partial \mathcal {B}_{0k}} (\Delta \varvec{T}\cdot \delta \hat{\varvec{g}} +\varvec{T}\cdot \Delta \delta \hat{\varvec{g}} ) \,\mathrm {d}A, \end{aligned}$$

(70)

which includes both sticking and sliding. However, when sticking occurs, $\hat{\varvec{g}}$ becomes $\hat{\varvec{g}}{^n}$ since $\omega =0$ in Eq. (35). Equation (70) then reduces to

$$\begin{aligned} \Delta \delta \Pi _\mathrm {c}&= \displaystyle \delta \mathbf {x}_e^\mathrm {T}\, \mathbf {k}_{kk} \,\Delta \mathbf {x}_e + \delta \mathbf {x}_e^\mathrm {T}\, \mathbf {k}_{k\bar{\ell }} \,\Delta \mathbf {x}_{\bar{e}} + \delta \mathbf {x}_{\bar{e}}^\mathrm {T}\, \mathbf {k}_{\bar{\ell }k} \,\Delta \mathbf {x}_{e} \nonumber \\&\quad + \delta \mathbf {x}_{\bar{e}}^\mathrm {T}\, \mathbf {k}_{\bar{\ell }\bar{\ell }} \,\Delta \mathbf {x}_{\bar{e}}, \end{aligned}$$

(71)

where $\bar{e}\in \mathcal {E}_\ell$ denotes the master elements that contain the previous interacting point $\hat{\varvec{\xi }}{^n}$, and

$$\begin{aligned} \mathbf {k}_{kk}:= & {} \displaystyle \int _{\partial \mathcal {B}_{0k}} \mathbf {N}_e^T\,\epsilon \, \mathbf {N}_e \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{k\bar{\ell }}:= & {} \displaystyle -\int _{\partial \mathcal {B}_{0k}} \mathbf {N}_e^T\,\epsilon \, \mathbf {N}_{\bar{e}}(\hat{\varvec{\xi }}{^n}) \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{\bar{\ell } k}:= & {} \displaystyle -\int _{\partial \mathcal {B}_{0k}} \mathbf {N}_{\bar{e}}^T(\hat{\varvec{\xi }}{^n}) \,\epsilon \, \mathbf {N}_{{e}} \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{\bar{\ell } \bar{\ell }}:= & {} \displaystyle \int _{\partial \mathcal {B}_{0k}} \mathbf {N}_{\bar{e}}^T(\hat{\varvec{\xi }}{^n}) \,\epsilon \, \mathbf {N}_{\bar{e}}(\hat{\varvec{\xi }}{^n}) \,\mathrm {d}A, \end{aligned}$$

(72)

denote the tangent matrices. When sliding occurs, i.e. $\omega =1$, Eq. (70) becomes

$$\begin{aligned} \Delta \delta \Pi _\mathrm {c}= & {} \displaystyle \delta \mathbf {x}_e^\mathrm {T}\, \mathbf {k}_{kk} \,\Delta \mathbf {x}_e + \delta \mathbf {x}_e^\mathrm {T}\, \mathbf {k}_{k\hat{\ell }} \,\Delta \mathbf {x}_{\hat{e}} + \delta \mathbf {x}_e^\mathrm {T}\, \mathbf {k}_{k\bar{\ell }} \,\Delta \mathbf {x}_{\bar{e}} \nonumber \\&+\, \delta \mathbf {x}_{\hat{e}}^\mathrm {T}\, \mathbf {k}_{\hat{\ell } k} \,\Delta \mathbf {x}_{e} + \delta \mathbf {x}_{\hat{e}}^\mathrm {T}\, \mathbf {k}_{\hat{\ell }\hat{\ell }} \,\Delta \mathbf {x}_{\hat{e}} + \delta \mathbf {x}_{\hat{e}}^\mathrm {T}\, \mathbf {k}_{\hat{\ell }\bar{\ell }} \,\Delta \mathbf {x}_{\bar{e}},\nonumber \\ \end{aligned}$$

(73)

where $\hat{e}\in \mathcal {E}_\ell$ denotes the master elements that contain the current interacting point $\hat{\varvec{\xi }}{^{n+1}}$, and the tangent matrices are defined by

$$\begin{aligned} \mathbf {k}_{kk}:= & {} \displaystyle \int _{\partial \mathcal {B}_{0k}} \mathbf {N}_e^T\,\epsilon \, (\mathbf {N}_e - \varvec{a}_\alpha \,\mathbf {M}^\alpha _e) \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{k\hat{\ell }}:= & {} \displaystyle -\int _{\partial \mathcal {B}_{0k}} \mathbf {N}_e^T\,\epsilon \, (\mathbf {N}_{\hat{e}} + \varvec{a}_\alpha \,\mathbf {M}^\alpha _{\hat{e}}) \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{k\bar{\ell }}:= & {} \displaystyle -\int _{\partial \mathcal {B}_{0k}} \mathbf {N}_e^T\,\epsilon \, \varvec{a}_\alpha \,\mathbf {M}^\alpha _{\bar{e}} \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{\hat{\ell } k}:= & {} \displaystyle -\int _{\partial \mathcal {B}_{0k}} \left[ \mathbf {N}_{\hat{e}}^T\,\epsilon \, (\mathbf {N}_e - \varvec{a}_\alpha \,\mathbf {M}^\alpha _e) - \mathbf {N}^\mathrm {T}_{\hat{e},\alpha }\,\varvec{T}\,\mathbf {M}^\alpha _e \right] \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{\hat{\ell } \hat{l}}:= & {} \displaystyle \int _{\partial \mathcal {B}_{0k}} \left[ \mathbf {N}_{\hat{e}}^T\,\epsilon \, (\mathbf {N}_{\hat{e}} + \varvec{a}_\alpha \,\mathbf {M}^\alpha _{\hat{e}} ) - \mathbf {N}^\mathrm {T}_{\hat{e},\alpha }\,\varvec{T}\,\mathbf {M}^\alpha _{\hat{e}} \right] \,\mathrm {d}A,\nonumber \\ \mathbf {k}_{\hat{\ell }\bar{\ell }}:= & {} \displaystyle \int _{\partial \mathcal {B}_{0k}}\left[ \mathbf {N}_{\hat{e}}^T\,\epsilon \, \varvec{a}_\alpha \,\mathbf {M}^\alpha _{\bar{e}} - \mathbf {N}^\mathrm {T}_{\hat{e},\alpha }\,\varvec{T}\,\mathbf {M}^\alpha _{\bar{e}} \right] \,\mathrm {d}A, \end{aligned}$$

(74)

with

$$\begin{aligned} \mathbf {M}^\alpha _e:= & {} \displaystyle \frac{\partial {\xi ^\alpha }}{\partial {\mathbf {x}_e}} = -c^{\alpha \beta }\,(\varvec{c}_\beta - \varvec{m}_\beta )\cdot \mathbf {N}_e,\nonumber \\ \mathbf {M}^\alpha _{\hat{e}}:= & {} \displaystyle \frac{\partial {\xi ^\alpha }}{\partial {\mathbf {x}_{\hat{e}}}} = -c^{\alpha \beta }\, \nonumber \\&\Big [ (\varvec{g}- \varvec{g}^\mathrm {max}_{\tau })\cdot \mathbf {N}_{\hat{e},\beta } - \varvec{c}_\beta \cdot \mathbf {N}_{\hat{e}} - \varvec{d}_\beta ^\gamma \cdot \mathbf {N}_{\hat{e},\gamma } \Big ]\nonumber \\ \mathbf {M}^\alpha _{\bar{e}}:= & {} \displaystyle \frac{\partial {\xi ^\alpha }}{\partial {\mathbf {x}_{\bar{e}}}} = -c^{\alpha \beta }\, \varvec{m}_\beta \cdot \mathbf {N}_{\bar{e}}, \end{aligned}$$

(75)

where

$$\begin{aligned} \varvec{m}_\alpha := \displaystyle \mu \,\frac{\Vert \varvec{g}_{\mathrm {n}}\Vert }{\Vert \hat{\varvec{g}}^n_{\tau }\Vert } (\varvec{a}_\alpha - \tau _{\alpha \beta }\,\varvec{a}^\beta )~. \end{aligned}$$

(76)

For the two-half-pass algorithm, all the tangent matrices associated with the variation of the master surface, i.e. $\mathbf {k}_{\bar{\ell }k}$ and $\mathbf {k}_{\bar{\ell }\bar{\ell }}$ in Eq. (71); $\mathbf {k}_{\hat{\ell }k}$, $\mathbf {k}_{\hat{\ell }\hat{\ell }}$, and $\mathbf {k}_{\hat{\ell }\bar{\ell }}$ in Eq. (73), are not needed.