Spectral formulation of the elastodynamic boundary integral equations for bi-material interfaces

Abstract

A spectral formulation of the plane-strain boundary integral equations for an interface between dissimilar elastic solids is presented. The boundary integral equations can be written in two equivalent forms: (a) The tractions can be written as a space-time convolution of the displacement continuities at the interface (Budiansky and Rice, 1979) (b) The displacement discontinuities can be written as a space-time convolution of the tractions at the interface (Kostrov, 1966). Prior work on spectral formulation of the boundary integral equations has adopted the former as the starting point. The present work has for its basis the latter form based on a space-time convolution of the tractions. Tractions and displacement components are given a spectral representation in the spatial coordinate along the interface. The radiation damping term is then explicitly extracted to avoid singularities in the convolution kernels. With the spectral forms introduced, the space-time convolutions reduce to convolutions in time for each Fourier mode. Due to continuity of tractions at the interface, this leads to a simpler formulation and form of the convolution kernels in comparison to the formulation involving convolutions over the slip and opening history at the bi-material interface. The convolution kernels are validated by studying some model problems to which analytical solutions are known. When coupled with a

Modeling and simulation in tribology across scales: An overview

“…Examples of applications include: lubrication problems [163][164][165], electro-elastic contact modeling [166,167], thermo-mechanical coupling [168], and many others. Using BEM-type formulations has also been used to treat elasto-dynamic frictional problems [169,170], whereas complex geometries and boundary conditions would still require usage of FEM or equivalent formulations [171,172].…”

Spectral boundary integral equation method for simulation of 2D and 3D slip ruptures at bi‐material interfaces

“…Here, $$j\; = \;1,\;2$$ correspond to the in‐plane problem $$j\; = \;3$$ corresponds to the antiplane problem. The inplane and antiplane problems have been studied by Ranjith 41 and Ranjith, 42 respectively. We focus now on the 3D case by combining those solutions.…”

“…We focus now on the 3D case by combining those solutions. Starting from Equation (11) of Ranjith 41 and Equation (10) of Ranjith, 42 $$$$\begin{equation}\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\hat U_1^ \pm \left( {p;q} \right)}\\[6pt] {\hat U_2^ \pm \left( {p;q} \right)}\\[6pt] {\hat U_3^ \pm \left( {p;q} \right)} \end{array} } \right\} = \left[ { \def\eqcellsep{&}\begin{array}{ccc} { \pm \hat K_{11}^ \pm }&{\hat K_{12}^ \pm }&0\\[6pt] {\hat K_{21}^ \pm }&{ \pm \hat K_{22}^ \pm }&0\\[6pt] 0&0&{ \pm \hat K_{33}^ \pm } \end{array} } \right]\;\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\hat T}_1}\left( {p;q} \right)}\\[6pt] {{{\hat T}_2}\left( {p;q} \right)}\\[6pt] {{{\hat T}_3}\left( {p;q} \right)} \end{array} } \right\},\end{equation}$$$$with …”

“…This approach uses elastodynamic convolutions of the traction components of stress at the interface. In the present study, a new spectral formulation for 3D elastodynamic problems for fracture on planar interfaces between dissimilar elastic half‐spaces is proposed based on Ranjith's 2D formulation 41,42 . The proposed SBI method and frictional constitutive law are used to numerically study rupture propagation at a bi‐material interface.…”

“…Since the tractions are continuous at the interface, obtaining a relation between displacement discontinuities or displacements and tractions is easier compared to the approach proposed by Geubelle and Breitenfeld 39 and Breitenfeld and Geubelle 40 . This study presents a 3D spectral formulation by combining bi‐material in‐plane and antiplane shear solutions described by Ranjith 41,42 . The formulation is then coupled with a constitutive law for the interface to simulate dynamic rupture propagation.…”