Axisymmetric indentation of a periodically layered, viscoelastic half-space
The authors analyze an indentation problem for a viscoelastic half-space by an axisymmetric indenter. The displacement formulation of the indentation problem is written as: \(\frac{\partial }{\partial x_{j}}\{ \int_{0}^{t}G_{ijkh}(x,t-\tau )\frac{\partial }{\partial \tau }(\frac{ \partial u_{k}}{\partial x_{h}})(x,\tau )d\tau \}=0\), \(i=1,2,3\), in \(\Omega \times \lbrack 0,\infty )\), \(\frac{\partial u_{k}}{\partial x_{i}}\cdot \widehat{e}_{3}=\frac{\partial \phi }{\partial x_{i}}\), \(i=1,2\), on \(\Gamma _{1}(t)\times \lbrack 0,\infty )\), \(\int_{0}^{t}G_{i3kh}(x,t-\tau )\frac{ \partial }{\partial \tau }(\frac{\partial u_{k}}{\partial x_{h}})(x,\tau )d\tau =0\), \(i=1,2\), on \(\partial \Omega \times \lbrack 0,\infty )\), \( \int_{0}^{t}G_{33kh}(x,t-\tau )\frac{\partial }{\partial \tau }(\frac{ \partial u_{k}}{\partial x_{h}})(x,\tau )d\tau =0\) on \(\Gamma _{2}(t)\times \lbrack 0,\infty )\). Here \(\Omega =\mathbb{R}\times \mathbb{R}\times (0,\infty )\), the indenter's surface, is expressed as \(\{x_{3}=\phi (r)\}\), where \(r=\sqrt{x_{1}^{2}+x_{2}^{2}}\) is the radial distance of the surface from the \(x_{3}\)-axis, the indenter is in contact with the boundary surface \( \partial \Omega =\mathbb{R}\times \mathbb{R}\times \{0\}\) of the half-space over a time-varying contact region \(\Gamma _{1}(t)\) in the \(x_{1}-x_{2}\) plane, \(\Gamma _{2}(t)\) is the boundary surface of the half-space exterior to the contact region, and \(G_{ijkh}(x,t)\) are the components of a fourth-order tensor that characterizes the material linear viscoelastic response. The authors then consider a periodic, heterogeneous and viscoelastic medium with a reference cell \(Y=(0,l_{1})\times (0,l_{2})\times (0,l_{3})\) for some \(l_{1},l_{2},l_{3}>0\) and they assume that the relaxation moduli take have different values in the two components of the medium. They set \(G_{ijkh}^{\varepsilon }(x,t)=G_{ijkh}(x/\varepsilon ,t)\). They also introduce a standard two-scale asymptotic expansion of the displacement field \(u^{\varepsilon }(x,t)\) as: \(u^{\varepsilon }(x,t)=u^{0}(x,y,t)+\varepsilon u^{1}(x,y,t)+\varepsilon ^{2}u^{2}(x,y,t)+\ldots \) and they identify the problems obtained in the successive orders of \(\varepsilon \), using Laplace transform. The authors identify the first term of the asymptotic expansion, a cell problem, and the effective relaxation moduli. They then specialize their procedure to layered media composed of homogeneous, isotropic layers of small thickness, but having different viscoelastic properties and they prove that the corresponding homogenized material is transversely isotropic. In the case of a homogeneous, transversely isotropic and viscoelastic half-space, they draw computations to get the solution of the indentation problem. The paper ends with comparisons between the results obtained through the homogenization procedure and simulations obtained through the finite element method.
scientific article; zbMATH DE number 7604068
scientific article