Analytical solution of the dynamic contact problem of anisotropic materials indented with a rigid wavy surface

Abstract

This article establishes a dynamic wavy contact model of anisotropic materials subject to a rigid indenter, which moves at a constant speed. There is a gap between the indenter and the substrate prior to deformation. The Galilean transformation is introduced to make the equations of motion tractable. Applying the generalized Almansi’s theorem, general solutions for anisotropic governing equations are given for three root cases by using potential theory. The stated problem is reduced to dual series equations, which are solved exactly. Displacement and stress are given in the infinite series forms. Numerical tests have been done to examine the convergence of the infinite series. It is found that maximum magnitudes of the stress can be relieved by adjusting the relative velocity and the elastic constant ratio.

Appendices

Appendix 1

• 1. Coefficients $n_{m}$ $\,(m = 0,1,2,3)$ appearing in Eq. (19)

$$ n_{0} = C_{55} C_{44} C_{33} - C_{33} C_{45}^{2} , $$

(67)

$$ \begin{aligned} n_{1} & = (C_{11} - C_{66} V^{2} )C_{44} C_{33} + C_{55} C_{66} (1 - V^{2} )C_{33} + C_{55} C_{44} (C_{55} - C_{66} V^{2} ) \\ & \quad - C_{55} (C_{36} + C_{45} )^{2} - 2C_{16} C_{45} C_{33} - C_{45}^{2} (C_{55} - C_{66} V^{2} ) \\ & \quad + 2C_{45} (C_{13} + C_{55} )(C_{36} + C_{45} ) - (C_{13} + C_{55} )^{2} C_{44} , \\ \end{aligned} $$

(68)

$$ \begin{aligned} n_{2} & = (C_{11} - C_{66} V^{2} )C_{66} (1 - V^{2} )C_{33} + (C_{11} - C_{66} V^{2} )C_{44} (C_{55} - C_{66} V^{2} ) \\ & \quad - (C_{11} - C_{66} V^{2} )(C_{36} + C_{45} )^{2} + C_{55} C_{66} (1 - V^{2} )(C_{55} - C_{66} V^{2} ) - C_{16}^{2} C_{33} \\ & \quad - 2C_{16} C_{45} (C_{55} - C_{66} V^{2} ) + 2C_{16} (C_{13} + C_{55} )(C_{36} + C_{45} ) \\ & \quad - (C_{13} + C_{55} )^{2} C_{66} (1 - V^{2} ), \\ \end{aligned} $$

(69)

$$ n_{3} = (C_{11} - C_{66} V^{2} )C_{66} (1 - V^{2} )(C_{55} - C_{66} V^{2} ) - C_{16}^{2} (C_{55} - C_{66} V^{2} ). $$

(70)

• 2. Known functions $\gamma_{mn}\,(m,n = 1,2,3)$ appearing in Eq. (22)

$$ \begin{aligned} \gamma_{11} & = (C_{36} + C_{45} )C_{16} - (C_{13} + C_{55} )C_{66} (1 - V^{2} ), \\ \gamma_{12} & = (C_{36} + C_{45} )C_{45} - (C_{13} + C_{55} )C_{44} , \\ \end{aligned} $$

(71)

$$ \begin{aligned} \gamma_{21} & = - (C_{36} + C_{45} )(C_{11} - C_{66} V^{2} ) + (C_{13} + C_{55} )C_{16} , \\ \gamma_{22} & = - (C_{36} + C_{45} )C_{55} + (C_{13} + C_{55} )C_{45} , \\ \end{aligned} $$

(72)

$$ \begin{aligned} \gamma_{31} & = (C_{11} - C_{66} V^{2} )C_{66} (1 - V^{2} ) - C_{16}^{2} , \\ \gamma_{32} & = (C_{11} - C_{66} V^{2} )C_{44} + C_{55} C_{66} (1 - V^{2} ) - 2C_{16} C_{45} , \\ \gamma_{33} & = C_{55} C_{44} - C_{45}^{2} . \\ \end{aligned} $$

(73)

• 3. Known functions $\tau_{i}^{(xz)}$ and $\tau_{i}^{(yz)}$ $\,(i = 1,2,3)$ appearing in Eqs. (43) and (44)

$$ \begin{aligned} \tau_{i}^{(xz)} & = - \left( {C_{55} \upsilon_{i} + C_{45} \varpi_{i1} \upsilon_{i} + C_{55} \varpi_{i2} } \right), \\ \tau_{i}^{(yz)} & = - \left( {C_{45} \upsilon_{i} + C_{44} \varpi_{i1} \upsilon_{i} + C_{45} \varpi_{i2} } \right). \\ \end{aligned} $$

(74)

• 4. Known functions $\kappa_{a}$ and $\kappa_{b}$ appearing in Eq. (47)

$$ \kappa_{a} = \frac{1}{{2\left( {o_{a} \varpi_{12} + o_{b} \varpi_{22} + \varpi_{32} } \right)}}, $$

(75)

$$ \kappa_{b} = o_{a} \tau_{1}^{(zz)} + o_{b} \tau_{2}^{(zz)} + \tau_{3}^{(zz)} . $$

(76)

• 5. Known functions $\tau_{i}^{(xx)}\,(i = 1,2,3)$ appearing in Eq. (61)

$$ \tau_{i}^{(xx)} = - C_{11} - C_{16} \varpi_{i1} + C_{13} \varpi_{i2} \upsilon_{i} . $$

(77)

• 6. Known functions $U_{i}\,(i = 1,2,3)$ appearing in Eq. (63)

$$ U_{1} = - \sum\limits_{i = 1}^{3} {\lambdabar_{i} } , $$

(78)

$$ U_{2} = - \sum\limits_{i = 1}^{3} {\varpi_{i1} \lambdabar_{i} } , $$

(79)

$$ U_{3} = \sum\limits_{i = 1}^{3} {\varpi_{i2} \lambdabar_{i} } \upsilon_{i}. $$

(80)

.

• 7. Known functions $\sigma_{0}$, $\Upsilon^{(u)}$, $\Upsilon^{(v)}$, $\Upsilon^{(w)}$, $\Upsilon^{(xx)}$, $\Upsilon^{(zz)}$, $\Upsilon^{(xz)}$ and $\Upsilon^{(yz)}$ appearing in Eq. (64)

$$ \sigma_{0} = - \sum\limits_{i = 1}^{3} {\tau_{i}^{(xx)} \lambdabar_{i} } , $$

(81)

$$ \Upsilon^{(u)} = \frac{{o_{a} e^{{z_{1} }} + o_{b} e^{{z_{2} }} + e^{{z_{3} }} }}{{\kappa_{b} }}, $$

(82)

$$ \Upsilon^{(v)} = \frac{{o_{a} \varpi_{11} e^{{z_{1} }} + o_{b} \varpi_{21} e^{{z_{2} }} + \varpi_{31} e^{{z_{3} }} }}{{\kappa_{b} }}, $$

(83)

$$ \Upsilon^{(w)} = - \frac{{o_{a} \varpi_{12} e^{{z_{1} }} + o_{b} \varpi_{22} e^{{z_{2} }} + \varpi_{32} e^{{z_{3} }} }}{{\kappa_{b} }}, $$

(84)

$$ \Upsilon^{(xx)} = - \frac{{o_{a} \tau_{1}^{(xx)} e^{{z_{1} }} + o_{b} \tau_{2}^{(xx)} e^{{z_{2} }} + \tau_{3}^{(xx)} e^{{z_{3} }} }}{{\kappa_{b} }}, $$

(85)

$$ \Upsilon^{(zz)} = - \frac{{o_{a} \tau_{1}^{(zz)} e^{{z_{1} }} + o_{b} \tau_{2}^{(zz)} e^{{z_{2} }} + \tau_{3}^{(zz)} e^{{z_{3} }} }}{{\kappa_{b} }}, $$

(86)

$$ \Upsilon^{(xz)} = - \frac{{o_{a} \tau_{1}^{(xz)} e^{{z_{1} }} + o_{b} \tau_{2}^{(xz)} e^{{z_{2} }} + \tau_{3}^{(xz)} e^{{z_{3} }} }}{{\kappa_{b} }}, $$

(87)

$$ \Upsilon^{(yz)} = - \frac{{o_{a} \tau_{1}^{(yz)} e^{{z_{1} }} + o_{b} \tau_{2}^{(yz)} e^{{z_{2} }} + \tau_{3}^{(yz)} e^{{z_{3} }} }}{{\kappa_{b} }}. $$

(88)

Appendix 2

Refe [23] presented the following generalized Almansi’s theorem and its proof: Let $R$ be a region of the (x, y, z)-space such that a straight line parallel to the z-axis intersects the boundary of $R$ at no more than two points. Let $F_{n} (x,y,z)$ be a solution of

$$ \mathop \Pi \limits_{i = 1}^{n} \nabla_{i}^{2} F_{n} = \nabla_{1}^{2} \nabla_{2}^{2} \cdots \nabla_{n - 1}^{2} \nabla_{n}^{2} F_{n} = 0\quad in\,R, $$

(89)

where

$$ \nabla_{i}^{2} = \Lambda + c_{i}^{2} D^{2} ,\quad D = \frac{\partial }{\partial z}, $$

(90)

and the $c_{i}$ are constants. Then $F_{n}$ admits the representation

$$ F_{n} (x,y,z) = F_{n - 1} (x,y,z) + z^{m} F^{(n)} (x,y,z), $$

(91)

where $F_{n - 1}$ and $F^{(n)}$, respectively, satisfy

$$ \mathop \Pi \limits_{i = 1}^{n} \nabla_{i}^{2} F_{n - 1} = 0, $$

(92)

$$ \mathop \Pi \limits_{i = 1}^{n} \nabla_{i}^{2} F^{(n)} = 0, $$

(93)

and $m(0 \le m \le n - 1)$ is the number of the coefficients $c_{i}^{2}\,(i = 1,2, \ldots ,n - 1)$ which are equal to $c_{n}^{2}$.