Surface effects on cylindrical indentation of a soft layer on a rigid substrate

Abstract

In this paper, theoretical and numerical methods are combined to investigate the two-dimensional, cylindrical indentation of an elastic soft layer bonded on a rigid substrate. By incorporating the Gurtin–Murdoch’s theory of surface elasticity into the Kerr model, we account for surface effects on the indentation behavior of a soft layer. The governing differential relation between the surface pressure and displacement of the film–substrate system is derived. For an incompressible thin layer, the explicit solutions are derived, which are simple and easy-to-use. Finite element simulations are performed using an explicit restart algorithm to study the indentation properties of the layer–substrate system and to examine the validity and accuracy of the theoretical solutions. This work holds promise for applications in mechanical characterization of soft materials, such as biological tissues and cells.

Appendix: Derivation of Kerr-type governing differential equation

For the case of an elastic thin layer bonded to a rigid substrate (Fig. 1), the displacements and stresses in the layer under the plane-strain assumption are expressed as [31]

$$ \begin{aligned} \sigma_{xz} & = - \frac{1}{2}\left[ {zD^{2} \cos \left( {zD} \right) - D\sin \left( {zD} \right)} \right]X^{\prime}_{0} \\ &\quad - \left[ {\cos \left( {zD} \right) + \frac{1}{2}zD\sin \left( {zD} \right)} \right]X^{\prime}_{1} \\ & \quad - \frac{1}{2}\left[ {z\cos \left( {zD} \right) + \frac{1}{D}\sin \left( {zD} \right)} \right]X^{\prime}_{2} - \frac{1}{2}\frac{z}{D}\sin \left( {zD} \right)X^{\prime}_{3} , \\ \end{aligned} $$

(25)

$$ \begin{aligned} \sigma_{zz} & = \left[ {\cos \left( {zD} \right) + \frac{1}{2}zD\sin \left( {zD} \right)} \right]X^{\prime\prime}_{0} \\ &\quad + \left[ {\frac{3}{2D}\sin \left( {zD} \right) - \frac{1}{2}z\cos \left( {zD} \right)} \right]X^{\prime\prime}_{1} \\ & \quad + \frac{z}{2D}\sin \left( {zD} \right)X^{\prime\prime}_{2} + \frac{1}{2}\left[ {\frac{1}{{D^{3} }}\sin \left( {zD} \right) - \frac{z}{{D^{2} }}\cos \left( {zD} \right)} \right]X^{\prime\prime}_{3} , \\ \end{aligned} $$

(26)

$$ \begin{aligned} Eu_{x} & = \left( {1 - v^{2} } \right)\left[ D\cos \left( {zD} \right)X_{0} + \sin \left( {zD} \right)X_{1} \right. \\ &\quad \left. + \frac{1}{D}\cos \left( {zD} \right)X_{2} + \frac{1}{{D^{2} }}\sin \left( {zD} \right)X_{3} \right] \\ & \quad - \left( {1 + v} \right)\left[ \left[ {\cos \left( {zD} \right) + \frac{1}{2}zD\sin \left( {zD} \right)} \right]X^{\prime}_{0} \right. \\ &\quad \left. + \left[ {\frac{3}{2D}\sin \left( {zD} \right) - \frac{1}{2}z\cos \left( {zD} \right)} \right]X^{\prime}_{1} \right. \\ & \quad + \frac{z}{2D}\sin \left( {zD} \right)X^{\prime}_{2} + \left. {\frac{1}{2}\left[ {\frac{1}{{D^{3} }}\sin \left( {zD} \right) - \frac{z}{{D^{2} }}\cos \left( {zD} \right)} \right]X^{\prime}_{3} } \right], \\ \end{aligned} $$

(27)

$$ \begin{aligned} Eu_{z} & = \left( {1 - v^{2} } \right)\left[ \sin \left( {zD} \right)X^{\prime}_{0} - \frac{1}{D}\cos \left( {zD} \right)X^{\prime}_{1} \right.\\ &\quad \left. + \frac{1}{{D^{2} }}\sin \left( {zD} \right)X^{\prime}_{2} - \frac{1}{{D^{3} }}\cos \left( {zD} \right)X^{\prime}_{3} \right] \\ & \quad - \left( {1 + v} \right)\left\{ \frac{1}{2}\left[ {zD^{2} \cos \left( {zD} \right) - D\sin \left( {zD} \right)} \right]X_{0} \right.\\ &\quad \left. + \left[ {\frac{1}{2}zD\sin \left( {zD} \right) + \cos \left( {zD} \right)} \right]X_{1} \right. \\ & \quad \left. { + \frac{1}{2}\left[ {\frac{{\sin \left( {zD} \right)}}{D} + z\cos \left( {zD} \right)} \right]X_{2} + \frac{z}{2D}\sin \left( {zD} \right)X_{3} } \right\}, \\ \end{aligned} $$

(28)

where X_i (i = 0, 1, 2, 3) are unknown functions of x, the prime represents the derivative of the variables, and D = d/dx denotes the differential operator.

By neglecting the variation of surface tension with deformation [21, 22], the normal stress boundary condition on the upper surface ($z = 0$) is given by

$$ \sigma_{zz} (x,0) = - p(x) + \sigma D^{2} u_{z} (x,0). $$

(29)

Other stress and displacement boundary conditions on the upper ($z = 0$) and lower surfaces ($z = - h$) of the elastic layer are written as

$$ \begin{aligned}& u_{z} (x,0) = - w,\; \, \sigma_{xz} (x,0) = 0, \hfill \\& u_{x} (x, - h) = 0, \, u_{z} (x, - h) = 0. \hfill \\ \end{aligned} $$

(30)

Substitution of Eqs. (25–28) into (29) and (30) leads to

$$ X_{0}^{{\prime \prime }} = - p(x) - \sigma_{1} D^{2} w, $$

(31)

$$ X_{3}^{{\prime }} = E^{*} D^{3} w, $$

(32)

$$ X_{1}^{{\prime }} = 0, $$

(33)

$$ \begin{aligned} &\left( {1 - v^{2} } \right)\left[ D\cos \left( {hD} \right)X_{0} - \sin \left( {hD} \right)X_{1} + \frac{1}{D}\cos \left( {hD} \right)X_{2} \right.\\ &\quad \left. - \frac{1}{{D^{2} }}\sin \left( {hD} \right)X_{3} \right] \hfill \\ &\quad - \left( {1 + v} \right)\left\{ \left[ {\cos \left( {hD} \right) + \frac{1}{2}hD\sin \left( {hD} \right)} \right]X^{\prime}_{0} \right. \hfill \\ &\quad \left. + \frac{h}{2D}\sin \left( {hD} \right)X^{\prime}_{2} \right. \\ &\quad \left. + \frac{1}{2}\left[ { - \frac{1}{{D^{3} }}\sin \left( {hD} \right) + \frac{h}{{D^{2} }}\cos \left( {hD} \right)} \right]X^{\prime}_{3} \right\} = 0, \hfill \\ \end{aligned} $$

(34)

$$ \begin{aligned} & \left( {1 - v^{2} } \right)\left[ - \sin \left( {hD} \right)X^{\prime}_{0} - \frac{1}{{D^{2} }}\sin \left( {hD} \right)X^{\prime}_{2} \right.\\ &\quad \left. - \frac{1}{{D^{3} }}\cos \left( {hD} \right)X^{\prime}_{3} \right] \hfill \\ &\quad - \left( {1 + v} \right)\left\{ \frac{1}{2}\left[ { - hD^{2} \cos \left( {hD} \right) + D\sin \left( {hD} \right)} \right]X_{0} \right.\\ &\quad \left. + \left[ {\frac{1}{2}hD\sin \left( {hD} \right) + \cos \left( {hD} \right)} \right]X_{1} \right. \\ &\quad \left. + \frac{1}{2}\left[ { - \frac{{\sin \left( {hD} \right)}}{D} - h\cos \left( {hD} \right)} \right]X_{2} + \frac{h}{2D}\sin \left( {hD} \right)X_{3} \right\} = 0. \hfill \\ \end{aligned} $$

(35)

After eliminating X₀, X₁, X₂, and X₃, the relation between contact pressure p and surface displacement w is obtained as

$$ \begin{aligned} & \left[ \frac{1}{4}(1 + v)(3 - 4v)\sin (hD)\sin (hD) \right.\\ &\quad \left. - (1 - v^{2} )(1 - v) + \frac{1}{4}(1 + v)h^{2} D^{2} \right]w \hfill \\ &\quad = \frac{{(1 - v^{2} )}}{E}\left\{ \frac{1}{2}h(1 + v) - \sin (hD)\cos (hD) \right.\\ &\quad \left. \left[ {\frac{1}{2D}(1 + v)(3 - 4v)} \right] \right\}\left( {p + \sigma D^{2} w} \right). \hfill \\ \end{aligned} $$

(36)

The trigonometric functions in Eq. (36) can be expanded in power series of hD as

$$ \begin{aligned} \sin (hD)\cos (hD) & = hD - \frac{2}{3}(hD)^{3} + \frac{2}{15}(hD)^{5} - \frac{4}{315}(hD)^{7} \cdots , \hfill \\ \sin (hD)\sin (hD) & = (hD)^{2} - \frac{1}{3}(hD)^{4} + \frac{2}{45}(hD)^{6} - \frac{1}{315}(hD)^{8} \cdots . \hfill \\ \end{aligned} $$

(37)

From the series expansion in Eq. (36) and neglecting high-order terms, we can obtain the governing Eq. (1).