On the Stability of Surface Growth: The Effect of a Compliant Surrounding Medium

Abstract

Abstract In a recent paper Abeyaratne et al. (J. Mech. Phys. Solids 167:104958, 2022) concerning the stability of surface growth of a pre-stressed elastic half-space with surface tension, it was shown that steady growth is never stable, at least not for all wave numbers of the perturbations, when the growing surface is traction-free. On the other hand, steady growth was found to be always stable when growth occurred on a flat frictionless rigid support and the stretch parallel to the growing surface was compressive. The present study is motivated by these somewhat unexpected and contrasting results. In this paper the stability of a pre-compressed neo-Hookean elastic half-space undergoing surface growth under plane strain conditions is studied. The medium outside the growing body resists growth by applying a pressure on the growing surface. At each increment of growth, the incremental change in pressure is assumed to be proportional to the incremental change in normal displacement of the growing surface. It is shown that surface tension stabilizes a homogeneous growth process against small wavelength perturbations while the compliance of the surrounding medium stabilizes it against large wavelength perturbations. Specifically, there is a critical value of stretch,

λ cr

∈ ( 0 , 1 )

$\lambda _{\mathrm{cr}} \in (0,1)$ , such that growth is linearly stable against infinitesimal perturbations of arbitrary wavelength provided the stretch parallel to the growing surface exceeds

λ cr

$\lambda _{\mathrm{cr}}$ . This stability threshold,

λ cr

$\lambda _{\mathrm{cr}}$ , is a function of the non-dimensional parameter

σ κ /

G 2

$\sigma \kappa /G^{2}$ , which is the ratio between two length-scales

σ / G

$\sigma /G$ and

G / κ

$G/\kappa $ , where

G

$G$ is the shear modulus of the elastic body,

σ

$\sigma $ is the surface tension, and

κ

$\kappa $ is the stiffness of the surrounding compliant medium. It is shown that

( a )

$(a)$

λ cr

→ 1

$\lambda _{\mathrm{cr}} \to 1$ as

κ → 0

$\kappa \to 0$ and

( b )

$(b)$

λ cr

→

0 +

$\lambda _{\mathrm{cr}} \to 0^{+}$ as

κ → ∞

$\kappa \to \infty $ , thus recovering the results in Abeyaratne et al. (J. Mech. Phys. Solids 167:104958, 2022) pertaining to the respective limiting cases where growth occurs

( a )

$(a)$ on a traction-free surface and

( b )

$(b)$ on a frictionless rigid support. The results are also generalized to include extensional stretches.