Evolution of Thermoelastic Stresses in a Finite-Width Slab or Thick Cylinder With a Growing or Receding Boundary

Abstract

Semi-analytical thermoelastic stress solutions for a single phase, homogeneous, and finite-width slab or thick cylinder with a constant-velocity growing or receding boundary under unit-loading were derived. Initially, a semi-analytical solution for the heat equation for a slab with a growing or receding boundary was derived in the Laplace domain, and a series representation was then used to approximate the inverse Laplace transform in the time domain. Conformal mapping was then used to transform the slab solution to an annulus. The resulting semi-analytical solutions were used with elasticity relationships to determine the resulting transient thermoelastic stresses. All solutions allow for convection on the fixed boundary that is the opposite side for a plate and outer radius for the cylinder. Once derived, the semi-analytical stress predictions were compared to finite element simulations with excellent agreement. Given the changing thickness, both the thermal and stress states cannot reach true steady-state equilibrium, especially for faster growth or recession rates. Indeed, the temperature states and resulting stresses become somewhat linear with respect to time, reflecting the constant velocity of growth or recession. In practice, the resulting solutions can be used to determine transient stresses during machining, wear, erosion, corrosion, and/or additive manufacturing, especially for lower temperature solid-state methods such as cold-spray.