Thermoelastic Stresses in Slabs and Thick-Cylinders With a Growing or Receding Boundary Under Complex Thermal Loading

Thermoelastic stresses for a single phase, finite-width slab or thick cylinder with a constant-velocity growing or receding boundary under complex temperature histories were derived. Initially, the heat equation was solved for a single phase, homogeneous, and finite-width slab under unit loading (step change) with a growing or receding boundary in the Laplace space and a series approximation then used for the inverse in the time domain. Conformal mapping transformed the slab solution to a cylinder or annulus; convection is allowed on the fixed boundary of each system (opposite side for a slab and outer radius for cylinder). Generalization was achieved by using Duhamal’s and Laplace convolution theorems for complex thermal loading. Integral elasticity-equations provided the relationships between the thermal transients and ensuing stresses. Thermoelastic stresses were compared to finite-element simulations with very good agreement obtained, especially for low to moderate growth/recession velocities. Given the changing thickness, both thermal- and stress-states cannot reach steady conditions, especially when the growth or recession is higher. In such instances, the thermal- and stress-states tend to become linear with time, reflecting the constant velocity of growth/recession. The resulting solutions can be used to determine stresses during machining, wear, erosion, corrosion, and/or additive manufacturing, especially for lower temperature solid-state methods such as cold-spray.