1. Reference problem

1.1. Geometry

a: radius of the circular internal crack in the horizontal plane xoy

b: radius of the sphere, with b = 2.5.10-3 m.

The radius a varies according to the modeling.

1.2. Material properties

Young’s modulus	E = 2 1011Pa
Poisson’s ratio	\(\nu\) = 0.3
linear expansion coefficient	\(\alpha\) = 1.2 10-5°C-1

1.3. Boundary conditions and loads

UX = ur = 0 on the axis of revolution X = r = 0

UY = uz = 0 in the horizontal plane Y = z = 0, outside the lips a ≤ r ≤ b

The lips are supposed to be free of constraints (no partial closure of the crack).

Zero temperature at the surface of the sphere.

Zero reference temperature (temperature at which thermal deformations are considered zero).

Uniform and negative temperature T = - Tf on the lips of the crack, including the crack background. The stationary thermal problem (of the Dirichlet type) must be solved beforehand by finite elements on the same mesh as that intended for mechanical calculation. We take Tf = 100°C.