The dynamic sphere test problem

The Finite Deformation Dynamic Sphere Test Problem

“…For consistency, the types of boundary conditions and materials used for cases (a) to (f) in the present report correspond to those employed for cases (a) to (f) in [2]. Nonetheless, it should be remarked that the magnitude of the applied loads is bigger, in this exercise, because the ultimate goal is to obtain large deformations.…”

“…Finally, it is worth observing that nearly incompressible behavior for linear elastic materials yields to S rr ≡ S θθ which only assuming small deformation implies σ rr ≡ σ θθ as in [2]. Conversely, if nearly incompressible neo-hookean materials are employed the radial and tangential Cauchy stresses are always identical (see Cases (g), (i), (o), (p)).…”

“…Jump and smooth jump boundary conditions are considered for the outer surface of the shell whereas the inner surface is traction-free [2,3]. For jump boundary conditions, the stress at the outer surface is suddenly increased at time t = 0 from 0 to a nonzero constant and maintained at this level throughout the analysis.…”

“…where the coefficients of the C matrix are obtained following [2]. Along with the second Piola-Kirchhoff stress, the Kirchhoff stress, τ , and the Cauchy stress, σ, are used in this manuscript.…”

“…The inner surface radius, the outer surface radius and the material's density are kept constant for all the considered test cases and their values are r i = 10mm, r o = 20mm and ρ = 1000Kg/m 3 respectively (equal to the values used in [2,3]).…”