Analysis of Adiabatic Shear Bands by Numerical Methods

ρ mass density in the present configuration ρ0 mass density in the reference configuration J determinant of the deformation gradient Fiα components of deformation gradient xi coordinates of a point in the present configuration Xα coordinates of a point in the reference configuration Tiα components of the first Piola Kirchhoff stress tensor vi components of velocity bi components of body force per unit mass e specific internal energy Qα components of heat flux per unit area in the reference configuration r supply of internal energy per unit mass Ω region occupied by the body in the reference configuration @Ω boundary of Ω θ temperature of a material point Nα components of the outward unit normal to the boundary fi components of the present force per unit undeformed area t time φ empty set h prescribed heat flux per unit undeformed area _ θ time rate of change of θ c specific heat φi components of test function η scalar test function H collection of real valued functions whose first-order derivative is square integrable Sm Sθ tensor product between sets Sm and Sθ ψA finite element basis function for node A for displacement degree of freedom ζA finite element basis function for node A for temperature degree of freedom diA displacement of node A along xi-axis θA temperature of node A