Patent ID: 8155933

Claim:
A computerized method of modeling residual stresses during laser cutting, comprising the steps of: establishing a set of variables u, t, T, x, y, z, ρ, C p , K xx , K yy and K zz , wherein the variable t represents time, the variable T represents temperature, the variables x, y and z represent Cartesian axes, ρ represents a density of a substrate material, C p represents a specific heat capacity of the substrate material, u is a scanning speed of a laser beam used to cut the substrate material, and K xx , K yy and K zz respectively represent thermal conductivities of the substrate material along the x, y and z-axes; calculating temperature variation in the substrate material due to laser cutting as: ρ ⁢ ∂ ( C p ⁢ T ) ∂ t = [ ∂ ∂ x ⁢ ( K xx ⁢ ∂ T ∂ x ) + ∂ ∂ y ⁢ ( K yy ⁢ ∂ T ∂ y ) + ∂ ∂ z ⁢ ( K zz ⁢ ∂ T ∂ z ) ] + ρ ⁢ ⁢ u ⁢ ∂ ( C p ⁢ T ) ∂ y ; establishing boundary conditions for the calculation of temperature variation; applying a Fourier heat model to a differential control volume such that ρ ⁢ ∂ ( C p ⁢ T ) ∂ t + { L } T ⁢ { q } = u ⁢ ∂ ∂ y ⁢ ( ρ ⁢ ⁢ C p ⁢ T ) , wherein { L } = { ∂ ∂ x ∂ ∂ y ∂ ∂ z } is a vector operator and {q} is a heat flux vector, wherein {q}=−[D]{L} T , where [ D ] = [ K xx 0 0 0 K yy 0 0 0 K zz ] ; setting K xx , K yy and K zz equal to a constant thermal conductivity K such that ∂ ∂ t ⁢ ( ρ ⁢ ⁢ C p ⁢ T ) = { L } T ⁡ [ [ D ] ⁢ { L } ⁢ T ] + u ⁢ ∂ ∂ y ⁢ ( ρ ⁢ ⁢ C p ⁢ T ) ; calculating a strain energy due to thermal stresses within the substrate material as {δu} T ∫ vol [B] T [D][B]dv{u}={δu} T ∫ vol [B] T [D][ε th ]dv, wherein {δu} T is a vector representing a set of arbitrary virtual displacements, [ε th ] represents a thermal strain vector, [B] is a strain-displacement matrix and v represents a differential volume, wherein calculation of temperature variation and strain energy in the substrate material due to laser cutting is performed by discretization; and displaying numerical results of the calculated temperature variation and strain energy in the substrate material due to laser cutting.