XuNeedleman

XuNeedleman(σₘₐₓ, τₘₐₓ, Φₙ, Φₜ, Δₙˢ)

Xu-Needleman traction-separation law.^[Xu1993] This is a commonly used cohesive law for brittle fracture, however it is revertible and thus not suited for unloading. In 3D, it is isotropic within the cohesive plane.

Arguments

• σₘₐₓ::Float64: cohesive normal strength
• τₘₐₓ::Float64: cohesive tangential strength (in-plane strength)
• Φₙ::Float64: normal work of separation
• Φₜ::Float64: tangential work of separation
• Δₙˢ: normal separation after complete shear separation under the condition of zero normal tension

For convenience the following functions can be used to convert parameters:

• xu_needleman_Φₙ
• xu_needleman_Φₜ
• xu_needleman_σₘₐₓ
• xu_needleman_τₘₐₓ

material_response(m::XuNeedleman, Δ::Tensor{1,dim}) where dim

Return the traction vector and the traction tangent for the given separation jump Δ. The last entry of Δ is interpreted as normal separation, the first entries are interpreted as in-plane separations.

No MaterialState is needed for the stress computation, thus if a state is handed over to material_response, the same state is returned.

xu_needleman_Φₙ(σₘₐₓ, δₙ)

Compute the normal work of separation Φₙ for the Xu-Needleman cohesive law based on the cohesive normal strength σₘₐₓ and the characteristic normal separation δₙ.

xu_needleman_Φₜ(τₘₐₓ, δₜ)

Compute the tangential work of separation Φₜ for the Xu-Needleman cohesive law based on the cohesive tangential strength τₘₐₓ and the characteristic tangential separation δₜ.

xu_needleman_σₘₐₓ(Φₙ, δₙ) = Φₙ / (exp(1.0) * δₙ)

Compute the cohesive normal strength σₘₐₓ for the Xu-Needleman cohesive law based on the normal work of separation Φₙ and the characteristic normal separation δₙ.

xu_needleman_τₘₐₓ(Φₜ, δₜ)

Compute the cohesive tangential strength τₘₐₓ for the Xu-Needleman cohesive law based on the tangential work of separation Φₜ and the characteristic tangential separation δₜ.