mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-09-11 14:11:55 +00:00
Jukka Aho
103 lines
2.8 KiB
Julia
103 lines
2.8 KiB
Julia
# This file is a part of JuliaFEM.
|
|
# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
|
|
|
|
# Elasticity problems
|
|
|
|
abstract ElasticityProblem <: AbstractProblem
|
|
|
|
function get_unknown_field_name{P<:ElasticityProblem}(::Type{P})
|
|
return "displacement"
|
|
end
|
|
|
|
function get_unknown_field_type{P<:ElasticityProblem}(::Type{P})
|
|
return Vector{Float64}
|
|
end
|
|
|
|
function ElasticityProblem(dim::Int=3, elements=[])
|
|
return Problem{PlaneStressElasticityProblem}(dim, elements)
|
|
end
|
|
|
|
abstract PlaneStressElasticityProblem <: ElasticityProblem
|
|
|
|
function PlaneStressElasticityProblem(dim::Int=2, elements=[])
|
|
return Problem{PlaneStressElasticityProblem}(dim, elements)
|
|
end
|
|
|
|
""" Elasticity equations.
|
|
|
|
Formulation
|
|
-----------
|
|
|
|
Field equation is:
|
|
∂u/∂t = ∇⋅f - b
|
|
|
|
Weak form is: find u∈U such that ∀v in V
|
|
|
|
δW := ∫ρ₀∂²u/∂t²⋅δu dV₀ + ∫S:δE dV₀ - ∫b₀⋅δu dV₀ - ∫t₀⋅δu dA₀ = 0
|
|
|
|
where
|
|
|
|
ρ₀ = density
|
|
b₀ = displacement load
|
|
t₀ = displacement traction
|
|
|
|
References
|
|
----------
|
|
|
|
https://en.wikipedia.org/wiki/Linear_elasticity
|
|
https://en.wikipedia.org/wiki/Finite_strain_theory
|
|
https://en.wikipedia.org/wiki/Stress_measures
|
|
https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid
|
|
https://en.wikipedia.org/wiki/Strain_energy_density_function
|
|
https://en.wikipedia.org/wiki/Plane_stress
|
|
https://en.wikipedia.org/wiki/Hooke's_law
|
|
|
|
"""
|
|
function get_residual_vector{P<:ElasticityProblem}(problem::Problem{P}, element::Element, ip::IntegrationPoint, time::Number; variation=nothing)
|
|
|
|
basis = element(ip, time)
|
|
|
|
u = element("displacement", ip, time, variation)
|
|
|
|
r = zeros(Float64, problem.dim, length(element))
|
|
|
|
# internal forces
|
|
if haskey(element, "youngs modulus") && haskey(element, "poissons ratio")
|
|
dbasis = element(ip, time, Val{:grad})
|
|
gradu = element("displacement", ip, time, Val{:grad}, variation)
|
|
F = I + gradu # deformation gradient
|
|
|
|
young = element("youngs modulus", ip, time)
|
|
poisson = element("poissons ratio", ip, time)
|
|
mu = young/(2*(1+poisson))
|
|
lambda = young*poisson/((1+poisson)*(1-2*poisson))
|
|
if P == PlaneStressElasticityProblem
|
|
lambda = 2*lambda*mu/(lambda + 2*mu) # <- correction for 2d problems
|
|
end
|
|
E = 1/2*(F'*F - I) # strain
|
|
S = lambda*trace(E)*I + 2*mu*E
|
|
|
|
J = det(element, ip, time)
|
|
T = J^-1*F*S*F'
|
|
#ip["cauchy stress"] = T
|
|
ip["gl strain"] = E
|
|
|
|
r += F*S*dbasis
|
|
end
|
|
|
|
# external forces - volume load
|
|
if haskey(element, "displacement load")
|
|
b = element("displacement load", ip, time)
|
|
r -= b*basis
|
|
end
|
|
|
|
# external forces - surface traction force
|
|
if haskey(element, "displacement traction force")
|
|
T = element("displacement traction force", ip, time)
|
|
r -= T*basis
|
|
end
|
|
|
|
return vec(r)
|
|
end
|
|
|