mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-09-18 09:41:31 +00:00
Olli Väinölä
108 lines
3.1 KiB
Julia
108 lines
3.1 KiB
Julia
# This file is a part of JuliaFEM.
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# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
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include("elasticplastic.jl")
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# Elasticity problems
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abstract ElasticityProblem <: AbstractProblem
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abstract PlaneStressElasticityProblem <: ElasticityProblem
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function get_unknown_field_name{P<:ElasticityProblem}(::Type{P})
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return "displacement"
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end
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function get_unknown_field_type{P<:ElasticityProblem}(::Type{P})
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return Vector{Float64}
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end
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# 3D Elasticity problems
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function ElasticityProblem(dim::Int=3, elements=[])
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return Problem{ElasticityProblem}("elasticity problem", dim, elements)
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end
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# 2D Plane stress elasticity problems
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function PlaneStressElasticityProblem(dim::Int=2, elements=[])
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return Problem{PlaneStressElasticityProblem}("plane stress elasticity problem", dim, elements)
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end
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""" Elasticity equations.
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Formulation
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-----------
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Field equation is:
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∂u/∂t = ∇⋅f - b
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Weak form is: find u∈U such that ∀v in V
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δW := ∫ρ₀∂²u/∂t²⋅δu dV₀ + ∫S:δE dV₀ - ∫b₀⋅δu dV₀ - ∫t₀⋅δu dA₀ = 0
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where
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ρ₀ = density
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b₀ = displacement load
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t₀ = displacement traction
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References
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----------
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https://en.wikipedia.org/wiki/Linear_elasticity
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https://en.wikipedia.org/wiki/Finite_strain_theory
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https://en.wikipedia.org/wiki/Stress_measures
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https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid
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https://en.wikipedia.org/wiki/Strain_energy_density_function
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https://en.wikipedia.org/wiki/Plane_stress
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https://en.wikipedia.org/wiki/Hooke's_law
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"""
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function get_residual_vector{P<:ElasticityProblem}(problem::Problem{P}, element::Element, ip::IntegrationPoint, time::Number; variation=nothing)
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r = zeros(Float64, problem.dim, length(element))
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J = get_jacobian(element, ip, time)
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# internal forces
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if haskey(element, "youngs modulus") && haskey(element, "poissons ratio")
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u = element("displacement", time, variation)
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grad = element(ip, time, Val{:grad})
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gradu = grad*u
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# deformation gradient
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F = I + gradu
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# material
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young = element("youngs modulus", ip, time)
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poisson = element("poissons ratio", ip, time)
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mu = young/(2*(1+poisson))
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lambda = young*poisson/((1+poisson)*(1-2*poisson))
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if P == PlaneStressElasticityProblem
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lambda = 2*lambda*mu/(lambda + 2*mu) # <- correction for 2d problems
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end
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# strain
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E = 1/2*(F'*F - I)
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# stress
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S = lambda*trace(E)*I + 2*mu*E
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r += F*S*grad*det(J)
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end
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# external forces - volume load
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if haskey(element, "displacement load")
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basis = element(ip, time)
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b = element("displacement load", ip, time)
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r -= b*basis*det(J)
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end
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# external forces - surface traction force
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if haskey(element, "displacement traction force")
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basis = element(ip, time)
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T = element("displacement traction force", ip, time)
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JT = transpose(J)
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s = size(JT, 2) == 1 ? JT : cross(JT[:,1], JT[:,2])
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r -= T*basis*norm(s)
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end
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return vec(r)
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end
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