mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-09-20 01:59:59 +00:00
Olli Väinölä
305 lines
8.5 KiB
Julia
305 lines
8.5 KiB
Julia
using ForwardDiff
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"""
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Create a isotropic Hooke material matrix C
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More information: http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_isotropic.cfm
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https://en.wikipedia.org/wiki/Hooke's_law
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http://www.ce.berkeley.edu/~sanjay/ce231mse211/symidentity.pdf
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Parameters
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----------
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E: Float
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Elastic modulus
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ν: Float
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Poisson constant
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Returns
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-------
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Array{Float64, (6,6)}
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"""
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function stiffnessTensor(E, ν)
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a = 1 - ν
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b = 1 - 2*ν
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c = 1 + ν
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multiplier = E / (b * c)
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return Float64[a ν ν 0 0 0;
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ν a ν 0 0 0;
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ν ν a 0 0 0;
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0 0 0 b 0 0;
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0 0 0 0 b 0;
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0 0 0 0 0 b].*multiplier
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end
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# Creating functions for newton: xₙ₊₁ = xₙ - df⁻¹ * f and initial values
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function find_root!(f, df, x; max_iter=50, norm_acc=1e-10)
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converged = false
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for i=1:max_iter
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dx = df(x) \ -f(x)
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x += dx
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norm(dx) < norm_acc && (converged = true; break)
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end
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converged || error("no convergence!")
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x
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end
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type State
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C :: Array{Float64, 2}
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stress_y :: Float64
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stress :: Array{Float64, 1}
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strain :: Array{Float64, 1}
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end
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"""
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Equivalent tensile stress.
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More info can be found from: https://en.wikipedia.org/wiki/Von_Mises_yield_criterion
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Section: Reduced von Mises equation for different stress conditions
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Parameters
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----------
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σ: Array{Float64, 6}
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Stress in Voigt notation
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Returns
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-------
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Float
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"""
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function stress_eq(stress)
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stress_ten = [stress[1] stress[6] stress[5];
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stress[6] stress[2] stress[4];
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stress[5] stress[4] stress[3]]
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stress_dev = stress_ten - 1/3 * trace(stress_ten) * eye(3)
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s = vec(stress_dev)
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return sqrt(3/2 * dot(s, s))
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end
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"""
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Von Mises Yield criterion
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More info can be found from: http://csm.mech.utah.edu/content/wp-content/uploads/2011/10/9tutorialOnJ2Plasticity.pdf
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Parameters
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----------
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σ: Array{Float64, 6}
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Stress in Voigt notation
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k: Float64
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Material constant, Yield limit
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Returns
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-------
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Float
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"""
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function vonMisesYield(stress, stress_y)
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stress_eq(stress) - stress_y
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end
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"""
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Function for NLsolve. Inside this function are the equations which we want to find root.
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Ψ is the yield function below. Functions defined here:
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dσ - C (dϵ - dλ*dΨ/dσ) = 0
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σₑ(σ) - k = 0
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Parameters
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----------
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params: Array{Float64, 7}
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Array containing values from solver
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dϵ: Array{Float64, 6}
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Strain rate vector in Voigt notation
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C: Array{Float64, (6, 6)}
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Material tensor
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k: Float
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Material constant, yield limit
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Δt: Float
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time increment
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σ_begin:Array{Float64, 6}
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Stress vector in Voigt notation
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Returns
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-------
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Array{Float64, 7}, return values for solver
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"""
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function vonMisesRoot(params, dstrain, C, stress_y, stress_base)
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# Creating wrapper for gradient
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vm_wrap(stress_) = vonMisesYield(stress_, stress_y)
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dfds = ForwardDiff.gradient(vm_wrap)
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# Stress rate and total strain
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dstress = params[1:6]
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stress_tot = vec(stress_base) + params[1:6]
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# Calculating plastic strain rate
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dstrain_p = params[end] * dfds(stress_tot)
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# Calculating equations
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function_1 = dstress - C * (dstrain - dstrain_p)
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function_2 = vm_wrap(stress_tot)
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[vec(function_1); function_2]
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end
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"""
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Stress for ideal plastic von Mises material model
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Parameters
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----------
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dϵ: Array{Float64, 6}
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Strain rate vector in Voigt notation
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Δt: Float
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time increment
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σ: Array{Float64, 6}
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Last stress vector in Voigt notation
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C: Array{Float64, (6, 6)}
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Material tensor
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k: Float
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Material constant, yield limit
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Returns
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-------
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Tuple
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Plastic strain rate dϵᵖ and new stress vector σ
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"""
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function calculate_stress!(dstrain, mat::State, ::Type{Val{:vonMises}})
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stress = mat.stress
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C = mat.C
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stress_y = mat.stress_y
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# Test stress
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stress_tria = stress + C * dstrain
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# Calculating and checking for yield
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yield = vonMisesYield(stress_tria, stress_y)
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if isless(yield, 0.0)
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mat.stress = vec(stress_tria)
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else
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# Yielding happened
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# Creating functions for newton: xₙ₊₁ = xₙ - df⁻¹ * f and initial values
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initial_guess = Float64[vec(stress_tria - stress); 0.1]
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f(stress_) = vonMisesRoot(stress_, dstrain, C, stress_y, stress)
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df = ForwardDiff.jacobian(f)
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# Calculating root
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result = nlsolve(not_in_place(f, df), initial_guess).zero
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mat.stress += result[1:6]
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end
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end
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function calculate_stress(dstrain, stress, C, stress_y,
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::Type{Val{:vonMises}},
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::Type{Val{:ElasticPlasticProblem}})
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# Test stress
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stress_tria = stress + C * dstrain
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# Calculating and checking for yield
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yield = vonMisesYield(stress_tria, stress_y)
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if isless(yield, 0.0)
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# stress[i] = stress_tria[i]
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return 0.0
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else
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# Yielding happened
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# Creating functions for newton: xₙ₊₁ = xₙ - df⁻¹ * f and initial values
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x = [vec(stress_tria - stress); 0.0]
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f(stress_) = vonMisesRoot(stress_, dstrain, C, stress_y, stress)
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df = ForwardDiff.jacobian(f)
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# Calculating root
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# result = nlsolve(not_in_place(f, df), initial_guess).zero
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max_iter = 10
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converged = false
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for i=1:5
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dx = df(x) \ -f(x)
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x += dx
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# println(x)
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norm(dx) < 1e-10 && (converged = true; break)
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end
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converged || error("no convergence!")
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# stress[:] += x[1:6]
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return x[end]
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end
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end
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##################################################################################
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# ----- AFTER THIS POINT: VON MISES : PLANE STRESS IMPLEMENTATION ----- #
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##################################################################################
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"""
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http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_plane_stress.cfm
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"""
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function stiffnessTensorPlaneStress(E, ν)
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a = 1 - ν^2
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b = 1 - ν
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multiplier = E / a
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return Float64[1 ν 0;
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ν 1 0;
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0 0 b].*multiplier
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end
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# von mises: plane stress
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# https://andriandriyana.files.wordpress.com/2008/03/yield_criteria.pdf
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function stress_eq_plane_stress(stress)
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s1, s2, t12 = stress
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# Calculating principal stresses
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# http://www.engineersedge.com/material_science/principal_vonmises_stress__13418.htm
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se1 = (s1 + s2)/2 + sqrt(((s1 - s2)/2)^2 + t12^2)
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se2 = (s1 + s2)/2 - sqrt(((s1 - s2)/2)^2 + t12^2)
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return sqrt(se1^2 -se1*se2 + se2^2)
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end
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# https://andriandriyana.files.wordpress.com/2008/03/yield_criteria.pdf
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function vonMisesYieldPlaneStress(stress, stress_y)
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stress_eq_plane_stress(stress) - stress_y
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end
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function vonMisesRootPlaneStress(params, dstrain, C, stress_y, stress_base)
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# Creating wrapper for gradient
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vm_wrap(stress_) = vonMisesYieldPlaneStress(stress_, stress_y)
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dfds = ForwardDiff.gradient(vm_wrap)
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# Stress rate and total strain
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dstress = params[1:3]
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stress_tot = vec(stress_base) + params[1:3]
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# Calculating plastic strain rate
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dstrain_p = params[end] * dfds(stress_tot)
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# Calculating equations
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function_1 = dstress - C * (dstrain - dstrain_p)
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function_2 = vm_wrap(stress_tot)
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[vec(function_1); function_2]
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end
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function calculate_stress(dstrain, stress, C, stress_y,
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::Type{Val{:vonMises}},
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::Type{Val{:PlaneStressElasticPlasticProblem}})
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# Test stress
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dstress = C * dstrain
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stress_tria = stress + dstress
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# Calculating and checking for yield
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yield = vonMisesYieldPlaneStress(stress_tria, stress_y)
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if isless(yield, 0.0)
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return dstress, zeros(3)
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else
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info("yielded")
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# Yielding happened
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# Creating functions for newton: xₙ₊₁ = xₙ - df⁻¹ * f and initial values
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x = [vec(stress_tria - stress); 0.0]
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f(stress_) = vonMisesRootPlaneStress(stress_, dstrain, C, stress_y, stress)
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df = ForwardDiff.jacobian(f)
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# Calculating root
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results = find_root!(f, df, x)
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dstress = results[1:3]
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stress_tot = stress + dstress
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plastic_multiplier = results[end]
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vm_wrap(stress_) = vonMisesYieldPlaneStress(stress_, stress_y)
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dfds = ForwardDiff.gradient(vm_wrap)
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dep = plastic_multiplier * dfds(vec(stress_tot))
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info("II ", stress_tot)
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info(vm_wrap(stress_tot))
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return dstress, dep
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end
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end
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