mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-08-06 04:21:33 +00:00
396 lines
41 KiB
Plaintext
396 lines
41 KiB
Plaintext
|
{
|
|||
|
|
"cells": [
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"# Ideal plastic Von Mises material\n",
|
|||
|
|
"\n",
|
|||
|
|
"Author(s): Olli Väinölä <olli.vainola@student.oulu.fi>\n",
|
|||
|
|
"\n",
|
|||
|
|
"In this notebook is an small tutorial, how to create a Von Mises material without any hardening. Equations are formulated into rate depended form."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 1,
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": false
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# imports\n",
|
|||
|
|
"using PyPlot\n",
|
|||
|
|
"using ForwardDiff\n",
|
|||
|
|
"using NLsolve"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"Let's create a isotropic Hooke material."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 2,
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": false
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"6x6 Array{Float64,2}:\n",
|
|||
|
|
" 2.69231e5 1.15385e5 1.15385e5 0.0 0.0 0.0 \n",
|
|||
|
|
" 1.15385e5 2.69231e5 1.15385e5 0.0 0.0 0.0 \n",
|
|||
|
|
" 1.15385e5 1.15385e5 2.69231e5 0.0 0.0 0.0 \n",
|
|||
|
|
" 0.0 0.0 0.0 1.53846e5 0.0 0.0 \n",
|
|||
|
|
" 0.0 0.0 0.0 0.0 1.53846e5 0.0 \n",
|
|||
|
|
" 0.0 0.0 0.0 0.0 0.0 1.53846e5"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 2,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"\n",
|
|||
|
|
"\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"Create a isotropic Hooke material matrix C \n",
|
|||
|
|
"\n",
|
|||
|
|
"More information: # http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke_isotropic.cfm\n",
|
|||
|
|
"\n",
|
|||
|
|
"Parameters\n",
|
|||
|
|
"----------\n",
|
|||
|
|
" E: Float\n",
|
|||
|
|
" Elastic modulus\n",
|
|||
|
|
" ν: Float\n",
|
|||
|
|
" Poisson constant\n",
|
|||
|
|
"\n",
|
|||
|
|
"Returns\n",
|
|||
|
|
"-------\n",
|
|||
|
|
" Array{Float64, (6,6)}\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"function hookeStiffnessTensor(E, ν)\n",
|
|||
|
|
" a = 1 - ν\n",
|
|||
|
|
" b = 1 - 2*ν\n",
|
|||
|
|
" c = 1 + ν\n",
|
|||
|
|
" multiplier = E / (b * c)\n",
|
|||
|
|
" return Float64[a ν ν 0 0 0;\n",
|
|||
|
|
" ν a ν 0 0 0;\n",
|
|||
|
|
" ν ν a 0 0 0;\n",
|
|||
|
|
" 0 0 0 b 0 0;\n",
|
|||
|
|
" 0 0 0 0 b 0;\n",
|
|||
|
|
" 0 0 0 0 0 b].*multiplier\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Pick material values\n",
|
|||
|
|
"E = 200.0e3\n",
|
|||
|
|
"ν = 0.3\n",
|
|||
|
|
"C = hookeStiffnessTensor(E, ν)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": false
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"# Defining equations for the calculation\n",
|
|||
|
|
"\n",
|
|||
|
|
"Functions are defined for strain controller simulation"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 3,
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": false
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"calculate_stress (generic function with 1 method)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 3,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"Equivalent tensile stress. \n",
|
|||
|
|
"\n",
|
|||
|
|
"More info can be found from: https://en.wikipedia.org/wiki/Von_Mises_yield_criterion\n",
|
|||
|
|
" Section: Reduced von Mises equation for different stress conditions\n",
|
|||
|
|
"\n",
|
|||
|
|
"Parameters\n",
|
|||
|
|
"----------\n",
|
|||
|
|
" σ: Array{Float64, 6}\n",
|
|||
|
|
" Stress in Voigt notation\n",
|
|||
|
|
"\n",
|
|||
|
|
"Returns\n",
|
|||
|
|
"-------\n",
|
|||
|
|
" Float\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"function σₑ(σ)\n",
|
|||
|
|
" e1 = (σ[1] - σ[2])^2\n",
|
|||
|
|
" e2 = (σ[2] - σ[3])^2\n",
|
|||
|
|
" e3 = (σ[3] - σ[1])^2\n",
|
|||
|
|
" e4 = σ[4]^2 \n",
|
|||
|
|
" e5 = σ[5]^2\n",
|
|||
|
|
" e6 = σ[6]^2\n",
|
|||
|
|
" return sqrt((e1 + e2 + e3 + 6 * (e4 + e5 + e6)) / 2.)\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"Von Mises Yield criterion\n",
|
|||
|
|
"\n",
|
|||
|
|
"More info can be found from: http://csm.mech.utah.edu/content/wp-content/uploads/2011/10/9tutorialOnJ2Plasticity.pdf\n",
|
|||
|
|
"\n",
|
|||
|
|
"Parameters\n",
|
|||
|
|
"----------\n",
|
|||
|
|
" σ: Array{Float64, 6}\n",
|
|||
|
|
" Stress in Voigt notation\n",
|
|||
|
|
" k: Float64\n",
|
|||
|
|
" Material constant, Yield limit\n",
|
|||
|
|
"\n",
|
|||
|
|
"Returns\n",
|
|||
|
|
"-------\n",
|
|||
|
|
" Float\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"function vonMisesYield(σ, k)\n",
|
|||
|
|
" σₑ(σ) - k\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"Function for NLsolve. Inside this function are the functions where we want to find root.\n",
|
|||
|
|
"Ψ is the yield function below. Functions defined here:\n",
|
|||
|
|
"\n",
|
|||
|
|
" dσ - C (dϵ - dλ*dΨ/dσ) = 0\n",
|
|||
|
|
" σₑ(σ) - k = 0\n",
|
|||
|
|
"\n",
|
|||
|
|
"Parameters\n",
|
|||
|
|
"----------\n",
|
|||
|
|
" params: Array{Float64, 7}\n",
|
|||
|
|
" Array containing values from solver\n",
|
|||
|
|
" dϵ: Array{Float64, 6}\n",
|
|||
|
|
" Strain rate vector in Voigt notation\n",
|
|||
|
|
" C: Array{Float64, (6, 6)}\n",
|
|||
|
|
" Material tensor\n",
|
|||
|
|
" k: Float\n",
|
|||
|
|
" Material constant, yield limit\n",
|
|||
|
|
" Δt: Float\n",
|
|||
|
|
" time increment\n",
|
|||
|
|
" σ_begin:Array{Float64, 6}\n",
|
|||
|
|
" Stress vector in Voigt notation\n",
|
|||
|
|
"\n",
|
|||
|
|
"Returns\n",
|
|||
|
|
"-------\n",
|
|||
|
|
" Array{Float64, 7}, return values for solver\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"function G(params, dϵ, C, k, Δt, σ_begin)\n",
|
|||
|
|
" # Creating wrapper for gradient\n",
|
|||
|
|
" yield(pars) = vonMisesYield(pars, k)\n",
|
|||
|
|
" dfdσ = ForwardDiff.gradient(yield)\n",
|
|||
|
|
" \n",
|
|||
|
|
" # Stress rate\n",
|
|||
|
|
" dσ = (σ_begin - params[1:6]) / Δt\n",
|
|||
|
|
" \n",
|
|||
|
|
" # Calculating plastic strain rate\n",
|
|||
|
|
" dϵp = params[end] * dfdσ(params)\n",
|
|||
|
|
" \n",
|
|||
|
|
" # Calculating equations\n",
|
|||
|
|
" function_1 = dσ - C * (dϵ - dϵp[1:6])\n",
|
|||
|
|
" function_2 = yield(params)\n",
|
|||
|
|
" [function_1[:]; function_2]\n",
|
|||
|
|
"end\n",
|
|||
|
|
"\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"Function which calculates the stress. Also handles if any yielding happens\n",
|
|||
|
|
"\n",
|
|||
|
|
"Parameters\n",
|
|||
|
|
"----------\n",
|
|||
|
|
" dϵ: Array{Float64, 6}\n",
|
|||
|
|
" Strain rate vector in Voigt notation\n",
|
|||
|
|
" Δt: Float\n",
|
|||
|
|
" time increment\n",
|
|||
|
|
" σ: Array{Float64, 6}\n",
|
|||
|
|
" Last stress vector in Voigt notation\n",
|
|||
|
|
" C: Array{Float64, (6, 6)}\n",
|
|||
|
|
" Material tensor\n",
|
|||
|
|
" k: Float\n",
|
|||
|
|
" Material constant, yield limit\n",
|
|||
|
|
"\n",
|
|||
|
|
"Returns\n",
|
|||
|
|
"-------\n",
|
|||
|
|
" Tuple\n",
|
|||
|
|
" Plastic strain rate dϵᵖ and new stress vector σ\n",
|
|||
|
|
"\"\"\"\n",
|
|||
|
|
"function calculate_stress(dϵ, Δt, σ, C, k)\n",
|
|||
|
|
"\n",
|
|||
|
|
" # Test stress\n",
|
|||
|
|
" σ_tria = σ + Δt * C * dϵ\n",
|
|||
|
|
" \n",
|
|||
|
|
" # Calculating yield\n",
|
|||
|
|
" yield = vonMisesYield(σ_tria, k)\n",
|
|||
|
|
" \n",
|
|||
|
|
" if yield > 1\n",
|
|||
|
|
" \n",
|
|||
|
|
" # Yielding happened\n",
|
|||
|
|
" # Creating functions for newton: xₙ₊₁ = xₙ - df⁻¹ * f and initial values\n",
|
|||
|
|
" initial_guess = [σ[:]; 0.0]\n",
|
|||
|
|
" f(σ_) = G(σ_, dϵ, C, k, Δt, σ_tria)\n",
|
|||
|
|
" df = ForwardDiff.jacobian(f)\n",
|
|||
|
|
" \n",
|
|||
|
|
" # Calculating root \n",
|
|||
|
|
" result = nlsolve(not_in_place(f, df), initial_guess).zero\n",
|
|||
|
|
" \n",
|
|||
|
|
" σ = result[1:6] \n",
|
|||
|
|
" dλ = result[end]\n",
|
|||
|
|
" \n",
|
|||
|
|
" # these are not necessary to calculate for our calculation but why not?\n",
|
|||
|
|
" yield_f(σ_) = vonMisesYield(σ_, k)\n",
|
|||
|
|
" dfdσ_ = ForwardDiff.gradient(yield_f)\n",
|
|||
|
|
" dϵᵖ = dλ * dfdσ_(σ)\n",
|
|||
|
|
" else\n",
|
|||
|
|
" dϵᵖ = zeros(Float64, (6, 1))\n",
|
|||
|
|
" σ = σ_tria\n",
|
|||
|
|
" end\n",
|
|||
|
|
" return (dϵᵖ, σ)\n",
|
|||
|
|
"end"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"# Defining strain history"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 4,
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": false
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"linspace(-0.0,-0.0006,200)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 4,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"steps = 200\n",
|
|||
|
|
"\n",
|
|||
|
|
"ϵ_tot = zeros(Float64, (steps, 6))\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Adding only strain in x-axis and counting for the poisson effect\n",
|
|||
|
|
"ϵ_tot[:, 1] = linspace(0, 0.002, steps)\n",
|
|||
|
|
"ϵ_tot[:, 2] = linspace(0, 0.002, steps).*-ν\n",
|
|||
|
|
"ϵ_tot[:, 3] = linspace(0, 0.002, steps).*-ν"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {},
|
|||
|
|
"source": [
|
|||
|
|
"# Simulation\n",
|
|||
|
|
"\n",
|
|||
|
|
"Ok, we're good to go! Now we just need to define yield limit and the main loop.\n",
|
|||
|
|
"\n",
|
|||
|
|
"This simulation is not time dependent, but since it's already defined in the equations we'll give it value 1"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": false,
|
|||
|
|
"scrolled": false
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "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
|
|||
|
|
"text/plain": [
|
|||
|
|
"PyPlot.Figure(PyObject <matplotlib.figure.Figure object at 0x7f31a0ba76d0>)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"(0,250)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"ϵ_last = zeros(Float64, (6))\n",
|
|||
|
|
"ϵᵖ = zeros(Float64, (6))\n",
|
|||
|
|
"σ = zeros(Float64, (6, 1))\n",
|
|||
|
|
"Δt = 1.0\n",
|
|||
|
|
"σy = 200.0\n",
|
|||
|
|
"ss = zeros(Float64, steps)\n",
|
|||
|
|
"ee = zeros(Float64, steps)\n",
|
|||
|
|
"for i=1:steps\n",
|
|||
|
|
" ss[i] = σ[1]\n",
|
|||
|
|
" ee[i] = ϵ_last[1]\n",
|
|||
|
|
" dϵ = (reshape(ϵ_tot[i, :, :], (6, 1)) - ϵ_last) / Δt \n",
|
|||
|
|
" dϵᵖ, σ = calculate_stress(dϵ, Δt, σ, C, σy)\n",
|
|||
|
|
" ϵ_last += dϵ * Δt\n",
|
|||
|
|
"end\n",
|
|||
|
|
"PyPlot.plot(ee, ss)\n",
|
|||
|
|
"PyPlot.title(\"Stress-Strain curve\")\n",
|
|||
|
|
"PyPlot.xlabel(\"Strain\")\n",
|
|||
|
|
"PyPlot.ylabel(\"Stress\")\n",
|
|||
|
|
"PyPlot.ylim([0, 250])"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": null,
|
|||
|
|
"metadata": {
|
|||
|
|
"collapsed": true
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": []
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"metadata": {
|
|||
|
|
"kernelspec": {
|
|||
|
|
"display_name": "Julia 0.5.0-dev",
|
|||
|
|
"language": "julia",
|
|||
|
|
"name": "julia-0.5"
|
|||
|
|
},
|
|||
|
|
"language_info": {
|
|||
|
|
"file_extension": ".jl",
|
|||
|
|
"mimetype": "application/julia",
|
|||
|
|
"name": "julia",
|
|||
|
|
"version": "0.5.0"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"nbformat": 4,
|
|||
|
|
"nbformat_minor": 0
|
|||
|
|
}
|