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 1 ksteube 1811 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 ksteube 1316 % 4 jfenwick 2548 % Copyright (c) 2003-2009 by University of Queensland 5 ksteube 1811 % Earth Systems Science Computational Center (ESSCC) 6 7 gross 625 % 8 ksteube 1811 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 gross 625 % 12 ksteube 1811 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 ksteube 1316 14 ksteube 1811 15 gross 578 \section{Elastic Deformation} 16 \label{ELASTIC CHAP} 17 ksteube 1316 In this section we want to examine the deformation of a linear elastic body caused by expansion through a heat distribution. We want 18 a displacement field $u\hackscore{i}$ which solves the momentum equation 19 gross 578 \index{momentum equation}: 20 gross 579 \begin{eqnarray}\label{HEATEDBLOCK general problem} 21 gross 578 - \sigma\hackscore{ij,j}=0 22 \end{eqnarray} 23 where the stress $\sigma$ is given by 24 gross 579 \begin{eqnarray}\label{HEATEDBLOCK linear elastic} 25 gross 578 \sigma\hackscore{ij}= \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i}) 26 - (\lambda+\frac{2}{3} \mu) \; \alpha \; (T-T\hackscore{ref})\delta\hackscore{ij} \;. 27 \end{eqnarray} 28 In this formula $\lambda$ and $\mu$ are the Lame coefficients, $\alpha$ is the 29 temperature expansion coefficient, $T$ is the temperature distribution and $T_{ref}$ a reference temperature. Note that 30 jfenwick 2584 \eqn{HEATEDBLOCK general problem} is similar to \eqn{WAVE general problem} introduced in section~\Sec{WAVE CHAP} but the 31 gross 578 inertia term $\rho u\hackscore{i,tt}$ has been dropped as we assume a static scenario here. Moreover, in 32 comparison to the \eqn{WAVE stress} 33 gross 579 definition of stress $\sigma$ in \eqn{HEATEDBLOCK linear elastic} an extra term is introduced 34 woo409 757 to bring in stress due to volume changes trough temperature dependent expansion. 35 gross 568 36 gross 578 Our domain is the unit cube 37 gross 593 \begin{eqnarray} \label{HEATEDBLOCK natural location} 38 jfenwick 2584 \Omega=\{(x\hackscore{i}) | 0 \le x\hackscore{i} \le 1 \} 39 gross 578 \end{eqnarray} 40 On the boundary the normal stress component is set to zero 41 gross 579 \begin{eqnarray} \label{HEATEDBLOCK natural} 42 gross 578 \sigma\hackscore{ij}n\hackscore{j}=0 43 \end{eqnarray} 44 and on the face with $x\hackscore{i}=0$ we set the $i$-th component of the displacement to $0$ 45 gross 579 \begin{eqnarray} \label{HEATEDBLOCK constraint} 46 gross 578 u\hackscore{i}(x)=0 & \mbox{ where } & x\hackscore{i}=0 \; 47 \end{eqnarray} 48 For the temperature distribution we use 49 gross 579 \begin{eqnarray} \label{HEATEDBLOCK temperature} 50 T(x)= T\hackscore{0} e^{-\beta \|x-x^{c}\|}; 51 gross 578 \end{eqnarray} 52 gross 2371 with a given positive constant $\beta$ and location $x^{c}$ in the domain. 53 ksteube 1316 54 gross 2371 %Later in \Sec{MODELFRAME} we will use 55 % $T$ from a time-dependent temperature diffusion problem as discussed in \Sec{DIFFUSION CHAP}. 56 57 lgraham 1700 When we insert~\eqn{HEATEDBLOCK linear elastic} we get a second order system of linear PDEs for the displacements $u$ which is called 58 gross 578 the Lame equation\index{Lame equation}. We want to solve 59 this using the \LinearPDE class to this. For a system of PDEs and a solution with several components the \LinearPDE class 60 takes PDEs of the form 61 gross 625 \begin{equation}\label{LINEARPDE.SYSTEM.1 TUTORIAL} 62 gross 2513 -(A\hackscore{ijkl} u\hackscore{k,l})\hackscore{,j}=-X\hackscore{ij,j} \; . 63 gross 568 \end{equation} 64 gross 625 $A$ is of \RankFour and $X$ is of \RankTwo. We show here the coefficients relevant 65 ksteube 1316 for the we trying to solve. The full form is given in~\eqn{LINEARPDE.SYSTEM.1}. 66 gross 568 The natural boundary conditions \index{boundary condition!natural} take the form: 67 gross 625 \begin{equation}\label{LINEARPDE.SYSTEM.2 TUTORIAL} 68 n\hackscore{j} A\hackscore{ijkl} u\hackscore{k,l}=n\hackscore{j}X\hackscore{ij} \;. 69 gross 568 \end{equation} 70 gross 625 Constraints \index{constraint} take the form 71 \begin{equation}\label{LINEARPDE.SYSTEM.3 TUTORIAL} 72 gross 568 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 73 \end{equation} 74 gross 578 $r$ and $q$ are each \RankOne. 75 gross 625 We can easily identify the coefficients in~\eqn{LINEARPDE.SYSTEM.1 TUTORIAL}: 76 gross 578 \begin{eqnarray}\label{LINEARPDE ELASTIC COEFFICIENTS} 77 A\hackscore{ijkl}=\lambda \delta\hackscore{ij} \delta\hackscore{kl} + \mu ( 78 lgraham 1746 \delta\hackscore{ik} \delta\hackscore{jl} 79 + \delta\hackscore{il} \delta\hackscore{jk}) \\ 80 gross 578 X\hackscore{ij}=(\lambda+\frac{2}{3} \mu) \; \alpha \; (T-T\hackscore{ref})\delta\hackscore{ij} \\ 81 \end{eqnarray} 82 The characteristic function $q$ defining the locations and components where constraints are set is given by: 83 gross 579 \begin{equation}\label{HEATEDBLOCK MASK} 84 gross 578 q\hackscore{i}(x)=\left\{ 85 \begin{array}{cl} 86 1 & x\hackscore{i}=0 \\ 87 0 & \mbox{otherwise} \\ 88 \end{array} 89 \right. 90 \end{equation} 91 Under the assumption that $\lambda$, $\mu$, $\beta$ and $T\hackscore{ref}$ 92 jfenwick 2584 are constant we may use $Y\hackscore{i}=(\lambda+\frac{2}{3} \mu) \; \alpha \; T\hackscore{i}$. However, 93 gross 578 this choice would lead to a different natural boundary condition which does not set the normal stress component as defined 94 gross 579 in~\eqn{HEATEDBLOCK linear elastic} to zero. 95 gross 578 96 gross 625 Analogously to concept of symmetry for a single PDE, we call the PDE defined by~\eqn{LINEARPDE.SYSTEM.1 TUTORIAL} symmetric if 97 gross 578 \index{symmetric PDE} 98 gross 625 \begin{eqnarray}\label{LINEARPDE.SYSTEM.SYMMETRY TUTORIAL} 99 gross 578 A\hackscore{ijkl} =A\hackscore{klij} \\ 100 gross 568 \end{eqnarray} 101 ksteube 1316 This Lame equation is in fact symmetric, given the difference in $D$ and $d$ as compared to the scalar case. 102 The \LinearPDE class is notified of this fact by calling its \method{setSymmetryOn} method. 103 gross 568 104 gross 579 After we have solved the Lame equation we want to analyse the actual stress distribution. Typically the von--Mises stress\index{von--Mises stress} defined by 105 \begin{equation} 106 \sigma\hackscore{mises} = \sqrt{ 107 gross 2513 \frac{1}{2} ((\sigma\hackscore{00}-\sigma\hackscore{11})^2 108 lgraham 1746 + (\sigma\hackscore{11}-\sigma\hackscore{22})^2 109 + (\sigma\hackscore{22}-\sigma\hackscore{00})^2) 110 gross 2513 + 3( \sigma\hackscore{01}^2+\sigma\hackscore{12}^2+\sigma\hackscore{20}^2) } 111 gross 579 \end{equation} 112 ksteube 1316 is used to detect material damage. Here we want to calculate the von--Mises and write the stress to a file for visualization. 113 gross 568 114 ksteube 1316 \index{scripts!\file{diffusion.py}} 115 The following script, which is available in \file{heatedbox.py} in the \ExampleDirectory, solves the Lame equation 116 gross 579 and writes the displacements and the von--Mises stress\index{von--Mises stress} into a file \file{deform.xml} in the \VTK file format: 117 \begin{python} 118 from esys.escript import * 119 from esys.escript.linearPDEs import LinearPDE 120 from esys.finley import Brick 121 #... set some parameters ... 122 lam=1. 123 mu=0.1 124 alpha=1.e-6 125 xc=[0.3,0.3,1.] 126 beta=8. 127 T_ref=0. 128 T_0=1. 129 #... generate domain ... 130 mydomain = Brick(l0=1.,l1=1., l2=1.,n0=10, n1=10, n2=10) 131 x=mydomain.getX() 132 #... set temperature ... 133 T=T_0*exp(-beta*length(x-xc)) 134 #... open symmetric PDE ... 135 mypde=LinearPDE(mydomain) 136 mypde.setSymmetryOn() 137 #... set coefficients ... 138 C=Tensor4(0.,Function(mydomain)) 139 for i in range(mydomain.getDim()): 140 for j in range(mydomain.getDim()): 141 C[i,i,j,j]+=lam 142 C[j,i,j,i]+=mu 143 C[j,i,i,j]+=mu 144 msk=whereZero(x)*[1.,0.,0.] \ 145 +whereZero(x)*[0.,1.,0.] \ 146 +whereZero(x)*[0.,0.,1.] 147 sigma0=(lam+2./3.*mu)*alpha*(T-T_ref)*kronecker(mydomain) 148 mypde.setValue(A=C,X=sigma0,q=msk) 149 #... solve pde ... 150 u=mypde.getSolution() 151 #... calculate von-Misses stress 152 g=grad(u) 153 sigma=mu*(g+transpose(g))+lam*trace(g)*kronecker(mydomain)-sigma0 154 sigma_mises=sqrt(((sigma[0,0]-sigma[1,1])**2+(sigma[1,1]-sigma[2,2])**2+ \ 155 gross 2513 (sigma[2,2]-sigma[0,0])**2)/2. \ 156 +3*(sigma[0,1]**2 + sigma[1,2]**2 + sigma[2,0]**2)) 157 gross 579 #... output ... 158 saveVTK("deform.xml",disp=u,stress=sigma_mises) 159 \end{python} 160 161 \begin{figure} 162 jfenwick 2335 \centerline{\includegraphics[width=\figwidth]{figures/HeatedBlock}} 163 gross 579 \caption{von--Mises Stress and Displacement Vectors.} 164 \label{HEATEDBLOCK FIG 2} 165 \end{figure} 166 167 Finally the the results can be visualize by calling 168 \begin{python} 169 mayavi -d deform.xml -f CellToPointData -m VelocityVector -m SurfaceMap & 170 \end{python} 171 Note that the filter \text{CellToPointData} is applied to create smooth representation of the 172 von--Mises stress. \fig{HEATEDBLOCK FIG 2} shows the results where the vertical planes showing the 173 von--Mises stress and the horizontal plane shows the vector representing displacements. 174

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