# Elasticity

Consider an elastic plate with undeformed shape $\Omega\times ]-h,h[$ in $\R^3$, $\Omega\subset\R^2$.

By the deformation of the plate, we assume that a point $P(x_1,x_2,x_3)$ moves to ${\cal P}(\xi_1,\xi_2,\xi_3)$. The vector $\mathbf{u}=(u_1,u_2,u_3)=(\xi_1-x_1,\xi_2-x_2,\xi_3-x_3)$ is called the displacement vector.

By the deformation, the line segment $\overline{\mathbf{x},\mathbf{x}+\tau\Delta\mathbf{x}}$ moves approximately to $\overline{\mathbf{x}+\mathbf{u}(\mathbf{x}),\mathbf{x}+\tau\Delta\mathbf{x} +\mathbf{u}(\mathbf{x}+\tau\Delta\mathbf{x})}$ for small $\tau$, where $\mathbf{x}=(x_1,x_2,x_3),\, \Delta\mathbf{x} =(\Delta x_1,\Delta x_2,\Delta x_3)$.

We now calculate the ratio between two segments

\eta(\tau)=\tau^{-1}|\Delta\mathbf{x}|^{-1} \left(|\mathbf{u}(\mathbf{x}+\tau\Delta\mathbf{x}) -\mathbf{u}(\mathbf{x})+\tau\Delta\mathbf{x}|-\tau|\Delta\mathbf{x}|\right)

then we have (see e.g. NECAS2017, p.32)

\begin{eqnarray*} \lim_{\tau\to 0}\eta(\tau)=(1+2e_{ij}\nu_i\nu_j)^{1/2}-1, \quad 2e_{ij}=\frac{\p u_k}{\p x_i}\frac{\p u_k}{\p x_j}+\left(\frac{\p u_i}{\p x_j}+ \frac{\p u_j}{\p x_i}\right) \end{eqnarray*}

where $\nu_i=\Delta x_i|\Delta\mathbf{x}|^{-1}$. If the deformation is small, then we may consider that

(\p u_k/\p x_i)(\p u_k/\p x_i)\approx 0

and the following is called small strain tensor

\varepsilon_{ij}(u)=\frac{1}{2}\left(\frac{\p u_i}{\p x_j}+ \frac{\p u_j}{\p x_i}\right)

The tensor $e_{ij}$ is called finite strain tensor.

Consider the small plane $\Delta \Pi(\mathbf{x})$ centered at $\mathbf{x}$ with the unit normal direction $\matbhf{n}=(n_1,n_2,n_3)$, then the surface on $\Delta \Pi(\mathbf{x})$ at $\mathbf{x}$ is

(\sigma_{1j}(\mathbf{x})n_j, \sigma_{2j}(\mathbf{x})n_j, \sigma_{3j}(\mathbf{x})n_j)

where $\sigma_{ij}(\mathbf{x})$ is called stress tensor at $\mathbf{x}$. Hooke's law is the assumption of a linear relation between $\sigma_{ij}$ and $\varepsilon_{ij}$ such as

\sigma_{ij}(\mathbf{x})=c_{ijkl}(\mathbf{x})\varepsilon_{ij}(\mathbf{x})

with the symmetry $c_{ijkl}=c_{jikl}, c_{ijkl}=c_{ijlk}, c_{ijkl}=c_{klij}$.

If Hooke's tensor $c_{ijkl}(\mathbf{x})$ do not depend on the choice of coordinate system, the material is called isotropic at $\mathbf{x}$.

If $c_{ijkl}$ is constant, the material is called homogeneous. In homogeneous isotropic case, there is Lamé constants $\lambda, \mu$ (see e.g. NECAS2017, p.43) satisfying

\begin{eqnarray} \sigma_{ij}=\lambda\delta_{ij}\textrm{div}\mathbf{u}+2\mu \varepsilon_{ij} \end{eqnarray}

where $\delta_{ij}$ is Kronecker's delta.

We assume that the elastic plate is fixed on $\Gamma_D\times ]-h,h[,\, \Gamma_D\subset \p\Omega$. If the body force $\mathbf{f}=(f_1,f_2,f_3)$ is given in $\Omega\times]-h,h[$ and surface force $\mathbf{g}$ is given in $\Gamma_N\times]-h,h[, \Gamma_N=\p\Omega\setminus\overline{\Gamma_D}$, then the equation of equilibrium is given as follows:

We now explain the plain elasticity.

• Plain strain:

On the end of plate, the contact condition $u_3=0,\, g_3=$ is satisfied.

In this case, we can suppose that $f_3=g_3=u_3=0$ and $\mathbf{u}(x_1,x_2,x_3)=\overline{u}(x_1,x_2)$ for all $-h.

• Plain stress:

The cylinder is assumed to be very thin and subjected to no load on the ends $x_3=\pm h$, that is,

The assumption leads that $\sigma_{3i}=0$ in $\Omega\times ]-h,h[$ and $\mathbf{u}(x_1,x_2,x_3)=\overline{u}(x_1,x_2)$ for all $-h.

• Generalized plain stress:

The cylinder is subjected to no load at $x_3=\pm h$. Introducing the mean values with respect to thickness,

\overline{u}_i(x_1,x_2)=\frac{1}{2h} \int_{-h}^h u(x_1,x_2,x_3)dx_3

and we derive $\overline{u}_3\equiv 0$. Similarly we define the mean values $\overline{f},\overline{g}$ of the body force and surface force as well as the mean values $\overline{\varepsilon}_{ij}$ and $\overline{\sigma}_{ij}$ of the components of stress and strain, respectively.

In what follows we omit the overlines of $\overline{u}, \overline{f},\overline{g}, \overline{\varepsilon}_{ij}$ and $\overline{\varepsilon}_{ij}$. Then we obtain similar equation of equilibrium given in \eqref{eqn::elasticity} replacing $\Omega\times ]-h,h[$ with $\Omega$ and changing $i=1,2$. In the case of plane stress, $\sigma_{ij}=\lambda^* \delta_{ij}\textrm{div}u+2\mu\varepsilon_{ij}, \lambda^*=(2\lambda \mu)/(\lambda+\mu)$.

The equations of elasticity are naturally written in variational form for the displacement vector $\mathbf{u}(\mathbf{x})\in V$ as

\int_\Omega [2\mu\epsilon_{ij}(\mathbf{u})\epsilon_{ij}(\mathbf{v}) +\lambda \epsilon_{ii}(\mathbf{u})\epsilon_{jj}(\mathbf{v})] =\int_\Omega \mathbf{f}\cdot \mathbf{v} +\int_\Gamma \mathbf{g}\cdot \mathbf{v}, \forall \mathbf{v}\in V

where $V$ is the linear closed subspace of $H^1(\Omega)^2$.

Beam

Consider an elastic plate with the undeformed rectangle shape $]0,10[\times ]0,2[$. The body force is the gravity force $\mathbf{f}$ and the boundary force $\mathbf{g}$ is zero on lower and upper side. On the two vertical sides of the beam are fixed.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 // Parameters real E = 21.5; real sigma = 0.29; real gravity = -0.05; // Mesh border a(t=2, 0){x=0; y=t; label=1;} border b(t=0, 10){x=t; y=0; label=2;} border c(t=0, 2){ x=10; y=t; label=1;} border d(t=0, 10){ x=10-t; y=2; label=3;} mesh th = buildmesh(b(20) + c(5) + d(20) + a(5)); // Fespace fespace Vh(th, [P1, P1]); Vh [uu, vv]; Vh [w, s]; // Macro real sqrt2 = sqrt(2.); macro epsilon(u1, u2) [dx(u1), dy(u2), (dy(u1)+dx(u2))/sqrt2] // macro div(u,v) (dx(u) + dy(v)) // // Problem real mu = E/(2*(1+sigma)); real lambda = E*sigma/((1+sigma)*(1-2*sigma)); solve Elasticity ([uu, vv], [w, s]) = int2d(th)( lambda*div(w,s)*div(uu,vv) + 2.*mu*( epsilon(w,s)'*epsilon(uu,vv) ) ) + int2d(th)( - gravity*s ) + on(1, uu=0, vv=0) ; // Plot plot([uu, vv], wait=true); plot([uu,vv], wait=true, bb=[[-0.5, 2.5], [2.5, -0.5]]); // Movemesh mesh th1 = movemesh(th, [x+uu, y+vv]); plot(th1, wait=true); 

Beam 3D

Consider elastic box with the undeformed parallelepiped shape $]0,5[\times ]0,1[\times]0,1[$. The body force is the gravity force $\mathbf{f}$ and the boundary force $\mathbf{g}$ is zero on all face except one the one vertical left face where the beam is fixed.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 include "cube.idp" // Parameters int[int] Nxyz = [20, 5, 5]; real [int, int] Bxyz = [[0., 5.], [0., 1.], [0., 1.]]; int [int, int] Lxyz = [[1, 2], [2, 2], [2, 2]]; real E = 21.5e4; real sigma = 0.29; real gravity = -0.05; // Mesh mesh3 Th = Cube(Nxyz, Bxyz, Lxyz); // Fespace fespace Vh(Th, [P1, P1, P1]); Vh [u1, u2, u3], [v1, v2, v3]; // Macro real sqrt2 = sqrt(2.); macro epsilon(u1, u2, u3) [ dx(u1), dy(u2), dz(u3), (dz(u2) + dy(u3))/sqrt2, (dz(u1) + dx(u3))/sqrt2, (dy(u1) + dx(u2))/sqrt2] // macro div(u1, u2, u3) (dx(u1) + dy(u2) + dz(u3)) // // Problem real mu = E/(2*(1+sigma)); real lambda = E*sigma/((1+sigma)*(1-2*sigma)); solve Lame ([u1, u2, u3], [v1, v2, v3]) = int3d(Th)( lambda*div(u1, u2, u3)*div(v1, v2, v3) + 2.*mu*( epsilon(u1, u2, u3)'*epsilon(v1, v2, v3) ) ) - int3d(Th)( gravity*v3 ) + on(1, u1=0, u2=0, u3=0) ; // Display real dmax = u1[].max; cout << "max displacement = " << dmax << endl; // Movemesh real coef = 0.1/dmax; int[int] ref2 = [1, 0, 2, 0]; mesh3 Thm = movemesh3(Th, transfo=[x+u1*coef, y+u2*coef, z+u3*coef], label=ref2); Thm = change(Thm, label=ref2); // Plot plot(Th, Thm, wait=true, cmm="coef amplification = "+coef); 
3d Beam deformed and undeformed box

## Fracture Mechanics#

Consider the plate with the crack whose undeformed shape is a curve $\Sigma$ with the two edges $\gamma_1,\, \gamma_2$.

We assume the stress tensor $\sigma_{ij}$ is the state of plate stress regarding $(x,y)\in \Omega_{\Sigma}=\Omega\setminus \Sigma$. Here $\Omega$ stands for the undeformed shape of elastic plate without crack.

If the part $\Gamma_N$ of the boundary $\p\Omega$ is fixed and a load ${\cal L}=(\mathbf{f},\mathbf{g})\in L^2(\Omega)^2\times L^2(\Gamma_N)^2$ is given, then the displacement $\mathbf{u}$ is the minimizer of the potential energy functional

{\cal E}(\mathbf{v};{\cal L},\Omega_{\Sigma}) =\int_{\Omega_{\Sigma}} \{w(x,\mathbf{v})-\mathbf{f}\cdot \mathbf{v}\} -\int_{\Gamma_N}\mathbf{g}\cdot \mathbf{v}\

over the functional space $V(\Omega_{\Sigma})$,

V(\Omega_{\Sigma}) =\left\{ \mathbf{v}\in H^1(\Omega_{\Sigma})^2;\; \mathbf{v}=0\quad \hbox{ on } \Gamma_D=\p\Omega\setminus\overline{\Gamma_N}\right\},

where $w(x,\mathbf{v})=\sigma_{ij}(\mathbf{v})\varepsilon_{ij}(\mathbf{v})/2$,

If the elasticity is homogeneous isotropic, then the displacement $\mathbf{u}(x)$ is decomposed in an open neighborhood $U_k$ of $\gamma_k$ as in (see e.g. OHTSUKA2000)

$$\mathbf{u}(x) = \sum_{l=1}^2 K_l(\gamma_k) r_k^{1/2} S^C_{kl}(\theta_k) + \mathbf{u}_{k,R}(x) \quad \mbox{for }x\in \Omega_{\Sigma}\cap U_k,\, k=1,2 \label{eqn::SIF}$$

with $\mathbf{u}_{k,R} \in H^2(\Omega_\Sigma\cap U_k)^2$, where $U_k,\, k=1,2$ are open neighborhoods of $\gamma_k$ such that $\p L_1\cap U_1=\gamma_1,\, \p L_m\cap U_2=\gamma_2$, and

\begin{eqnarray} S^C_{k1}(\theta_k) & = & \frac 1 {4\mu} \frac 1 {(2\pi)^{1/2}} \left[ \begin{array}{c} [2\kappa-1]\cos(\theta_k/2)-\cos(3\theta_k/2)\\ -[2\kappa+1]\sin(\theta_k/2)+\sin(3\theta_k/2) \end{array}\right],\\ S^C_{k2}(\theta_k) & = & \frac 1 {4\mu} \frac 1 {(2\pi)^{1/2}} \left[ \begin{array}{c} -[2\kappa-1]\sin(\theta_k/2)+3\sin(3\theta_k/2)\\ -[2\kappa+1]\cos(\theta_k/2)+\cos(3\theta_k/2) \end{array}\right]. \nonumber \end{eqnarray}

where $\mu$ is the shear modulus of elasticity, $\kappa=3-4\nu$ ($\nu$ is the Poisson's ratio) for plane strain and $\kappa=\frac {3-\nu} {1+\nu}$ for plane stress.

The coefficients $K_1(\gamma_i)$ and $K_2(\gamma_i),$ which are important parameters in fracture mechanics, are called stress intensity factors of the opening mode (mode I) and the sliding mode (mode II), respectively.

For simplicity, we consider the following simple crack

\Omega=\{(x,y):\; -1<x<1, -1<y<1\},\qquad \Sigma=\{(x,y):\; -1\le x\le 0, y=0\}

with only one crack tip $\gamma=(0,0)$. Unfortunately, FreeFem++ cannot treat crack, so we use the modification of the domain with U-shape channel (see U-shape example) with $d=0.0001$. The undeformed crack $\Sigma$ is approximated by

\begin{eqnarray*} \Sigma_d&=&\{(x,y):\; -1\le x\le -10*d, -d\le y\le d\}\\ &&\cup\{(x,y):\; -10*d\le x\le 0, -d+0.1*x\le y\le d-0.1*x\} \end{eqnarray*}

and $\Gamma_D=$R in U-shape figure.

In this example, we use three technique:

• Fast Finite Element Interpolator from the mesh Th to Zoom for the scale-up of near $\gamma$.

• After obtaining the displacement vector $\mathbf{u}=(u,v)$, we shall watch the deformation of the crack near $\gamma$ as follows,

 1 2 mesh Plate = movemesh(Zoom, [x+u, y+v]); plot(Plate); 
• Adaptivity is an important technique here, because a large singularity occurs at $\gamma$ as shown in \eqref{eqn::SIF}.

The first example creates mode I deformation by the opposed surface force on B and T in the vertical direction of $\Sigma$, and the displacement is fixed on R.

In a laboratory, fracture engineers use photoelasticity to make stress field visible, which shows the principal stress difference

\begin{eqnarray} \sigma_1-\sigma_2=\sqrt{(\sigma_{11}-\sigma_{22})^2+4\sigma_{12}^2} \end{eqnarray}

where $\sigma_1$ and $\sigma_2$ are the principal stresses.

In opening mode, the photoelasticity make symmetric pattern concentrated at $\gamma$.

(Crack Opening, $K_2(\gamma)=0$)

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 // Parameters real d = 0.0001; int n = 5; real cb = 1, ca = 1, tip = 0.0; real E = 21.5; real sigma = 0.29; // Mesh border L1(t=0, ca-d){x=-cb; y=-d-t;} border L2(t=0, ca-d){x=-cb; y=ca-t;} border B(t=0, 2){x=cb*(t-1); y=-ca;} border C1(t=0, 1){x=-ca*(1-t)+(tip-10*d)*t; y=d;} border C21(t=0, 1){x=(tip-10*d)*(1-t)+tip*t; y=d*(1-t);} border C22(t=0, 1){x=(tip-10*d)*t+tip*(1-t); y=-d*t;} border C3(t=0, 1){x=(tip-10*d)*(1-t)-ca*t; y=-d;} border C4(t=0, 2*d){x=-ca; y=-d+t;} border R(t=0, 2){x=cb; y=cb*(t-1);} border T(t=0, 2){x=cb*(1-t); y=ca;} mesh Th = buildmesh(L1(n/2) + L2(n/2) + B(n) + C1(n) + C21(3) + C22(3) + C3(n) + R(n) + T(n)); plot(Th, wait=true); cb=0.1; ca=0.1; mesh Zoom = buildmesh(L1(n/2) + L2(n/2) + B(n) + C1(n) + C21(3) + C22(3) + C3(n) + R(n) + T(n)); plot(Zoom, wait=true); // Fespace fespace Vh(Th, [P2, P2]); Vh [u, v]; Vh [w, s]; fespace zVh(Zoom, P2); zVh Sx, Sy, Sxy, N; // Problem real mu = E/(2*(1+sigma)); real lambda = E*sigma/((1+sigma)*(1-2*sigma)); solve Problem ([u, v], [w, s]) = int2d(Th)( 2*mu*(dx(u)*dx(w) + ((dx(v)+dy(u))*(dx(s)+dy(w)))/4) + lambda*(dx(u) + dy(v))*(dx(w) + dy(s))/2 ) -int1d(Th, T)( 0.1*(1-x)*s ) +int1d(Th, B)( 0.1*(1-x)*s ) +on(R, u=0, v=0) ; // Loop for (int i = 1; i <= 5; i++){ mesh Plate = movemesh(Zoom, [x+u, y+v]); //deformation near gamma Sx = lambda*(dx(u) + dy(v)) + 2*mu*dx(u); Sy = lambda*(dx(u) + dy(v)) + 2*mu*dy(v); Sxy = mu*(dy(u) + dx(v)); N = 0.1*1*sqrt((Sx-Sy)^2 + 4*Sxy^2); //principal stress difference if (i == 1){ plot(Plate, bw=1); plot(N, bw=1); } else if (i == 5){ plot(Plate, bw=1); plot(N, bw=1); break; } // Adaptmesh Th = adaptmesh(Th, [u, v]); // Solve Problem; } 
Fig. 13: Crack open displacement (COD) and Principal stress difference in the first mesh Fig. 14: COD and Principal stress difference in the last adaptive mesh

It is difficult to create mode II deformation by the opposed shear force on B and T that is observed in a laboratory. So we use the body shear force along $\Sigma$, that is, the $x$-component $f_1$ of the body force $\mathbf{f}$ is given by

f_1(x,y)=H(y-0.001)*H(0.1-y)-H(-y-0.001)*H(y+0.1)

where $H(t)=1$ if $t>0$; $= 0$ if $t<0$.

Crack Sliding, $K_2(\gamma)=0$

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 // Parameters real d = 0.0001; int n = 5; real cb = 1, ca = 1, tip = 0.0; real E = 21.5; real sigma = 0.29; // Mesh border L1(t=0, ca-d){x=-cb; y=-d-t;} border L2(t=0, ca-d){x=-cb; y=ca-t;} border B(t=0, 2){x=cb*(t-1); y=-ca;} border C1(t=0, 1){x=-ca*(1-t)+(tip-10*d)*t; y=d;} border C21(t=0, 1){x=(tip-10*d)*(1-t)+tip*t; y=d*(1-t);} border C22(t=0, 1){x=(tip-10*d)*t+tip*(1-t); y=-d*t;} border C3(t=0, 1){x=(tip-10*d)*(1-t)-ca*t; y=-d;} border C4(t=0, 2*d){x=-ca; y=-d+t;} border R(t=0, 2){x=cb; y=cb*(t-1);} border T(t=0, 2){x=cb*(1-t); y=ca;} mesh Th = buildmesh(L1(n/2) + L2(n/2) + B(n) + C1(n) + C21(3) + C22(3) + C3(n) + R(n) + T(n)); plot(Th, wait=true); cb=0.1; ca=0.1; mesh Zoom = buildmesh(L1(n/2) + L2(n/2) + B(n) + C1(n) + C21(3) + C22(3) + C3(n) + R(n) + T(n)); plot(Zoom, wait=true); // Fespace fespace Vh(Th, [P2, P2]); Vh [u, v]; Vh [w, s]; fespace zVh(Zoom, P2); zVh Sx, Sy, Sxy, N; fespace Vh1(Th,P1); Vh1 fx = ((y>0.001)*(y<0.1))-((y<-0.001)*(y>-0.1)); // Problem real mu = E/(2*(1+sigma)); real lambda = E*sigma/((1+sigma)*(1-2*sigma)); solve Problem ([u, v], [w, s]) = int2d(Th)( 2*mu*(dx(u)*dx(w) + ((dx(v) + dy(u))*(dx(s)+ dy(w)))/4) + lambda*(dx(u) + dy(v))*(dx(w) + dy(s))/2 ) -int2d(Th)( fx*w ) +on(R, u=0, v=0) ; // Loop for (int i = 1; i <= 3; i++){ mesh Plate = movemesh(Zoom, [x+u, y+v]); //deformation near gamma Sx = lambda*(dx(u) + dy(v)) + 2*mu*dx(u); Sy = lambda*(dx(u) + dy(v)) + 2*mu*dy(v); Sxy = mu*(dy(u) + dx(v)); N = 0.1*1*sqrt((Sx-Sy)^2 + 4*Sxy^2); //principal stress difference if (i == 1){ plot(Plate, bw=1); plot(N, bw=1); } else if (i == 3) { plot(Plate, bw=1); plot(N, bw=1); break; } // Adaptmesh Th=adaptmesh(Th, [u, v]); // Solve Problem; } 
Fig. 15: (COD) and Principal stress difference in the first mesh Fig. 16: COD and Principal stress difference in the last adaptive mesh

## References#

[NECAS2017] NECAS, Jindrich et HLAVÁCEK, Ivan. Mathematical theory of elastic and elasto-plastic bodies: an introduction. Elsevier, 2017.

[OHTSUKA2000] OHTSUKA, K. Theoretical and Numerical analysis of energy release rate in 2D fracture. INFORMATION, 2000, vol. 3, p. 303-315.