Applied Elasticity & Plasticity

Cheatsheet Content

1. Stress and Strain Tensors Stress Tensor ($\sigma_{ij}$) Definition: Represents the internal forces acting within a deformable body. Symmetric Tensor: $\sigma_{ij} = \sigma_{ji}$ (3x3 matrix with 6 independent components). Matrix form: $$ \sigma_{ij} = \begin{pmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{pmatrix} $$ Traction Vector ($\vec{T}$): Force per unit area acting on a surface with normal $\vec{n}$. $\vec{T} = \sigma \cdot \vec{n}$ or $T_i = \sigma_{ij} n_j$ Principal Stresses: Eigenvalues of the stress tensor. Represent normal stresses on planes where shear stresses are zero. Principal Directions: Eigenvectors corresponding to principal stresses. Maximum Shear Stress: $\tau_{max} = \frac{\sigma_{max} - \sigma_{min}}{2}$ Von Mises Equivalent Stress ($\sigma_v$): Used in plasticity theory to predict yielding. $\sigma_v = \sqrt{\frac{1}{2}[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2]}$ or for 3D state: $\sigma_v = \sqrt{\frac{1}{2}[(\sigma_{xx}-\sigma_{yy})^2 + (\sigma_{yy}-\sigma_{zz})^2 + (\sigma_{zz}-\sigma_{xx})^2 + 6(\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)]}$ Strain Tensor ($\epsilon_{ij}$) Definition: Represents the deformation of a material. Symmetric Tensor: $\epsilon_{ij} = \epsilon_{ji}$ (3x3 matrix with 6 independent components). Components: $\epsilon_{xx} = \frac{\partial u}{\partial x}$, $\epsilon_{yy} = \frac{\partial v}{\partial y}$, $\epsilon_{zz} = \frac{\partial w}{\partial z}$ (Normal strains) $\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}$, $\gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}$, $\gamma_{zx} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}$ (Engineering shear strains) $\epsilon_{xy} = \frac{1}{2}\gamma_{xy}$, $\epsilon_{yz} = \frac{1}{2}\gamma_{yz}$, $\epsilon_{zx} = \frac{1}{2}\gamma_{zx}$ (Tensorial shear strains) Principal Strains: Eigenvalues of the strain tensor. Tensor Transformations Stress Transformation: $\sigma'_{ij} = Q_{ik} Q_{jl} \sigma_{kl}$, where $Q$ is the rotation matrix. Strain Transformation: $\epsilon'_{ij} = Q_{ik} Q_{jl} \epsilon_{kl}$ 2. Equilibrium Equations For a continuum, in the absence of body forces: $\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} = 0$ $\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} = 0$ $\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z} = 0$ With body forces $F_x, F_y, F_z$: $\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0$ $\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0$ $\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z} + F_z = 0$ 3. Indices and Tensors Kronecker Delta ($\delta_{ij}$) Definition: $\delta_{ij} = 1$ if $i=j$, and $\delta_{ij} = 0$ if $i \neq j$. Properties: $\delta_{ii} = 3$ (in 3D) $\delta_{ij} A_j = A_i$ (substitution property) $\delta_{ij} \delta_{jk} = \delta_{ik}$ Permutation Symbol ($\epsilon_{ijk}$) Definition: $\epsilon_{ijk} = 1$ if $(i,j,k)$ is an even permutation of $(1,2,3)$ $\epsilon_{ijk} = -1$ if $(i,j,k)$ is an odd permutation of $(1,2,3)$ $\epsilon_{ijk} = 0$ if any two indices are repeated Properties: $\epsilon_{ijk} \epsilon_{klm} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$ (epsilon-delta identity) $\epsilon_{ijk} \epsilon_{ijk} = 6$ Tensor Orders Zeroth-order tensor (scalar): Invariant under coordinate transformation (e.g., mass, temperature). First-order tensor (vector): Transforms with one rotation matrix (e.g., force, displacement, $\vec{a} = a_i \vec{e}_i$). Second-order tensor: Transforms with two rotation matrices (e.g., stress, strain, $\sigma_{ij}$). Fourth-order tensor: Transforms with four rotation matrices (e.g., elasticity tensor $C_{ijkl}$). 4. Airy Stress Function ($\Phi$) Used for 2D elasticity problems (plane stress/plane strain). Stress components derived from $\Phi$: $\sigma_{xx} = \frac{\partial^2 \Phi}{\partial y^2}$ $\sigma_{yy} = \frac{\partial^2 \Phi}{\partial x^2}$ $\tau_{xy} = -\frac{\partial^2 \Phi}{\partial x \partial y}$ Compatibility Equation (biharmonic equation): $\nabla^4 \Phi = \frac{\partial^4 \Phi}{\partial x^4} + 2\frac{\partial^4 \Phi}{\partial x^2 \partial y^2} + \frac{\partial^4 \Phi}{\partial y^4} = 0$ General solution for polynomial $\Phi(x,y) = A_nx^n + B_ny^n + C_nx^ky^m...$ can be used. For polar coordinates $(\rho, \theta)$: $\sigma_{\rho\rho} = \frac{1}{\rho}\frac{\partial \Phi}{\partial \rho} + \frac{1}{\rho^2}\frac{\partial^2 \Phi}{\partial \theta^2}$ $\sigma_{\theta\theta} = \frac{\partial^2 \Phi}{\partial \rho^2}$ $\tau_{\rho\theta} = -\frac{\partial}{\partial \rho}\left(\frac{1}{\rho}\frac{\partial \Phi}{\partial \theta}\right)$ Biharmonic equation in polar coordinates: $\left(\frac{\partial^2}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial}{\partial \rho} + \frac{1}{\rho^2}\frac{\partial^2}{\partial \theta^2}\right) \left(\frac{\partial^2 \Phi}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial \Phi}{\partial \rho} + \frac{1}{\rho^2}\frac{\partial^2 \Phi}{\partial \theta^2}\right) = 0$ Common solutions for $\Phi$ in polar coordinates (e.g., for a hole in a plate or concentrated load): $\Phi = A \ln \rho + B \rho^2 \ln \rho + C \rho^2 \cos(2\theta) + D \rho^4 \cos(2\theta) + E \rho \theta \sin\theta + F \rho \cos\theta + G \theta$ 5. Constitutive Laws (Hooke's Law for Isotropic Materials) Relates stress and strain. Generalized Hooke's Law: $\epsilon_x = \frac{1}{E}[\sigma_x - \nu(\sigma_y + \sigma_z)]$ $\epsilon_y = \frac{1}{E}[\sigma_y - \nu(\sigma_x + \sigma_z)]$ $\epsilon_z = \frac{1}{E}[\sigma_z - \nu(\sigma_x + \sigma_y)]$ $\gamma_{xy} = \frac{\tau_{xy}}{G}$, $\gamma_{yz} = \frac{\tau_{yz}}{G}$, $\gamma_{zx} = \frac{\tau_{zx}}{G}$ Relationship between elastic constants: $G = \frac{E}{2(1+\nu)}$ Plane Stress: $\sigma_z = \tau_{xz} = \tau_{yz} = 0$. $\epsilon_x = \frac{1}{E}(\sigma_x - \nu \sigma_y)$ $\epsilon_y = \frac{1}{E}(\sigma_y - \nu \sigma_x)$ $\epsilon_z = -\frac{\nu}{E}(\sigma_x + \sigma_y)$ Plane Strain: $\epsilon_z = \gamma_{xz} = \gamma_{yz} = 0$. $\sigma_z = \nu(\sigma_x + \sigma_y)$ $\epsilon_x = \frac{1}{E}[(1-\nu^2)\sigma_x - \nu(1+\nu)\sigma_y]$ $\epsilon_y = \frac{1}{E}[(1-\nu^2)\sigma_y - \nu(1+\nu)\sigma_x]$ 6. Compatibility Equations Ensure that a given strain field corresponds to a continuous, single-valued displacement field. For 2D (plane strain/stress): $\frac{\partial^2 \epsilon_{xx}}{\partial y^2} + \frac{\partial^2 \epsilon_{yy}}{\partial x^2} = \frac{\partial^2 \gamma_{xy}}{\partial x \partial y}$ For 3D (Saint-Venant's compatibility equations): $\frac{\partial^2 \epsilon_{xx}}{\partial y^2} + \frac{\partial^2 \epsilon_{yy}}{\partial x^2} = \frac{\partial^2 \gamma_{xy}}{\partial x \partial y}$ $\frac{\partial^2 \epsilon_{yy}}{\partial z^2} + \frac{\partial^2 \epsilon_{zz}}{\partial y^2} = \frac{\partial^2 \gamma_{yz}}{\partial y \partial z}$ $\frac{\partial^2 \epsilon_{zz}}{\partial x^2} + \frac{\partial^2 \epsilon_{xx}}{\partial z^2} = \frac{\partial^2 \gamma_{zx}}{\partial z \partial x}$ $2\frac{\partial^2 \epsilon_{xx}}{\partial y \partial z} = \frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{yz}}{\partial x} + \frac{\partial \gamma_{zx}}{\partial y} + \frac{\partial \gamma_{xy}}{\partial z}\right)$ (and cyclic permutations) 7. MATLAB for Elasticity Problems Eigenvalue/Eigenvector Calculation: % Stress tensor (example) sigma = [sxx, sxy, sxz; sxy, syy, syz; sxz, syz, szz]; [eigenvectors, eigenvalues] = eig(sigma); principal_stresses = diag(eigenvalues); % Sort principal stresses if needed [principal_stresses, idx] = sort(principal_stresses, 'descend'); principal_directions = eigenvectors(:, idx); Matrix Operations (e.g., $c = Ax$, $D = AB$): A = [A11, A12, A13; ...]; % Example matrix A x = [x1; x2; x3]; % Example vector x c = A * x; % Matrix-vector multiplication B = [B11, B12, B13; ...]; % Example matrix B D = A * B; % Matrix-matrix multiplication Index Notation (e.g., $c_i = A_{ij}x_j$, $D_{ik} = A_{ij}B_{jk}$): % For c_i = A_ij x_j (equivalent to A*x) c_matlab = A * x; % MATLAB handles this directly % For D_ik = A_ij B_jk (equivalent to A*B) D_matlab = A * B; % MATLAB handles this directly % Manual index notation (for understanding, not efficient in MATLAB) c_manual = zeros(size(A,1), 1); for i = 1:size(A,1) for j = 1:size(A,2) c_manual(i) = c_manual(i) + A(i,j) * x(j); end end D_manual = zeros(size(A,1), size(B,2)); for i = 1:size(A,1) for k = 1:size(B,2) for j = 1:size(A,2) % A_ij B_jk, j is the summation index D_manual(i,k) = D_manual(i,k) + A(i,j) * B(j,k); end end end 8. Strain Rosette Analysis Equations for normal strains at angles $\theta_a, \theta_b, \theta_c$: $\epsilon_a = \epsilon_{xx} \cos^2 \theta_a + \epsilon_{yy} \sin^2 \theta_a + \gamma_{xy} \sin \theta_a \cos \theta_a$ $\epsilon_b = \epsilon_{xx} \cos^2 \theta_b + \epsilon_{yy} \sin^2 \theta_b + \gamma_{xy} \sin \theta_b \cos \theta_b$ $\epsilon_c = \epsilon_{xx} \cos^2 \theta_c + \epsilon_{yy} \sin^2 \theta_c + \gamma_{xy} \sin \theta_c \cos \theta_c$ For 0-60-120 rosette: $\epsilon_{xx} = \epsilon_0$ $\epsilon_{yy} = \frac{1}{3}(2\epsilon_{60} + 2\epsilon_{120} - \epsilon_0)$ $\gamma_{xy} = \frac{2}{\sqrt{3}}(\epsilon_{60} - \epsilon_{120})$ Principal Strains: $\epsilon_{1,2} = \frac{\epsilon_{xx}+\epsilon_{yy}}{2} \pm \sqrt{\left(\frac{\epsilon_{xx}-\epsilon_{yy}}{2}\right)^2 + \left(\frac{\gamma_{xy}}{2}\right)^2}$ Maximum Shear Strain: $\gamma_{max} = \epsilon_1 - \epsilon_2$ Principal Stresses from Principal Strains (Plane Stress): $\sigma_1 = \frac{E}{1-\nu^2}(\epsilon_1 + \nu \epsilon_2)$ $\sigma_2 = \frac{E}{1-\nu^2}(\epsilon_2 + \nu \epsilon_1)$ 9. Polar Coordinates (Transformation) Displacement components: $u_x = u_r \cos\theta - u_\theta \sin\theta$, $u_y = u_r \sin\theta + u_\theta \cos\theta$ Strain components in polar coordinates: $\epsilon_{rr} = \frac{\partial u_r}{\partial_r}$ $\epsilon_{\theta\theta} = \frac{1}{r}\frac{\partial u_\theta}{\partial\theta} + \frac{u_r}{r}$ $\gamma_{r\theta} = \frac{1}{r}\frac{\partial u_r}{\partial\theta} + \frac{\partial u_\theta}{\partial r} - \frac{u_\theta}{r}$ Stress components in polar coordinates: $\sigma_{rr} = \sigma_{xx} \cos^2\theta + \sigma_{yy} \sin^2\theta + 2\tau_{xy} \sin\theta \cos\theta$ $\sigma_{\theta\theta} = \sigma_{xx} \sin^2\theta + \sigma_{yy} \cos^2\theta - 2\tau_{xy} \sin\theta \cos\theta$ $\tau_{r\theta} = (\sigma_{yy} - \sigma_{xx}) \sin\theta \cos\theta + \tau_{xy} (\cos^2\theta - \sin^2\theta)$ 10. Torsion of Non-Circular Sections For a rectangular section $b \times h$ ($b \ge h$): Maximum shear stress occurs at midpoint of long sides: $\tau_{max} = \frac{T}{\alpha b h^2}$ Angle of twist per unit length: $\phi = \frac{T}{\beta b h^3 G}$ (where $\alpha, \beta$ are shape factors from tables) For thin-walled closed sections (Bredt's formula): Shear flow $q = \frac{T}{2 A_m}$, where $A_m$ is the mean enclosed area. Shear stress $\tau = \frac{q}{t} = \frac{T}{2 A_m t}$ Angle of twist per unit length: $\phi = \frac{T}{4 A_m^2 G} \oint \frac{ds}{t}$ For thin-walled open sections: $\tau_{max} = \frac{T t_{max}}{J}$ $J = \frac{1}{3} \sum b_i t_i^3$ (for multiple rectangles) Angle of twist per unit length: $\phi = \frac{T}{G J}$