# Solid Mechanics ### Dear god the single hardest class that I've ever taken Thankfully, the course [textbook](http://solidmechanics.org/contents.php) is readily available online. > This course provides an introduction to the mechanics of solids from a continuum perspective. Topics covered in this course include: vector and tensor analysis, coordinate systems and calculus in curvilinear coordinate systems, kinematics (motion, deformation and strain), stress and momentum balance, energy principles and balance laws, linear isotropic and anisotropic elasticity, thermoelasticity, method of solutions for 2-D and 3-D linear elastic boundary value problems, applications to simple structures. This is the ceiling on the complexity of information that I have a reasonable grasp on. This course started pretty slow and then accelerated pretty rapidly. ### Lecture 1: Scalars Fields (*in this context*) are variables which depend on multiple variables, typically position. In PDE's, most students are introduced to temperature fields but fields can be any kind of variable. Temperature or pressure, which are scalars, are simplest but velocity or stress can also depend on location. $$\tikz \draw (0pt,0pt) -- (20pt,6pt);$$ $$ \begin{align\*} & \bold{a} = a\_1\bold{e}\_1+a\_2\bold{e}\_2+a\_3\bold{e}\_3 \\ & a\_i = \left<\bold{a},\bold{e}\_i\right> \\ & \left | \bold{e}\_i \right | = 1\\ & \bold{x} = \overbrace{x\_1\bold{e}\_1+x\_2\bold{e}\_2+x\_3\bold{e}\_3 = x\_i\bold{e}*i}^{\texttt{Summation Convention}} \\ & \delta*{ij} = {\bf e}\_i\cdot {\bf e}*j = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}\\ & \left | \bold{a} \right | = \sqrt{a\_i a\_i} \\ & \epsilon*{ijk} = \\ & \bold{n} = \frac{\bold{a} \times \bold{b}}{\left | \bold{a} \times \bold{b} \right |} \end{align\*} $$ Scalar Field. Vector Field Index Notation Algebraic Vector Operations #### Addition #### Products Norm Scalar Dot Vector Cross Outer Product Kronecker Delta Permutation Symbol \---- Epsilon Delta Identity $$ \epsilon\_{i j k} \epsilon\_{i m n}=\delta\_{j m} \delta\_{k n}-\delta\_{j n} \delta\_{k m} $$ Notice that the two Levi-Civita symbols share an index, $$i$$, as a dummy index. The appearance of a dummy index indicates a contraction, a reduction in rank. On the left-hand side the Kronecker Delta symbols are of rank 2 whereas on the right, rank 3 tensors appear. Scalar Triple Product ### Lecture 2: Vectors ### Lecture 3: Tensors ### Lecture 4: Eigenvalues ### Lecture 5: Tensor Calc ### Lecture 6: Curvilinear Coordinates ### Lecture 7: Kinematics ### Lecture 8: Polar Spherical Coordinates ### Lecture 9: Stretch and Right Cauchy Green Deformation ### Lecture 10: Motion, Deformation, and Strain ### Lecture 11: Strain Measures ### Lecture 12: Force and Momentum Balance ### Lecture 13: Conservation of Mass ### Lecture 14: Proof of Cauchy's Theorem from BLM ### Lecture 15: Traction and Stress from Undeformed Configuration ### Lecture 16: Elastic Material Behavior ### Lecture 17: Material Symmetry ### Lecture 18: Elastic Constants ### Lecture 19: Strain Energy and Thermo Elasticity ### Lecture 20: Linear Thermo-Elastic Boundary Value Problems ### Lecture 21: Thermo-Elastic BVP in Cylindrical Coordinates ### Lecture 22: ### Force Mechanical interaction (push or pull) between * parts of a body * body and environment Contact Force Act on a surface due to contact with environment or other parts of the body Body Force Exerted through the interior of a body, due to environment or itself * gravity * electromagnetism * self gravitation Traction Stress vector. the second definition is due to newton's second law $$ {\bf T}({\bf n},{\bf y}, t) = \lim\limits\_{dA \rightarrow 0} \frac{d{\bf {p}}}{dA} = -{\bf T}(-{\bf n},{\bf y}, t) $$ Momentum Balance Laws Newton Euler Equations Balance of Linear Momentum $$ {\bf P}(V,t) = \frac{d}{dt} \left\[{\bf \ell}(V,t)\right] ; \quad {\bf \ell}(V,t)= \int\limits\_V {\bf v}({\bf y},t)\rho({\bf y},t)dV $$ $$ {\bf M}(V,t,{\bf p}) = \frac{d}{dt} \left\[{\bf h}(V,t,{\bf p})\right] ; \quad {\bf h}(V,t,{\bf p})= \int\limits\_V \left({\bf y}-{\bf p}\right)\times{\bf v}({\bf y},t)\rho({\bf y},t)dV $$ Velocity Velocity Gradient Rate of Deformation Spin Rate of Volume Change Acceleration Conservation of Mass Global Version of Newton Euler Equations Cauchy's Theorem or the Existence of Stress Proof of Cauchy's Theorem Physical Interpretation of Cauchy Stress Principal Stresses Hydrostatic Stress Deviatoric Stress Von Mises Effective Stress Traction and Stress wrt. Undeformed Configuration Rarely do we have the outcome, but the initial configuration and what happens to it. 1st Piola-Kirchoff Stress Tensor Stress Traction Relation Localization on V0 Other Stress measures Elastic Material Behavior * Kinematics, Small Strain * Momentum Balance * Knowns and Unkonwns * Constitutive Laws, Thermodynamics, Energy Balance of Energy Material Linearity Voight/Nye Representation Material Symmetry Special Cases Monoclinic Orthotropic Transverse Isotropy/Hexagonal Symmetry Cubic Symmetry Global/Sample and Material Reference Frames Isotropy Elastic Constants Strain Energy Density Linear Elastic Linear Elastic Isotropic Strain Energy Decomposition Thermoelasticity * Thermal Strain * Isotropic Thermal Strain Isotropic Thermoelasticity --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. 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