add notes, slides, lecture notes on fea

uni/mmme/3086_computer_modelling_techniques/fea.md

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+---
+author: Akbar Rahman
+date: \today
+title: 
+tags: []
+uuid: 8f32c8c0-5ed4-49ab-aca8-e5e058262580
+lecture_slides: [ ./lecture_slides/fea ]
+lecture_notes: [ ./lecture_notes/fea ]
+exercise_sheets: []
+---
+
+# MMME3086 Content
+
+1. Direct and Energy based formulation of 1d elements (*stiffness matrices*)
+1. Assembly of stiffness matrices to form the *global stiffness matrix*
+1. 2d pin jointed structures
+1. continuum elements
+1. structural elements
+1. practical FEA guidelines
+
+## Coursework
+
+- Worth 35% of module
+- Will be set on 2023-11-09
+- Will be due on 2023-11-23
+
+# Background (Slides 0101)
+
+- domain is discretised into *finite elements*
+- each element is defined by its corners (*nodes*)
+- typical shapes for elements are triangular/quadrilaterals in 2d problems or
+  tetrahedral/hexahedral in 3d problems
+- for each element, the behaviour is described by the displacements of the nodes and material law
+  (stress strain relationships)
+
+  - this is usually expressed as the *stiffness* of the element
+
+- elements are assembled in a mesh and the requirements of continuity and equilibrium between
+  neighbouring elements are satisfied
+
+  - this assembly process results in a large system of simultaneous equations
+
+- boundary conditions are applied to assembly of elements, to yield a unique solution to the
+  overall system
+- solution matrices are sparsely populated
+- equations are solved numerically to compute the displacements at each node
+
+  - the displacements of each node can be used to obtain the stresses in each element
+
+- finite element method is suitable for practical engineering stress analysis of complex geometries
+- to obtain good accuracy in regions of rapidly changing variables, a large number of small (*fine*)
+  elements must be used
+
+## Basic Overview of the Steps Required for FEA
+
+1. Discretise the domain
+1. Write the element stiffness matrices
+1. Assemble the global stiffness matrix
+1. Apply boundary conditions
+1. Solve matrix
+1. Post-processing---e.g. obtaining additional information like reaction forces and element stresses
+
+# Stress Analysis Fundamentals (Slides 0102)
+
+## Uniaxial Loading
+
+- The engineering definitions for engineering stress and strain assumes that stress is uniform,
+  but this is rarely true over large areas.
+- However the definition gets more useful for small elements:
+
+  \begin{equation}
+  \sigma = \lim_{\delta A \rightarrow 0} \sigma_\text{engineering}
+  \end{equation}
+
+- For uniaxial loading situations, *engineering (nominal) strain* is:
+
+  \begin{equation}
+  \varepsilon_\text{engineering}  = \frac{\Delta L}{L_0}
+  \end{equation}
+
+- For uniaxial loading situations, *engineering (nominal) stress* is:
+
+  \begin{equation}
+  \sigma_\text{engineering} = \frac{F}{A_0}A
+  \end{equation}
+
+## Multi-Axial (3D) Stress and Strain Definitions
+
+- In Cartesian axes system, are six components of stresses on the elements:
+
+  - *direct stresses* $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$ (tensile/compressive stresses)
+    caused by forces normal to the area
+  - *shear stresses* $\sigma_{xy}$, $\sigma_{xz}$, $\sigma_{yz}$ caused by shear forces parallel to
+    the area
+
+    - the first subscript refers to the direction of the outward normal to the plane the stress is
+      acting on
+    - the second subscript refers to the direction of the stress
+
+- The stress and strain vectors can be expressed as the following:
+
+  \begin{equation}
+  \pmb \sigma = \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{xy} \\ \sigma_{xz} \\ \sigma_{yz} \end{bmatrix}
+  \pmb \varepsilon = \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \varepsilon_{zz} \\ \varepsilon_{xy} \\ \varepsilon_{xz} \\ \varepsilon_{yz} \end{bmatrix}
+  \end{equation}
+
+  where:
+
+  \begin{align*}
+  \sigma_{xx} &= \frac{\partial u_x}{\partial x} \\
+  \sigma_{xx} &= \frac{\partial u_x}{\partial x} \\
+  \sigma_{xx} &= \frac{\partial u_x}{\partial x} \\
+  \varepsilon_{xy} &= \frac12 \left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right) \\
+  \varepsilon_{xz} &= \frac12 \left(\frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x}\right) \\
+  \varepsilon_{yz} &= \frac12 \left(\frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y}\right)
+  \end{align*}
+
+- In computation mechanics, the stress and strain relation is modelled as:
+
+  \begin{equation}
+  \pmb \sigma = \pmb d \pmb \varepsilon
+  \end{equation}
+
+### Stress-Strain Relationship (Hooke's Law)
+
+- Hooke's law can be used to create the following stress-strain relations for isotropic linear
+  elastic materials with thermal strain:
+
+  \begin{align}
+  \varepsilon_{xx} &= \frac 1 E \left( \sigma_{xx} - \nu\left(\sigma_{yy} + \sigma_{zz}\right)\right) + \alpha\Delta T \\
+  \varepsilon_{yy} &= \frac 1 E \left( \sigma_{yy} - \nu\left(\sigma_{xx} + \sigma_{zz}\right)\right) + \alpha\Delta T \\
+  \varepsilon_{zz} &= \frac 1 E \left( \sigma_{zz} - \nu\left(\sigma_{yy} + \sigma_{xx}\right)\right) + \alpha\Delta T \\
+  \varepsilon_{xy} &= \frac{\sigma_{xy}}{2\mu} \\
+  \varepsilon_{xz} &= \frac{\sigma_{xz}}{2\mu} \\
+  \varepsilon_{yz} &= \frac{\sigma_{yz}}{2\mu} \\
+  \mu &= \frac{E}{2(1+\nu)}
+  \end{align}
+
+  where $E$ is Young's modulus, $\nu$ is Poisson's ratio, $\alpha$ is the coefficient of thermal expansion,
+  $\Delta T$ is the temperature change, and $\mu$ is the shear modulus
+
+# Energy Methods (Slides 0103)
+
+## Stability
+
+![I'm not sure entirly what the significance of this slide is.](./images/vimscrot-2023-10-31T15:36:02,741218558+00:00.png)
+
+- Strain energy is released upon the removal of applied loads and the body returns to undeformed state:
+
+  \begin{equation}
+  U = \frac12 \pmb \sigma \pmb \sigma \times V
+  \end{equation}
+
+  - If the material behaviour is non-linear, it can be generalised to:
+
+    \begin{equation}
+    U = \int_\nu\int_\varepsilon \sigma \mathrm{d}\varepsilon \mathrm{d}V
+    \end{equation}
+
+- Work done by external forces can be expressed as:
+
+  \begin{equation}
+  W = \sum_i F_iu_i
+  \end{equation}
+
+  where $i$ is any point where force $F_i$ causes displacement $u_i$
+
+- The *total potential energy* can be expressed as:
+
+  \begin{equation}
+  \text{TPE} = U - W
+  \end{equation}
+
+  - The principle of minimum TPE states that when the body is in equilibrium, TPE must be 'stationary'
+    with respect to the variables of the problem
+  - The equilibrium is **stable if the TPE is minimum**
+  - In most FE problems, the displacement $u$ is chosen as the unknown variables of the problem:
+
+    \begin{equation}
+    \frac{\partial(\text{TPE})}{\partial u} = 0
+    \end{equation}
+
+# Mathematical Background (Slides 0104)
+
+See lecture slides,
+[MMME1026 Notes](https://notes.alv.cx/permalink?uuid=16edb140-9946-4759-93df-50cad510fe31),
+and [lecture slides](./lecture_slides/numerical_methods)/[notes](./lectures_notes/numerical_methods) on numerical methods.
+
+# Simple 1D Finite Elements (Slides 0105)
+
+## A Simple Uniaxial 1D Pin-Jointed Element
+
+See lecture slides.