9.1 Linear elasticity

Linear elasticity is a fundamental concept in solid mechanics, describing how materials deform under applied forces. It assumes a linear relationship between stress and strain, simplifying complex material behavior for engineering applications. This theory forms the basis for structural analysis and design in various fields.

The chapter covers stress-strain relationships, Hooke's law, and equilibrium equations. It explores plane stress and strain conditions, variational formulations, and numerical methods like FEM. Advanced topics include anisotropic materials and coupled field problems, providing a comprehensive overview of linear elasticity theory and its applications.

Fundamentals of linear elasticity

Stress and strain tensors

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• Stress tensor represents the internal forces acting on a material point
  • Normal stresses act perpendicular to the surface (e.g., tension or compression)
  • Shear stresses act parallel to the surface (e.g., torsion or bending)
• Strain tensor describes the deformation of a material point relative to its original configuration
  • Normal strains represent the elongation or contraction along principal axes
  • Shear strains represent the angular distortion between two orthogonal planes
• Cauchy's stress theorem relates the stress vector on any plane to the stress tensor components
• Small strain theory assumes infinitesimal deformations and neglects higher-order terms

Hooke's law for isotropic materials

• Hooke's law establishes a linear relationship between stress and strain for elastic materials
• For isotropic materials, the elastic properties are independent of the direction
  • Young's modulus ($E$) relates normal stress to normal strain
  • Shear modulus ($G$) relates shear stress to shear strain
  • Poisson's ratio ($\nu$) characterizes the transverse contraction under axial loading
• The constitutive equation for isotropic linear elasticity is given by $\sigma_{ij} = \lambda \varepsilon_{kk} \delta_{ij} + 2\mu \varepsilon_{ij}$
  • $\lambda$ and $\mu$ are Lamé constants, related to $E$ and $\nu$
  • $\delta_{ij}$ is the Kronecker delta (1 if $i=j$, 0 otherwise)

Equilibrium equations and boundary conditions

• Equilibrium equations ensure the balance of forces and moments in a deformable body
  • In the absence of body forces, $\frac{\partial \sigma_{ij}}{\partial x_j} = 0$
  • Body forces (e.g., gravity, electromagnetic fields) can be included as $\frac{\partial \sigma_{ij}}{\partial x_j} + f_i = 0$
• Boundary conditions specify the loading or displacement constraints on the body's surface
  • Traction boundary conditions prescribe the surface forces (e.g., pressure, shear)
  • Displacement boundary conditions prescribe the displacements or rotations (e.g., fixed support, symmetry)
• Compatibility equations ensure the continuity and single-valuedness of displacements
  • For small strains, the compatibility equations are given by $\varepsilon_{ij,kl} + \varepsilon_{kl,ij} = \varepsilon_{ik,jl} + \varepsilon_{jl,ik}$

Plane stress and plane strain

Assumptions and simplifications

• Plane stress assumes that the stress components perpendicular to the plane are zero (e.g., thin plates)
  • $\sigma_{zz} = \sigma_{xz} = \sigma_{yz} = 0$
  • Reduces the 3D problem to a 2D one in the $xy$-plane
• Plane strain assumes that the strain components perpendicular to the plane are zero (e.g., long cylinders)
  • $\varepsilon_{zz} = \varepsilon_{xz} = \varepsilon_{yz} = 0$
  • Simplifies the 3D problem to a 2D one in the $xy$-plane
• Both assumptions lead to a reduction in the number of independent variables and equations

Airy stress function approach

• The Airy stress function $\phi(x,y)$ is a scalar potential that automatically satisfies the equilibrium equations
• Stress components are obtained from the second derivatives of $\phi$:
  • $\sigma_{xx} = \frac{\partial^2 \phi}{\partial y^2}$, $\sigma_{yy} = \frac{\partial^2 \phi}{\partial x^2}$, $\sigma_{xy} = -\frac{\partial^2 \phi}{\partial x \partial y}$
• The compatibility equation in terms of $\phi$ is given by the biharmonic equation: $\nabla^4 \phi = 0$
• Boundary conditions are expressed in terms of $\phi$ and its derivatives

Compatibility equations and boundary conditions

• Compatibility equations ensure that the strain components are consistent with the displacement field
  • For plane stress and plane strain, the compatibility equation reduces to $\frac{\partial^2 \varepsilon_{xx}}{\partial y^2} + \frac{\partial^2 \varepsilon_{yy}}{\partial x^2} = 2\frac{\partial^2 \varepsilon_{xy}}{\partial x \partial y}$
• Boundary conditions for the Airy stress function approach involve prescribing values of $\phi$ and its derivatives
  • Traction boundary conditions are expressed in terms of $\frac{\partial \phi}{\partial x}$, $\frac{\partial \phi}{\partial y}$, and $\frac{\partial^2 \phi}{\partial x \partial y}$
  • Displacement boundary conditions are more complex and involve integrals of $\phi$ and its derivatives

Variational formulation

Principle of minimum potential energy

• The principle of minimum potential energy states that the equilibrium configuration of an elastic body minimizes the total potential energy
• The total potential energy $\Pi$ is the sum of the strain energy $U$ and the potential energy of external loads $W$
  • $\Pi = U + W = \int_V \frac{1}{2} \sigma_{ij} \varepsilon_{ij} dV - \int_V f_i u_i dV - \int_S t_i u_i dS$
  • $f_i$ are body forces, $t_i$ are surface tractions, and $u_i$ are displacements
• The equilibrium configuration satisfies the variational principle $\delta \Pi = 0$

Weak form of equilibrium equations

• The weak form of the equilibrium equations is obtained by applying the principle of virtual work
• Virtual work principle: $\int_V \sigma_{ij} \delta \varepsilon_{ij} dV = \int_V f_i \delta u_i dV + \int_S t_i \delta u_i dS$
  • $\delta \varepsilon_{ij}$ and $\delta u_i$ are virtual strains and virtual displacements, respectively
• The weak form relaxes the continuity requirements on the solution and allows for approximate solutions
• It serves as the basis for the finite element method

Galerkin method and finite element discretization

• The Galerkin method is a weighted residual technique for solving the weak form of the equilibrium equations
• The domain is discretized into finite elements, and the displacement field is approximated using shape functions
  • $u_i(x) \approx \sum_{j=1}^n N_j(x) u_i^j$
  • $N_j(x)$ are the shape functions, and $u_i^j$ are the nodal displacements
• The weak form is transformed into a system of algebraic equations: $\mathbf{K}\mathbf{u} = \mathbf{f}$
  • $\mathbf{K}$ is the global stiffness matrix, $\mathbf{u}$ is the nodal displacement vector, and $\mathbf{f}$ is the nodal force vector
• The finite element solution converges to the exact solution as the element size decreases

Numerical methods for linear elasticity

Finite element method (FEM) basics

• FEM is a numerical technique for solving boundary value problems in engineering and physics
• The domain is discretized into a finite number of elements (e.g., triangles, quadrilaterals, tetrahedra)
• The primary unknown (e.g., displacement) is approximated within each element using shape functions
• The element equations are assembled into a global system of equations, which is solved for the nodal unknowns
• FEM is particularly well-suited for problems with complex geometries and heterogeneous material properties

Isoparametric elements and shape functions

• Isoparametric elements use the same shape functions to interpolate both the geometry and the primary unknown
• The element geometry is mapped from a reference element (e.g., unit square or cube) to the physical element
  • $x_i(\xi, \eta) = \sum_{j=1}^n N_j(\xi, \eta) x_i^j$
  • $\xi$ and $\eta$ are the natural coordinates in the reference element
• Common isoparametric elements for linear elasticity include:
  • Linear and quadratic triangles (2D)
  • Bilinear and biquadratic quadrilaterals (2D)
  • Linear and quadratic tetrahedra (3D)
  • Trilinear and triquadratic hexahedra (3D)
• Higher-order shape functions provide better accuracy but increase computational cost

Element stiffness matrix and load vector assembly

• The element stiffness matrix $\mathbf{K}^e$ relates the nodal displacements to the nodal forces for a single element
  • $\mathbf{K}^e = \int_{V_e} \mathbf{B}^T \mathbf{D} \mathbf{B} dV$
  • $\mathbf{B}$ is the strain-displacement matrix, and $\mathbf{D}$ is the constitutive matrix
• The element load vector $\mathbf{f}^e$ accounts for the external forces acting on the element
  • $\mathbf{f}^e = \int_{V_e} \mathbf{N}^T \mathbf{b} dV + \int_{S_e} \mathbf{N}^T \mathbf{t} dS$
  • $\mathbf{N}$ is the matrix of shape functions, $\mathbf{b}$ are the body forces, and $\mathbf{t}$ are the surface tractions
• The global stiffness matrix and load vector are assembled by summing the contributions from all elements
  • $\mathbf{K} = \sum_{e=1}^{n_e} \mathbf{K}^e$, $\mathbf{f} = \sum_{e=1}^{n_e} \mathbf{f}^e$
  • $n_e$ is the total number of elements in the mesh

FEM implementation considerations

Mesh generation and refinement strategies

• Mesh generation is the process of discretizing the domain into finite elements
• Structured meshes have a regular topology and are easier to generate but less flexible
  • Suitable for simple geometries (e.g., rectangles, cubes)
  • Quadrilateral and hexahedral elements are commonly used
• Unstructured meshes have an irregular topology and can conform to complex geometries
  • Triangular and tetrahedral elements are commonly used
  • Delaunay triangulation and advancing front methods are popular algorithms
• Adaptive mesh refinement (AMR) techniques adjust the mesh based on error indicators
  • $h$-refinement: Subdividing elements in regions with high errors
  • $p$-refinement: Increasing the polynomial order of shape functions in high-error elements
  • $hp$-refinement: Combining $h$- and $p$-refinement for optimal performance

Boundary condition application techniques

• Essential (Dirichlet) boundary conditions prescribe the values of the primary unknown on the boundary
  • Displacement boundary conditions in linear elasticity
  • Implemented by modifying the global stiffness matrix and load vector
    • Elimination approach: Removing rows and columns corresponding to constrained degrees of freedom
    • Penalty method: Adding large values to the diagonal entries of constrained degrees of freedom
• Natural (Neumann) boundary conditions prescribe the values of the derivative of the primary unknown on the boundary
  • Traction boundary conditions in linear elasticity
  • Implemented by modifying the global load vector
    • Adding the equivalent nodal forces due to the applied tractions
• Mixed (Robin) boundary conditions involve a combination of the primary unknown and its derivative
  • Implemented using a combination of the techniques for essential and natural boundary conditions

Efficient linear solver algorithms for FEM

• The global system of equations in FEM is typically large, sparse, and symmetric positive definite (SPD)
• Direct solvers (e.g., Gaussian elimination, Cholesky decomposition) are robust but computationally expensive for large systems
  • Suitable for small to medium-sized problems or as a preconditioner for iterative solvers
• Iterative solvers (e.g., conjugate gradient, GMRES) are more efficient for large systems but may require preconditioning for convergence
  • Krylov subspace methods are widely used for SPD systems
  • Multigrid methods are effective for problems with multiple scales
• Parallel computing techniques (e.g., domain decomposition, message passing) can significantly accelerate the solution process
  • Partitioning the mesh and distributing the workload among multiple processors
  • Scalable parallel solvers (e.g., PETSc, Trilinos) are available for large-scale simulations

Verification and validation

Analytical solutions for simple geometries

• Analytical solutions are available for linear elasticity problems with simple geometries and boundary conditions
  • Infinite plate with a circular hole under uniaxial tension (Kirsch problem)
  • Thick-walled cylinder under internal and external pressure (Lamé problem)
  • Cantilever beam with end load (Euler-Bernoulli beam theory)
• These solutions serve as benchmarks for verifying the accuracy of numerical methods
• Comparing the FEM results with the analytical solutions helps assess the discretization error and convergence rate

Convergence analysis and error estimation

• Convergence analysis studies how the numerical solution approaches the exact solution as the mesh is refined
• A priori error estimates provide theoretical bounds on the discretization error based on the element size and shape function order
  • For linear elements, the error in the $L^2$ norm is proportional to $h^2$, where $h$ is the element size
  • For quadratic elements, the error in the $L^2$ norm is proportional to $h^3$
• A posteriori error estimates use the computed solution to assess the local and global errors
  • Residual-based error estimators measure the residual of the equilibrium equations
  • Recovery-based error estimators compare the computed solution with a smoothed or enhanced solution
• Adaptive mesh refinement relies on error estimators to identify regions that require further refinement

Comparison with experimental results

• Validation involves comparing the numerical results with experimental data to assess the model's predictive capability
• Common experimental techniques for measuring displacements and strains in linear elasticity include:
  • Strain gauges: Measuring local strains through changes in electrical resistance
  • Digital image correlation (DIC): Measuring full-field displacements and strains using optical images
  • Interferometry: Measuring displacements using the interference of light waves
• Validation metrics quantify the agreement between numerical and experimental results
  • Correlation coefficients, root-mean-square error, and normalized errors are commonly used
• Discrepancies between simulations and experiments can be attributed to modeling assumptions, material property uncertainties, and measurement errors
  • Sensitivity analysis and uncertainty quantification techniques can help identify the sources of discrepancies and improve the model

Advanced topics in linear elasticity

Anisotropic and inhomogeneous materials

• Anisotropic materials have direction-dependent elastic properties
  • Examples include composites, single crystals, and textured polycrystals
  • The constitutive matrix $\mathbf{D}$ has up to 21 independent constants (triclinic symmetry)
  • Orthotropic and transversely isotropic materials are common special cases
• Inhomogeneous materials have spatially varying elastic properties
  • Examples include functionally graded materials and geological formations
  • The constitutive matrix $\mathbf{D}$ becomes a function of position $\mathbf{D}(\mathbf{x})$
• FEM can handle anisotropic and inhomogeneous materials by assigning different material properties to each element
  • The element stiffness matrix is computed using the local constitutive matrix
  • Homogenization techniques can be used to derive effective properties for heterogeneous materials

Thermoelasticity and coupled field problems

• Thermoelasticity studies the interaction between thermal and mechanical fields in elastic bodies
  • Temperature changes induce thermal strains and stresses
  • The constitutive equation is modified to include the thermal expansion term: $\sigma_{ij} = C_{ijkl}(\varepsilon_{kl} - \alpha \Delta T \delta_{kl})$
  • $\alpha$ is the coefficient of thermal expansion, and $\Delta T$ is the temperature change
• Coupled field problems involve the simultaneous solution of multiple physics fields
  • Examples include piezoelectricity (mechanical-electrical), poroelast