EN 224 Linear Elasticity

K.-S. Kim, Sem I, 2014

Robert Hooke first stated the law in 1660 as a Latin anagram "ceiiinosssttuu" (published 1676). He published the solution of his anagram in 1678 as: ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force"). Brief History of Experimental Linearized Elasticity The historical information presented here has been taken from The experimental foundations of solid mechanics by J.F. Bell in Handbuch der Physiks, Volume VIa/1.

1678 : Robert Hooke Discovers that force is a linear function of elongation based on experiments on long, thin wires and springs. His anagram for this law was "ceiiinosssttuu" (published 1676) which was deciphered as "Ut tensio sic vis" in his 1678 paper. 1720 : Jordan Ricatti Proposes that elastic properties of a body could be inferred from the frequency of vibration. The first experimental study of elastic E-moduli. 1729 : Pieter Van Musschenbroek Publishes the first book showing testing machines for tension, compression, and flexure. 1766 : Leonhard Euler Introduces the concept of "Young's modulus" eighty years before Thomas Young popularized Euler's concepts of the "height of the modulus" and the "weight of the modulus". 1780 : Charles Augustin Coulomb First to measure the shear modulus in the modern sense. 1787 : Ernst Chladni Calculates ratios of the velocity of sound in air to that in various solids. This work provided a major impetus for 19th century continuum mechanics. 1807 : Thomas Young Publishes his "Lectures on Natural Philosophy". This work led to the popularization of the "height of the modulus". The units were in feet. 1809 : Jean Baptiste Biot First direct measurement of the velocity of sound in a solid. 1813 : Alphonse Duleau First quasi-static experiments for small deformation linear elasticity (by design). This work provided experimental evidence for numerous theoretical developments in elasticity, including St Venant's principle and the theoretical work of Cauchy, Poisson and Navier. 1841 : Guillaume Wertheim Presents first definitive study of elastic properties of solids under various conditions to the French Academy. This study included results from Jean Victor Poncelet, Thomas Tregold, Antoin Masson, Felix Savart among others. Linear plots of stress versus strain begin to be widely used. 1848 : Guillaume Wertheim First experiments showing that the Poisson's ratio of a solid does not have the constant value of 0.25. 1859 : Gustav Robert Kirchhoff First measurement of Poisson's ratio independent of the elastic modulus and specimen diameter. 1869 : Marie Alfred Cornu First direct optical measurement of Poisson's ratio. 1882 : Woldemar Voigt Performs experiments to prove the isotropy or otherwise of solids. 1904 : Arnulph Malloc Devises a method to determine the quasi-static bulk modulus based on the theory of linear elasticity. 1908 : Eduard August Gruneisen The Poisson's ratio is first determined experimentally as ratio of lateral and longitudinal strains. Uses Malloc's method to determine the compressibility of solids. Brief Early History of Theoretical Linearized Elasticity

1687 : Isaac Newton Publishes "Principia" which provide the laws of motion : inertia, conservation of momentum, and balance of forces, though inertia and momentum remained undefined. 1684 : Gottfried Wilhelm Leibniz Finds the relation between bending moment and the moment of inertia of a linear elastic beam. 1691-1704 : James Bernoulli Derives the general equations of equilibrium using different methods : balance of forces, balance of moments, and the principle of virtual work. Finds that the stress (force/area) as a function of strain characterizes a material and thus proposes the first true stress-strain relation and a material property. 1713 : Antoine Parent Determines the position of the neutral fiber and postulates the existence of shear stresses. 1736 : Leonhard Euler Publishes "Mechanics" where he defines a mass-point and acceleration. Also introduces vectors. Most of the equations in mechanics in use today can be traced to the work of Euler. 1742 : John Bernoulli First to refer all positions to a single, rectangular Cartesian co-ordinate system. 1743 : Jean le Rond d'Alembert First to derive a partial differential equation as the statement of a law of motion.

1750-1758 : Leonhard Euler Formulates the principles of conservation of linear momentum and moment of momentum. Distinguishes mass from inertia. 1773 : Charles Augustin de Coulomb Proved that shear stresses exist in a bending beam. 1788 : Joseph Louis Lagrange Publishes "Mechanique Analitique" which contains much of the mechanics known until that time. 1821 : Claude-Louis Navier 1822 : Augustin Louis Cauchy Discovers the stress principle - relating the total forces and total moment to internal and external tractions. Cartesian co-ordinate system. This is basically the first description of the stress tensor. Cauchy also presented the equations of equilibrium and showed that the stress tensor is symmetric. 1833 : Siméon Denis Poisson Publishes statement and proof that a system of pairwise equilibriated and central forces exerts no torque. This is fundamental to the principle of conservation of moment of momentum. 1855 : Adhémar Jean Claude Barré de Saint-Venant 1918 : Emmy Noether (Symmetry of Physical Laws and Conservation Laws) 1951 : Eshelby (Configurational force field elasticity) Singular solutions (Green, BPN, Taylor, Westergaard, England, Mindlin, etc.) : Punch (Hertz, Snedon) : Nonlocal elasticity (Cosserat, Eringen) : Poro-elasticity (Biot) Homogenization (Hashin, Strikman, Willis) : Anisotropy (Stroh, Barnett, Dundurs, Ting) Texts/structures (Love, Timoshenko, Von Karman, Muskhelishivili, Sternberg, Gurtin) Computation (FEM, Boundary Element Methods) Inverse Method Elasticity (FPM) EN 224 Linear Elasticity Course Outline K.-S. Kim, Sem I, 2014

0. Introduction and Mathematical Preliminaries

1. Review of the Field Equations of Linear Elasticity 1.1 Kinematics of Deformable Solids 1.2 Kinetics of Deformable Solids 1.3 Constitutive Models and for Elastic Materials 1.4 Summary of Linearized Field Equations; Boundary and Initial Value Problems 2. Theorems of Linear Elasticity 2.1 Superposition 2.2 Existence and Uniqueness Theorems 2.3 Reciprocal Theorem 2.4 Energy and Work Theorems 2.4.1. Principle of Minimum Potential Energy 2.4.2. Principle of Minimum Complementary Energy 2.4.3. Principle of Virtual Work 2.5 Noether’s Theorems (Lie symmetry) 3. Basic One-Dimensional Problems* (Reading chapter) 3.1 Properties of Axi-symmetric Linear Elastic Fields 3.2 Saint Venant Solutions of Beam Deformations 4. 3D Static Boundary Value Problems 5.1 Papkovich Neuber Potentials 5.2 Singular Solutions for an Infinite Solid 5.3 Solutions for 3D dislocation loops in an infinite solid 5.4 The Boundary Element Method 5.5 Eigenstrains 5.6 Eshelby Inclusion Problems 5.7 Singular Solutions for the Half Space 5.8 Contact Problems 5a. 2D Static Boundary Value Problems 4a.1. Field Equations and Boundary Conditions 4a.2. Plane Strain and Plane Stress Approximations 4a.3. Stress Equations of Compatibility and Levi’s Theorem 4a.4 Solution of Plane Problems using Airy Stress Functions 5b. Complex Variable Methods for Plane Elastostatics 4b.1 Review of Complex Variables 4b.2 Complex Variable Representation of Plane Elastostatic Solutions 4b.3 Boundary Conditions on Complex Potentials 4b.4 Series Solutions for Complex Potentials 4b.5 Application of the Cauchy Integral Formula 4b.6 Conformal Mapping 6. Elasticity theory for anisotropic materials 6.1 General Principles: Constitutive law and field equations for anisotropic materials 6.2 Anti-plane shear solutions for anisotropic materials 6.3 The Stroh representation for general plane deformation of anisotropic materials 6.4 Solutions to selected boundary value problems for anisotropic materials 7. Inhomogeneity and Lie symmetry in linear elasticity 7.1. Noether’s theorem revisited 7.2. Eshelby’s inhomogeneity and configurational force fields 7.3. Lie symmetry and conservation integrals 8. Emergence of elasticity and homogenization 8.1. Kinematic & static homogenization and linearization 8.2. Mean field theory 8.3. Homogenization bound theorems 8.4. Scaling theory 9. Elasticity theory for inverse problems 9.1. Loss of reciprocity 9.2. Spectral auxiliary field decompositions 9.3. Field projection methods 10. Computational (multi-physical) linear elasticity 10.1. Variational approximations: Finite element methods (for inverse ptoblems) 10.2. Boundary element methods (for inverse problems) 10.3. Phase field method for multi-physics linear elesticity