Fundamentals of Elasticity Theory Professor M. H. Sadd Reference: Elasticity Theory Applications and Numerics, M.H. Sadd, Elsevier/Academic Press, 2009

Value of Elasticity Theory Theory of Elasticity Based Upon Principles of Continuum Mechanics, Elasticity Theory Formulates Stress Analysis Problem As Mathematical Boundary-Value Problem for Solution of Stress, Strain and Displacement Distribution in an Elastic Body. Governing Field Equations Model Physics Inside Region (Same For All Problems) Boundary Conditions Describe Physics on Boundary (Different For Each Problem) R Su St Value of Elasticity Theory Develops “Exact” Analytical Solutions For Problems of Limited Complexity Provides Framework for Understanding Limitations of Strength of Materials Models Establishes Framework for Developing Linear Finite Element Modeling Generates Solutions for Benchmark Comparisons with FEA Solutions

Deformation and Strain Two-Dimensional Theory u(x,y) u(x+dx,y) v(x,y) v(x,y+dy) dx dy A B C D A' B' C' D'   x y Strain Displacement Relations Three-Dimensional Theory

Deformation and Strain Example

Rigid Body Motion Two-Dimensional Example C D dy vo A dx B uo x Zero Strains!

Strain Compatibility Compatibility Equation Undeformed Configuration 2 3 1 4 Undeformed Configuration Deformed Configuration Continuous Displacements Discontinuous Displacements Discretized Elastic Solid x y Compatibility Equation

Strain Compatibility Example

Body and Surface Forces Sectioned Axially Loaded Beam Surface Forces: T(x) S Cantilever Beam Under Self-Weight Loading Body Forces: F(x)

Traction and Stress Traction Vector P3 P2 F n A p P1 (Externally Loaded Body) (Sectioned Body) Traction Vector Note that ordinary elasticity theory does not include nor allow concentrated moments to exist at a continuum point

Stress Components x z y y x yx z xy xz zy yz zx

Stress Transformation x3 x1 x2 x1 x2 x3 Three-Dimensional Transformation x y x'  y' Two-Dimensional Transformation

Stress Transformation Example u

Principal Stresses and Directions Ii = Fundamental Invariants Roots of the characteristic equation are the principal stresses s1 s2 s3 Corresponding to each principal stress is a principal direction n1 n2 n3 that can be used to construct a principal coordinate system y (General Coordinate System) 1 3 2 (Principal Coordinate System) n1 n3 n2 x z y x yx z xy xz zy yz zx 1 3 2

Equilibrium Equations Body Forces

Equilibrium Equation Example

Hooke’s Law Isotropic Homogeneous Materials = Lamé’s constant  = shear modulus or modulus of rigidity E = modulus of elasticity or Young’s modulus v = Poisson’s ratio

Orthotropic Materials (Three Planes of Material Symmetry) Nine Independent Elastic Constants for 3-D Four Independent Elastic Constants for 2-D

Physical Meaning of Elastic Constants  (Simple Tension)  (Pure Shear) p (Hydrostatic Compression)

Relations Among Elastic Constants

Typical Values of Elastic Constants

Basic Formulation Fundamental Equations (15) - Strain-Displacement (6) - Compatibility (3) - Equilibrium (3) - Hooke’s Law (6) Fundamental Unknowns (15) - Displacements (3) - Strains (6) - Stresses (6) Typical Boundary Condtions Displacement Conditions Mixed Conditions Traction Conditions R S Su St T(n) u

Basic Problem Formulations Problem 1 (Traction Problem) Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the tractions are prescribed over the surface of the body. Problem 2 (Displacement Problem) Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the displacements are prescribed over the surface of the body. Problem 3 (Mixed Problem) Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the tractions are prescribed over the surface St and the distribution of the displacements are prescribed over the surface Su of the body. Displacement Conditions Mixed Conditions Traction Conditions R S Su St T(n) u

Basic Boundary Conditions Coordinate Boundary Examples y=Ty r xy=Tx r r  x   r y xy=Ty r x y x=Tx (Cartesian Coordinate Boundaries) (Polar Coordinate Boundaries) Non-Coordinate Boundary Example n = unit normal vector x y

Boundary Condition Examples Fixed Condition u = v = 0 Traction Free Condition x y a b S Traction Condition l (Coordinate Surface Boundaries) (Non-Coordinate Surface Boundary)

Symmetry Boundary Conditions Rigid-Smooth Boundary Condition Symmetry Line y x

Example Solution – Beam Problem sx - Contours

Saint-Venant’s Principle The Stress, Strain and Displacement Fields Due to Two Different Statically Equivalent Force Distributions on Parts of the Body Far Away From the Loading Points Are Approximately the Same. P/2 P/2 P x x xy y x xy y x y y Stresses Approximately Equal

Strain Energy Strain Energy = Energy Stored Inside an Elastic Solid Due to the Applied Loadings One-Dimensional Case dx u dz  dy x y z Three-Dimensional Case

Principle of Virtual Work The virtual displacement ui = {u, v, w} of a material point is a fictitious displacement such that the forces acting on the point remain unchanged. The work done by these forces during the virtual displacement is called the virtual work. Virtual Strain Energy = Virtual Work Done by Surface and Body Forces Change in Potential Energy (UT-W) During a Virtual Displacement from Equilibrium is Zero.