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**Fundamentals of Elasticity Theory**

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

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**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

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**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

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**Deformation and Strain Example**

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**Rigid Body Motion Two-Dimensional Example**

C D dy vo A dx B uo x Zero Strains!

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**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

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**Strain Compatibility Example**

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**Body and Surface Forces**

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

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**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

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

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**Stress Transformation**

x3 x1 x2 x1 x2 x3 Three-Dimensional Transformation x y x' y' Two-Dimensional Transformation

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**Stress Transformation Example**

u

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**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

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**Equilibrium Equations**

Body Forces

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**Equilibrium Equation Example**

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**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

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**Orthotropic Materials (Three Planes of Material Symmetry)**

Nine Independent Elastic Constants for 3-D Four Independent Elastic Constants for 2-D

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**Physical Meaning of Elastic Constants**

(Simple Tension) (Pure Shear) p (Hydrostatic Compression)

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**Relations Among Elastic Constants**

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**Typical Values of Elastic Constants**

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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

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**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

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**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

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**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)

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**Symmetry Boundary Conditions**

Rigid-Smooth Boundary Condition Symmetry Line y x

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**Example Solution – Beam Problem**

sx - Contours

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**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

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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

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**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.

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