Theory of Elasticity
Introduction
• Concerned with determining stress, strain, and displacement distribution in an elastic solid
under the influence of external forces
• Using continuum mechanics, formulation establishes a mathematical boundary value
problem model – set of governing partial differential field equations with particular boundary
conditions
Engineering Applications
Aeronautical/Aerospace Engineering - stress, fracture, and fatigue analysis in aero structures.
Civil Engineering - stress and deflection analysis of structures including rods, beams, plates, and
shells; geomechanics involving the stresses in soil, rock, concrete, and asphalt materials.
Materials Engineering - to determine the stress fields in crystalline solids, around dislocations
and in materials with microstructure.
Mechanical Engineering - analysis and design of machine elements, general stress analysis,
contact stresses, thermal stress analysis, fracture mechanics, and fatigue.
Subject also provides basis for advanced studies in inelastic material behavior including plasticity
and viscoelasticity, and to computational stress analysis using finite/boundary element methods.
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
, Basic Methods of Stress & Deflection Analysis
Mechanics of Materials (Strength of Materials)
Simplified analysis based upon the use of assumptions related to the geometry of the
deformation, e.g., plane sections remain plane. See Appendix D in text for review.
Theory of Elasticity
General approach using principles of continuum mechanics. Develops mathematical boundary-
value problems for solution to the stress, strain and displacement distributions in a given body.
Computational Methods: Finite Element, Boundary Element, and Finite Difference
Each method discretizes body under study into many computational elements or cells. Solution is
then determined over each element or cell. Computers are used to handle detailed calculations.
Experimental Stress Analysis
Numerous techniques such as photoelasticity, strain gages, brittle coatings, fiber optic sensors,
Moire' holography, etc. have been developed to experimentally determine the stress, strain or
displacements at specific locations in models or actual structures and machine parts.
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
, Chapter 1 Mathematical Preliminaries
Common Variable Types in Elasticity
Elasticity theory is a mathematical model of material deformation. Using principles of
continuum mechanics, it is formulated in terms of many different types of field
variables specified at spatial points in the body under study. Some examples include:
Scalars - Single magnitude
mass density , temperature T, modulus of elasticity E, . . .
Vectors – Three components in three dimensions
displacement vector u ue1 ve 2 we 3 , e1, e2, e3 are unit basis vectors
Matrices – Nine components in three dimensions
stress matrix
x xy xz
[]
yx y yz
zx zy z
Other – Variables with more than nine components
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
, Index/Tensor Notation
With the wide variety of variables, elasticity formulation makes use of a tensor
formalism using index notation. This enables efficient representation of all
variables and governing equations using a single standardized method.
Index notation is a shorthand scheme whereby a1
a11 a12 a13
a whole set of numbers or components can be a i
a 2 , a ij
a 21 a 22 a 23
represented by a single symbol with subscripts a3
a 31 a 32 a 33
In general a symbol aij…k with N distinct indices represents 3N distinct numbers
Addition, subtraction, multiplication and equality of index symbols are defined in
the normal fashion; e.g.
a1 b1 a11 b11
a12 b12 a13 b13
ai bi a b2 , aij bij
a b21 a 22 b22 a 23 b23
2 21
a3 b3
a31 b31
a32 b32 a33 b33
a1
a11 a12
a13 a1b1
a1b2 a1b3
ai
a 2 , aij
a
21
a 22 a 23
ai b j
ab a 2 b2 a 2 b3
2 1
a3
a31 a32
a33 a3b1
a3b2 a3b3
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Introduction
• Concerned with determining stress, strain, and displacement distribution in an elastic solid
under the influence of external forces
• Using continuum mechanics, formulation establishes a mathematical boundary value
problem model – set of governing partial differential field equations with particular boundary
conditions
Engineering Applications
Aeronautical/Aerospace Engineering - stress, fracture, and fatigue analysis in aero structures.
Civil Engineering - stress and deflection analysis of structures including rods, beams, plates, and
shells; geomechanics involving the stresses in soil, rock, concrete, and asphalt materials.
Materials Engineering - to determine the stress fields in crystalline solids, around dislocations
and in materials with microstructure.
Mechanical Engineering - analysis and design of machine elements, general stress analysis,
contact stresses, thermal stress analysis, fracture mechanics, and fatigue.
Subject also provides basis for advanced studies in inelastic material behavior including plasticity
and viscoelasticity, and to computational stress analysis using finite/boundary element methods.
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
, Basic Methods of Stress & Deflection Analysis
Mechanics of Materials (Strength of Materials)
Simplified analysis based upon the use of assumptions related to the geometry of the
deformation, e.g., plane sections remain plane. See Appendix D in text for review.
Theory of Elasticity
General approach using principles of continuum mechanics. Develops mathematical boundary-
value problems for solution to the stress, strain and displacement distributions in a given body.
Computational Methods: Finite Element, Boundary Element, and Finite Difference
Each method discretizes body under study into many computational elements or cells. Solution is
then determined over each element or cell. Computers are used to handle detailed calculations.
Experimental Stress Analysis
Numerous techniques such as photoelasticity, strain gages, brittle coatings, fiber optic sensors,
Moire' holography, etc. have been developed to experimentally determine the stress, strain or
displacements at specific locations in models or actual structures and machine parts.
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
, Chapter 1 Mathematical Preliminaries
Common Variable Types in Elasticity
Elasticity theory is a mathematical model of material deformation. Using principles of
continuum mechanics, it is formulated in terms of many different types of field
variables specified at spatial points in the body under study. Some examples include:
Scalars - Single magnitude
mass density , temperature T, modulus of elasticity E, . . .
Vectors – Three components in three dimensions
displacement vector u ue1 ve 2 we 3 , e1, e2, e3 are unit basis vectors
Matrices – Nine components in three dimensions
stress matrix
x xy xz
[]
yx y yz
zx zy z
Other – Variables with more than nine components
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
, Index/Tensor Notation
With the wide variety of variables, elasticity formulation makes use of a tensor
formalism using index notation. This enables efficient representation of all
variables and governing equations using a single standardized method.
Index notation is a shorthand scheme whereby a1
a11 a12 a13
a whole set of numbers or components can be a i
a 2 , a ij
a 21 a 22 a 23
represented by a single symbol with subscripts a3
a 31 a 32 a 33
In general a symbol aij…k with N distinct indices represents 3N distinct numbers
Addition, subtraction, multiplication and equality of index symbols are defined in
the normal fashion; e.g.
a1 b1 a11 b11
a12 b12 a13 b13
ai bi a b2 , aij bij
a b21 a 22 b22 a 23 b23
2 21
a3 b3
a31 b31
a32 b32 a33 b33
a1
a11 a12
a13 a1b1
a1b2 a1b3
ai
a 2 , aij
a
21
a 22 a 23
ai b j
ab a 2 b2 a 2 b3
2 1
a3
a31 a32
a33 a3b1
a3b2 a3b3
Elasticity Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
