chapter 2 deformation: displacements & strain
DESCRIPTION
Elasticity Theory, Applications and Numerics M. H Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Deformation ExampleTRANSCRIPT
Chapter 2 Deformation: Displacements & Strain
Examples of Continuum Motion & Deformation
(Undeformed Element) (Rigid Body Rotation)
(Horizontal Extension) (Shearing Deformation) (Vertical Extension)
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Deformation Example
(Deformed) (Undeformed)
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Small Deformation Theory
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Two Dimensional Geometric Deformation
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Strain-Displacement Relations
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A B
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Strain Tensor
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Example 2-1: Strain and Rotation ExamplesDetermine the displacement gradient, strain and rotation tensors for the following displacement field: 32 ,, CxzwByzvyAxu , where A, B, and C are arbitrary constants. Also calculate the dual rotation vector = (1/2)(u).
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Strain Transformation
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Two-Dimensional Strain Transformation
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Principal Strains & Directions0]det[ 32
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(General Coordinate System) (Principal Coordinate System) No Shear Strains
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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Spherical and Deviatoric Strains
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ijijij eee ˆ~ . . . Spherical Strain Tensor
. . . Deviatoric Strain Tensor
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Compatibility ConceptNormally we want continuous single-valued displacements;
i.e. a mesh that fits perfectly together after deformation
Undeformed State
Deformed State
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Mathematical Concepts Related to Deformation Compatibility
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Strain-Displacement Relations
Given the Three Displacements:We have six equations to easily determine the six strains
Given the Six Strains:We have six equations to determine three displacement components. This is an over-determined system and in general will not yield continuous single-valued displacements unless the strain components satisfy some additional relations
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
2
3
1
4
(b) Undeformed Configuration
2
3
1
4
(c) Deformed Configuration Continuous Displacements
2
3
1
4
(d) Deformed Configuration Discontinuous Displacements
(a) Discretized Elastic Solid
Physical Interpretation of Strain Compatibility
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Compatibility EquationsSaint Venant Equations in Terms of Strain
Guarantee Continuous Single-Valued Displacements in Simply-Connected Regions
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Examples of Domain Connectivity
(a) Two-Dimensional Simply Connected
(b) Two-Dimensional Multiply Connected
(c) Three-Dimensional Simply Connected
(d) Three-Dimensional Simply Connected
(e) Three-Dimensional Multiply Connected
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Curvilinear Strain-Displacement RelationsCylindrical Coordinates
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island