slides chapter 2 deformation displacements & strain
DESCRIPTION
MCE 571 Theory of Elasticity ITRANSCRIPT
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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
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Deformation Example
(Deformed) (Undeformed)
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
![Page 3: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/3.jpg)
Small Deformation Theory
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
![Page 4: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/4.jpg)
Two Dimensional Geometric Deformation
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Strain-Displacement Relations
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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
![Page 6: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/6.jpg)
Strain Transformation
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
![Page 7: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/7.jpg)
Two-Dimensional Strain Transformation
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
![Page 8: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/8.jpg)
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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. . . Deviatoric Strain Tensor
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
![Page 10: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/10.jpg)
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
![Page 11: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/11.jpg)
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
![Page 12: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/12.jpg)
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(b) Undeformed Configuration
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(c) Deformed Configuration Continuous Displacements
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(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
![Page 13: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/13.jpg)
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
![Page 14: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/14.jpg)
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
![Page 15: Slides Chapter 2 Deformation Displacements & Strain](https://reader033.vdocument.in/reader033/viewer/2022061610/563dbbbd550346aa9aafd79f/html5/thumbnails/15.jpg)
Curvilinear Strain-Displacement RelationsCylindrical Coordinates
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Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island