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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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