MECH593 Introduction to Finite Element Methods

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MECH593 Introduction to Finite Element Methods
Finite Element Analysis of 3-D Problems
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3-D Linear Elasticity
Constitutive Relationship
Isotropic materials
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3-D Linear Elasticity
Kinematics
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3-D Linear Elasticity
Equilibrium
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3-D Linear Elasticity
Governing Equation
Weak Form
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3-D Tetrahedral Elements
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Formulation of 4-Node Tetrahedral Element
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Formulation of 4-Node Tetrahedral Element
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Formulation of 4-Node Tetrahedral Element
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3-D Brick Elements
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Formulation of 8-Node Brick Element
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Thermal Stress and Thermal Strain
Case 1
Case 2
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Deformation Due to Thermal Mismatch
E1 , a1
E2 , a2
a1 lt a2 , DT gt 0
DL1 La1DT DL2 La2DT
DL DL2 - DL1 L(a2 - a1)DT LDaDT
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Thermal Stress and Thermal Strain Governing
Equations
Total Strain
Hookes Law
Strain-Deformation
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Thermo-Elasticity Analysis
Plane Strain Analysis
Plane Stress Analysis
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Finite Element Model of Thermal Stress and
Thermal Strain 1-D Bar
Equilibrium in terms of displacement
Example
Weak form
where
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Equations of Plane Thermo-Elasticity
Governing Equations (Static Equilibrium)
Strain-Deformation (Small Deformation)
Constitutive Relation (Linear Elasticity)
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4-Node Rectangular Element for Plane Stress
Analysis
Let
Choose
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4-Node Rectangular Element for Plane Stress
Analysis
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Evaluation of Strains
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Evaluation of Stresses
Plane Strain Analysis
Plane Stress Analysis
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Thermal Variation Induced by Deformation
Example thermoelastic damping in resonators
100 MHz
10 kHz
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Thermal Variation Induced by Deformation
Modified thermal conduction equation
Where To is the initial temperature.
Approaches (1) iterative method
(2) fully coupled method