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Chapter 5: Boundary value problems in solid mechanics
Recall governing equations:
Example:
(pages 250-255, Timoshenko & Goodier)Pure bending of a prismatic cantilever beam:
Note: For a cross section at z = c:
Note: For the lateral surfaces of the beam:
Boundary Value Problem
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Simplifying Assumptions:
These systems of PDEs, cannot be solved analytically, in general. We use numerical methods and use simplifying assumptions.
Read Example 27 from the textbook.(See. 101 from Timoshenko & Goodier)
Example: 2D Plane Problems
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To understand how to obtain numerical solutions to complicated 2D/3D problems in general, let's first study some 1-D problems where we can usually obtain exact solutions.
One-dimensional (1D) little BVP
Alternative derivation:
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Example. Column under self weight:
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Strong Forms and Weak forms
Weak forms
Weak form of the Problem Statement.
This relies on the fundamental theorem of Calculus of Variations:
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Proof of Fundamental theorem of calculus of variations
Restriction on the choice of Function spaces
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Possible choices for function spaces:
Example of Approximate solution to Weak form:
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Weak Form: Method of weighted Residuals (MWR):
Weak Form: Principle of Virtual Work (PVW):
Substituting the material model:
Weak form problem statements:
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Boundary Conditions
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Examples of Boundary Conditions & Approximation Function Spaces
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Variational (Energy-based) methods
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Energy weak form:
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Vainberg's Theorem: Existence of an Energy form (Variational form)
Vainberg's Theorem
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Example:
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Hamilton's Principle (for Dynamics)
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Weak form in 3D
One can show that the Vainberg's Theorem is satisfied for the above weak form.Energy form in 3D
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