By Felippa C.
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Extra info for Advanced finite element methods
8. 9. 10. §5. 1. Direct Tensor Notation . . . . . 2. Matrix Notation . . . . . . 1. The Strain-Displacement Equations . . . 2. Constitutive Equations . . . . . 3. Internal Equilibrium Equations . . . 1. Surface Compatibility Equations . . . 2. Surface Equilibrium Equations . . . 1. Grad-Div Direct Tensor Notation . . . 2. Full Notation . . . . . . 1. 1. Step 1: Choose Master Field(s) . . . 2. Step 2: Choose Weak Connections . . . 3. Step 3: Construct a First Variation .
Out of print. K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, 1972 (2nd expanded edition 1981). With an encyclopœdic coverage of the title subject, this is a good reference monograph. Drawbacks are the flat exposition style (there is no unifying, lifting theme a la Lanczos) and the cost (since it is out of print — Washizu passed away in 1985 — used copies, if found on the Internet, go for over $150). R. Weinstock, Calculus of Variations, with Applications to Physics and Engineering, McGraw-Hill, 1952.
For matrices the dot is omitted. Some authors also omit the dot for tensors. Four-dimensional tensors are represented by underlined boldface uppercase symbols. 2). Operators that map vectors to vectors are usually represented by boldface uppercase symbols. An ubiquitous operator is nabla: ∇, which should be boldface except that the symbol is not available in bold. Applied to a scalar function, say φ, it produces its gradient: ∂φ ∂x ∂φ1 ∂φ . e for the scalar σi j e ji , but this is unnecessary as it is easily expressed in terms of : by transposing the second tensor.
Advanced finite element methods by Felippa C.