Abstract
In this Chapter we consider the virtual work equation of a static problem and its relationship to the variational method and energy principle.
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Notes
- 1.
The continuity of ϕ gives an essential condition for ensuring the equivalence between the strong form and the weak form. We must be particularly careful of this requirement if we apply a numerical method such as the finite element method to a problem in which the continuity condition is not completely satisfied, such as a stress concentration problem at a crack tip.
- 2.
Let f be a general function that is bounded ( | | f | | < ∞) but may not be smooth, whereas ϕ is a sufficiently smooth function where ϕ(t) → 0 if t → ± ∞. Then a generalized derivativef ′ can be defined by
$$\qquad \left \langle {f}^{{\prime}},\,\phi \right \rangle ={ \int\nolimits \nolimits }_{-\infty }^{\infty }{f}^{{\prime}}(t)\,\phi (t)\,dt = f(t)\,\phi (t){\Bigr |}_{ -\infty }^{\infty }-{\int\nolimits \nolimits }_{-\infty }^{\infty }f(t)\,\phi {(t)}^{{\prime}}(t)\,dt = -\left \langle f,\,{\phi }^{{\prime}}\right \rangle.$$ - 3.
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© 2012 Springer-Verlag Berlin Heidelberg
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Ichikawa, Y., Selvadurai, A.P.S. (2012). Virtual Work Equation, Variational Methods and Energy Principles. In: Transport Phenomena in Porous Media. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25333-1_4
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DOI: https://doi.org/10.1007/978-3-642-25333-1_4
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25332-4
Online ISBN: 978-3-642-25333-1
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