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where
This important theorem, called the theorem or principle of minimum potential energy, will be used in Chapter 7 as our starting point in the formulation of mathematical models for beams, plates and shells.
Given the potential energy and the space of admissible functions, it is possible to determine the strong form. This is illustrated by the following example.
Example 1.2 Let us determine the strong form corresponding to the potential energy defined by
with
Since u minimizes
(1.39)
Therefore we have
where the last two terms are zero because
and, substituting this into eq. (1.40), we get
(1.41)
Since this holds for all
(1.42)
minimizes the potential energy defined by eq. (1.38). This is the strong form of the problem.
Remark 1.2 The procedure in Example 1.2 is used in the calculus of variations for identifying the differential equation, known as the Euler9‐Lagrange10 equation, the solution of which maximizes or minimizes a functional. In this example the solution minimizes the potential energy on the space
Remark 1.3 Whereas the strain energy is always positive, the potential energy may be positive, negative or zero.
1.3 Approximate solutions
The trial and test spaces defined in the preceding section are infinite‐dimensional, that is, they span infinitely many linearly independent functions. To find an approximate solution, we construct finite‐dimensional subspaces denoted, respectively, by
