Finite Element Analysis. Barna Szabó

Figure 1.2 Exercise 1.3: The function

Remark 1.1 defined on by eq. (1.20) satisfies the properties of linear forms listed in Section A.1.2 if the following inequality is satisfied:

      (1.35)

      1.2.2 The principle of minimum potential energy

Theorem 1.2 The function that satisfies for all minimizes the quadratic functional⁸ , called the potential energy;

      (1.36)

      on the space .

      Proof: For any , we have:

      (1.37)

      where unless . Therefore any admissible nonzero perturbation of u will increase .

      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

(1.38)

      with .

      Since u minimizes ,