Figure 1.

Maxwell–Betti reciprocal theorem. (a) Linear elastic materials versus (b) nonlinear elastic materials according to the Maxwell–Betti reciprocal theorem. (a) A linear elastic material that is subjected to two different forces, $F_A\ne F_B$. At instant $t_1$, the material exhibited a displacement at point B of $u_B$ when subjected to a force $F_A$ at point A. According to the Maxwell–Betti reciprocal theorem, we expect that the material deforms at instant $t_2$ such that its displacement at point A due to an applied force $F_B$ at point B is $u_A=F_A{u}_B/F_B$ (thus, $F_B{u}_A=F_A{u}_B$). (b) A nonlinear elastic material that is subjected to two different forces, $F_A\ne F_B$. This material disobeys the Maxwell–Betti reciprocal theorem such that $F_B{u}_A\ne F_A{u}_B$.