Analytical Problems Regarding the Fundamental Equations of Isothermal Static Elasticity

Abstract

In the last chapter necessary conditions for the existence of derivatives of the solution of the basic problem of static elasticity with respect to a paramater θ, to which the external forces are assumed proportional, have been exhibited. It has been seen that if u _r has derivatives of arbitrarily high order with respect to θ at θ = 0, it is possible, formally, to construct the power series

$$ {u_{r}} = \sum\limits_{{h = 1}}^{\infty } {u_{r}^{{\left( h \right)}}\frac{{{\theta ^{{\left( h \right)}}}}}{{h!}}} , $$

((5.1))

where u ^(h)_r is the solution of a linear problem analogous to that of the linear theory of elasticity, with certain known terms expressed as functions of u ⁽¹⁾_r , u ⁽²⁾_r , ..., u ^(h-1)_r for h ≥ 2, while u_r⁽¹⁾ is simply the solution of the basic problem for the case of small strain. Therefore, if the expansion (5.1) is justified, the integration of the fundamental problem of large strain may be reduced to one of determining the solutions of certain linear problems of small deformation problems which in concrete cases are finite in number, since for practical applications some finite number of terms in (5.1) will be taken.