A DISCUSSION ON “REPRESENTING GENERAL SOLUTION OF EQUATIONS IN THEORY OF ELASTICITY BY HARMONIC FUNCTIONS”

But,generally,it is impossible to determine P,Q,R,φ,ψ,A,B from(4)unless the domain ofelasticity is convex in x-direction,y-direction,and z-direction.This problem was resolved byEubanks and Sternberg in 1956.And,they gave examples which illustrat that solutions of Eq.(4)may not exist in nonconvex regions.So the solution stated in1]is only effective in three-directconvex region,for instance,sphere,cube and so on.And it may not hold true if it is structures whichare commonly used in engineering such as the sheel and the I-shaped beam.It is well known that in 1932 and 1934,Papkovich and Neuber proposed a general solution forEq.(1).It has the form

A DISCUSSION ON "REPRESENTING GENERAL SOLUTION OF EQUATIONS IN THEORY OF ELASTICITY BY HARMONIC FUNCTIONS"

If body force is absent, the displacement of the linear theory of elasticity are given by E/1-2v ·ae/ax + EΔu=0,E/1-2v ·ae/ay + EΔu=0,E/1-2v ·ae/az + EΔw=0.