# Jian-zhong Zhao's blog

## Saint-Venant's Principle: Experimental and Analytical

Mathematical provability , then classication, of Saint-Venant's Principle are discussed. Beginning with the simplest case of Saint-Venant's Principle, four problems of elasticity are discussed mathematically. It is concluded that there exist two categories of elastic problems concerning Saint-Venant's Principle: Experimental Problems, whose Saint-Venant's Principle is established in virtue of supporting experiment, and Analytical Problems, whose Saint-Venant's decay is proved or disproved mathematically, based on fundamental equations of linear elasticity.

## Saint-Venant's Principle: Rationalized and Rational

The problem of statement of Saint-Venant's Principle is concerned.  Statement of Boussinesq or Love  is ambiguous so that its interpretations are in contradiction with each other. Rationalized Statement of Saint-Venant’s Principle of elasticity  is suggested to rule out the ambiguity of Statements of Boussinesq and Love. Rational Saint-Venant's Principle is suggested to fit and guide  applications of the principle  to  fields of continuum physics and cover the analogical case as well as the non-analogical case discovered and discussed in this paper .

## Zanaboni Theory and Saint-Venant's Principle: Updated

Zanaboni Theory is mathematically analyzed in this paper. The conclusion is that Zanaboni Theorem is invalid and  not a proof of  Saint-Venant's Principle;  Discrete Zanaboni Theorem and Zanaboni's energy decay are inconsistent with Saint-Venant's decay; the inconsistency, discussed here, between Zanaboni Theory   and Saint-Venant's Principle  provides more proofs that Saint-Venant's Principle is not generally true.

## Zanaboni Theorem and Saint-Venant's Principle

Violating the law of energy conservation, Zanaboni Theorem is invalid
and Zanaboni's proof is wrong. Zanaboni's mistake of " proof " is analyzed.
Energy Theorem for Zanaboni Problem is suggested and proved.
Equations and conditions are established in this paper for Zanaboni Problem,
which are consistent with , equivalent or identical to each other. Zanaboni
Theorem is, for its invalidity , not a mathematical formulation or
proof of Saint-Venant's Principle.

## Saint-Venant's Principe of the Problem of the Cylinder: Modified

The Statement of Modied Saint-Venant's Principle is suggested. The
axisymmetrical deformation of the infinite circular cylinder loaded by an
equilibrium system of forces on its near end is discussed and its formulation of Modied Saint-Venant's Principle is established. It is evident that
finding solutions of boundary-value problems is a precise and pertinent
approach to establish Saint-Venant type decay of elastic problems.               T

## Saint-Venant's Principe of the " Cavity in Cylinder " Problem

The problem of a cylinder with a small spherical cavity loaded by an
equilibrium system of forces is suggested and discussed and its formulation of Saint-Venant's Principle is established. It is evident that finding
solutions of boundary-value problems is a precise and pertinent approach
to establish Saint-Venant type decay of elastic problems.

## Special Saint-Venant's Principe of the “ Hole in Plate” Problem

The problem of the
infinite plate with a central hole loaded by an equilibrium system of forces is
generalized and its formulation of Special Saint-Venant's Principle is
established. It is essential to develop mathematical theories of Special
Saint-Venant's Principle one by one if Elasticity has to be constructed to be
rational, logical, rigorous and secure mechanics.

## Saint-Venant's Principe of the Problem of the Cylinder

The problem of the
infinite axisymmetrical circular cylinder loaded by an equilibrium system of
forces on its near end is discussed and its formulation of Special
Saint-Venant's Principle is established. It is essential to develop
mathematical theories of Special Saint-Venant's Principle one by one if
Elasticity has to be constructed to be rational, logical
rigorous and secure mechanics.

## Zanaboni Theorem Is Invalid：Re-review

Saint-Venant’s Principle in elasticity  has its over 100 year’s history. Boussinesq  and Love  announced general statements of Saint-Venant’s Principe. The early authors made  important contribution to the principle. Zanaboni “proved” a theorem trying to concern Saint-Venant’s Principle, but in the present paper we will prove that Zanaboni’s theorem is false.

## The Study by Mr. Wu etc Concerning Saint-Venant’s Principle is Insignificant

The paper titled "The Study by Mr. Wu etc  Concerning Saint-Venant’s Principle is Insignificant" reviewed the series of four works of “proof of Saint-Venant’s Principle” published by Wu etc .The four articles proved neither the discrete Saint-Venant type decay of section-forces of the finite elements in the chain model of discretization of the cylinder  nor  the discrete Saint-Venant-type decay of section-displacements, posed in their papers.