文档介绍：Closed-Form Solutions
David Roylance
Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
February 21, 2001
Introduction
During most of its historical development, the science of Mechanics of Materials relied principally
on closed-form (putational) mathematical theorists. Much of their work represents
mathematical intuition and skill of a very high order, challenging even for advanced researchers
of today. This theory is taught primarily in graduate subjects, but is outlined here both to
provide some background that will be useful in the Module on Fracture and as a preliminary
introduction to these more advanced subjects.
Governing equations
We have earlier shown (see Module 9) how the spatial gradients of the six Cauchy stresses are
related by three equilibrium equations that can be written in pseudovector form as
LTσ= 0 (1)
These are augmented by six constitutive equations which can be written for linear elastic mate-
rials as (see Module 11)
σ= D (2)
and six kinematic or strain-displacement equations (Module 8)
= Lu (3)
These fifteen equations must be satisfied by the fifteen independent functions (three displace-
ments u, six strains , and six stresses σ). These functions must also satisfy boundary conditions
on displacement
u = uˆ on Γu (4)
where Γu is the portion of the boundary on which the displacements u = uˆ are prescribed. The
remainder of the boundary must then have prescribed tractions T = Tˆ, on which the stresses
must satisfy Cauchy’s relation:
ˆ
σnˆ= T on ΓT (5)
1
In the familiar cantilevered beam shown in Fig. 1, the region of the beam at the wall constitutes
Γu, having specified (zero) displacement and slope. All other points on the beam boundary
make up ΓT , with a load of P at the loading point A and a specified load of zero elsewhere.
Figure 1: Cantilevered beam.
With structures such as the beam that have simple geometries, solutions can be obta