ABSTRACT
In this section we consider prismatic (multiply or simply connected) bodies made from 1isotropic, linearly elastic materials and we choose the x axis to coincide with the locus of
the centroid of their cross sections. The bodies are originally in a stress-free, strain-free o state of mechanical and thermal equilibrium at the uniform temperature T . Subsequently,
the bodies are subjected to specified specific body forces and to specified boundary conditions and reach a second state of mechanical and thermal equilibrium at the uniform
o 1temperature T in which the component of displacement u vanishes while their other two 2 3components of displacement are functions of only x and x . We say that these bodies are
in a state of plane strain. Referring to the strain-displacement relations (2.16) the components of strain of these bodies are
Referring to the stress-strain relations for a state of plane strain (3.50), we see that
Substituting relations (7.2) into the equilibrium equations (2.69), we have
2 3Inasmuch as the components of stress are only functions of x and x , referring to relations (7.3), we see that in order to have a state of plane strain in a body the distribution of the specific body force must have the following form:
(7.5a)
(7.5b) (7.5c)
(7.6)
The stress distribution (7.2) when substituted into the traction-stress relations (2.73) must give components of traction which when evaluated at the points of the boundary of the body where components of traction are specified give the specified components of traction. Referring to Fig. 7.1 the unit vector outward normal to the lateral surfaces of a prismatic body is . Consequently, using relations (7.2), from relations (2.73) we find that the components of traction acting on the lateral surfaces of a prismatic body, in a state of plane strain, must have the following form:
1 1 The unit vector normal to the end surfaces (x 0 and x = L) of a prismatic body is . Consequently, using relations (2.73) we find that the components of traction acting on the end surfaces of a prismatic body in a state of plane strain, must have the following form:
We limit our discussion to bodies on the lateral surfaces of which the components of traction
and ` are specified. That is, we do not consider
boundary value problems involving bodies having one or more components of displacement specified on one or more of their lateral surfaces.
