Page 228 - Mechanical Behavior of Materials

228 Chapter 5 Stress–Strain Relationships and Behavior

5.21 Consider a pressure vessel that is a thin-walled tube with closed ends and wall thickness t.
The volume enclosed by the vessel is determined from the inner diameter D and length L by
2
V e = π D L/4. The ratio of a small change in the enclosed volume to the original volume
can be found by obtaining the differential dV e and then dividing by V e , which gives

dV e dD dL
= 2 +
V e D L
Verify this expression. Then derive an equation for dV e /V e as a function of the pressure p
in the vessel, the vessel dimensions, and elastic constants of the isotropic material. Assume
that L is large compared with D, so that the details of the behavior of the ends are not
important.
5.22 Consider a thin-walled spherical pressure vessel of inner diameter D and wall thickness t.
Derive an equation for the ratio dV e /V e , where V e is the volume enclosed by the vessel, and
dV e is the change in V e when the vessel is pressurized. Express the result as a function of
pressure p, vessel dimensions, and elastic constants of the isotropic material.
5.23 A block of isotropic material is stressed in the x- and y-directions as shown in Fig. P5.23.
The ratio of the magnitudes of the two stresses is a constant, so that σ y = λσ x .

(a) Determine the stiffness in the x-direction, E = σ x /ε x , as a function of only λ and the
elastic constants E and ν of the material.

(b) Compare this apparent modulus E with the elastic constant E as obtained from a
uniaxial test, and comment on the comparison. (Suggestion: Assume that ν =0.3 and
consider λ values of −1, 0, and +1.)

Figure P5.23

5.24 A sample of isotropic material is subjected to a compressive stress σ z and is confined so that
it cannot deform in either the x-or y-directions, as shown in Fig. P5.24.
(a) Do stresses occur in the material in the x- and y-directions? If so, obtain equations for
σ x and for σ y , each as functions of only σ z and the elastic constant ν for the material.

(b) Determine the stiffness E = σ z /ε z in the direction of the applied stress σ z in terms of
only the elastic constants E and ν for the material. Is E equal to the elastic modulus

E as obtained from a uniaxial test? Why or why not?
(c) What happens if Poisson’s ratio for the material approaches 0.5?