Page 225 - Mechanical Behavior of Materials

Problems and Questions 227

strength. Determine the following: (a) stress in the length direction, (b) strain in the length
direction, (c) strain in the transverse direction, (d) length while under load, and (e) diameter
while under load.
5.12 Employ Eq. 5.26(a) and (b) as follows:
(a) Obtain an expression for the ratio ε y / ε x as a function of stresses and the elastic
constants for the material. Under what conditions is the negative of this ratio equal
to Poisson’s ratio ν?
(b) Obtain an expression for the ratio σ x / ε x as a function of stresses and the elastic
constants for the material. Under what conditions is this ratio equal to the elastic
modulus E?
5.13 For the special case of plane stress, σ z = τ yz = τ zx = 0, proceed as follows:
(a) Write the resulting simplified version of Hooke’s law, Eqs. 5.26 and 5.27.
(b) Then invert the simplified forms of Eq. 5.26(a) and (b) to obtain relationships that give
the stresses σ x and σ y , each as a function of strains and materials constants only.
(c) Also derive the equation that gives ε z as a function of the other two strains and materials
constants.
5.14 Strains are measured on the surface of a brass alloy part as follows: ε x = 1600×10 −6 ,
ε y = 1300×10 −6 , and γ xy = 1500×10 −6 . Estimate the in-plane stresses σ x , σ y , and τ xy, and
also the strain ε z normal to the surface. (Assume that the gages were bonded to the metal when
there was no load on the part, that there has been no yielding, and that no loading is applied
directly to the surface, so that σ z = τ yz = τ zx =0.)
5.15 Strains are measured on the surface of a polycarbonate plastic part as follows: ε x = 0.022, ε y
= −0.0158, and γ xy = 0.0096. Estimate in-plane stresses σ x , σ y , and τ xy, and also the strain
ε z normal to the surface. (The same assumptions apply as for Prob. 5.14.)
5.16 Strains are measured on the surface of a low alloy steel part as follows: ε x =−1750×10 −6 ,
ε y = 900×10 −6 , and γ xy = 600×10 −6 . Estimate in-plane stresses σ x , σ y , and τ xy,
and also the strain ε z normal to the surface. (The same assumptions apply as for
Prob. 5.14.)
5.17 Strains are measured on the surface of a mild steel part as follows: ε x = 250 × 10 −6 ,
ε y =−950 × 10 −6 , and γ xy = 400 × 10 −6 . Estimate in-plane stresses σ x , σ y , and τ xy ,
and also the strain ε z normal to the surface. (The same assumptions apply as for
Prob. 5.14.)
5.18 Strains are measured on the surface of a titanium alloy part as follows: ε x = 3300 ×
10 −6 , ε y = 110 × 10 −6 , and γ xy = 650 × 10 −6 . Estimate the in-plane stresses σ x , σ y , and
τ xy , and also the strain ε z normal to the surface. (The same assumptions apply as for
Prob. 5.14.)
5.19 A plate of metal is subjected to stresses σ x = 186 MPa and σ y = 152 MPa. The strains
that occur as a result of these stresses are measured to be ε x = 1900 × 10 −6 and ε y =
1250 × 10 −6 . No yielding occurs in the plate, that is, the behavior is elastic. Estimate the
elastic modulus E and Poisson’s ratio ν for the metal. What type of metal is it?
5.20 A thin-walled spherical vessel contains a pressure p and has inner radius r and wall
thickness t. It is made of an isotropic material that behaves in a linear-elastic manner.
Determine the each of following as a function of the pressure, geometric dimensions, and
material constants involved: (a) change in radius, r, and (b) change in wall thickness, t.