Page 208 - Mechanical Behavior of Materials

Section 5.3 Elastic Deformation 209

ν(1 + ν) 0.343(1 + 0.343) −6
ε x =− σ z =− (−75 MPa) = 266 × 10 Ans.
E 130,000 MPa

E 130,000 MPa

E = = = 147,300 MPa Ans.
1 − ν 2 1 − 0.343 2

Discussion The compressive σ z results in negative (that is, compressive) values for σ y and
ε z , but positive ε x , as expected from the physical situation. The apparent elastic modulus E
2

is larger than the elastic modulus E, differing by the ratio E /E = 1/(1 − ν ); specifically,

E /E = 1.133 in this case. This is explained by noting that E is the ratio of stress to strain
only for the uniaxial case, and ratios of stress to strain for other states of stress and strain are
determined by behavior obeying the three-dimensional form of Hooke’s law.

5.3.4 Volumetric Strain and Hydrostatic Stress

In stressed bodies, small volume changes occur that are associated with normal strains. Shear strains
are not involved, as they cause no volume change, only distortion. Consider a rectangular solid, as
in Fig. 5.10, where there are normal strains in three directions. The dimensions L, W, and H change
by infinitesimal amounts, dL, dW, and dH, respectively, so that the normal strains are

dL dW dH
ε x = , ε y = , ε z = (5.31)
L W H

Figure 5.10 Volume change due to normal strains.