Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

1
εz = [ σ z − νσ ( x + σ y )] + α∆T
E
1
−6
=
[0 MPa − (0.33)( −135.96 MPa − 360 MPa)] + (23 × 10 /C)(45 ° ° C)
3
73 × 10 MPa
−6
= 3,277 × 10 mm/mm
Ans.
(c) Change in Volume of the Block
Calculate the initial volume of the block as
6 3
V = (250 mm)(90 mm)(250 mm) = 5.625 × 10 mm
Then, calculate the dilatation e from the three normal stresses and Equation (13.22):
1−
2 ν
e = ( σ x + σ y + σ z )
E
1−
2(0.33)
=
( −135.96 MPa − 360 MPa + 0)
3
73 × 10 MPa
−6
= 795.073 × 10
The change in volume of the block is thus
− 6 6 3 3
∆ V = eV = (795.073 × 10 )(5.625 × 10 mm ) = 4,472 mm Ans.
13.9 Generalized Hooke’s Law
for orthotropic materials
y
y
ε
σ
= x x
E
x
x
ε
ν σ
= x
y –
E
τ xy
τ xy
τ
γ xy
xy =
G
FIGURE 13.15 Isotropic
material.
σ x
For a homogeneous, isotropic material, three material properties—the elastic modulus E,
shear modulus G, and Poisson’s ratio ν—are sufficient to describe the relationship of stress
to strain. Further, of these three properties, only two are independent: For an isotropic
material, a tensile normal stress causes an elongation in the direction of the stress and a
contraction in the direction perpendicular to the stress (Figure 13.15), whereas shear
stresses cause only shearing deformations. These types of deformations exist regardless of
the direction of the stress.
For some materials, three material properties are not sufficient to describe the behavior
of the material. For example, wood has three mutually perpendicular planes of material
symmetry: (a) a plane parallel to the grain, (b) a plane tangential to the grain, and (c) a
radial plane. Solids having material properties that are different in three mutually perpendicular
directions at a point within their body are known as orthotropic materials. Such
materials have three natural axes that are mutually perpendicular.
Figure 13.16 shows an orthotropic material subjected to a normal stress that acts in the
direction of a natural axis of the material. The 1 and 2 directions in this figure represent
natural axes of the material. Each of these directions has its own elastic modulus E 1 and
E 2 , and its own unique Poisson’s ratio. For normal strain in the 2 direction resulting from
loading in the 1 direction, Poisson’s ratio is denoted ν 12 . Similarly, Poisson’s ratio ν 21 gives
the normal strain in the 1 direction that results from loading in the 2 direction. Just like an
isotropic material, the orthotropic material elongates in the direction of the stress and contracts
in the perpendicular direction. However, if the same magnitude of stress were applied
perpendicular to the vertical natural axis of the material (i.e., in the 2 direction), the deformations
in the horizontal and vertical directions would be much different.
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