Page 224 - Mechanical Behavior of Materials

224 Chapter 5 Stress–Strain Relationships and Behavior

and similarly for the y- and z-directions and the yz and zx planes. There are two independent elastic
constants: the elastic modulus E, and Poisson’s ratio ν. The shear modulus G is related to these by

E
G = (5.68)
2(1 + ν)
The volumetric strain is the sum of the normal strains, ε v = ε x + ε y + ε z . For the isotropic,
homogeneous case, volumetric strain is related to the applied stresses by

1 − 2ν
ε v = σ x + σ y + σ z (5.69)
E
This equation indicates that the volume change is zero for ν = 0.5. Values for virtually all materials
lie within the limits ν = 0 and 0.5, usually between ν = 0.2 and 0.4.
Some materials, notably fibrous composites, are significantly anisotropic. A particular case
of anisotropy that is often encountered is orthotropy, in which the material has symmetry about
three orthogonal planes. Such a material has nine independent elastic constants. There is a different
value of the elastic modulus for each orthogonal direction, E X , E Y , and E Z , and also three
independent values of Poisson’s ratio and of shear modulus, ν XY , G XY , etc., corresponding to the
three orthogonal planes. These constants are defined only on the special X-Y-Z coordinate axes
that are parallel to the planes of symmetry in the material. If the orientation of the coordinate axes
change, the elastic constants change.
For in-plane loading of sheets and plates of composite materials with unidirectional fibers, the
elastic constants can be estimated from those of the reinforcement and matrix materials:
E r E m
E X = V r E r + V m E m , E Y = (5.70)
V r E m + V m E r
In this case, X is the fiber (reinforcement) direction, Y is the transverse direction, and V r , V m are
the volume fractions of reinforcement and matrix, respectively.

NEW TERMS AND SYMBOLS

(a) Terms
anelastic strain perfectly plastic
bulk modulus, B Poisson’s ratio, ν
elastic (Young’s) modulus, E recovery
generalized Hooke’s law relaxation
homogeneous rheological model
hydrostatic stress, σ h shear modulus, G
isotropic steady-state creep
linear elasticity tensile viscosity, η
linear hardening thermal expansion coefficient, α
linear viscoelasticity transient creep
orthotropic volumetric strain, ε v