Page 192 - Mechanical Behavior of Materials

Section 5.2 Models for Deformation Behavior 193

Figure 5.2 Relationship of models to stress, strain, and strain rate, in a bar of material.

deformation increases with time. But an increasing fraction of the applied force is needed to pull
against the spring as x increases, so that less force is available to the dashpot, and the rate of

deformation decreases. The deformation approaches the value P/k if the force is maintained for
a long period of time. If the applied force is removed, the spring, having been extended, now pulls
against the dashpot. This results in all of the deformation being recovered at infinite time.
Rheological models may be used to represent stress and strain in a bar of material under
axial loading, as shown in Fig. 5.2. The model constants are related to material constants that are
independent of the bar length L or area A. For elastic deformation, the constant of proportionality
between stress and strain is the elastic modulus, also called Young’s modulus, given by
σ
E = (5.2)
ε

Substituting the definitions of stress and strain, and also employing P = kx, yields the relationship
between E and k:

kL
E = (5.3)
A
For the plastic deformation model, the yield strength of the material is simply

P o
σ o = (5.4)
A

For the steady-state creep model, the material constant analogous to the dashpot constant c is called
1
the coefficient of tensile viscosity and is given by
σ
η = (5.5)
˙ ε

1
In fluid mechanics, viscosities are defined in terms of shear stresses and strains, η τ = τ/ ˙γ ,where η = 3η τ relates
values of tensile and shear viscosity for an ideal incompressible material.