Page 226 - Mechanical Behavior of Materials

226 Chapter 5 Stress–Strain Relationships and Behavior

(a) Plot the stress–strain response for loading to a strain of ε = 0.010. Of this total strain,
how much is elastic, and how much is plastic?
(b) Also plot the response following (a) if the strain now decreases until it reaches zero.
5.3 For the elastic, linear-hardening rheological model of Fig. 5.3(c), how is the behavior affected
by changing E 2 while E 1 remains constant? You may wish to enhance your discussion by
including a sketch showing how the σ-ε path varies with E 2 .
5.4 A brass alloy is represented by the elastic, linear hardening model of Fig. 5.3(c), with constants
of E 1 = 100 GPa, E 2 = 3 GPa, and σ o = 500 MPa. Plot the stress–strain response for loading
toastrainof ε = 0.025. Of this total strain, how much is elastic, and how much is plastic?
◦
5.5 At 500 C, a material has an elastic modulus of E= 70 GPa and a tensile viscosity of η = 1200
GPa-s. Assuming that the elastic, steady-state creep model of Fig. 5.5(a) applies, determine
the response to a stress of 50 MPa maintained for 5 minutes and then removed. Plot both the
strain versus time and stress versus strain for a total time of interval of 10 minutes.
5.6 Consider the strain in a transient creep model, as during the time span 0–1–2 in Fig. 5.5(b).
Starting from Eq. 5.15, derive Eq. 5.17 that gives the strain at time 2.
5.7 A polymer has constants for the elastic, transient creep model of Fig. 5.5(b) of E 1 =2GPa,
5
E 2 = 4 GPa, and η =10 GPa-s. Determine and plot the strain versus time response for a stress
of 50 MPa applied for one day.
5.8 Consider relaxation under constant strain ε of a model with a spring and dashpot in series, as

m
in Fig. 5.6, but let the dashpot behave according to the nonlinear equation ˙ε = Bσ , where B
and m are material constants, with m being typically in the range 3 to 7. Derive an equation

for σ as a function of ε , time t, and the various model constants.
5.9 A polymer is used for shrink-on banding to keep cardboard boxes from popping open during
shipment of merchandise. The tension in the banding is observed to have dropped to 90% of
its initial value after three months. Estimate how long it will take for the tension to drop to
50% of its original value. The polymer may be assumed to behave according to an elastic,
steady-state creep model, as in Fig. 5.6.

Section 5.3 2
5.10 Consider a material of length 300 mm, with a rectangular cross section, 40 mm wide by 3 mm
thick. When subjected to a tensile load of 3000 N, assume that the length is found to increase
by 3 mm and the width to decrease by 0.15 mm. Also assume that, upon removal of the load,
the length and width are measured again and are found to have returned to approximately
their original values. Determine the following: (a) stress in the length direction, (b) strain in
the length direction, (c) strain in the width direction, (d) elastic modulus, (e) Poisson’s ratio,
and (f) shear modulus.
5.11 A bar of brass alloy is 200 mm long and has a circular cross section of diameter 50 mm. It
is subjected to a tensile load of 500 kN, which gives a stress well below the material’s yield

2
For metal alloys, where specific values are not available, the elastic constants E and ν may generally be approximated
as being the same as for the corresponding pure metal, as from Table 5.2. See the discussion in Section 5.3.2.