The relation between the stress a(t) and strain 7(t) of a material is often repr

Question

The relation between the stress a(t) and strain 7(t) of a material is often represented by the mechanical models shown in Fig. 9.17. The models consist of elastic elements represented by springs in which sigma(t) and gamma(t) are related by a = G gamma (Fig. 9.17(a)), and viscous elements represented by dashpots in which sigma = eta gamma (Fig. 9.17(b)). Answer the following questions. Consider the model shown in Fig. 9.17(c) in which the spring and the dashpot are connected in series. This model is called the Maxwell model. Show that sigma(t) and gamma(t) are related by the following differential equation sigma/G + sigma/eta = gamma Calculate the relaxation modulus G(t) for the above model. Calculate gamma(t) when sigma(t) is changed as follows sigma(t) = {0 t

Explanation / Answer

Let us consider the spring´s model

sigma = G gamma

and dashpot´s model

sigma = eta d gamma/dt,

____________________________

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the figure c. The model can be represented by the following equation:

if gamma= strain, then

d gamma / dt =(1/G) d sigma/dt + sigma/eta.

Under this model, if the material is put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied

______________________________________

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.