Question

A copper wire upon loading instantaneously increases in length to $l$, and then continues to elongate gradually. Upon unloading, the wire retracts to length $l$. According to the Maxwell model, which one of the options given correctly relates the total strain $E$, the applied stress $S$, the modulus $G$, the material’s resistance to flow $\eta$, and the elapsed time $t$ between loading and unloading?

• A. $E = (S/G) - (S/\eta)t$
• B. $E = (S/G) \times (S/\eta)t$
• C. $E = (S/G) + (S/\eta)t$
• D. $E = (S/G) / (S/\eta)t$

Hint:

In Maxwell’s model, total strain = elastic strain $+$ viscous strain. Elastic part is instantaneous, viscous part grows with time.

Answer 1

The Correct Option is C

Step 1: Recall the Maxwell model.
The Maxwell model of viscoelasticity combines a spring (elastic element) and a dashpot (viscous element) in series. The total strain is the sum of the elastic strain and the viscous strain.
Step 2: Write expression for elastic strain.
Elastic strain due to the spring is given by: \[ E_{elastic} = \frac{S}{G} \]
Step 3: Write expression for viscous strain.
Viscous strain due to the dashpot is proportional to time: \[ E_{viscous} = \frac{S}{\eta} t \]
Step 4: Total strain.
The total strain in the Maxwell model is: \[ E = E_{elastic} + E_{viscous} \] Substituting values: \[ E = \frac{S}{G} + \frac{S}{\eta}t \]

Step 5: Match with options.
This matches with option (C). Final Answer: \[ \boxed{E = \frac{S}{G} + \frac{S}{\eta}t} \]