For some viscoelastic polymers that are subjected to stress relaxation tests, the stress decays with time according to $$ \sigma(t)=\sigma(0) \exp \left(-\frac{t}{\tau}\right) $$ where \(\sigma(t)\) and \(\sigma(0)\) represent the timedependent and initial (i.e., time \(=0\) ) stresses, respectively, and \(t\) and \(\tau\) denote elapsed time and the relaxation time; \(\tau\) is a timeindependent constant characteristic of the material. A specimen of a viscoelastic polymer whose stress relaxation obeys Equation \(15.10\) was suddenly pulled in tension to a measured strain of \(0.6\); the stress necessary to maintain this constant strain was measured as a function of time. Determine \(E_{r}(10)\) for this material if the initial stress level was \(2.76\) MPa (400 psi), which dropped to \(1.72 \mathrm{MPa}\) (250 psi) after \(60 \mathrm{~s}\).

Viscoelastic polymers exhibit both viscous and elastic properties when undergoing deformation. These materials behave somewhat like viscous liquids, which flow under a constant stress, and somewhat like elastic solids, which quickly snap back to their original shape once the stressing force is removed. However, viscoelastic polymers will slowly return to their original shape, illustrating a blend of these behaviors.

The nature of this response is heavily dependent on factors like time, temperature, and the rate of applied load. For example, consider pulling a piece of chewing gum; stretch it slowly, and it flows easily, but try to snap it quickly, and it breaks like a brittle solid. This analogy helps in understanding the behavior at a fundamental level. The response to stress over time can again be captured in terms of two key phenomena: stress relaxation and creep.

In stress relaxation, if a viscoelastic polymer is rapidly deformed to a fixed strain and held there, the stress required to maintain that strain will decrease over time. This is because the polymer chains need time to untangle and align themselves in the direction of the stress, effectively diminishing the internal resistance to the deformation.

Capturing these characteristics through mathematical models can be complex, but an exponential decay function often approximates the stress relaxation behavior well.

The relaxation time constant (\tau) is a pivotal parameter in understanding viscoelastic materials. It essentially represents the time required for the stress in the material to decrease to 1/e (about 36.8%) of its original value after a constant deformation has been applied.

Mathematically, if \( \sigma(t) \) describes the stress at time \( t \) and \( \sigma(0) \) is the initial stress when \( t = 0 \), then the stress relaxation can often be expressed as \( \sigma(t) = \sigma(0) \exp \left(-\frac{t}{\tau}\right) \). Here, the negative sign in the exponent indicates a decrease in stress over time.

By observing how the stress changes over a specific time interval, one can calculate the relaxation time constant for a material, as shown in the solution provided. Knowing \( \tau \) is crucial as it helps to predict how the material will behave under long-term stress applications, which is particularly important in structural and biomedical applications, where polymers must endure sustained loads.

The relaxed modulus (\( E_r(t) \)) provides a measure of the material's stiffness at a specific elapsed time after undergoing a stress relaxation process. It is calculated using the stress value at that particular time divided by the constant strain applied.

The formula \( E_r(t) = \frac{\sigma(t)}{\epsilon} \) takes into account the constantly decreasing stress value \( \sigma(t) \) as the polymer relaxes, while \( \epsilon \) represents the fixed strain it was subjected to. In the context of the solved exercise, the relaxed modulus is evaluated at a chosen time (10 seconds) post the initial deformation.

From the exercise, if you know the initial stress, the strain level, and the time constant, you can predict the stress at any later time using the stress relaxation equation. Once the stress at the time of interest is calculated, as seen in the solution, the relaxed modulus is then easily found. This modulus is a snapshot of how stiff the material is at that point in time during the relaxation process, providing valuable information for designers and engineers when considering the long-term performance of polymer-based components.