For many viscous materials, the viscosity \(\eta\) may be defined in terms of the expression $$ \eta=\frac{\sigma}{d \epsilon / d t} $$ where \(\sigma\) and \(d \epsilon / d t\) are, respectively, the tensile stress and the strain rate. A cylindrical specimen of a soda-lime glass of diameter 5 \(\mathrm{mm}(0.2\) in.) and length \(100 \mathrm{~mm}\) (4 in.) is subjected to a tensile force of \(1 \mathrm{~N}\left(0.224 \mathrm{lb}_{\mathrm{f}}\right)\) along its axis. If its deformation is to be less than \(1 \mathrm{~mm}(0.04\) in.) over a week's time, using Figure 13.7, determine the maximum temperature to which the specimen may be heated.

Viscosity is closely related to how materials respond to forces, which is where the concept of tensile stress comes into play. In simple terms, tensile stress measures how much force is applied over a certain area on a material. Think of it like spreading out your body weight on a bed of nails; the more nails there are, the less it hurts, because your weight is distributed over a larger area.

To find the tensile stress (\tau) given a force (F) applied to a cylindrical object, like our glass specimen in the exercise, we use the formula \tau = \(\frac{F}{A}\), where A is the cross-sectional area. For a cylinder, this area is A = \(\pi r^2\), with r being the radius of the cylinder. Mathematically, this relationship allows us to see how changing dimensions of the object or the magnitude of the force affects the stress experienced by the material.Applying this to our example, with a force of 1 newton and a radius of 2 mm, we calculated that the tensile stress was 0.159 N/mm². This understanding is crucial because tensile stress is a key parameter that can influence how materials deform under load. It is a foundational concept for materials engineering and physics.

When dealing with the deformation of materials under stress, the strain rate is another essential factor. It's basically the speed at which a material changes shape. The faster the strain rate, the more quickly the material deforms.

In our exercise, strain rate was calculated from the change in length (d Epsilon) over a specific time period (d t). The formula we used was \frac{d Epsilon}{d t} = \(\frac{\Delta L / L}{t}\), where \Delta L is the change in length, L is the original length, and t is the time over which the change occurs.By calculating the strain rate as 0.02 strain/week for our glass specimen, we gained insight into how fast it was deforming under the applied stress. Understanding the strain rate is particularly important in assessing whether a material can withstand certain conditions over time without failing, making it a fundamental concept for structural analysis and design.

The deformation of a cylindrical specimen, such as the soda-lime glass in our exercise, is a practical application of the concepts of tensile stress and strain rate in measuring how materials respond to forces. Deformation refers to the change in shape or size of an object due to applied forces and can be temporary or permanent.

In practical scenarios, controlling the deformation is crucial, especially when precision is required. For the soda-lime glass specimen, we want the deformation to be less than 1 mm over a week's time to ensure its integrity. By calculating the maximum allowable temperature using the established relationship between stress, strain rate, and viscosity, engineers and designers can predict how various conditions like temperature changes will affect the structure of materials.

The formula ta = \(\frac{sigma}{d Epsilon / d t}\) allows us to calculate how viscous forces counteract the deformation. In our case, a viscosity of 7.95 N·week/mm² was determined, setting the baseline for how much the glass cylinder can bend without breaking under specified conditions.

Understanding cylindrical specimen deformation is paramount in fields like construction, manufacturing, and material science, where it's necessary to know how materials will behave when subjected to various forces.