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Mechanical properties are fundamental in understanding how materials behave under different types of forces and loads. These properties include elasticity, plasticity, toughness, and hardness, among others. In this exercise, two essential properties are Young's modulus ( E ) and Poisson's ratio ( u).

Young's modulus ( E ) is a measure of the stiffness of a material. It quantifies the relationship between stress (force per unit area) and strain (deformation over original length) in the elastic deformation region. For steel in this problem, E is given as 200 GPa, indicating a very stiff material that does not deform easily under applied stress.

Poisson's ratio ( u) refers to the ratio of lateral strain to axial strain in a material subjected to uniaxial stress. It indicates how much a material will expand or contract in directions perpendicular to the direction of compression or stretching. Here, for steel, u is 0.30, suggesting moderate lateral expansion when stretched longitudinally. Together, these properties help us understand how the steel tank will behave when internal pressure is applied.

Thin-walled pressure vessels are structures such as tanks or pipes where the wall thickness is considerably small compared to the vessel's radius. A general rule of thumb is that the thickness ( t) should be less than one-tenth of the radius ( R) of the vessel. In this exercise, the tank's wall thickness is 6 mm, and the inside radius is 300 mm, qualifying it as a thin-walled structure.

Due to their geometry, thin-walled pressure vessels experience stress mainly in two directions: circumferential (or hoop) stress and longitudinal stress. Hoop stress acts along the circumference, trying to "burst" the vessel open, while longitudinal stress acts along the length of the vessel. In cylindrical vessels, hoop stress ( \sigma_y) is typically higher than longitudinal stress ( \sigma_x), and is given by the formula:

\[ \sigma_y = \frac{pR}{t} \]

Where p is the internal pressure. This helps in understanding the different forces at play in thin-walled vessels and is crucial for safe design and analysis.

Stress-strain relations describe how materials deform under stress, essentially linking mechanical stress with the resulting strain. This relation is central to understanding the design and analysis of engineering systems.

For materials like steel, we use Hooke's Law in the context of axial loading, which implies the stress-strain relation as follows:

\[ \epsilon_y = \frac{1}{E} \left( \sigma_y - u \sigma_x \right) \]

Where \epsilon_y is strain in the circumferential direction, E is Young’s modulus, and u is Poisson’s ratio. \sigma_y and \sigma_x are the hoop and longitudinal stresses, respectively.

This equation allows engineers to compute the strain in a material, given known stresses and material properties. By rearranging this equation, as done in the original solution, one can solve for unknown variables like internal pressure when other parameters are available. This practical use of stress-strain relationships is key in calculating the pressures that thin-walled vessels can safely withstand. Understanding and applying these relationships helps ensure that the designs of such vessels are both efficient and reliable.