A cylindrical specimen of a hypothetical metal alloy is stressed in compression. If its original and final diameters are \(30.00\) and \(30.04 \mathrm{~mm}\), respectively, and its final length is \(105.20 \mathrm{~mm}\), compute its original length if the deformation is totally elastic. The elastic and shear moduli for this alloy are \(65.5\) and \(25.4\) GPa, respectively.

Poisson's ratio, symbolized by ug, is a measure that tells us how much a material will expand or contract in directions perpendicular to the direction of compression or stretching. Consider a rubber band being stretched; not only does it become longer, but also thinner. The same occurs in reverse when you compress a material - it can expand outwards. Poisson's ratio is calculated using the formula: \[u = \frac{E}{2G} - 1\], where \(E\) stands for the elastic modulus and \(G\) represents the shear modulus. A common value of Poisson's ratio for many materials is around 0.3, indicating that materials typically shrink by about 0.3 times their stretching amount transversely when stretched longitudinally.

The term 'longitudinal strain' refers to the deformation experienced by a material along its length when subjected to stress. It is a dimensionless quantity defined as the change in length per unit length and can be positive (elongation) or negative (contraction). Mathematically, it's represented as:\[\varepsilon_{longitudinal} = \frac{L_{final} - L_{initial}}{L_{initial}}\]. In the context of the exercise, longitudinal strain has a crucial part in finding the original length of the cylinder before its compression. Since materials respond to stress differently, accurately calculating longitudinal strain is essential for engineers when designing structures and predicting their behavior under various load conditions.

The shear modulus, or modulus of rigidity, symbolized as \(G\), describes how a material reacts to shearing forces. It quantifies the material's response by measuring the ratio of shear stress to the shear strain. The formula is given by:\[G = \frac{{\text{shear stress}}}{{\text{shear strain}}}\]. It helps us understand the relationship between the force applied parallel to a surface and the deformation it causes from the original shape. A higher shear modulus value indicates a material that resists shear deformation well, which is desired in applications where rigidity is necessary, such as in the beams of a building to maintain structural integrity.

Elastic modulus, also known as Young's modulus and represented by \(E\), is a fundamental property of materials reflecting their stiffness and is defined as the ratio of tensile stress to longitudinal (tensile) strain. The expression for calculating it is:\[E = \frac{{\text{tensile stress}}}{{\text{tensile strain}}}\]. It gives us an insight into how much a material will deform under a certain load and is crucial for guaranteeing that structures won't fail under stress. The elastic modulus plays a pivotal role in determining the degree of compression or extension possible before a material reaches its elastic limit and suffers permanent deformation.