Consider a brick of material thrown into the ocean at its deepest point. The Ocean is seven miles...

Question:

Consider a brick of material thrown into the ocean at its deepest point. The Ocean is seven miles deep where there are 5,280 feet/mile and each cubic foot of water weighs 60 lbs. Thus, the hydrostatic pressure on the brick is 2,217,600 pounds per square foot or 15,400 pounds per square inch [psi].

The brick is made of polyester elastomer where E = 0.03 * {eq}10^{6} {/eq} psi and Poisson's ratio is 0.3. The initial dimensions of the brick are exactly 2.000" by 3.000" by 6.000".

What are the final dimensions (to three decimal places) of the polyester elastomer brick when it is on the bottom of the ocean detailed above?

Poisson's Ratio.

Imagine that a steel rod is subjected to the action of an axial tensile force. The rod will experience a longitudinal elongation or axial strain. In the same way, there is also a lateral strain of the rod. If we divide the negative of the lateral strain by the axial strain, we obtain Poisson's reaction.

{eq}\text{Known data:}\\ P = -15400\,\dfrac{lb}{in^2}\\ E = 0.03\times{10^6}\,\dfrac{lb}{in^2}\\ \nu = 0.3\\ a = 2.000\,in\\ b = 3.000\,in\\ c = 6.000\,in\\ \text{Unknowns:}\\ a' = ? \\ b' = ?\\ c' = ?\\ {/eq}

Water exerts a great pressure on the block, that is, a great compression force at each point on the surface of the block that is expressed by a negative sign. The unitary deformation or tensil strain of each of the dimensions of the block is defined by the following relationship:

{eq}\epsilon_x = \epsilon_y = \epsilon_z = \dfrac{P}{E}(1-2\nu) = \dfrac{-15400\,lb/in^2}{0.03\times{10^6}\,lb/in^2}\left [ 1- 2(0.3) \right ] = -0.2053\\ {/eq}

The final dimension a' is the length a plus the longitudinal deformation x.

{eq}a' = a + \epsilon_x a = a\left [ 1 + \epsilon_x \right ]\\ a' = 2.000\,in\left [ 1 + (-0.2053) \right ] = 1.589\,in\\ \color{blue}{a' = 1.589\,in}\\ {/eq}

The final dimensions b' and c' are respectively:

{eq}b' = b\left [ 1 + \epsilon_y \right ]\\ b' = 3.000\,in\left [ 1 + (-0.2053) \right ] = 2.384\,in\\ \color{blue}{b' = 2.384\,in}\\ {/eq}

{eq}c' = c\left [ 1 + \epsilon_y \right ]\\ c' = 6.000\,in\left [ 1 + (-0.2053) \right ] = 4.768\,in\\ \color{blue}{c' = 4.768\,in}\\ {/eq}