Show that $$ \lambda+2 \mu=\frac{Y(1-\sigma)}{(1+\sigma)(1-2 \sigma)}=B+\frac{4}{3} \mu . $$ Show that when an external body force per unit volume \(\mathbf{F}_{1 e}\) is present, the equation for the elastic displacement (7.5.5) becomes $$ (\lambda+\mu) \nabla \nabla \cdot e+\mu \nabla \cdot \nabla_{e}+\mathbf{F}_{1 e}=\rho_{\varphi} \frac{\partial^{2} e}{\partial l^{2}} $$ (This equation becomes an equation for elastic equilibrium when \(\varphi\) is not a function of time. Its solution \(\rho(x, y, z)\) must then satisfy specified conditions at the boundary of a body. The stress and strain at each point of the body can be calculated knowing \(\theta\). We thus have formulated the problem of elastic equilibrium in a fundamental way.)

The Lame parameters, named after the French mathematician Gabriel Lamé, play a significant role in the study of the mechanics of continuous media, especially in materials subject to stress and strain. Think of them as ingredients in a recipe that describes how a material deforms under various forces.

Lamé's first parameter, often denoted by the Greek letter \(\lambda\), and the second parameter, \(\mu\), also known as the shear modulus, define the relationship between stress (force per unit area) and strain (deformation) in a linear elastic material. While the shear modulus quantifies the material's response to shear stress (like the feeling of resistance when you try to slide a book along a table), Lamé's first parameter is associated with volumetric changes, such as when a sponge is squeezed and it changes its volume but not its shape.

Young's modulus, symbolized by \(Y\), is a measure of the stiffness of an elastic material. You can imagine it as a score for how much a material resists changing shape when you stretch or compress it. A material with a high Young's modulus is like a strict teacher – it doesn't budge easily, think a chunk of steel. On the other hand, a low Young's modulus is akin to a lenient teacher, one who's more flexible, like a rubber band.

It's described by the ratio of stress (force per unit area) to strain (proportional change in length) in the range where the material remains elastic. That means within these limits, the material will return to its original shape and size after the force is removed. In the relationship we're looking at, the Young's modulus is interrelated with Lamé's parameters and the Poisson's ratio through a formula that connects all these material constants in a concise manner.

The Poisson ratio, which we represent with the Greek letter \(\sigma\), is like the social behavior of materials. When a rod of material is stretched and it gets longer, it often gets skinnier in the other directions. Conversely, if you squash it, it will likely bulge out to the sides. This interaction between dimensions in response to stress is what the Poisson ratio measures.

The value of \(\sigma\) is a dimensionless quantity typically ranging from 0 to 0.5 for most materials. A rubber band that gets thinner as it stretches has a high Poisson ratio, close to 0.5, whereas a cork that doesn't change its diameter much has a Poisson ratio close to 0. This concept is pivotal in understanding how materials deform in three-dimensional space when forces are applied.

The bulk modulus, denoted by \(B\), is like the material's response to a group hug – it measures how resistant a substance is to uniform compression from all sides. It's a thermodynamic parameter that gives you a sense of how incompressible a material is. For instance, water is more incompressible compared to air, meaning it has a higher bulk modulus.

In a mathematical sense, the bulk modulus is defined as the ratio of volumetric stress to the volumetric strain for a material. When related to the other material constants, it provides a comprehensive picture of the elastic properties of a material. The equation given in the problem ties the bulk modulus with the shear modulus and Lamé's first parameter in a balance that surfaces during elastic deformations.

Elastic displacement refers to the change in position of points in a material when it is subjected to forces. Picture it this way: when you poke a blob of putty, the indents are like the material is displaced elastically. Displacement is mapped by a vector field within the material, representing the new positions of the points within the material's body as a result of applied forces.

Moreover, when analyzing problems of elastic equilibrium such as in the given exercise, we consider the absence of motion over time \(\varphi\) – meaning we're looking at the material in a 'freeze-frame' where everything holds still. In this instance, the material's elastic displacement is governed by the equilibrium equations derived, which also factors in any body forces \(\mathbf{F}_{1 e}\) that might exist, such as gravitational forces acting on all parts of the body.