A uniform tensile stress \(f_{x x}\) is applied to the ends of a rectangular flat plate of thickness 1\. Concurrently the sides of the plate are constrained by applying a uniform tensile stress \(f_{y y}\) 76 Introduction to the Theory of Elasticity \(\mathrm{~ m o n m e n m e n}\) of such magnitude that the width w' of the plate remains constant. Show that the plate Young's modulus \(Y_{p} \equiv f_{x x} / \epsilon_{x x}\) is \(Y /\left(1-\sigma^{2}\right)\) and that the plate Poisson's ratio \(\sigma_{p} \equiv-\epsilon_{u z} / \epsilon_{x x}\) is \(\sigma /(1-\sigma) .\)

Hooke's Law is a fundamental principle that forms the basis of understanding material properties in elastic regions. It states that, within the elastic limit, the strain in a material is directly proportional to the applied stress.

In simpler terms, envision a spring: If you pull it gently, it stretches, and when you let go, it returns to its original shape. Hooke's Law is the mathematical description of this behavior for any elastic material. The formal representation for a one-dimensional state of stress can be expressed using the formula:
\[ F = k \times x \]
where \( F \) is the force applied to the material, \( k \) is the spring constant (or stiffness), and \( x \) is the extension or compression of the material (strain).

Now, considering a three-dimensional material under stress, Hooke's law becomes more complex and is represented in terms of stress \( (\tau) \) and strain \( (\theta) \) as:
\[ \theta = \frac{\tau}{E} \]
Where \( E \) is the Young's modulus of the material, describing its stiffness. Exploring the relationship between these variables helps us predict how a material will react under different loading conditions.

Material strain represents the deformation experienced by a body resulting from stress. Strain is a dimensionless quantity, often expressed as a percentage. It is defined as the change in dimension divided by the original dimension.

For instance, consider a bar with an initial length \( L_0 \). When a force is applied to it, it stretches to a new length \( L \). The strain, \( \text{\varepsilon} \), can be quantified as:
\[ \text{\varepsilon} = \frac{L - L_0}{L_0} \]
This is called 'longitudinal strain'. There's also 'shear strain,' which occurs when the applied stress causes a change in shape but not volume, and 'volumetric strain,' associated with a change in the body's volume.

Strain tells us how much a material can deform under stress before it fails or is permanently deformed. This concept is crucial for ensuring structures can withstand the loads they are designed for without experiencing failure or excessive deformation that may lead to reduced functionality or collapse.

The stress-strain relationship is pivotal in understanding material behavior in structural engineering and material science. It describes how a material deforms as stress is applied to it and is graphically represented by the stress-strain curve.

In the initial portion of this curve, the relationship between stress (force per unit area) and strain (deformation relative to original length) typically follows a linear path. This linear relationship indicates the material's elastic behavior, where it will return to its original shape after the removal of the applied stress.

The slope of the linear portion of the stress-strain curve is the 'modulus of elasticity' or Young's modulus \( (Y) \). A steeper slope means a stiffer material. The point where the material no longer behaves elastically and begins to deform permanently is known as the yield point. Beyond this, the material undergoes plastic deformation.

Understanding the stress-strain relationship guides the design of materials and structures for various applications, ensuring they can sustain operational stresses without failing. It also assists in predicting material behavior under different types of loading and conditions, which is crucial for safety and performance.