1. Need for stress-strain relation

Consider a body clamped at one part of the boundary and acted upon by an external load on another part of its boundary as shown in Figure 1. Any general point in the reference configuration (denoted by position vector X) is displaced by u(X,t) as shown in the figure.

A body experiences changes when forces are applied to it. When part of a body is fixed, and the other part is loaded, the body deforms, meaning its shape changes. The position of a point in the original shape of the body, denoted as 'X', shifts to a new position given by 'u(X,t)', which shows how much it moves and how time affects this movement. This relationship is crucial for understanding how materials respond to forces.

Think of a rubber band. When you pull one end while the other is held tight, it stretches and deforms. The point where you measure the rubber band’s stretch from is similar to 'X', and how much it stretches away from that point is comparable to 'u(X,t)'.

Usually, we are interested in knowing the final deformed shape of the body and the stress generated in it. The knowledge of stress is important because it helps us to check if it is within the prescribed limits so that the body does not fail or fracture.

Understanding stress is crucial because it informs us whether the material can handle the applied forces without breaking. Engineers and designers need to ensure that the stress does not exceed certain limits; otherwise, the material might fail or fracture. This concept is essential for safety in buildings, bridges, and other infrastructures.

Consider a bridge: engineers must calculate the stress on different parts of the bridge to ensure it can safely support traffic. If the stress exceeds limits, it could lead to the bridge collapsing, much like how a weight placed too heavily on a thin shelf could cause it to snap.

In order to find stress in the body, we can use the following equations of equilibrium derived earlier. We have nine unknown components of stress matrix in three equations here. Using symmetry of stress matrix, the number of unknowns reduces to six.

To analyze the stress within a body, we apply equilibrium equations which consider the balance of forces. Initially, there are nine unknown components of stress that we must solve, but using the symmetrical properties of the stress matrix, we simplify this to six unknowns. This mathematical reduction is key for solving real-world problems efficiently.

Imagine trying to balance a book on your head. You might initially think about how much pressure is on each corner of the book. By recognizing the symmetry (the book is balanced evenly), you can reduce the complexity of your calculations and focus on just a few points while maintaining balance.

Thus, this system cannot be solved without the help of additional equations. These additional equations are given by stress-strain relation.

When we set up the equations to solve for stress using equilibrium, we find that we have more unknowns than equations. To resolve this issue, we require additional relationships known as stress-strain relations to link stress and strain directly, allowing us to express one in terms of the other.

Think of a puzzle: if you have more pieces than the slots available, you need additional rules or information about how the pieces fit together. Similarly, the stress-strain relationship provides the necessary connections to solve the mathematical puzzles in structural analysis.

If we can now in some way express stress in terms of strain, then indirectly, we have actually expressed stress in terms of displacement.

By deriving a relationship between stress (the internal forces) and strain (the deformations), we simplify our equations to handle just the displacements of the body. This transformation makes it easier to analyze how a material will respond under load, paving the way to solve the equilibrium equations.

Imagine a sponge. When you press it (apply stress), it deforms (strain). By knowing how much it squishes for a given pressure, you can predict how much it will squish under different pressures, allowing you to design things like sponges for specific uses based on their stress-strain characteristics.

Thus, the stress-strain relation is necessary to get the deformed solution of a body from equilibrium equations. Often, the stress-strain relation is also called constitutive relation.

The relationship between stress and strain is pivotal in material science and solid mechanics because it allows engineers to predict how materials will behave under various loading conditions. It’s one of the fundamental principles that guide the design of safe and effective materials and structures.

Think of a tuning fork. When you strike it, it vibrates. The stress and strain relationship illustrates how the material responds to being struck, and engineers use this principle to understand how materials can vibrate in musical instruments, which is vital for creating the right sounds.

--