2.5. Representation in a coordinate system

Let us write the LMB equation in (e₁,e₂,e₃) coordinate system. For the first three terms, σ will be represented by a matrix. When we take the derivative of a matrix by a scalar, we take the derivative of each term in the matrix. Then we need to multiply by e in (e₁,e₂,e₃) coordinate system.(e.g., in this coordinate system, e will simply be [100]ᵀ).

In this chunk, we are discussing how to express the Linear Momentum Balance (LMB) equation using a coordinate system defined by three basis vectors e₁, e₂, and e₃. The stress tensor σ is represented as a matrix, which allows us to analyze stress in a structured manner. When we differentiate this matrix with respect to a scalar, we apply the derivative to each individual entry of the matrix. After differentiating, we adjust the matrix according to the coordinates in our system, specifically multiplying by the basis vector e that corresponds to the direction we are interested in.

Think of a grid on a map where each point represents a different location with specific traffic data (e.g., cars per hour). Just as you can analyze traffic at a specific point by breaking down the map into grid squares (e₁, e₂, e₃), we analyze stress in a material by breaking it down using the matrix form of the stress tensor.

We get in total three scalar equations from the three rows of the above matrix equation, i.e., (21), (22), (23). Equation (19) is the tensor form of the Linear Momentum Balance and equations (21), (22) and (23) represent it in (e₁,e₂,e₃) coordinate system.

From the matrix representation of the LMB equation, we derive three separate scalar equations corresponding to each row of the matrix. These equations, labeled as (21), (22), and (23), provide a more straightforward understanding of how forces balance along each of the three coordinate directions defined by e₁, e₂, and e₃. Each scalar equation expresses the force balance along a specific axis, making it easier to analyze and solve problems in solid mechanics.

Consider how a team of engineers would individually calculate the load on each beam of a bridge (e₁ for length, e₂ for width, e₃ for height). By breaking down the overall force into components, they can determine if each beam can support the load without failing, just like how we evaluate forces in our scalar equations.

The first of these (equation (21)) is actually just the balance of forces along e₁. This is because all the terms present in this equation act in e₁ direction. Similar interpretation can be made for the remaining two equations (22) and (23).

Equation (21) focuses purely on the force balance in the e₁ direction, meaning all forces acting on the system are evaluated along the x-axis if we associate the e₁ vector with that direction. This principle applies similarly to equations (22) and (23), where they analyze the force balance along e₂ and e₃. By separately considering each direction, we can simplify the analysis of complex systems into manageable parts.

Think of balancing a seesaw. When a child sits on one side, you need to know exactly how much weight is on both sides (forces along the horizontal line) to ensure it stays level. Each side represents a direction similar to how we analyze forces along the e₁, e₂, and e₃ axes.

These equations contain derivatives in space. Thus, we would also need some boundary conditions to solve them. The solution thus obtained would give us the distribution of stress in the body.

The scalar equations derived from the tensor form of LMB contain spatial derivatives, which means the distribution of stress in the material is dependent on the conditions at the boundaries of that material. To find a solution to these equations, we must specify boundary conditions that define how the material interacts with its environment, such as fixed points or applied loads. Introducing these conditions helps us calculate the stress values accurately across the material.

Imagine a water tank where the pressure will be different at various heights (boundary conditions). Knowing the pressure at the top (how full it is) allows you to predict pressures at lower levels accurately. Similarly, knowing the boundary conditions of a solid object helps us understand the stress distribution within it.

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