2.2. Trigonometric formula for σ and τ

Now, our goal is to calculate the normal and shear components of traction on planes whose normals are perpendicular to e . The blue line in Figure 2 shows a general plane of such kind. The normal to this plane is represented by n and assume that it makes an angle α with e axis.

In this chunk, we articulate the main objective of this section, which is to calculate the normal and shear components of traction acting on a specific type of plane. This plane has its normal vector (denoted as n) perpendicular to the third principal stress direction, represented by the e vector. Additionally, this normal vector n is inclined at an angle (α) from another direction, specifically the e1 axis. Understanding this setup is crucial as it establishes the basis for the subsequent calculations of stress components.

Imagine you are driving a car up a hill. The direction of the car is analogous to the e axis, while the angle you steer represents the angle α. Just like the car's direction affects the slope's incline, the angle α between the stress components will influence the resulting stresses acting on a surface.

To get to this arbitrary plane n, we can rotate our e1 plane by α about the e3 axis.

This statement discusses the method for achieving the desired orientation of the plane by rotating the reference e1 plane by the angle α around the e3 axis. This rotation allows us to align the plane exactly with the general plane defined by the normal vector n. Understanding the geometric interpretation of this rotation is important, as it helps visualize how angular adjustments affect the orientations of the stress components we want to calculate.

Think about twirling a steering wheel. As you turn it (representing the rotation about the e3 axis), the direction you are pointing (or driving the car) changes. This simulates how we can adjust our plane's alignment by rotating the reference plane to match our intended normal.

Now, σ will be given by (4) To get τ, we need to first represent the direction along which it is acting which we denote by n⊥.

In this segment, we introduce the notation for normal stress (σ) and shear stress (τ). The equation provided indicates how σ is computed for the specified plane. To calculate τ, we need to understand the direction in which τ acts, represented by n⊥, which is orthogonal to n. This implies that while σ measures the stress along the normal direction, τ captures the stress acting tangentially along the plane's surface.

Imagine a painter applying paint on a canvas. The paint directly applied on the surface represents the normal stress (σ), while the paint layered along the edge or the side represents shear stress (τ). Just like the two actions are different yet related, normal and shear stresses quantify different aspects of how a material bears load.

If we look at Figure 2, we can find the angle that n⊥ makes with all the three axes. The representation of n⊥ will thus be.

This chunk highlights the importance of identifying the angle that the orthogonal direction (n⊥) makes with the three coordinate axes. By determining these components, we ensure that we are correctly calculating shear stress, accounting for how the applied normal and shear stresses interact based on their orientations. Understanding these angles is imperative to correctly evaluate the stress components acting on the chosen plane.

Visualize trying to lift a box at an angle. The direction in which you pull represents equally the n and n⊥ components. Just as you need to consider both the vertical and horizontal forces to find the effective lifting force, we must analyze the angles to compute shear stresses correctly.

Upon doing some algebraic manipulation in equation (4), we get (7) Further, using the following trigonometric identities: cos(2α) = cos²(α)−sin²(α)=2cos²(α)−1=1−2sin²(α), sin(2α) = 2sin(α)cos(α), we obtain.

Here, we focus on the algebraic manipulation needed to express σ and τ in terms of angle α using specific trigonometric identities. By applying these identities, we can convert the relationships between the normal and shear components into a more usable form. This underscores the significance of trigonometric functions in resolving shear and normal stresses based on their angles, which are ultimately foundational to mechanics.

Consider a seesaw. The relationships between the angles at which you push down on one side and the consequential lift on the opposite side symbolize the trigonometric identities: the interplay of forces represents the algebra we've done to correlate the components. Just as the seesaw dynamics depend on angles, the stress relationships also pivot on the calculations done via trigonometric functions.

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