A material point in equilibrium has 1 independent component of shear stress in the xz plane. a)True b)- False

b) False

In the context of mechanics of materials, when we refer to a material point in equilibrium with respect to shear stress, it means that the net effect of all the stresses acting on the body results in no acceleration of the material point.

Equilibrium conditions require that the sum of forces and moments acting on a point (or on the whole body) must be equal to zero. This applies to both normal stresses (which act perpendicular to a surface) and shear stresses (which act parallel to a surface).

For a material point in equilibrium, considering shear stresses in three dimensions, we typically have three planes on which shear stresses can act: xy, xz, and yz planes. Each of these planes has an associated pair of shear stresses due to Newton's third law. For instance, the shear stress acting on the xz plane (τ_xz) will have a corresponding shear stress on the zx plane (τ_zx), and these are equal in magnitude but opposite in direction.

Therefore, having only a single component of shear stress on the xz plane without an accompanying component on the zx plane would violate the equilibrium conditions, as it would result in a net moment that is not counteracted, thus moving the material point out of equilibrium. Therefore, shear stresses always come in equal and opposite pairs.

Extra: Stress at a point in a material is described by a stress tensor, which in a three-dimensional case is a 3x3 symmetrical matrix. This tensor fully defines the state of stress at a point, including normal and shear stress components.

- Normal stress components (σ_xx, σ_yy, σ_zz) act perpendicular to the planes defined by the coordinates (x, y, z). - Shear stress components (τ_xy, τ_xz, τ_yx, τ_yz, τ_zx, τ_zy) give the stress acting parallel to these planes.

Each off-diagonal shear stress element in the tensor comes in pairs which are equal based on the symmetry of the tensor (τ_xy = τ_yx, τ_xz = τ_zx, τ_yz = τ_zy). This reflects physical reality as per action and reaction forces; for a body to be in equilibrium, these pairs must exist to avoid unbalanced torques.

Understanding stress equilibrium is crucial in fields like structural engineering, materials science, and mechanical engineering, where it's important to ensure that structures or components do not fail under applied loads.