In the mechanics of materials, a shear stress acting across a plane is always accompanied by an equal shear stress on a plane normal (perpendicular) to it, forming a pair of complementary shears that maintain rotational equilibrium. State whether this assertion is correct.

Introduction:
The statement tests the concept of complementary shear stresses, a fundamental requirement for rotational equilibrium in a stressed element in solid mechanics.

Given Data / Assumptions:

• Continuum, static equilibrium, small deformations.
• No body couples or special anisotropic effects that violate standard Cauchy stress assumptions.

Concept / Approach:
For an elemental square (or cube) under shear stress on one face, rotational equilibrium requires a balancing shear on the adjacent, mutually perpendicular face. These are called complementary shear stresses and have equal magnitude and opposite sense to prevent net moment on the element.

Step-by-Step Solution:
1) Consider a tiny element with shear tau acting on the horizontal plane.2) The shear produces a couple tending to rotate the element.3) To maintain zero net moment, an equal shear tau must act on the vertical (normal) plane, in the opposite sense.4) Hence, shear across one plane is necessarily accompanied by an equal shear on the plane normal to it.

Verification / Alternative check:
From equilibrium of moments on the differential element: tau_xy = tau_yx.

Why Other Options Are Wrong:
False: contradicts moment equilibrium.
Depends on material only: this arises from equilibrium, not constitutive law.
True only at yield: validity is independent of yielding; it holds in elastic and plastic ranges provided equilibrium applies.

Common Pitfalls:
Assuming complementary shear arises from Hooke's law; it actually comes from balance of moments and the symmetry of the Cauchy stress tensor in absence of body couples.

Final Answer:
True