Simple shear displaces one coordinate direction in proportion to a transverse coordinate while preserving volume. Its deformation gradient has a single off-diagonal component . Simple shear flow describes the corresponding velocity field rather than the finite deformation.
For the reciprocal-Jacobian quadratic energy under , has off-diagonal entries and equal to one and all others zero. Every unit reference normal in the first two coordinate directions therefore maximizes nominal traction. The Cauchy stress tensor has upper-left block . Its largest absolute eigenvalue is , with spatial normal proportional to . Pulling this back by gives reference normal proportional to and plane .
For , direct multiplication gives . Thus length-preserving line directions form the union of two planes in direction space: and . Their intersection is the third coordinate axis. This concerns segment length rather than pointwise fixed lines; a segment can move and turn while keeping its length.
The two nontrivial principal stretches are , with . The maximizing reference direction is proportional to . Consequently it tends to as tends to zero from above. At zero itself all stretches equal one, so every direction maximizes length; the limit selects a direction from a degenerate endpoint. This follows from the largest eigenvector of the right Cauchy-Green deformation tensor.
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Simple shear is a type of deformation that occurs in materials when they are subjected to shear stress. In this mode of deformation, layers of material slide past each other while the overall volume remains constant. It is characterized by the parallel displacement of adjacent layers, which results in an angular distortion of the material. In a simple shear scenario, one side of an object is moved in one direction while the opposite side is held in place or moved in the opposite direction, creating a shear strain.