4. Shear stress:
Shear stress is the stress which is applied parallel or
tangential to a face of a material; it differs from normal
stress which is applied perpendicularly. Shear stress, tends
to cause deformation of a material by slippage along a
plane or planes parallel to the imposed stress and may
occur in both liquids and solids. Shear stress in most
engineering texts is t (tau). t =F/A
Shear strain :
It is the components of a strain at a point that produce changes in
shape of a body (distortion) without a volumetric change.That is,
the tangent of the angular change in orientation of two initially
perpendicular lines.
5. Shear stress strain curve:
.
If the applied load consists of two equal and
opposite parallel forces which do not share
the same line of action, then there will be a
tendency for one part of the body to slide
over, or shear from the other part.
In the figure , if the section LM is parallel to
the forces and has an area A, then the
average Shear Stress .t=F/A
We can start by looking at an
element of material undergoing
a shear stress (the red arrows)
6. shear stress
The upper face has a force directed parallel,equal
and opposite to the bottom face.These two
forces equal in magnitude but opposite in
direction generate a couple on the element.Since
the element is in equilibrium.The two forces
acting on the left and right faces produce a
couple that is equal in magnitude to the couple
produced by the forces acting on the top and
bottom faces. This will always be the case
when an element in under shear.
The shear acting on the four faces of the element
cause a deformation of the element. If the lower
left corner is considered stationary, we can look
at how much the upper left corner of the element
moves. The upper left corner has moved a
distance δ in the positive x direction. If we look at
the angle made by the deformation and the
positive y-axis. This is not the angle θ’.The sine of
this angle is δ x/L.
shear strain
7. shear strain
If δx is very small with respect to L, which is
generally the case,then the value of the angle in
radians is approximately equal to the sign of the
angle δ x/L .γ(gamma) is the symbol for the
shear strain, so we have γ y = δ x/L. It is labeled
with an xy subscript because we are looking at
the shear strain in the xy plane I have labeled it
with a y subscript because it is the angle made
with the y-axis.γ y = δ x/L. The shear is actually
the difference between the original angle
between the side along the y-axis and the side
along the x-axis and the angle after loading θ’, γ y
= δ x/L. The angle θ’ is determined by using θ
'=π/2−γ y=π/2−δ x/L. If there is also a
deformation above or below the x-axis, it must
also be included to solve for θ’.θ ' = π/2−γ xy.
8. Modulus of Rigidity:
For Elastic materials it is found that within certain limits, Shear
Strain is proportional to the Shear Stress producing it.
The ratio of shear stress to shear strain is called modulus of
rigidity.
Modulus of Rigidity G = Shear stress/Shear strain= t/Y
