## Shear Properties of Polymers

The shear properties of polymers are
important in many applications, particulalry when polymers are used for structural
parts. The simplest case
is a homogeneous isotropic body. For this case, the mechanical response depends on only two constants,
the *shear modulus* *G* and the *Poisson ratio* *ν.*

The figure below shows a simple shear
deformation of a solid body. The body has the shape of a cube which
has been deformed by a force
*F*, acting in a plane *A _{0}* parallel to
the deformation.

### Shear Deformation of a Solid

For small strains, the deformation of the
cube is elastic, that is, the deformation is homogenous and after removal of the deforming load the
specimen returns to its original size and shape. Furthermore, the
shear stress (*τ*) is
directly proportional to the shear strain (*γ*), that is, the material obeys
*Hooke's law*:

*τ* = *G γ*

where *G* is the *shear modulus*.
It is the slope of the stress-strain curve, i.e., the ratio between an incremental increase in applied stress, Δ*τ*, and
an incremental deformation, Δ*γ*. *G* is a measure for the stiffness of a material. The reciprocal of
the shear modulus is the *shear compliance, J*, defined by

*γ* =
*J τ*

In a *stress-strain experiment*, a sample is
sheared at a constant shear rate and the stress, *τ*, is measured as
function of strain, *γ*.^{1} In most cases, the stress-strain response of
a shear sample is reported in terms of nominal or engineering
stress and strain. The *nominal or engineering shear* *stress*
is defined as the force divided by the initial (undeformed) area of
the surface in which the shear force acts:

*τ _{e}* =

*F*/

*A*

_{0}and the e*ngineering shear strain* is
defined as the angle of deformantion, tan *θ*, which is equal to the amount of deformation, Δ*x*,
to the hight of the cube, *h*:

*γ _{e}* = tan

*θ*= Δ

*x*/

*h*

For isotropic materials and elastic
deforamtion where Hooke's law is valid, the shear modulus and the
tensile modulus can be related through the *Poison's ratio* *ν*:

*E* = 2 (1 + *ν*) *G*

Thus, only two of the three properties have to be measured.

##### References & Notes

The shear modulus

*G*is often measured with a torsion pendulum tester or DMA by subjecting the specimen to an oscillatory deformation. The complex shear modulus G* is given by the expression|G*| = (G'

^{2}+ G''^{2})^{1/2}The storage modulus G' in this equation represents the elastic behavior and is a function of shear and strain amplitude and phase angle.

M.D. Lechner, K. Gehrke, E.H. Nordmeier,

*Makromolekulare Chemie*, 1993L.H. Sperling,

*Introduction to Physical Polymer Science*, New York 1992