In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain: where The derived SI unit of shear modulus is the pascal, although it is usually expressed in gigapascals or in thousand pounds per square inch. Its dimensional form is M1L−1T−2, replacing force by mass times acceleration.
Explanation
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law:
Young's modulusE describes the material's strain response to uniaxial stress in the direction of this stress,
the shear modulusG describes the material's response to shear stress. These moduli are not independent, and for isotropic materials they are connected via the equations.
The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force. In the case of an object shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in different directions. In this case, one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value. One possible definition of a fluid would be a material with zero shear modulus.
The shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals. Several models exist that attempt to predict the shear modulus of metals. Shear modulus models that have been used in plastic flow computations include:
the MTS shear modulus model developed by and used in conjunction with the Mechanical Threshold Stress plastic flow stress model.
the Steinberg-Cochran-Guinan shear modulus model developed by and used in conjunction with the Steinberg-Cochran-Guinan-Lund flow stress model.
the Nadal and LePoac shear modulus model that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.
MTS model
The MTS shear modulus model has the form: where is the shear modulus at, and and are material constants.
SCG model
The Steinberg-Cochran-Guinan shear modulus model is pressure dependent and has the form where, μ0 is the shear modulus at the reference state, p is the pressure, and T is the temperature.
NP model
The Nadal-Le Poac shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form: where and μ0 is the shear modulus at absolute zero and ambient pressure, ζ is a material parameter, m is the atomic mass, and f is the Lindemann constant.
