|Derivations from other quantities:||G = τ / γ|
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:
- = shear stress;
- is the force which acts
- is the area on which the force acts
- in engineering, = shear strain. Elsewhere,
- is the transverse displacement
- is the initial length
Shear modulus' derived SI unit is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch (kpsi).
The shear modulus is always positive.
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 modulus describes the material's response to linear strain (like pulling on the ends of a wire or putting a weight on top of a column),
- the bulk modulus describes the material's response to uniform pressure (like the pressure at the bottom of the ocean or a deep swimming pool)
- the shear modulus describes the material's response to shearing strains (like cutting it with dull scissors).
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 (such as friction). In the case of an object that's 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.
Influences of selected glass component additions on the shear modulus of a specific base glass.
In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave, is controlled by the shear modulus,
- G is the shear modulus
- is the solid's density.
Shear modulus of metals
Shear modulus of copper as a function of temperature. The experimental data
are shown with colored symbols.
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 (and possibly that of alloys). 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 (MTS) plastic flow stress model.
- the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.
- the Nadal and LePoac (NP) shear modulus model that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.
MTS shear modulus model
The MTS shear modulus model has the form:
where µ0 is the shear modulus at 0 K, and D and T0 are material constants.
SCG shear modulus model
The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form
where, µ0 is the shear modulus at the reference state (T = 300 K, p = 0, η = 1), p is the pressure, and T is the temperature.
NP shear modulus model
The Nadal-Le Poac (NP) 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:
and µ0 is the shear modulus at 0 K and ambient pressure, ζ is a material parameter, kb is the Boltzmann constant, m is the atomic mass, and f is the Lindemann constant.
- ^ Nic, M.; Jirat, J.; Kosata, B., eds. (2006–). "shear modulus, G". IUPAC Compendium of Chemical Terminology (Online ed.). doi:10.1351/goldbook.S05635. ISBN 0-9678550-9-8. http://goldbook.iupac.org/S05635.html.
- ^ McSkimin, H.J.; Andreatch, P. (1972). "Elastic Moduli of Diamond as a Function of Pressure and Temperature". J. Appl. Phys. 43 (7): 2944–2948. Bibcode 1972JAP....43.2944M. doi:10.1063/1.1661636.
- ^ a b c d e Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. Boston: McGraw-Hill. ISBN 0-07-013441-3.
- ^ Material properties
- ^ Spanos, Pete (2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber". Rubber World. http://www.thefreelibrary.com/Cure+system+effect+on+low+temperature+dynamic+shear+modulus+of...-a0111451108.
- ^ Shear modulus calculation of glasses
- ^ Overton, W.; Gaffney, John (1955). "Temperature Variation of the Elastic Constants of Cubic Elements. I. Copper". Physical Review 98 (4): 969. Bibcode 1955PhRv...98..969O. doi:10.1103/PhysRev.98.969.
- ^ a b Nadal, Marie-Hélène; Le Poac, Philippe (2003). "Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation". Journal of Applied Physics 93 (5): 2472. Bibcode 2003JAP....93.2472N. doi:10.1063/1.1539913.
- ^ March, N. H., (1996), Electron Correlation in Molecules and Condensed Phases, Springer, ISBN 0-306-44844-0 p. 363
- ^ Varshni, Y. (1970). "Temperature Dependence of the Elastic Constants". Physical Review B 2 (10): 3952. Bibcode 1970PhRvB...2.3952V. doi:10.1103/PhysRevB.2.3952.
- ^ Chen, Shuh Rong; Gray, George T. (1996). "Constitutive behavior of tantalum and tantalum-tungsten alloys". Metallurgical and Materials Transactions A 27 (10): 2994. Bibcode 1996MMTA...27.2994C. doi:10.1007/BF02663849.
- ^ Goto, D. M.; Garrett, R. K.; Bingert, J. F.; Chen, S. R.; Gray, G. T. (2000). "The mechanical threshold stress constitutive-strength model description of HY-100 steel". Metallurgical and Materials Transactions A 31 (8): 1985–1996. doi:10.1007/s11661-000-0226-8.
- ^ Guinan, M; Steinberg, D (1974). "Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements". Journal of Physics and Chemistry of Solids 35 (11): 1501. Bibcode 1974JPCS...35.1501G. doi:10.1016/S0022-3697(74)80278-7.
|Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.|