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Shear modulus | |
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Common symbols | G, S |

SI unit | pascal |

Derivations from other quantities | G = τ / γ G = E / 2(1+n) |

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:^{ [1] }

- Explanation
- Shear waves
- Shear modulus of metals
- MTS model
- SCG model
- NP model
- Shear relaxation modulus
- See also
- References

where

- = shear stress
- is the force which acts
- is the area on which the force acts
- = shear strain. In engineering , elsewhere
- is the transverse displacement
- is the initial length

The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch (ksi). Its dimensional form is M^{1}L^{−1}T^{−2}, replacing *force* by *mass* times *acceleration*.

Material | Typical values for shear modulus (GPa) (at room temperature) |
---|---|

Diamond ^{ [2] } | 478.0 |

Steel ^{ [3] } | 79.3 |

Iron ^{ [4] } | 52.5 |

Copper ^{ [5] } | 44.7 |

Titanium ^{ [3] } | 41.4 |

Glass ^{ [3] } | 26.2 |

Aluminium ^{ [3] } | 25.5 |

Polyethylene ^{ [3] } | 0.117 |

Rubber ^{ [6] } | 0.0006 |

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
*E*describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height), - the Poisson's ratio
*ν*describes the response in the directions orthogonal to this uniaxial stress (the wire getting thinner and the column thicker), - the bulk modulus
*K*describes the material's response to (uniform) hydrostatic pressure (like the pressure at the bottom of the ocean or a deep swimming pool), - the
**shear modulus***G*describes the material's response to shear stress (like cutting it with dull scissors). These moduli are not independent, and for isotropic materials they are connected via the equations .^{ [7] }

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 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.

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,

where

- G is the shear modulus
- is the solid's density.

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.^{ [11] }

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
^{ [12] }and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model.^{ [13] }^{ [14] } - the Steinberg-Cochran-Guinan (SCG) shear modulus model developed by
^{ [15] }and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model. - the Nadal and LePoac (NP) shear modulus model
^{ [10] }that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.

The MTS shear modulus model has the form:

where is the shear modulus at , and and are material constants.

The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form

where, μ_{0} is the shear modulus at the reference state (*T* = 300 K, *p* = 0, η = 1), *p* is the pressure, and *T* is the temperature.

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:

where

and μ_{0} 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.

The **shear relaxation modulus** is the time-dependent generalization of the shear modulus ^{ [16] }:

- .

In physics, the **Navier–Stokes equations**, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

**Linear elasticity** is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

A **shear stress**, often denoted by **τ**, is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

**Hemorheology**, also spelled **haemorheology**, or **blood rheology**, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

The **bulk modulus** of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting *relative* decrease of the volume. Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear, and Young's modulus describes the response to linear stress. For a fluid, only the bulk modulus is meaningful. For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.

In seismology, **S-waves**, **secondary waves**, or **shear waves** are a type of elastic wave and are one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves.

In physics, a **perfect fluid** is a fluid that can be completely characterized by its rest frame mass density and *isotropic* pressure *p*.

The **Sauerbrey equation** was developed by the German Günter Sauerbrey in 1959, while working on his doctoral thesis at the Technical University of Berlin, Germany. It is a method for correlating changes in the oscillation frequency of a piezoelectric crystal with the mass deposited on it. He simultaneously developed a method for measuring the characteristic frequency and its changes by using the crystal as the frequency determining component of an oscillator circuit. His method continues to be used as the primary tool in quartz crystal microbalance (QCM) experiments for conversion of frequency to mass and is valid in nearly all applications.

**Nonlinear acoustics** (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics and elasticity. These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.

In astrophysics, the **Tolman–Oppenheimer–Volkoff (TOV) equation** constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modelled by general relativity. The equation is

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

**Volume viscosity** is a material property relevant for characterizing fluid flow. Common symbols are or . It has dimensions, and the corresponding SI unit is the pascal-second (Pa·s).

**Viscoplasticity** is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.

The **viscosity** of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.

In solid mechanics, the **Johnson–Holmquist damage model** is used to model the mechanical behavior of damaged brittle materials, such as ceramics, rocks, and concrete, over a range of strain rates. Such materials usually have high compressive strength but low tensile strength and tend to exhibit progressive damage under load due to the growth of microfractures.

The **Adams–Williamson equation**, named after L. H. Adams and E. D. Williamson, is an equation used to determine density as a function of radius, more commonly used to determine the relation between the velocities of seismic waves and the density of the Earth's interior. Given the average density of rocks at the Earth's surface and profiles of the P-wave and S-wave speeds as function of depth, it can predict how density increases with depth. It assumes that the compression is adiabatic and that the Earth is spherically symmetric, homogeneous, and in hydrostatic equilibrium. It can also be applied to spherical shells with that property. It is an important part of models of the Earth's interior such as the Preliminary reference Earth model (PREM).

The **acoustoelastic effect** is how the sound velocities of an elastic material change if subjected to an initial static stress field. This is a non-linear effect of the constitutive relation between mechanical stress and finite strain in a material of continuous mass. In classical linear elasticity theory small deformations of most elastic materials can be described by a linear relation between the applied stress and the resulting strain. This relationship is commonly known as the generalised Hooke's law. The linear elastic theory involves second order elastic constants and yields constant longitudinal and shear sound velocities in an elastic material, not affected by an applied stress. The acoustoelastic effect on the other hand include higher order expansion of the constitutive relation between the applied stress and resulting strain, which yields longitudinal and shear sound velocities dependent of the stress state of the material. In the limit of an unstressed material the sound velocities of the linear elastic theory are reproduced.

Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last work.

**K-epsilon (k-ε) turbulence model** is the most common model used in Computational Fluid Dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two equation model that gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.

**SST turbulence model** is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.

- ↑ IUPAC ,
*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) " shear modulus,*G*". doi : 10.1351/goldbook.S05635 - ↑ 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. - 1 2 3 4 5 Crandall, Dahl, Lardner (1959).
*An Introduction to the Mechanics of Solids*. Boston: McGraw-Hill. ISBN 0-07-013441-3.CS1 maint: multiple names: authors list (link) - ↑ Rayne, J.A. (1961). "Elastic constants of Iron from 4.2 to 300 ° K".
*Physical Review*.**122**(6): 1714–1716. Bibcode:1961PhRv..122.1714R. doi:10.1103/PhysRev.122.1714. - ↑ Material properties
- ↑ Spanos, Pete (2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber".
*Rubber World*. - ↑ [Landau LD, Lifshitz EM.
*Theory of Elasticity*, vol. 7. Course of Theoretical Physics. (2nd Ed) Pergamon: Oxford 1970 p13] - ↑ 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. - 1 2 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): 3952–3958. 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" (PDF).
*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. - ↑ Rubinstein, Michael, 1956 December 20- (2003).
*Polymer physics*. Colby, Ralph H. Oxford: Oxford University Press. p. 284. ISBN 019852059X. OCLC 50339757.CS1 maint: multiple names: authors list (link)

Conversion formulae | |||||||
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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. | |||||||

Notes | |||||||

There are two valid solutions. | |||||||

Cannot be used when | |||||||

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