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Concept# Bulk modulus

Summary

The bulk modulus ( or ) of a substance is a measure of the resistance of a substance to bulk compression. 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 stress, and Young's modulus describes the response to normal (lengthwise stretching) 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. The reciprocal of the bulk modulus at fixed temperature is called the isothermal compressibility.
The bulk modulus (which is usually positive) can be formally defined by the equation
where is pressure, is the initial volume of the substance, and denotes the derivative of pressure with respect to volume. Since the volume is inversely proportional to the density, it follows that
where is the initial density and denotes the derivative of pressure with respect to density. The inverse of the bulk modulus gives a substance's compressibility. Generally the bulk modulus is defined at constant temperature as the isothermal bulk modulus, but can also be defined at constant entropy as the adiabatic bulk modulus.
Strictly speaking, the bulk modulus is a thermodynamic quantity, and in order to specify a bulk modulus it is necessary to specify how the pressure varies during compression: constant-temperature (isothermal ), constant-entropy (isentropic ), and other variations are possible. Such distinctions are especially relevant for gases.
For an ideal gas, an isentropic process has:
where is the heat capacity ratio. Therefore, the isentropic bulk modulus is given by
Similarly, an isothermal process of an ideal gas has:
Therefore, the isothermal bulk modulus is given by
When the gas is not ideal, these equations give only an approximation of the bulk modulus.

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Bulk modulus

The bulk modulus ( or ) of a substance is a measure of the resistance of a substance to bulk compression. 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 stress, and Young's modulus describes the response to normal (lengthwise stretching) stress. For a fluid, only the bulk modulus is meaningful.

Shear modulus

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: 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 of the area. The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi).

Elastic modulus

An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region: A stiffer material will have a higher elastic modulus. An elastic modulus has the form: where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter.

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The lattice dynamics of solid He-4 has been explored using pulsed NMR methods to study the motion of He-3 impurities in the temperature range (0.05-0.20 K) where experiments have revealed anomalies attributed to superflow or unexpected viscoelastic properties of the solid He-4 lattice. We report the results of measurements of the nuclear spin-lattice and spin-spin relaxation times that measure the fluctuation spectrum at high and low frequencies, respectively, of the He-3 motion that results from quantum tunneling in the He-4 matrix. The measurements were made for He-3 concentrations 16 < x(3) < 2000 ppm. For He-3 concentrations x(3) = 16 and 24 ppm, large changes are observed for both the spin-lattice relaxation time T-1 and the spin-spin relaxation time T-2 at temperatures close to those for which the anomalies are observed in measurements of torsional oscillator responses and the shear modulus. These changes in the NMR relaxation rates were not observed for higher He-3 concentrations.

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