Volumetric elastic modulus,

Munns, R., Greenway, H., Stelter, T.L. Kuo, J. (1983). Turgor pressure, volumetric elastic modulus, osmotic volume and ultrastructure of Chlorella emersoni grown at high and low external NaCl. Journal of Experimental Botany, 34,144-55. 

For other cases, the uptake or internal production of osmotically active solutes can control cell growth, including effects that involve the reversible cell extension caused by changes in internal hydrostatic pressure and are quantitatively described by the volumetric elastic modulus e (Eq. 1.17). For instance, when reversible or elastic changes in volume exist, then (1 )dPidt should be added to the right-hand side of Equation 2.33. Using Equation... 

TMA measures the mechanical response of a polymer looking at (1) expansion properties including the coefficient of linear expansion, (2) tension properties such as measurement of shrinkage and expansion under tensile stress, i.e., elastic modulus, (3) volumetric expansion, i.e., specific volume, (4) single-fiber properties, and (5) compression properties such as measuring the softening or penetration under load. 

TMA measures the mechanical responses of a polymer as a function of temperature. Typical measurements include (1) expansion properties, i.e., the expansion of a material leading to the calculation of the linear expansion coefficient (2) tension properties, i.e., the shrinkage and expansion of a material under tensile stress e.g., elastic modulus (3) dilatometric properties, i.e., the volumetric expansion within a confining medium e.g., specific volume ... 

Fluids that show elasticity to some extent are termed viscoelastic fluids, and some polymer solutions demonstrate such behavior. Elasticity is the tendency of a substance or body to return to its original form, after the applied stress that caused strain (i.e., a relative volumetric change in the case of a polymer solution) has been removed. The elastic modulus (Pa) is the ratio of the applied stress (Pa) to strain (-). The relaxation time (s) of a viscoelastic fluid is defined as the ratio of its viscosity (Pa s) to its elastic modulus. 

Comparing the calculated and experimental stress-strain diagrams for real plastic foams, we will take Eo in Eq. (77) as the elasticity modulus of the polymer base Y and y are foam and polymer base densities (volumetric weights), respectively ... 

In order to solve the coupled THM problem, mechanical properties should be considered for the rock and a constitutive law is also necessary to account for permeability changes induced by deformations. A simple way that permits coupled calculations is using the following properties for the rock Elastic modulus 36800 MPa Poisson ratio 0.2 and thermal expansion 0.00002. As heating of the rock takes place, volumetric expansion in the confined rock induces changes in porosity. In reality, fractures will tend to- close. These porosity/aperture changes can be transformed into permeability variations. A exponential law has been used to model permeability variations ... 

The modulus of elasticity, E, and the modulus of rigidity, G, as defined above, apply under longitudinal and shear forces, respectively. When a hydrostatic force is applied, a third elastic modulus, the modulus of compressibility or bulk modulus, K, is used. It is the reciprocal of compressibility, /J, and is defined as the ratio of the hydrostatic pressure, cri,> to the volumetric strain, AV/Vo ... 

Materials typically expand when heated. The expansion is characterized by the linear or volumetric thermal expansion coefficient. The three primary types of materials expand differently. Polymers expand more than metals, and metals expand more than ceramics. For many materials thermal expansion is related to the melting temperature of the material, also a relation between the thermal expansion coefficient of polymers and their elastic modulus is given. 

While several simplifying assumptions needed to be made so as to derive an analytical model, the model captures all relevant physical processes. Specifically, it employed thermodynamic equilibrium conditions for temperature, pressure, and chemical potential to derive the equation of state for water sorption by a single cylindrical PEM pore. This equation of state yields the pore radius or a volumetric pore swelling parameter as a function of environmental conditions. Constitutive relations for elastic modulus, dielectric constant, and wall charge density must be specified for the considered microscopic domain. In order to treat ensemble effects in equilibrium water sorption, dispersion in the aforementioned materials properties is accounted for. 

The equivalent elastic modulus, Gip, for the mechanically interlocked interphase is calculated by assuming an isostress condition for the interphase, illustrated in Pig. 23.13. In other words, the adhesive and the adherend surface (projections) are subjected to the same level of shear stress, tjp, at the interphase, while the composite interfacial shear strain, yjp, is a volumetric weighted average of the adhesive and adherend strains, that is,... 

This approach can be illustrated by unsaturated polyesters based on an almost equimolar combination of maleate and phthalate of propylene glycol, crosslinked by styrene (45 wt%) (Mortaigne el al., 1992). Six samples differing by the prepolymer molar mass were analyzed. The chain-ends concentration, b, was determined by volumetric analysis of alcohols and acids in the initial reactive mixture. Then, the system was cured, elastic measurements were made in the rubbery state at Tg + 30° C, and the shear modulus G was plotted against chain-ends concentration (Fig. 14.7). The following relationship was obtained ... 

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

For the case where cr, = cTj = cTj which is known as hydrostatic stress (the situation when pressure is applied on a glass embedded in a material of low elastic constants - glass piece in steatite or in AgCl in a high pressure cell or simply embedded in a liquid) - then there are no shear strains. The hydrostatic stress is simply the pressure, P and the volumetric strain is Cm so that bulk modulus is also defined as... 

Elastomers are cross-linked macromolecules above the glass transition temperature. The cross-link density is the fundamental engineering quantity which, for instance, determines the modulus of elasticity. Usually, it is measured during vulcanization of well-defined rubber samples in a vulcameter by the moment necessary to perform a given torsional shear of the test sample. Swelling experiments can be performed alternatively, but are problematic for filled elastomers. Such measurements are based on the assumption that the measured quantity does not vary over the sample volume. Inhomogeneous cross-link densities can be determined from the surface hardness, but volumetric resolution is achieved by conventional methods only after cutting the sample. 

Equation (3) is the equation of equilibrium of the porous medium. In this equation, it is assumed that the medium is non-linearly elastic, and G (Pa) and A (Pa) are Lame s constants of elasticity and P is the coefficient of volumetric thermal expansion of the solid matrix. G and A, and also A d the bulk modulus can also be expressed as functions of the... 

Where eJ is the elastic volumetric strain and efj components of the elastic deviatoric strain. Kq and Po are respectively the initial bulk modulus and shear modulus of undamaged material. The scalar variable d characterises the isotropic damage. The... 

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