Elastic, defined moduli

The ideal elastic element is represented by a spring that obeys Hooke s law (with a defined modulus of elasticity) (Hooke element). The elastic deformation is instantaneous. In a dynamic experiment the stress is in phase with the deformation ... 

Interfacial relaxation methods are typically based on a perturbation of the equilibrium state of an interface by small changes of the interfacial area. The ratio of the amplitudes of surface tension and relative area changes gives the modulus of elasticity , defined as... 

In a direct analogy to the tensile case, a Hooke s law of shear may be used to define the elastic shear modulus ... 

Fig. 6a presents the concept of cylindrical shell discretisation with the use of shell - plate finite elements having elastic properties and radially oriented constrains, as shown. On the basis the experimentally defined modulus of elasticity of grain the elastic characteristics of the bulk solids flexibility C (Fig. 6b) for the cylindrical elastic shell ring was determined... 

The stiffness of the fill mass is defined as its resistance to stress-induced deformation and is usually expressed in terms of a modulus of elasticity, shear modulus or... 

In the figure below a series of terms is defined in order to help describe and define viscosity and viscoelasticity. The diagram represents a simple three - dimensional volume of fluid [or solid] and shear stress strain and shear rate are described as well as the parameters for viscosity and elasticity [shear modulus]. The origin of normal forces is also shown, but this is not common to all fluids and is likely to occur in certain cases where polymers are exhibiting entanglement coupling or filled systems are colliding with each other. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

Such nonequilihrium surface tension effects ate best described ia terms of dilatational moduh thanks to developments ia the theory and measurement of surface dilatational behavior. The complex dilatational modulus of a single surface is defined ia the same way as the Gibbs elasticity as ia equation 2 (the factor 2 is halved as only one surface is considered). 

Table 10-56 gives values for the modulus of elasticity for nonmetals however, no specific stress-limiting criteria or methods of stress analysis are presented. Stress-strain behavior of most nonmetals differs considerably from that of metals and is less well-defined for mathematic analysis. The piping system should be designed and laid out so that flexural stresses resulting from displacement due to expansion, contraction, and other movement are minimized. This concept requires special attention to supports, terminals, and other restraints. 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

The effective length is defined as the measured center distance plus the outside circumference of one of the inspection pulleys. This measurement method accounts for the modulus of elasticity, or stretch ability, and dimensional variations among belts with the same cross-section. 

The generalized stress-strain curve for plastic shown in Fig. 2-7 serves to define several useful qualities that include the tensile strength, modulus (modulus of elasticity) or stiffness (initial straight line slope of... 

The constant is called the modulus of elasticity (E) or Young s modulus (defined by Thomas Young in 1807 although the concept was used by others that included the Roman Empire and Chinese-BC), the elastic modulus, or just the modulus. This modulus is the straight line slope of the initial portion of the stress-strain curve, normally expressed in terms such as MPa or GPa (106 psi or Msi). A... 

Frequently, a product becomes loaded when it is subjected to a defined deflection. The actual load then is a result of the structural reaction of the product to the applied strain. Unlike directly applied loads, strain-induced loads are dependent on the modulus of elasticity and, with TPs, will generally decrease in magnitude over time. Many assembly and thermal stresses could be the result of strain-induced loads. They include metal insert press fits in the plastic and clamping or screw attachments. 

Although hardness is a somewhat nebulous term, it can be defined in terms of the tensile modulus of elasticity. From a more practical side, it is usually characterized by a combination of three measurable parameters (1) scratch resistance (2) abrasion or mar resistance and (3) indentation under load. To measure scratch resistance or hardness, an approach is where a specimen is moved laterally under a loaded diamond point. The hardness value is expressed as the load divided by the width of the scratch. In other tests, especially in the paint industry, the surface is scratched with lead pencils of different hardnesses. The hardness of the surface is defined by the pencil hardness that first causes a visible scratch. Other tests include a sand-blast spray evaluation. 

It is further assumed that the mesophase layer consists of a material having progressively variable mechanical properties. In order to match the respective properties of the two main phases bounding the mesophase, a variable elastic modulus for the mesophase may be defined, which, for reasons of symmetry, depends only on the radial distance from the fiber-mesophase surface. In other words, it is assumed that the mesophase layer consists of a series of elementary peels, whose constant mechanical properties differ to each other by a quantity (small enough) defined by the law of variation of Ej(r). 

When the pressure at some point in a fluid is changed, the new condition takes a finite time to be transmitted to some other point in the fluid because the state of each intervening element of fluid has to be changed. The velocity of propagation is a function of the bulk modulus of elasticity e, where e is defined by the relation ... 

The two mechanical properties measured most frequently using indentation techniques are the hardness, H, and the elastic modulus, E. A t5pical load-displacement curve of an elastic-plastic sample during and after indentation is presented in Fig. 30, which also serves to define some of the experimental quantities involved in the measurement. 

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