Isotropic homogeneous

Mephisto is devoted to predict the ultrasonic scans (A,B or C-scans) for a priori knowledge of the piece and the defects within. In the present version Mephisto only deals with homogeneous isotropic materials. The piece under test can be planar, cylindrical or have a more complex geometry. The defects can be either planar (one or several facets), or volumetric (spherical voids, side drilled holes, flat or round bottom holes). 

Let us consider a thin homogeneous isotropic beam of thickness 2s. We assume that the beam mid-line coincides with the segment (0,1) of the axis X. At the point y = 1/2, the beam has an inclined cut as a segment having the angle a with the vertical line, 0 < tana < 2e). We look for the function % = (W, w) of horizontal displacements W(x) and vertical displacements w(x) provided that the external forces g(x),f(x) are given. The condition of clamped edges... 

Laminates. Two or more layers of material bonded together form a laminated composite. Common examples of laminates are in automobile windshields (laminated glass) and bimetal thermostats (9). In both cases homogeneous, isotropic layers of materials are bonded together to form nonhomogeneous composite laminates (see Laminates). 

The shock-compression pulse carries a solid into a state of homogeneous, isotropic compression whose properties can be described in terms of perfect-crystal lattices in thermodynamic equilibrium. Influences of anisotropic stress on solid materials behaviors can be treated as a perturbation to the isotropic equilibrium state. ... 

Fracture is caused by higher stresses around flaws or cracks than in the surrounding material. However, fracture mechanics is much more than the study of stress concentration factors. Such factors are useful in determining the influence of relatively large holes in bodies (see Section 6.3, Holes in Laminates), but are not particularly helpful when the body has sharp notches or crack-like flaws. For composite materials, fracture has a new dimension as opposed to homogeneous isotropic materials because of the presence of two or more constituents. Fracture can be a fracture of the individual constituents or a separation of the interface between the constituents. 

Although vitreous silica is nominally a homogeneous isotropic amorphous material, and should normally remain so during its service life, it is in fact in a metastable condition. The tendency to revert to crystalline forms with attendant deterioration in mechanical durability places severe limitations on the range of applications. Figure 18.2 illustrates the polymorphic forms of silica, and the dimensional changes accompanying each transition. 

Given these differences between rigid and flexible conduit, let us examine the differences between steel and RTR pipe, both of which are, of course, flexible conduits. First, steel pipe is by definition constructed from a material, steel, that for our purposes is a homogeneous isotropic substance. Therefore, steel pipe can be considered to have the same material properties in all directions that is, it is equally strong in both the hoop and longitudinal directions [Fig. 4-2(b)]. 

Fig. 4-2(b) Material properties of relatively homogeneous isotropic steel pipe. 

A study of the effect of the mesophase layer on the thermomechanical behaviour and the transfer mechanism of loads between phases of composites will be presented in this study. Suitable theoretical models shall be presented, where the mesophase is taken into consideration as an additional intermediate phase. To a first approximation the mesophase material is considered as a homogeneous isotropic one, while, in further approximations, more sophisticated models have been developed, in which the mesophase material is considered as an inhomogeneous material with progressively varying properties between inclusions and matrix. Thus, improvements of the basic Hashin-Rosen models have been incorporated, making the new models more flexible and suitable to describe the real behaviour of composites. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

An attempt has been made by Spiering et al. [39,40] to relate the magnitude of the interaction parameter F(x) as derived from experiment to the elastic interaction between HS and LS ions via an image pressure [47]. To this end, the metal atoms, inclusive of their immediate environments, in the HS and LS state are considered as incompressible spheres of radius /"h and Tl, respectively. The spheres are embedded in an homogeneous isotropic elastic medium, representing the crystal, which is characterized by specific values of the bulk modulus K and Poisson ratio a where 0 < a < 0.5. The change of molecular volume A Fas determined by X-ray diffraction may be related to the volume difference Ar = Ph — of the hard spheres by ... 

The mechanism of conduction is most easily understood by the study of conduction through homogeneous isotropic solids, because in this case convection is not present. As a simple illustration of heat transfer by conduction, let a flat parallel-sided plate of a uniform solid material, whose flat faces are maintained at temperatures Tt and T2 respectively (Tj > T2) be considered (Figure 3.15). Heat would be transferred from the face at the higher temperature (Tj) to that at the lower temperature (T2). Let the rate of this transfer be dQjdt, and the area of the plate perpendicular to the direction of heat flow be S. If L is the plate thickness, then it is found that dQ/dt is proportional to (Tt - T2) S/L. In other words,... 

While the theoretical value (based on homogeneous, isotropic turbulence) of the Smagorinsky coefficient cs amounts to 0.165 (Mason and Callen, 1986), in many simulation studies lower values for cs proved to result in a better reproduction of experimental data. This may have to do with the abundant presence of shear flows in process equipment. Derksen (2003) reported that varying cs values in the range 0.08-0.14 does not have a large impact on the simulation results. A value of 0.12 is recommended. 

In a homogeneous isotropic medium in which D= tE and B= pH, a complex field vector is defined as... 

The observations on which thermodynamics is based refer to macroscopic properties only, and only those features of a system that appear to be temporally independent are therefore recorded. This limitation restricts a thermodynamic analysis to the static states of macrosystems. To facilitate the construction of a theoretical framework for thermodynamics [113] it is sufficient to consider only systems that are macroscopically homogeneous, isotropic, uncharged, and large enough so that surface effects can be neglected, and that are not acted on by electric, magnetic or gravitational fields. The only mechanical parameter to be retained is the volume V. For a mixed system the chemical composition is specified in terms of the mole numbers Ni, or the mole fractions [Ak — 1,2,..., r] of the chemically pure components of the system. The quantity V/(Y j=iNj) is called the molar... 

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

Table 1. Elasticity and rigidity constants for a homogeneous isotropic elastic medium...

In a homogeneous isotropic elastic medium it is possible to split acoustic waves in independent longitudinal and transverse waves, each travelling at a speed cL and cT, respectively. As 2 is greater than or equal to zero, cT is lower than or equal to cl/ J2. 

In homogeneous isotropic turbulence, the two-point velocity correlation function can be expressed (Pope 2000) in terms of the longitudinal (/) and transverse (g) auto-correlation functions ... 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

Hughes, T. J. R., L. Mazzei, A. A. Oberai, and A. A. Wray (2001b). The multiscale formulation of large eddy simulation Decay of homogeneous isotropic turbulence. 

Use of these assumptions is necessary to limit the number of variable parameters that must be considered in the equations. Calculation of the response of an aquifer that is not homogeneous, isotropic, or infinite in extent becomes very complex. Many complex situations are better suited to sophisticated computer simulations. 

The electromagnetic fields (x,r) and H(x, t) associated with scattering from a microsphere satisfy Maxwell s equations. For a homogeneous, isotropic linear material the time-harmonic electrical held E and the magnetic held H satisfy vector wave equations, which in SI units are (Bohren and Huffman, 1983)... 

In contrast, most models of turbulent flame speeds assume homogeneous isotropic turbulence with no mean flow or strain. This leads to substantial differences between model predictions and experimental observations [4]. 

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