Material isotropic

A material is mechanically isotropic if all of its mechanical properties are the same in all spatial directions. The elasticity tensor must thus remain unchanged by arbitrary rotations of the material or the coordinate system. Its components must be invariant with respect to rotations. 

This invariance property can be used to show that the elasticity matrix has the simple form  

All components not specified vanish, so there are only two independent parameters, Cn and Ci2- 

The following relations between these parameters and the more familiar Young s modulus E, Poisson s ratio 1/, and shear modulus G hold  

Apart from E, G, and u, the so-called Lame s elastic constants A and /u are sometimes used. Their relation to the other elastic constants is 

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Second corner reflection The first corner reflection appears as usual when the transducer is coupled to the probe at a certain distance from the V-butt weld. The second corner reflection appears if the transducer is positioned well above the V-hutt weld. If the weld is made of isotropic material the wavefront will miss (pass) the notch without causing any reflection or diffraction (see Fig. 3(a)) for this particular transducer position. In the anisotropic case, the direction of the phase velocity vector will differ from the 45° direction in the isotropic case. Moreover, the direction of the group velocity vector will no longer be the same as the direction of the phase velocity vector (see Fig. 3(b), 3(c)). This can be explained by comparing the corresponding slowness and group velocity diagrams. 

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). 

For an isotropic material, all orientations are equally probable and all such products that have an odd number of Tike direction cosines will vanish upon averaging-. This restricts the nonvanishing tensor elements to those such as xVaaa abba - Similarly for the elements Such orientational averaging is crucial in... 

Trends in the field of economics are the centralization of the powder fabrication to enable production on a large scale and the manufacture of low quahty anisotropic materials by a much less expensive technology. An example of the latter is the introduction of alignment during pressing of the raw material mixture in the fabrication route of isotropic materials. 

When a complex magnetisation mode is desired, isotropic materials are preferred. Lateral magnetisation, always in multipole, is only appHed to isotropic materials. 

Fig. 16. Self-similar crack propagation in an isotropic material. The crack propagates in a direction perpendicular to the cycHc loading axis (mode I loading).

When an isotropic material is subjected to planar shock compression, it experiences a relatively large compressive strain in the direction of the shock propagation, but zero strain in the two lateral directions. Any real planar shock has a limited lateral extent, of course. Nevertheless, the finite lateral dimensions can affect the uniaxial strain nature of a planar shock only after the edge effects have had time to propagate from a lateral boundary to the point in question. Edge effects travel at the speed of sound in the compressed material. Measurements taken before the arrival of edge effects are the same as if the lateral dimensions were infinite, and such early measurements are crucial to shock-compression science. It is the independence of lateral dimensions which so greatly simplifies the translation of planar shock-wave experimental data into fundamental material property information. 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

Wallace [15], [16] gives details on effects of nonlinear material behavior and compression-induced anisotropy in initially isotropic materials for weak shocks, and Johnson et ai. [17] give results for infinitesimal compression of initially anisotropic single crystals, but the forms of the equations are the same as for (7.10)-(7.11). From these results it is easy to see where the micromechanical effects of rate-dependent plastic flow are included in the analysis the micromechanics (through the mesoscale variables and n) is contained in the term y, as given by (7.1). 

Spruce soundboards have a Young s modulus anisotropy of about (11.6 GPa/0.71 GPa) = 16. A replacement material must therefore have a similar anisotropy. This requirement immediately narrows the choice down to composites (isotropic materials like metals or polymers will probably sound awful). 

Equations la and lb are for a simple two-phase system such as the air-bulk solid interface. Real materials aren t so simple. They have natural oxides and surface roughness, and consist of deposited or grown multilayered structures in many cases. In these cases each layer and interface can be represented by a 2 x 2 matrix (for isotropic materials), and the overall reflection properties can be calculated by matrix multiplication. The resulting algebraic equations are too complex to invert, and a major consequence is that regression analysis must be used to determine the system s physical parameters. ... 

The classical relationship between the shear modulus G, and the tensile modulus, E, for an isotropic material is... 

The change in shape of a material when it is subjected to a change in temperature is determined by the coefficient of thermal expansion, aj- Normally for isotropic materials the value of aj will be the same in all directions. For convenience this is often taken to be the case in plastics but one always needs... 

The convention normally used is that direct stresses and strains have one suffix to indicate the direction of the stress or strain. Shear stresses and strains have two suffices. The first suffix indicates the direction of the normal to the plane on which the stress acts and the second suffix indicates the direction of the stress (or strain). Poisson s Ratio has two suffices. Thus, vi2 is the negative ratio of the strain in the 2-direction to the strain in the 1-direction for a stress applied in the 1-direction (V 2 = — il for an applied a ). v 2 is sometimes referred to as the major Poisson s Ratio and U2i is the minor Poisson s Ratio. In an isotropic material where V21 = i 2i. then the suffices are not needed and normally are not used. 

It is also worthy of note that large values of Poisson s Ratio can occur in a laminate. In this case a peak value of over 1.5 is observed - something which would be impossible in an isotropic material. Large values of Poisson s Ratio are a characteristic of unidirectional fibre composites and arise due to the coupling effects between extension and shear which were referred to earlier. 

It is also important to note that although the laminae [ 45] indicates that Ex = Ey =. GN/m, this laminate is not isotropic or even quasi-isotropic. As shown in Chapter 2, in an isotropic material, the shear modulus is linked to the other elastic properties by the following equation... 

In an isotropic material subjected to a uniaxial stress, failure of the latter type is straightforward to predict. The tensile strength of the material will be known from materials data sheets and it is simply a question of ensuring that the applied uniaxial stress does not exceed this. 

If an isotropic material is subjected to multi-axial stresses then the situation is slightly more complex but there are well established procedures for predicting failure. If a,i and Oy are applied it is not simply a question of ensuring that neither of these exceed ar- At values of and Oy below oj there can be a plane within the material where the stress reaches ot and this will initiate failure. 

A plastic composite is made up of three layers of isotropic materials as follows ... 

Bodies with temperature-dependent isotropic material properties are not homogeneous when subjected to a temperature gradient, but still are isotropic. 

The inherent anisotropy (most often only orthotropy) of composite materials leads to mechanical behavior characteristics that are quite different from those of conventional isotropic materials. The behavior of isotropic, orthotropic, and anisotropic materials under loadings of normal stress and shear stress is shown in Figure 1-4 and discussed in the following paragraphs. 

For isotropic materials, application of normal stress causes extension in the direction of the stress and contraction in the perpendicular... 

If there is an infinite numtser of planes of material property symmetry, then the foregoing relations simplify to the isotropic material relations with only two independent constants in the stiffness matrix ... 

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... 

If the bulk modulus were negative, a hydrostatic pressure would cause expansion of a cube of isotropic material Finally, for isotropic materials, Poisson s ratio is restricted to the range... 

Show that the determinant inequality In Equation (2.48) tor orthotropic materials correctly reduces to v< 1/2 for isotropic materials. 

Show that Equation (2.54) reduces for isotropic materials to known bounds on v. 

For plane stress on isotropic materials, the strain-stress relations are... 

Then, obviously the maximum principal stress is lower than the largest strength. However, 02 is greater than Y, so the lamina must fail under the imposed stresses (perhaps by cracking parallel to the fibers, but not necessarily). The key observation is that strength is a function of orientation of stresses relative to the principal material coordinates of an orthotropic lamina. In contrast, for an isotropic material, strength is independent of material orientation relative to the imposed stresses (the isotropic material has no orientation). 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

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