Materials constants isotropic 98

For those not familiar with this type information recognize that the viscoelastic behavior of plastics shows that their deformations are dependent on such factors as the time under load and temperature conditions. Therefore, when structural (load bearing) plastic products are to be designed, it must be remembered that the standard equations that have been historically available for designing steel springs, beams, plates, cylinders, etc. have all been derived under the assumptions that (1) the strains are small, (2) the modulus is constant, (3) the strains are independent of the loading rate or history and are immediately reversible, (4) the material is isotropic, and (5) the material behaves in the same way in tension and compression. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

The membrane consists of a stack of thin films of a certain thickness, d with characteristic isotropic and homogeneous material constants c, and k (see Fig. 3.2). The composition of the stack locally depends on the x,y-coordinates. The local membrane thickness is defined as... 

Ki = a y/a is the stress intensity factor, and F, the material constant, both of which depend on the degree of anisotropy of the composite controlled by the composite elastic moduli in the longitudinal and transverse directions, El and Ej, in-plane Poisson ratio, vlt, and Glj. For a perfectly isotropic material, jr/8(l + Vlt) 0.3. Also, the material parameters, i and < 2 are given by ... 

From here on we dispense with vectorial notation, on the assumption that the material is isotropic. Note that on integrating over the field at constant entropy the temperature of the system necessarily increases, so that, in contrast to the earlier treatment, ao(T, V) now changes during integration. In fact, since the entropy has the functional dependence S = S T, V, q) we must first invert this relation to specify T = T(S, V, q), so that the polarizability is written out as ao(S(T, V, q), V). We then integrate (5.7.12) by parts to find... 

Material Constants, Elastic wave velocities have been obtained for oil shale by ultrasonic methods for various modes of propagation. Elastic constants can be inferred from these data if the oil shale is assumed to be a transversely isotropic solid (9). This is a reasonable approximation considering the bedded nature of the rock. Many of the properties of oil shale depend on the grade (kerogen content), which in turn is correlated with the density ( 10). The high pressure behavior of oil shale under shock loading has been studied in gas-gun impact experiments (11). 

For an isotropic body, there are only two stress components that are independent of the other. This means that while different loadings and strains can be imposed, the material constants relating stress and strain are not all unique. There are six common elastic constants (Table 6.1), including Poisson s ratio, defined as the ratio of the lateral strain accompanying a longitudinal strain, V = -eiilsn- Since only two are unique for an isotropic body, each elastic constant can be expressed as a function of any other pair these expressions are tabulated in Table 6.1 in terms of Poisson s ratio (Mott and Roland, 2009). 

Using the unidirectional stiffness properties of the composite material (HTA/6376) in Table 11.1, the laminates were modelled with one orthotropic solid element per ply in the thickness direction, leading to 16 elements through-thickness for both the splice plate and skin plate. As before, the titanium bolts were modelled with isotropic material properties, with material constants Eb = 110 GPa, Vb = 0.29. Linear 8-node hexa-hedral brick elements with a reduced integration scheme were used for the laminates and bolts. This element formulation was used to reduce the cost of the analysis and size of the output files, which were very large. 

If the charges are isotropically deformable the proportionality factor a is a characteristic material constant independent of direction it is termed the polarizability of the substance and has volume dimensions. To determine polarizability experimentally, we start with the dielectric constant of the substance to be investigated a is related under certain generally valid suppositions to e by the Clausius-Mosotti equation... 

An orthotropic material is called transversely isotropic when one of its principal planes is a plane of isotropy, i.e. at every point there is a plane on which the mechanical properties are the same in all directions [2]. Unidirectional carbon fibers packed in a hexagonal array with a relatively high volume fraction can be considered transversely isotropic, with the 2-3 plane normal to the fibers as the plane of isotropy (Figure 22.2). For a transversely isotropic material, it should be noted that the subscripts 2 and 3 (for a 2-3 plane of symmetry) in the material constants are interchangeable. Hence... 

Here, the first term is referred to as the first-order electro-optic effect (Pochels effect), and the second term is referred to as the second-order electro-optic effect (Kerr effect). The coefficients rij/, and Ry/, are ternary and quaternary tensor quantities known as the Pochels constant (first-order electro-optic constant) and Kerr constant (second-order electro-optic constant), respectively. As Table 7.1.3 shows, a second-order electro-optic effect is present in materials, including isotropic materials such as glass, whereas first-order electro-optic effects are only observed in piezoelectric crystals. In Table 7.1.3, electro-optic effects are present in crystals belonging to point groups. 

E. H. Twizell and R. W. Ogden, Non-Linear Optimization of the Material Constants in Ogden s Stress Deformation Function for Incompressible Isotropic Elastic Materials J. Austral. Math. Soc., Sen B 24, 424 434 (1983). 

Table 2.3 contains an overview of the elastic constants for some metals and ceramics. As can be seen, the anisotropy factor of tungsten is 1.0, so it is (almost) isotropic even as a single crystal. For most other materials, almost isotropic properties can only be found in a polycrystalline state. The direction dependence of Young s modulus for selected materials is plotted in figure 2.10. 

For all of the cases of substrate curvature induced by film stress that were considered in Section 2.1, it was assumed that both the film and substrate materials were isotropic. This provided a basis for a relatively transparent discussion of curvature phenomena and it led to results which have proven to be broadly useful. However, there are situations for which some understanding of the influence of material anisotropy of the film material or the substrate material, or perhaps of both materials, is important to know. Therefore, in this section, representative results on the influence of material anisotropy on substrate curvature are included. Results are established for two particular curvature formulas. In the first case, the film is presumed from the outset to be very thin compared to the substrate. Furthermore, the substrate is assumed to be isotropic and the film is considered to be generally anisotropic. In the second case discussed, no restriction is placed on the thickness of the film relative to the substrate, but both materials are assumed to be anisotropic. However, to obtain expressions for curvature which are not too complex to be interpretable, attention is limited to cases for which both the film and substrate materials are orthotropic, that their axes of orthotropy aligned with each other and that one axis of or-thotropy of each material is normal to the film-substrate interface. There is no connection between the values of orthotropic elastic constants of the two materials. Consideration of these two cases illustrates the most useful approaches for anisotropic materials generalizations for cases of greater complexity are evident. 

How many material constants are needed to characterize a linear elastic homogeneous isotropic material How many material constants are needed to characterize a linear elastic homogeneous anisotropic material ... 

If every arbitrary section plane is a plane of isotropy, the material is isotropic there only two independent constants remain ... 

Each of the above elements can be viewed as being transversely isotropic about the common fiber direction, say x, thus characterized by five material constants, say E, E22, G23, Gi2, and V12, where %2 — X3 is the plane of isotropy and V12 is the Poisson s ratio that expresses the strain 62/ i. due to For in-plane problems, say within the Xi — X2 plane, G23 is globally irrelevant. Similarly, only two thermal and two hydral expansional coefficients are of interest, namely i, 2 (=as) and jSi,... 

This frictionless assumption is often appropriate for very stiff materials where adhesive forces are relatively unimportant, but it is often not the case for softer materials such as elastomers, where adhesive forces play a very important role. In these cases, a full-friction boundary condition, where sliding of the two surfaces is not allowed, is often more appropriate, In many important cases (contact of a very thick, incompressible elastic layer, for example) there is little or no practical difference for these two boundary conditions. Nevertheless, in the discussion that follows, we are careful to indicate that boundary condition (frictionless or full-friction) that formally applies in each case. In all cases we assume that the contacting materials are isotropic and homogeneous, each being characterized by two independent elastic constants. 

Since these material constants are defined for a perfectly isotropic and homogeneous material within an ideally elastic, small strain regime, application of such mechanical concepts to single molecules should only be done with reservations. 

On the other hand, rubber can deform elastically up to extensions of as much as 7. As rubber came into use as an engineering material during World War n, a need arose to express Hooke s law for large deformations. Using the Finger deformation tensor, we can come up with the result quite easily. If the stress at any point is linearly proportional to deformation and if the material is isotropic (i.e., has the same proportionality in all directions), then the extra stress due to tteformation should be determined by a constant times the deformation. 

Because the material is assumed to be macroscopically isotropic it is necessary to determine only two material constants. The most convenient constants are the bulk modulus K and shear modulus G, since from (1.13) we have ... 

In order to determine the effective elasticity matrix E, it is necessary in general to apply six linearly independent loadings defined by linear displacement and constant traction boundary conditions. For a material such as that considered here, which is assumed to be macroscopically isotropic, two constants completely define the macroscopic response, and these can be obtained from a single test. In particular, the two material constants used are the bulk and shear moduli, whose effective values are given by (1.39). 

Due to the defective microstructure of the SWRAl composite no attempts have been made in this case to compare the experimentally determined elastic and materials constants with theoretical calculations of these constants which may be e.g. based on the rule of mixtures. This has been, however, possible with the non-porous and uniform GFRE composite. The calculations have been based on data obtained for the whole composite and its separate components. According to [5,13j the following relations hold for the relations between the constants of a composite consisting of an isotropic matrix reinforced with continuous isotropic fibres and the constants of the separate constituents ... 

In an isotropic, stationary medium, the material constants cr, e, and pt are uniform and constant scalars. The first pair of Maxwell s equations may then be stated ... 

The thermoelastic law [2] for isotropic, homogeneous materials allows to relate the resultant peak to peak temperature change AT [K] to the peak to peak amplitude of the periodic change in the sum of the principal stresses Aci=Aai+Acr2 [Pa] at the same point being k the thermoelastic constant [Pa ] of the test material equal to ... 

Infrared ellipsometry is typically performed in the mid-infrared range of 400 to 5000 cm , but also in the near- and far-infrared. The resonances of molecular vibrations or phonons in the solid state generate typical features in the tanT and A spectra in the form of relative minima or maxima and dispersion-like structures. For the isotropic bulk calculation of optical constants - refractive index n and extinction coefficient k - is straightforward. For all other applications (thin films and anisotropic materials) iteration procedures are used. In ellipsometry only angles are measured. The results are also absolute values, obtained without the use of a standard. 

The Z-direction is perpendicular to the page. For simplicity the material is assumed to be isotropic, ie same properties in all directions. However, in some cases for plastics and almost always for fibre composites, the properties will be anisotropic. Thus E and v will have different values in the x, y and z direction. Also, it should also be remembered that only at short times can E and v be assumed to be constants. They will both change with time and so for long-term loading, appropriate values should be used. 

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