Elastic behavior isotropic

Elastic Behavior. In the following discussion of the equations relevant to the design of thick-walled hoUow cylinders, it should be assumed that the material of which the cylinder is made is isotropic and that the cylinder is long and initially free from stress. It may be shown (1,2) that if a cylinder of inner radius, and outer radius, is subjected to a uniform internal pressure, the principal stresses in the radial and tangential directions, and <7, at any radius r, such that > r > are given by... 

In general, there are three kinds of moduli Young s moduli E, shear moduli G, and bulk moduli K. The simplest of all materials are isotropic and homogeneous. The distinguishing feature about isotropic elastic materials is that their properties are the same in all directions. Unoriented amorphous polymers and annealed glasses are examples of such materials. They have only one of each of the three kinds of moduli, and since the moduli are interrelated, only two moduli are enough to describe the elastic behavior of isotropic substances. For isotropic materials... 

We will discuss some preliminary results, which have been performed recently l01). In Fig. 39a the results for polymer No. 2d of Table 10 are shown, which were obtained by torsional vibration experiments. At low temperatures the step in the G (T) curve and the maximum in the G"(T) curve indicate a p-relaxation process at about 120-130 K. Accordingly the glass transition is detected at about 260 K. At 277 K the nematic elastomer becomes isotropic. This phase transformation can be seen only by a very small step in G and G" in the tail of glass transition region, which is shown in more detail in Fig. 39 b. From these measurements we can conclude, that the visco-elastic properties are largely dominated by the properties of the polymer backbone the change of the mesogenic side chains from isotropic to liquid crystalline acts only as a small disturbance and in principle the visco-elastic behavior of the elastomer... 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

Poisson s ratio is conventionally always positive and is less than 0.5 in magnitude. For a homogeneous isotropic material, these two constants (E and o) completely specify the elastic behavioral response.1... 

Rheological studies of PET nanocomposites are not ample, but show very interesting features. In the low frequency range, the nanocomposites display a more elastic behavior than that of PET. It appears that there are some physical network structures formed due to filler interactions, collapsed by shear force, and after all the interactions have collapsed, the melt state becomes isotropic and homogeneous. Linear viscoelastic properties of polycaprolactone and Nylon-6 [51] with MMT display a pseudo-solidlike behavior in the low frequency range of... 

In the isotropic state, as in classical networks, the elastic behavior is governed by... 

A general treatment of the stress-strain relations of rubberlike solids was developed by Rivlin (1948, 1956), assuming only that the material is isotropic in elastic behavior in the unstrained state and incompressible in bulk. It is quite surprising to note what far-reaching conclusions follow from these elementary propositions, which make no reference to molecular structure. 

Moving to a larger scale, let us now look at the influence of microstructure on elastic behavior. As indicated in the previous section, the elastic constants are a fundamental property of single crystals through the geometry and stiffness of the atomic bonds. Thus, one may expect elastic behavior to be controlled simply by the choice of material. By using composite materials, however, one can control the final set of elastic properties with some precision, i.e., by mixing phases with different elastic constants. Clearly, it is useful to be able to predict the elastic constants of a composite from those of its constituents. This has been accomplished for many types of composite microstructures. For this section, however, the emphasis will be on (elastically) isotropic composites, i.e., composites contain-... 

Figure 3.9 Composite sphere assemblage used by Hashin (1983) for exact solutions to the elastic behavior of statistically isotropic composites.

In the above analysis of slow viscous flow, the materials were assumed to be incompressible. This is not, however, necessary to derive the flow equations. If the compressibility is included, two viscosity coefficients are needed. This is analogous to the need for two constants to describe the elastic behavior of isotropic materials. 

As only two elastic constants are needed to describe the linear elastic behavior of isotropic materials, there are various relationships between the four engineering elastic constants. Some of these are given below. 

Viscoeiasticity. As already noted, the time-dependent properties of polymer-based materials are due to the phenomenon of viscoelasticity (qv), a combination of solid-like elastic behavior with liquid-like flow behavior. During deformation, equations 3 and 6 above applied to an isotropic, perfectly elastic solid. The work done on such a solid is stored as the energy of deformation that energy is released completely when the stresses are removed and the original shape is restored. A metal spring approximates this behavior. 

Erdogan and Ratwani( ) presented an analytical solution based on a one-dimensional model for calculating stresses in a stepped lap joint. One adherend was treated as isotropic and the second as orthotropic, and linear elastic behavior was assumed. The thickness variation of the stresses in both the adherends and the adhesive was neglected. 

In the rheology of condensed phases, the elastic modulus G is often used as the sole characteristic of elastic behavior. In isotropic media mechanics, it has been established that for solid-like bodies, the modulus G 2/5 of the Young s modulus [10]. 

In solving viscoelastic stress analysis problems, assumptions on the material properties are often essential as gathering accurate time dependent data for viscoelastic properties is difficult and time consuming. Thus, one often only has properties for shear modulus, G(t) or Young s modulus, E(t), but not both. Yet of course for even the simplest assumption of a homogeneous, isotropic viscoelastic material, two independent material properties are required for solution of two or three dimensional stress analysis problems. Consequently, three assumptions relative to material properties are frequently encountered in viscoelastic stress analysis. These are incompressibility, elastic behavior in dilatation and synchronous shear and bulk moduli. Each of the common assumptions defines a particular value for either the bulk modulus or Poisson s ratio as follows. 

More recently, Bedoui et al [221,222] showed that a classical inclu-sion/matrix model (Figure 1.17) associated with a differential scheme provides satisfactory results for predicting the elastic behavior of isotropic semicrystalline polymers. 

Bedoui F, Dimii J and Regnier G (2006) Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers, Acta Mater 54 1513-1523. 

The two Lame constants occurring in Equations (22) through (26) are one possible choiee of elastic constants which can be used in the case of isotropie materials. Depending on the application in question, other elastic constants can be more advantageous, e.g. the tensile modulus (Yoimg s modulus) E (imits [GPa]), the shear modulus G (imits [GPa]), the bulk modulus K (imits [GPa]) and the Poisson ratio V (dimensionless). Some of these constants are preferable from the practical point of view, since they can be relatively easily determined by standard test procedures E and G ), while others are preferable from the theoretical point of view, e.g. for micromechanical calculations (G and K). Note, however, that even in the case of isotropic materials always two of these elastic constants are needed to determine the elastic behavior completely. 

In a similar way as the Young modulus E and the Poisson ratio V are connected to the uniaxial extension test, the shear modulus G and the bulk modulus K are connected to simple shear and isotropic deformation (i.e. dilatation or compression). Note that, accidentally, it turns out that the shear modulus G equals the second Lame constant ju. Since for isotropic materials only two of the elastic constants are independent, the knowledge of any pair of them is sufficient to calculate the other constants and thus to describe the elastic behavior of isotropic materials completely. For easy reference in this chapter we list the most important interrelations between the elastic constants ... 

The second material property adopted for tibia was transversely isotropic property of the elastic behavior. The stiffness matrix for cortical bone was assumed to have 5 independent constants as follows [5] ... 

In this research, by using a real geometric model of tibia, according to its complex and unique geometry, the effects of different mechanical properties of tibia on the stress analysis under a transversal impact load has been investigated. The maximum stress was seen in the case of viscoelastic model of tibia while the minimum was found with the transversely isotropic property. In agreement with previous studied [7 and 8], the maximum amount of stress reached by the transversely isotropic model of tibia was closer to the results of theoretical and experimental works by other researchers. The dependency of the viscoelastic material property to the time caused the maximum stress to be seen in the last increment of the impact cycle. But, for the elastic behavior of tibia, the maximum stress was seen in the increment with the maximum applied force. The stress relaxation was seen by a reduction in the maximum amount of stress just after the impact load was over because of the constant strain rate in the tibial shaft. 

Isotropic elastic properties, E, the Young s modulus, and t>, the Poisson s ratio can be used to represent the elastic behavior of the solder alloy. The temperature-dependent elastic behavior of two lead-free solder alloys Sn-Ag-Cu and Sn-Ag are presented in this section. For example, E is given by ... 

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