Linear Elastic Moduli

If a crystal is subjected to small strain elastic deformation it is convenient to imagine the energetics of the strained solid in terms of the linear theory of elasticity. As we noted in chap. 2, the stored strain energy may be captured via the elastic strain energy density which in this context is a strictly local quantity of the form 

For our present purposes, we note that the correspondence of interest is given by 

On the recognition that the expansion is about the reference state and hence that the linear term in the expansion vanishes (i.e. (d ioi/UfyOre/ = 0), eqn (5.85) reduces to 

It is worth noting that this expression is generic as a means of extracting the elastic moduli from microscopic calculations. We have not, as yet, specialized to any particular choice of microscopic energy functional. The physical interpretation of this result is recalled in fig. 5.2. The basic observations may be summarized as follows. First, we note that the actual energy landscape is nonlinear. One next adopts the view that even in the presence of finite deformations, there is a 

Within the pair potential setting where the total energy is given by Etot = 5 Hmn Veffirmn), the elastic moduli are given by 

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As previously noted, this chapter has been concerned mainly with those models for the creep of ceramic matrix composite materials which feature some novelty that cannot be represented simply by taking models for the linear elastic properties of a composite and, through transformation, turning the model into a linear viscoelastic one. If this were done, the coverage of models would be much more comprehensive since elastic models for composites abound. Instead, it was decided to concentrate mainly on phenomena which cannot be treated in this manner. However, it was necessary to introduce a few models for materials with linear matrices which could have been developed by the transformation route. Otherwise, the discussion of some novel aspects such as fiber brittle failure or the comparison of non-linear materials with linear ones would have been incomprehensible. To summarize those models which could have been introduced by the transformation route, it can be stated that the inverse of the composite linear elastic modulus can be used to represent a linear steady-state creep coefficient when the kinematics are switched from strain to strain rate in the relevant model. 

Here P Is a static pressure field and Is the shear modulus of elasticity. Tlie shear modulus Is related to Che linear elasticity modulus as... 

The Rheometric Scientific RDA II dynamic analy2er is designed for characteri2ation of polymer melts and soHds in the form of rectangular bars. It makes computer-controUed measurements of dynamic shear viscosity, elastic modulus, loss modulus, tan 5, and linear thermal expansion coefficient over a temperature range of ambient to 600°C (—150°C optional) at frequencies 10 -500 rad/s. It is particularly useful for the characteri2ation of materials that experience considerable changes in properties because of thermal transitions or chemical reactions. 

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

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... 

When two linear-elastic materials (though with different moduli) are mixed, the mixture is also linear-elastic. The modulus of a fibrous composite when loaded along the fibre direction (Fig. 25.1a) is a linear combination of that of the fibres, Ef, and the matrix, E, ... 

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

Now, for the special case of a linear elastic material this is readily expressed in terms of the stress, Oc, on the material and its modulus, E. 

Paul [3-4] was apparently the first to use the bounding (variational) techniques of linear elasticity to examine the bounds on the moduli of multiphase materials. His work was directed toward-analvsis of the elastic moduli of alloyed metals rath, tha tow5 rdJ ber-reW composite materials. Accordiriglyrthe treatment is for an js 6pjc composite material made of different isotropic constituents. The omposifeTnaterial is isotropic because the alloyed constituents are uniformly dispersed and have no preferred orientation. The modulus of the matrix material is... 

The mechanical properties can be studied by stretching a polymer specimen at constant rate and monitoring the stress produced. The Young (elastic) modulus is determined from the initial linear portion of the stress-strain curve, and other mechanical parameters of interest include the yield and break stresses and the corresponding strain (draw ratio) values. Some of these parameters will be reported in the following paragraphs, referred to as results on thermotropic polybibenzoates with different spacers. The stress-strain plots were obtained at various drawing temperatures and rates. 

In the region where the relationship between stress and strain is linear, the material is said to be elastic, and the constant of proportionality is E, Young s modulus, or the elastic modulus. 

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 linear visco-elastic range ends when the elastic modulus G starts to fall off with the further increase of the strain amplitude. This value is called the critical amplitude yi This is the maximum amplitude that can be used for non-destructive dynamic oscillation measurements... 

In classic terms, the elastic modulus of a material is the stress divided by the strain (i.e., the slope) of the linear portion of its force versus elongation curve at low strain. In this region, the material is assumed to behave in a Hookean fashion, i.e., stress and strain are linearly proportional, as illustrated in Fig. 8.5 a). Most polymers do not behave in this manner. 

Poison s ratio is used by engineer s in place of the more fundamental quality desired, the bulk modulus. The latter is in fact determined by r for linearly elastic systems—h ncc the widespread use of v engineering equation for large deformations, however, where the Strain is not proportional to the stress, a single value of the hulk modulus may still suffice even when the value of y is not- constant,... 

The experimental determination of RBA, however, is difficult but some attempts have been made and these include direct observation, measurements of electrical conductivity, shrinkage energy, gas adsorption and light scattering. The linear elastic response of paper has been explained in terms of various micromechanical models which take into account both fibre and network properties, including RBA. An example of one which predicts the sheet modulus, Es is given below ... 

The elastic modulus is constant at small stresses and strains. This linearity gives us Hooke s Law1, which states that the stress is directly proportional to the strain. 

The spring is elastically storing energy. With time this energy is dissipated by flow within the dashpot. An experiment performed using the application of rapid stress in which the stress is monitored with time is called a stress relaxation experiment. For a single Maxwell model we require only two of the three model parameters to describe the decay of stress with time. These three parameters are the elastic modulus G, the viscosity r and the relaxation time rm. The exponential decay described in Equation (4.16) represents a linear response. As the strain is increased past a critical value this simple decay is lost. 

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

Fig.5. The elastic modulus G (co) and dissipative modulus G (co) for linear top) and three-arm-star branched (bottom) polyisoprene from [5]. Note the broad range of relaxation times indicated by the width of the peak in the star-polymer...

This measures the curvature about an axis perpendicular to the dispersion plane, i.e. the cylindrical curvature, and it may be necessary to rotate the wafer through 90° to get the orthogonal component. This may be related to absolute stress in the wafer with knowledge of the wafer thickness, diameter and elastic modulus. The most accurate method is to measure a number of points on a wafer and use a linear regression formula for the average curvature. 

TMA measures the mechanical response of a polymer looking at (1) expansion properties including the coefficient of linear expansion, (2) tension properties such as measurement of shrinkage and expansion under tensile stress, i.e., elastic modulus, (3) volumetric expansion, i.e., specific volume, (4) single-fiber properties, and (5) compression properties such as measuring the softening or penetration under load. 

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