Predicting the viscoelastic behavior of polymer nanocomposites

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6 Predicting the viscoelastic behavior of polymer nanocomposites A. Beyle and C. C. Ibeh , Pittsburg State University, USA Abstract: Prediction of elastic moduli and some physical properties of heterogeneous media on the basis of known characteristics of constituents, their volumetric concentrations, shapes and orientation of inclusions and the type of lattice formed by the set of inclusions, is a developed area of the theory. This chapter is devoted to expansion of the theory to viscoelastic systems mostly made from viscoelastic polymeric matrices and elastic fillers. The method is based on elasto-viscoelastic analogy, in which the elastic moduli of constituents participating in the formulae for the effective moduli are replaced by corresponding integral operators of the theory of viscoelasticity. The next step is the decoding of the functions of operators, i.e. in replacing the expressions containing several operators by one equivalent operator with effective parameters. The difference between application of the theory to composites and nanocomposites is discussed. The prediction of elastic and viscoelastic behavior of composites and nanocomposites of geometrically similar structures is proposed. Prediction of the failure for composites and nanocomposites has to be done differently. Prediction of viscoelastic behavior is described for composites made from polymeric matrices and spherical, short cylindrical inclusions, platelets, hollow spherical and cylindrical inclusions, and spherical voids. Influence of stiffness ratio of matrix and inclusion on the effective properties is analyzed. Key words: composites, nanocomposites, effective properties, viscoelasticity, creep, loss modulus, storage modulus, integral operators of viscoelasticity, syntactic foams, rigid fillers.

6.1

Specific features of nanoparticles and nanocomposites

Mechanical properties of heterogeneous systems depend on such factors as properties of constituents, concentration of constituents, shape of fillers, orientation of fillers, etc. Existing theories used for calculations of the effective elastic properties are not sensitive to the absolute sizes of inclusions, and only their volumetric concentration is important. In the case of nanoparticles, some modifications are possible due to the influence of the interfacial interaction between phases on the bulk matrix properties; this role increases in nano-structured systems due to the high surface-to-volume ratio of nano-inclusions. However, the classical analysis still describes the main effects in the case of elastic properties of nanocomposites. Deviations in experimental data from theoretical ones are mostly related to the effect of aggregation of nanoparticles. The theoretical prediction of 184 © Woodhead Publishing Limited, 2011

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viscoelastic properties is less developed despite the principle of elastic-viscoelastic analogy. In this chapter, it is planned to demonstrate the results of theoretical prediction of viscoelastic behavior and to illustrate them by our experimental data as well as via published literature data. Some nanoparticles have irregular shape. However many types of nanoparticles have quasi-classical shapes, and are spherical, cylindrical-long fibers or cylindricalplane disks. Such nanoparticles as nanoclay and nanographene are platelets; carbon nanofibers have cylindrical shape; silicon carbide nanoparticles can be considered as quasi-spherical ones. Carbon nanotubes are considered as hollow cylinders whereas buckyballs are hollow spheres. These classical shapes will be analyzed using elastic-viscoelastic analogy and Rabotnov’s algebra of resolvent integral operators of viscoelasticity. Nanoparticles have excellent mechanical and physical properties and very high ratio of the total surface of matrix-inclusions boundaries to the total volume of nanocomposite. However, the improvement of properties in comparison to the matrix properties is not very high as expected by many researchers. The causes of such contradictions are discussed and illustrated by the results and data of a modeling approach. Starting from Griffith’s classical works of the 1920s it becomes clear that the huge difference between theoretical strength of materials and their real strength (two to three decimal orders) can be reduced if the sizes of the solid body are decreased. Following Griffith’s ideas the first high-strength glass fibers were produced industrially in the 1940s. The idea of building strong bulk materials using thin strong fibers and binding matrix (initially polymeric, later metallic, ceramic, cement, etc.), i.e. the idea of composites, was taken from the architecture occurring in natural materials: wood, bones, etc. are the natural composites. Progress in technology resulted in the ability to make nano-sized fillers, the individual strength of which approaches to the theoretical value. However, the expectations that the properties of nanocomposites would be much higher than the properties of conventional composites are not realized. Some progress has been achieved in dynamic applications and in other areas but it is not very pronounced. Information about nanocomposites’ properties, technology, and applications can be found in multiple sources, for example in references 1 to 3. In this situation the critical review of the existing methods for prediction of the mechanical behavior of heterogeneous materials could be useful. This chapter is devoted to applicability of the methods of prediction of viscoelastic behavior of heterogeneous materials for the particular case of nanocomposites. All mechanical properties of heterogeneous systems depend on properties of constituents, on their concentrations, on shape of fillers, on fillers’ orientation, on type of spatial lattice, etc. Existing theories used for calculations of the effective elastic properties are non-sensitive to the absolute sizes of the inclusions, and only their volumetric concentration is important. In the case of nano­­­particles, some modifications are possible due to the influence of the interfacial interaction between phases on the bulk matrix properties; this effect increases

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in nano-structured systems due to their high surface-to-volume ratio. Really, volume of the individual inclusion, VI is related to the total volume V of composite as

[6.1]

where c is the volume concentration of inclusions and NI the total number of the identical inclusions in the volume V. The ratio of the total surface S of all inclusions to the total volume of composite can be written as:

[6.2]

where SI is the outer surface area of the individual inclusion, ξI the dimensionless inclusion’s form factor:

[6.3]

The values of the form factor for different types of inclusions are presented in Table 6.1. According to Eq. 6.2 both form factor and especially the decreasing volume of the individual inclusion play significant roles in elevated surface effects in nanocomposites. The simplest model, which takes into account surface effects, includes three phases: inclusions, matrix, and thin layer of modified matrix separated matrix and inclusions (see, for example, reference 4). In the case of thermoplastic matrix this layer is formed due to lower mobility of the macro­ molecular segments near the solid surface; thickness of the layer is a few segments. Table 6.1  Form factors ξI for different types of inclusions Shape type

Sizes ratio

Form factor ξI

Spherical Cubical Cylindrical Cylindrical Cylindrical Cylindrical Circular platelet Circular platelet Circular platelet Circular platelet Square platelet Square platelet Square platelet Square platelet

Any Any L / R = 10 L / R = 100 L / R = 1000 L / R → ∞ D / h = 10 D / h = 100 D / h = 1000 D / h → ∞ L / h = 10 L / h = 100 L / h = 1000 L / h → ∞

4.836 6 6.942 13.732 29.321

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∞

10.28 40.55 184.9

∞

11.14 43.95 200.4

∞

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In the case of thermosets the kinetics of chemical reactions near the surfaces of inclusions are different from the kinetics of reactions in bulk and as a result the macromolecular structure and properties are different. In some cases, when inclusions are acting as nucleators of physical or morphological transitions, the thickness of the modified layer of matrix can be comparable with the size of inclusions. Stepwise change of the matrix properties is an idealized model replacing continuous monotonic change of the matrix properties from the surface of inclusion to bulk properties. Unfortunately there is not enough data on the effective thickness of modified matrix layers as well as on properties of such modified layers. The classical models of composites taking into account properties of matrix and inclusions, shapes of inclusions, their orientations, type of spatial lattice, etc., still describe the main effects in the case of elastic properties of nanocomposites. Differences between experimental data and theoretical ones are mostly related to the effect of aggregation of nanoparticles, to the shape deviation from the ideal one mostly as waviness, to the nonuniformity of the nanoparticles’ distribution over the volume of nanocomposites, to imperfections in orientation, lattice, etc. The aggregation of nanoparticles is the biggest obstacle in nanocomposite technology. During transportation and storage nanoparticles can accumulate electrical charges, and nanoparticles with opposite charges can form the aggregate much more easily than the conventional fillers simply due to smaller masses. The number of nanoparticles Nn providing the same volume concentration c as a number Nc of the conventional particles is inversed proportional to the cube of their sizes ratio:

[6.4]

Because conventional fillers have sizes mostly in the range 1 µm to 1 mm but nanoparticles mostly in the range of 10 to 100 nm, then the number of nanoparticles replacing the same mass of the conventional fillers has to be from 103 to 1015 bigger. Taking into account that the distances between neighbor particles are proportional to their sizes (if volume concentration is the same) and that hydrodynamic resistance is proportional to the size of the particle, the probability of nanoparticle aggregation during technologically mixing them with matrix in liquid form (before solidification of nanocomposite) is very high. The presence of aggregates is worse than the presence of voids. They are not only stress concentrators for the matrix; they can carry very low tensile and shear stresses and practically in this aspect are no better than voids; but the main negative effect is that they are working as levers opening cracks in the matrix. Sonication technology allows the destruction of the aggregates but it also changes the polymeric matrix.5

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6.2

Viscoelasticity of polymer matrix

Viscoelasticity can be described with the help of creep curve, relaxation curve, spectrum of retardance times, spectrum of relaxation times, and dependencies of complex modulus or compliance on frequency. Each of the above-mentioned characteristics can be calculated if one of them is known (in the case of known time spectra it is necessary to know also instantaneous and long time moduli). Usually behavior of polymeric matrix is described with the help of the generalized Maxwell (Fig. 6.1a) or generalized Kelvin-Voigt (Fig. 6.1b) model. These models are equivalent, i.e. the set of characteristics (elastic moduli Ei and viscosities ηi) of one model can be recalculated from the whole set of characteristics of another model. It is necessary to mention that Ek in one model is not the same as Ek in another model; similar can be noted with respect to ηk! In addition to the strains related to the stresses, the strains related to the thermal expansion, to the swelling due to humidity change, and to the chemical or physical shrinkage due to chemical reactions or physical transitions have to be added, as

6.1  Generalized Maxwell (a) and Kelvin-Voigt (b) models of mechanical behavior of polymeric matrix. For ideally solidified polymer the viscosity η0 → ∞ (i.e. this damping element can be removed from the model).

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well as in some cases strains related to the influence of electrical and magnetic fields which must also be taken into account. If free strains (not related to the stresses) are not satisfied in the equations of strains compatibility, the stresses appear to compensate this mismatch and the whole system of equations of mechanics of the particular continuum must be solved for the determination of stresses and strains. All parameters of the models described above are dependent on temperature, on depth of chemical reaction, on humidity, etc. Especially sensitive is the viscosity element η0. In a technological process of mixing matrix with fillers, the matrix material is in the liquid form and η0 is finite. After the depth of the chemical reaction of solidification (initiated by adding some curing agent or by rising the temperature) is rising, the viscosity η0 is rising exponentially and in some critical value of the depth of chemical reaction, the gel point is achieved and the viscosity η0 becomes infinitely high. The reaction is continued after the gel point but it is connected with further changes of the properties of other elements participating in the models shown in Fig. 6.1. In general, all viscous elements in the models shown in Fig. 6.1 are more sensitive to the temperature change and to the depth of chemical reaction than the elastic elements. Change of viscoelastic properties with temperature is often studied with the help of Dynamic Mechanical Thermal Analysis (DMTA). Polymeric matrices are the most technologically convenient ones but their own mechanical characteristics are not impressive: the Young’s modulus of the majority of polymeric matrices at room temperature is in the range 0.1–10 GPa (for comparison: steel has Young’s modulus 210 GPa; the best carbon fibers, carbon nanofibers, and carbon nanotubes have Young’s modulus 1000 GPa). The tensile strength of polymeric matrices is in the range 1–200 MPa (for comparison: good-quality steel has tensile strength over 1000 MPa, the strength of the best carbon fibers is approaching 6000 MPa, the strength of carbon nanofibers is in the range 10 000–50 000 MPa but the theoretical strength of carbon materials is about 150 GPa). Polymeric matrices are characterized by pronounced viscoelastic behavior. The constitutive law for viscoelastic bodies can be written in the most general form (see Rabotnov6 for example) as:

[6.5]

Here εij is strain tensor, is tensor of free strains except thermal expansion, αij is tensor of coefficients of linear thermal expansion, is tensor-functional, and σkl is stress tensor. Functional connects the current value of some function f(t), not to the current value of the function g(t) as it is used in parametric form of functions, but connects its value to the whole history of the function g(τ):

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[6.6]

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In the theory of viscoelasticity the functional is written usually in the form of the Volterra integral operator:

[6.7]

where the kernel Ψ(t – τ) describes the fading memory about previous actions on the material. It has a dimension of the inversed time. In the particular case of the kernel:

[6.8]

the Volterra integral operator describes the mechanical behavior of the models shown in Fig. 6.1. Expression 6.7 can be rewritten in the alternative symbolic form:

[6.9]

where Π × is an operator. Expression 6.7 can be also rewritten as:

[6.10]

where Λ is a dimensionless function. To find the expression for σ (t) from Eq. 6.7, it is necessary to solve a Volterra integral equation of the second kind. The solution has the following general form:

[6.11]

Here Φ (t – τ) is the resolvent of the Volterra integral equation of the second kind and ϒ  × is the operator of the solution. Formally, the relationship between two operators can be written as:

[6.12]

In the more general case:

[6.13]

where operators Π × (λ) and ϒ  × (λ) are mutually resolvent operators depending on the parameter λ because the kernels are dependent on λ now:

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[6.14]

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The rules of Rabotnov's algebra of resolvent operators6–8 are as follows:

[6.15]

[6.16]



[6.17]

Remembering that the product of Volterra operators:

[6.18]

with corresponding kernels L(t – τ) and M(t – τ) is the operator with kernel:

[6.19]

It means that the square of the resolvent operators in the left side of Eq. 6.17 containing iterated kernel (Eq. 6.19) can be replaced by an operator containing a single kernel as shown on the right side of Eq. 6.17. Resolvent operators represent the majority of operators used in the theory of viscoelasticity. There are some additional Rabotnov’s algebra rules following from those above:

[6.20]

[6.21] In the particular case:

[6.22]

[6.23]





[6.24]

[6.25]

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The more complicated case of mutual resolvents of the sum of operators is not written here due to the lack of space but it is used in calculations of some numerical examples shown in the following sections of the chapter. Let’s demonstrate an application of the above formulae for the case of a threeelement model (Fig. 6.1b). The version of the Kelvin-Voigt model containing only elements E1, E2,η2 can be described with a particular form of Eq. 6.7: [6.26] Using Eq. 6.15 (from right side to the left side) and taking into account that in this case:

it is possible to find easily the inverse form:

[6.27]

derivation of which is much longer if traditional methods are used. Let’s denote module-operators as: In shear:

[6.28]



[6.29]

In tension-compression:

Then the corresponding compliance-operators will have the form: In shear:

[6.30]

In tension-compression:

[6.31]

Here, µi ≥ 0; λi < 0, where i has to be replaced by G or E, correspondingly. There is a set of formulae for calculation of the effective elastic properties of different types of composites via properties of constituents and their volume concentrations. The change of viscoelastic properties of composite or nanocomposite with concentration of fillers can be calculated from the change of corresponding elastic properties. If elastic moduli are replaced in the formulae by instantaneous viscoelastic moduli, the calculated results give the instantaneous

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effective moduli of composite. If elastic moduli are replaced in the formulae by long-time viscoelastic moduli, the calculations results give the long-time effective moduli of composite. The remaining problem is to find the way for predicting transient properties. Volterra proved that the solution of the viscoelastic problem can be obtained from the solution of the corresponding elastic problem by replacing elastic moduli by corresponding viscoelastic operators. The complicated expressions containing several operators can nowadays be numerically estimated using appropriate software. However, for analytical approach and for simplification of the numerical calculations it is reasonable to use Rabotnov’s algebra of resolvent operators6 to convert some expressions with operators into one operator with shifted parameters, or at least to decrease a number of operators participating in the expressions for the effective characteristics. Some examples of derivation can be found in the literature.9–13 Another way to come to viscoelastic characteristics’ dependencies on volumetric concentrations of fillers is to replace elastic characteristics such as moduli by complex viscoelastic characteristics14, 15 and to convert the corresponding algebraic expression containing many complex numbers to the standard form of complex number using trivial operations such as:

[6.32]

The attempt to replace elastic moduli by complex viscoelastic moduli done in the work16 does not look very attractive because the hypothesis about constant Poisson’s ratio in viscoelastic process was used, which is a very rough one. Choice of the particular form of operator of viscoelasticity is subjective: it depends on the experience of the researcher, on the precision of experimental determination of characteristics and the required precision of the prediction of viscoelastic behavior, etc. The most popular types of operators are: (a) Operator (Eq. 6.8) containing sum of exponents (b) Abel operator of fractional derivative with kernel: [6.33]



 here Γ (x) is gamma function, which is a generalization of the factorial on w non-integer values of argument:

[6.34]

(c) Rabotnov’s operator of fractional exponent with kernel:

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[6.35]

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(d) Rzhanitsin’s operator with kernel:

[6.36]

(e) Koltunov’s operator with kernel:

[6.37]

(f ) Sum of Rabotnov’s operators. The most popular operators are sum of exponents and Rabotnov’s operator. Sum of exponents may contain different numbers of viscoelastic characteristics, which are necessary to determine experimentally (three, five, seven, etc. depending on the number of elements in the model in Fig. 6.1). A bigger time range is asking to take into account more and more terms. Precision and statistical determination of the viscoelastic characteristics drop down with the rise of the number of exponents. Rabotnov’s operator contains four constants only and successfully approximates experimental data in a wide time range but its application requires more complicated calculations. Before application of elastic-viscoelastic analogy for calculation of the effective viscoelasticity of nanocomposites, let’s demonstrate it for the calculation of the viscoelastic Poisson’s ratio of polymer matrix. For an isotropic body the Poisson’s ratio can be calculated via any pair from three elastic moduli: Young’s modulus E, shear modulus G, and bulk modulus K according to the formulae:



[6.38]

Let us use, for example, the third formula. If the compliances operators in tension and in shear are: [6.39] Then using Eq. 6.21 it can be found that:

[6.40]

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If complex moduli on tension and shear are known, the complex Poisson’s ratio can be calculated:

[6.41]

For isotropic polymers, the commonly used simplifying hypothesis is that the volume change is happening as a result of pure elastic deformation, but the shape change is a result of viscoelastic deformation. This hypothesis is valid only if no orientation occurs during deformation and the polymer remains isotropic after it. For thermosets this hypothesis works and it is validated by multiple experiments, but for thermoplastics this hypothesis is too rough because of the pronounced orientation effects. If this hypothesis is used, then the first two formulae (6.38) are more convenient than the third one. It is enough to know viscoelastic properties on tension or on shear only and two elastic characteristics in the case of using the hypothesis of pure elastic bulk strain. Operators describing shear and tensile/ compression properties are interrelated in this case:

[6.42]

The operator form is the following: In the case of known viscoelastic properties on shear:



[6.43]

In the case of known viscoelastic properties on tension/compression:



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[6.44]

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The complex form is the following: In the case of known viscoelastic properties on shear:



[6.45]

In the case of known viscoelastic properties on tension/compression:

[6.46]

Experimental data on creep of typical polymeric matrix (epoxy) obtained at different levels of stresses show that up to 70–80% of static strength, the viscoelastic behavior of polymeric matrix is linear, i.e. strains are increasing with stresses linearly at any time but at higher stresses strain dependence on stress becomes nonlinear (Fig. 6.2). Viscoelasticity of new polymeric fibers (polyaramid, polyparaphenylenebenzobisoxazole (PBO), ultra high molecular weight polyethylene (UHMWPE), etc.) is more nonlinear with stresses. These fibers are characterized also by very high anisotropy. Different versions of viscoelastic operators describe transient process of creep differently but the short-time behavior and long-time behavior are described similarly. The main characteristic of short-time behavior is instantaneous modulus (such as E0) and long-time behavior is described with help of protracted modulus (such as E∞). These concepts are illustrated by Fig. 6.3. Creep curves allow finding the main parameters of viscoelasticity (see Fig. 6.3). During creep, the deformation in the perpendicular direction is also increasing; moreover, the Poisson’s ratio is increasing with time (Fig. 6.4). Such increase (about 10%) means that the shear modulus during the creep is changed by about 50–60% if change of the volume is purely elastic. Ignoring change of the Poisson’s ratio means in this case that the shear creep is ignored also. Or it means that the creep on tension has happened due to the volumetric creep only, which is absurd. It is the reason why the hypothesis about constant Poisson’s ratio used in some simplified methods of the solution of viscoelasticity problems (see for example16) is a very rough one. Using data from Fig. 6.2 and Eq. 6.44 it is possible to predict the change of the Poisson’s ratio with time during the creep. The results of the calculations are shown in Fig. 6.5. They are close to experimental data shown in Fig. 6.4. Using

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6.2  Creep curves of malein-epoxy matrix at different levels of stresses (shown in MPa under each curve).11

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6.3  Determination of instantaneous and long-time moduli from the creep curve.

6.4  Change of the Poisson's ratio with time during the creep of the malein-epoxy matrix.11

the same data and Eq. 6.46 the complex Poisson’s ratio dependencies on the frequency were calculated and plotted in Fig. 6.6 and 6.7. Systematic study of Poisson’s effect in viscoelasticity was undertaken by Lakes and colleagues (see, for example, references 17 and 18).

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6.5  Calculated change of the Poisson's ratio with time during the creep on tension. E0 / E∞ = 1.5. Retardance time in the tensile creep τEcr = 200 hours.

6.6  Dependence of the real part of the complex Poisson's ratio on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here E0 / E∞ = 1.5.

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6.7  Dependence of the imaginary part of the complex Poisson's ratio on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here E0 / E∞ = 1.5.

6.3

Viscoelasticity of polymers filled by quasi-spherical nanoparticles

Many nanoparticles made from substances such as oxides and carbides of different chemical elements have irregular shapes, which could be statistically equated to the spherical shape. If conventional fillers content can reach several tens of percents of the whole volume of composites, the maximal volume content of nanofillers is usually about several percents. The solution of the problem of calculation of effective elastic properties of a matrix filled by small volume fraction of spherical inclusions is well known (see, for example Christensen’s book14):



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[6.47]

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where K is bulk/hydrostatic modulus, G is shear modulus; indices M and I denote matrix and inclusion, correspondingly; c is volume concentration of inclusions. Effective properties are denoted by putting the corresponding property in angular brackets. Poisson’s ratio ν can be used in this formula instead of shear modulus taking into account the well-known formula:

[6.48]

and:



[6.49]

Formula 6.47 is valid in a wide range of volume concentrations. In the case when inclusions are much stiffer than matrix under hydrostatic pressure, Expression 6.47 degenerates into:



[6.50]

Examples of the calculation of the effective bulk modulus for the epoxy-solid glass microspheres and epoxy-silicon carbide nanofillers according to Eq. 6.44 as well as for the degenerated case (Eq. 6.50) are shown in Fig. 6.8. The majority of modern fillers can be considered with respect to polymeric matrices as practically approaching absolutely rigid inclusions (Fig. 6.8a). It is possible to note that the Poisson’s ratio of matrix material provides a bigger effect than the relative stiffness of inclusions (Fig. 6.8b). Replacing elastic moduli by viscoelastic operators in Eq. 6.47 and using Rabotnov algebra of resolvent operators, it is possible to derive the final formula. Because it is too bulky, here the simplified formula is shown, which is derived using two assumptions: the volumetric strain of the matrix is ideally elastic and the filler is much stiffer than matrix:



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[6.51]

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6.8  Relative increase of the effective bulk modulus of the material filled by spherical inclusions as a function of the volume concentration c of inclusions: (a) composite made from polymeric matrix (epoxy) (νM = 0.382) filled by glass solid microspheres (solid line), by magnesium oxide or low grade silicon carbide or carbon micro or nanospheres (dashed line) or by absolutely rigid micro or nanospheres (dotted line); (b) composites made from absolutely rigid spherical inclusions in the matrices having different Poisson's ratio. © Woodhead Publishing Limited, 2011

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Here νM0 is the instantaneous Poisson’s ratio of matrix, ϒ ×EM is the viscoelastic operator describing behavior of matrix under unidirectional tension/compression, and λEM and µEM are the parameters participating in this operator (see Eq. 6.29 and 6.31). The results of calculations of the volume creep are presented in Fig. 6.9. Creep of materials containing small volume fractions of inclusions is shown in Fig. 6.9a, and containing big volume fractions of inclusions in Fig. 6.9c. Filling materials with solid particles increases their stiffness and as a result the instantaneous strains are decreasing with fillers concentrations. However, the creep is increasing more clearly if creep curves are made dimensionless by dividing full strain by instantaneous strain at the same level of filling (Fig. 6.9b, 6.9d). The effect is small but nevertheless it is remarkable: with increase of concentration of rigid particles the bulk creep is increasing! It is explained by the increase of the total part of the zones in the matrix that is subjected to the stress concentration and because the shear stresses are pronounced. In the framework of the used hypotheses the creep is provided by shear only; these zones contribute more and more to the total volume creep. At small concentration of the fillers the effect is practically proportional to the filler concentrations (Fig. 6.9a, 6.9b) but at higher concentrations the dependence of the effect on concentration becomes nonlinear (Fig. 6.9c, 6.9d). Complex effective bulk modulus can be calculated as:



[6.52]

where:

[6.53]

and E"M, E"M, and EM0 are storage, loss, and instantaneous moduli of matrix on tension/compression, respectively; νM0 is instantaneous Poisson’s ratio of matrix. Numerical examples of calculations are shown in Fig. 6.10 and 6.11. Data in Fig. 6.10 support the conclusion made from the analysis of Fig. 6.9. At high frequency the behavior of viscoelastic material always corresponds to instantaneous reaction on load in creep but behavior at low frequency is similar to long-time creep behavior.

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6.9  Effective bulk creep function 〈JK〉 (volume creep curve divided by the hydrostatic stress) of material containing different volume percentages of absolutely rigid spherical particles; τEcr is the strain retardance time in the creep at unidirectional tension/compression; here EM ∞ / EM 0 = 0.6; 〈K0〉 is the instantaneous bulk modulus of matrix.

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6.9  Continued.

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6.10  Dependencies of the bulk storage modulus 〈K’〉 (real part of the complex bulk modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM ∞ / EM 0 = 0.6 and KM is the instantaneous bulk modulus of matrix.

6.11  Dependencies of the bulk loss modulus 〈K”〉 (imaginary part of the complex bulk modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM ∞ / EM 0 = 0.6 and KM is the instantaneous bulk modulus of matrix.

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Effective shear modulus for the case of small volume concentration of inclusions can be found using Eq. 6.14:



[6.54]

For the case of significantly stiffer inclusions the formula (Eq. 6.54) degenerates into:

[6.55]

Examples of calculations using Eq. 6.54 and 6.55 are shown in Fig. 6.12. In Christensen’s book,14 the method of calculation of the effective shear modulus at medium and high concentration of inclusions is shown. It is based on a three-phase concentric spherical cell model. Unfortunately, the final formulae are bulky. Here we use another approach based on Vanin’s book,9 where he

6.12  Relative increase of the shear modulus of composite made from polymeric matrix (epoxy) filled by glass microspheres (GI /GM = 25), magnesium oxide or low grade silicon carbide or carbon micro or nanospheres (GI /GM = 50) or by absolutely rigid micro or nanospheres (GI /GM → ∞) as function of the volume concentration c of inclusions.  according to Eq. 6.14 and 6.54–6.55 derived for small concentrations;  according to Eq. 6.9 and 6.65–6.66 derived for medium concentrations; in the last case it is necessary to use also Eq. 6.68 and 6.47–6.50.

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derived a more compact formula for the effective Young’s modulus of concentrated solid suspension. From Vanin’s formula and known effective bulk modulus (Eq. 6.47), the effective shear modulus can be found easily. The viscoelastic operator can be found on the basis of Eq. 6.54, 6.44, and 6.23:

[6.56]

By a similar method it is possible to derive the formula for the inversed operator of shear compliance:



[6.57]

 Operator 6.57 gives shear compliance function and creep curves, which are dependent on filler concentration (see Fig. 6.13). Complex shear modulus:

[6.58]

can be found by replacing the shear modulus of matrix and Poisson’s ratio of matrix in Eq. 6.55 by corresponding complex values and by using Eq. 6.46, 6.32,

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6.13  Effective shear creep function 〈JG〉 (shear strain creep curve divided by the shear stress) of matrix filled by different volume percentages of absolutely rigid spherical particles; τEcr is the strain retardance time in the creep at unidirectional tension/compression. Here EM∞ / EM0 = 0.6 and GM0 is the instantaneous shear modulus of the matrix.

6.38, 6.42, 6.48, and 6.49, taking into account that the bulk modulus of matrix can be expressed with the help of instantaneous Young’s modulus and Poisson’s ratio. The final formula for the shear storage modulus is:

[6.59]

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The final formula for the loss modulus is:

[6.60]

The numerical examples of calculations of the dependencies of the components of complex modulus on frequency and on concentration of inclusions are shown in Fig. 6.14 and 6.15.

6.14  Dependencies of the effective shear storage modulus 〈G´〉 (real part of the complex shear modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and GM0 is instantaneous shear modulus of matrix. Shear storage modulus approaches the instantaneous one at infinite frequency.

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6.15  Dependencies of the effective shear loss modulus G” (imaginary part of the complex shear modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and GM0 is instantaneous shear modulus of matrix.

From two known effective elastic characteristics of an isotropic body the other ones can be easily calculated using well-known formulae:

[6.61] [6.62]



For Young’s modulus the final formula following from Eq. 6.47, 6.54, and 6.62 can be written as:

[6.63]

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For the rigid inclusions this formula degenerates into:

[6.64] A more complicated solution, which is valid for slightly bigger volume concentrations of inclusions, is given by Vanin.9 He derived a direct formula for the effective Young’s modulus instead of using solution for the shear modulus:

[6.65]

For the absolutely rigid inclusions this formula degenerates into: [6.66] The results of calculations are shown in Fig. 6.16. In a similar way to that done for the shear, it is also possible to derive formulae for the effective viscoelastic module-operator of the filled material for tension/ compression and the inversed viscoelastic compliance-operator. Unfortunately, the final formulae are too bulky to be printed here. An example of calculations using the compliance-operator is shown in Fig. 6.17. Complex Young’s modulus can be calculated via previously found complex bulk modulus (Eq. 6.52) and complex shear modulus (Eq. 6.58–6.60):



[6.67]

Numerical examples of calculations are shown in Fig. 6.18 and 6.19. It is necessary to mention that viscoelastic behavior on shear and on tension/ compression are similar, which is natural because the bulk creep of the matrix was

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6.16  Relative increase of the Young's modulus of composite made from polymeric matrix (epoxy) filled by glass microspheres (GI /GM = 25), by micro or nanospheres made from material with (GI /GM = 50) MgO, low grades of carbon, silicon carbide, etc.) or by absolutely rigid micro or nanospheres (GI /GM → ∞) as function of the volume concentration c of According to Eq. 6.14 and 6.63–6.64 derived for small inclusions. according to Eq. 6.9 and 6.65–6.66 derived for concentrations; medium concentrations.

ignored in the model, but this similarity is not absolute because the stress concentration in matrix near inclusions on shear and on tension/compression are different. Other effective elastic properties can be easy calculated from two known effective elastic characteristics of an isotropic body using well-known formulae:



[6.68]

The results of calculations of Poisson’s ratio change with concentration of inclusions are shown in Fig. 6.20. This is the only characteristic which may

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6.17  Tension/compression creep function JE (creep curve divided by the stress) of composite made from matrix filled by different volume percentages of absolutely rigid spherical particles; τEcr is the strain retardance time in the creep at unidirectional tension/compression. Here EM∞ / EM0 = 0.6 and EM0 is the instantaneous Young's modulus of the matrix.

6.18  Dependencies of the effective storage modulus 〈E´〉 (real part of the complex modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and EM0 is the instantaneous shear modulus of the matrix. © Woodhead Publishing Limited, 2011

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6.19  Dependencies of the effective loss modulus 〈E”〉 (imaginary part of the complex modulus) of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6 and EM0 is the instantaneous shear modulus of the matrix.

6.20  Change of Poisson's ratio of composite made from polymeric matrix (epoxy) filled by glass solid microspheres (GI /GM = 25), by micro or nanospheres made from material withGI /GM = 50 (MgO, low grades of carbon, silicon carbide, etc.) or by absolutely rigid micro or nanospheres (GI /GM → ∞) as function of the volume concentration c of According to Eq. 6.68, and Eq. 6.14 and 6.63–6.64 inclusions. according to Eq. 6.68 and derived for small concentrations; Eq. 6.9 and 6.65–6.66 derived for medium concentrations; in both cases Eq. 6.47–6.50 are also used.

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change non-monotonically with fillers concentration. Results of calculations of viscoelastic Poisson’s effect are shown in Fig. 6.21 and 6.22. It is interesting that the frequency dependence of the imaginary part of the Poisson’s ratio is practically independent of fillers concentration.

6.21  Dependencies of the real part 〈ν´〉 of the effective complex Poisson's ratio of polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6.

6.22  Dependencies of the effective imaginary part 〈ν”〉 of the complex Poisson's ratio of composite made from polymeric matrix filled by rigid spheres with volume concentration c on frequency ω (made dimensionless by multiplication by the retardance time in the creep τEcr at tension). Here EM∞ / EM0 = 0.6. © Woodhead Publishing Limited, 2011

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6.4

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Viscoelasticity of polymers filled by platelet-shape nanoparticles

Let’s consider two cases of platelets orientation in the matrix: parallel and random. Bulk modulus is calculated according to the same formula (Eq. 6.47) as for spherical inclusions and it is independent of inclusions orientation. Along the plane of inclusions, the Young’s modulus can be roughly calculated on the basis of the primitive model ‘rectangular parallelepiped inclusion in rectangular parallelepiped cell’, neglecting Poisson’s effect. Before applying it to platelet inclusion, let’s verify the model on cubical inclusions arranged in cubical lattice. For simplicity of the verification let’s consider inclusions as absolutely rigid. The upper estimation of Young’s modulus for this particular case gives:

[6.69]

The lowest estimation of Young’s modulus for this particular case gives

[6.70]

The results of calculations according to Eq. 6.69 and 6.70 are compared in Fig. 6.23 with previous calculations according to Vanin’s formula (Eq. 6.66) for

6.23  Dependencies of the effective Young's modulus of composite on concentration of absolutely rigid inclusions; cubic inclusions, upper estimation according to Eq. 6.69, cubic inclusions, lowest estimation according to Eq. 6.70, spherical inclusions according to Eq. 6.66. © Woodhead Publishing Limited, 2011

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spherical inclusions. The results for spherical inclusions are very close to the results of the lowest estimation using the model described above. Calculations for the platelet inclusions were done according to the lowest estimate for the case of cubic lattice of platelet inclusions and for the case of the lattice in which the individual cell has the shape similar to inclusion. For the cubical lattice of platelet inclusions oriented parallel to each other, the formula for the effective Young’s modulus is:



[6.71]

Here κ is the ratio of thickness of the plate to its size in plane. Inversed ratio is called aspect ratio. Dependence of the effective modulus on the ratio of Young’s moduli of inclusion and matrix for platelets with different aspect ratio is shown in Fig. 6.24. It is following from these data that the use of majority platelet fillers for polymer matrices (ratio of moduli 25–50) is practically equivalent to the use of absolutely rigid platelets. For this case the formula (Eq. 6.71) degenerates into:



[6.72]

6.24  Dependence of the effective Young's modulus 〈E1〉 in the direction parallel to the plane of the platelet inclusions on the ratio of the Young's moduli of inclusion to the matrix and on the aspect ratio of inclusions.

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In the case of the lattice having cells with a shape similar to the shape of inclusions, the effective Young’s modulus is calculated according the same formula (Eq. 6.70) as for cubical inclusions, i.e. the result is independent of the aspect ratio and effectiveness of the reinforcement is much lower than in the case of the cubic lattice. In reality the random package of the inclusions in the matrix can be considered as some intermediate case between these two types of lattices. It is necessary to mention that the cubic lattice for the platelets with a high aspect ratio can be realized only at small volumetric concentrations c ≤ κ. Increase of the aspect ratio leads to higher effectiveness of the reinforcement but in the very high aspect ratios the problem of keeping plane shape of the inclusions is encountered. Due to insufficient bending and torsion stiffness of very thin platelet inclusions, their shape can be distorted by matrix flow during the technological process, by shrinkage of matrix, etc. Effectiveness of the reinforcement by distorted inclusions is much lower than the effectiveness of ideally plane platelets. For randomly oriented platelet inclusions it is possible to derive the formula for the effective Young’s modulus:



[6.73]

which for the big ratios of Young’s moduli degenerates into:



[6.74]

This formula is valid only for the small concentration of inclusions, especially if the aspect ratio is big. The high theoretical effectiveness of reinforcement by platelet inclusions mentioned in many works is obtained for the extrapolation of the aspect ratio to infinity. For the finite aspect ratio, the effectiveness is not so high. Viscoelasticity of platelet inclusions reinforced material is mostly determined by viscoelasticity of polymer matrix as it follows from the formulae derived for absolutely rigid inclusions.

6.5

Viscoelasticity of polymers filled by nanofibers

There are several models for describing finite length fiber-reinforced materials. None of these models is rigorous and they are based on the use of some simplified hypotheses. One of the best models made by Russel19 is based on three

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hypotheses: that the concentration of short fibers is small; the shape of the fibers is approximated by prolate ellipsoid; and all fibers are oriented parallel with each other. The final formulae are bulky (the formula for longitudinal Young’s modulus can be found in Eq. 6.14). For the prediction of viscoelastic behavior, the simplified formula for the case of much stiffer inclusion than the matrix is used here:



[6.75]

Here κ is the ratio of the diameter of short fiber to its length. Inversed ratio (length to diameter) is called ‘aspect ratio’. Nevertheless, the more general formula was used in numerical calculations of the effect of the presence of aligned short fibers. The results of calculations are presented in Fig. 6.25 and 6.26. The qualitative effect of the relative stiffness of inclusions for the case of short fibers (Fig. 6.25) is the same as for platelet inclusions (Fig. 6.24). The effective longitudinal modulus depends on the relative stiffness nonlinearly and it approaches the asymptote corresponding to absolutely rigid short inclusions. The picture is dramatically changed only for the case of straight continuous fibers. Dependence of the effective longitudinal modulus on the aspect ratio is also nonlinear (Fig. 6.26).

6.6

Viscoelasticity of polymers filled by buckyballs and nanotubes

Hollow fillers are used widely in composite and nanocomposite technology providing light weight, good thermal insulation properties, better resistance to dynamic loads, etc. Syntactic foams use hollow microspheres/microballoons; hollow fibers have been used for reinforcing from ancient time; hollow nanospheres, called buckyballs, have been used for the last couple of decades. Other hollow nano-size objects called fullerenes are used also. The biggest number of investigations in nanofillers is done in nanotubes. Syntactic foams are used widely in many industries such as electrical machinery and shipbuilding. The behavior of solid glass spheres in polymeric matrices and especially carbon spherical particles under hydrostatic pressure is close to the behavior of absolutely rigid inclusions. As a result, hollow glass spheres having a half or less of the mass of solid spheres give almost the same reinforcing effect (Fig. 6.27). Even glass hollow microspheres with 10% of the mass of a solid microsphere still have a reinforcing effect. For carbon hollow nanospheres this ‘quasi-neutral’ limit is much lower. Only microspheres with very thin walls act as voids. Similar conclusions can be done with respect to the effective Young’s modulus (Fig. 6.28).

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6.25  Dependence of the effective Young's modulus 〈E1〉 of the material made from matrix with Young's modulus EM and parallel fibrous inclusions having Young's modulus EI on the relative stiffness for different aspect ratio (length to diameter). Calculations are based on Russel formulae.14, 19 These results and the following discussion were presented in references 20 and 21. Volume concentration of inclusions is 5% (a) and 10% (b).

For continuous straight fibers, replacement of solid fibers by hollow fibers does not change the longitudinal effective Young’s modulus if the mass concentration of fibers is the same. For short hollow fibers the result of such replacement is different (see Fig. 6.29).

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6.26  Dependence of the effective Young's modulus 〈E1〉 of the material made from matrix with Young's modulus EM and parallel fibrous inclusions having Young's modulus EI on the aspect ratio (length to diameter) for inclusions having different relative stiffness. Calculations are based on Russel formulae14, 19 and were presented in references 20 and 21. Volume concentration of inclusions is 5%. The asymptotic line for the case EI /EM = 200 is shown for comparison.

This phenomenon can be understood if we return to the results of calculations shown in Fig. 6.25. If the solid fiber is replaced by two hollow fibers with the same total mass, their effective stiffness drops twice but it is still sufficiently high and the effect of reinforcement by one short hollow fiber is comparable to the effect of reinforcement by one solid fiber. This is due to the significant nonlinearity of the curves in Fig. 6.25. However, because the number of fibers and their volume concentration in a nanocomposite is doubled, the total reinforcing effect is almost doubled. It is the reason why nanotubes are more promising than solid nanofibers. When the thickness of the walls of fibers is decreased more, the effect is decreased. Moreover, there is the optimal thickness of the walls of nanotubes providing the maximal reinforcing effect. The real optimum will be shifted toward more thick walls than is shown in Fig. 6.29 because too thin nanotubes behave not as hollow rods but as shells and can be deformed more easily and by different ways (local wall bending or buckling, etc.).

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6.27  Dependence of the effective bulk modulus 〈K〉 of material made from matrix with bulk modulus KM and hollow spherical inclusions on the volume concentration of inclusions. Here m = 1 – (r0 /r1)3 is the volume concentration of the solid phase in single inclusion. Calculations are made for vinyl-ester matrix and glass microballoons.

6.28  Dependence of the effective Young's modulus 〈E〉 of material made from matrix with Young's modulus EM and hollow spherical inclusions on the volume concentration of inclusions. Here m = 1 – (r0 /r1)3 is the volume concentration of the solid phase in single inclusion. Calculations are made for vinyl-ester matrix and glass microballoons.

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6.29  Dependence of the effective longitudinal Young's modulus 〈E1〉 of short hollow fibers with the same mass concentration as solid fibers on the concentration of solid phase in the cross-section of the fiber m = 1 – (r0 /r1)2. Initial volume concentration of solid fibers and aspect ratio are shown in the figure. Other values used in calculations: EI / EM = 300; νI = 0.25; νM = 0.35.

6.7

Viscoelasticity of nanoporous polymers

In the case of spherical voids Eq. 6.47 degenerates into: [6.76]

The corresponding viscoelastic operator can be found by replacing Poisson’s ratio by Eq. 6.44 and by using Eq. 6.20); the result has the following form:



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[6.77]

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In the case of spherical voids Eq. 6.54 degenerates into: [6.78] Vanin’s formula (Eq. 6.65) degenerates into:

[6.79] Eq. 6.63 degenerates into:

[6.80] The results of calculations of the effective elastic properties are shown in Fig. 6.30–6.33. It is necessary to mention that at concentrations of voids of 40–50% their shape begins to change and after 60% converts to a polyhedron,22 for which all these formulae are incorrect. Moreover, due to some simplifications

6.30  Relative drop of the effective bulk modulus 〈K〉 of porous material as a function of the volume concentration c of spherical voids. Calculations are made for different values of matrix Poisson's ratio νM.

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6.31  Relative drop of the shear modulus of porous material as a function of the volume concentration c of spherical voids: according to Eq. 6.14 and 6.78 derived for small concentrations; according to Eq. 6.9 and Eq. 6.79, 6.68 and 6.76 derived for medium concentrations.

6.32  Relative drop of the Young's modulus of porous material as a function of the volume concentration c of spherical voids: according to Eq. 6.14 and Eq. 6.78, 6.62 and 6.76 derived for small concentrations; according to Eq. 6.9 and Eq. 6.79, derived for medium concentrations; according to Eq. 6.80.

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6.33  Change of Poisson's ratio of porous material as a function of the volume concentration c of spherical voids: according to Eq. 6.14 and Eq. 6.68, 6.78 and 6.76 derived for small concentrations; according to Eq. 6.9 and Eq. 6.68, 6.79 and 6.76 derived for medium concentrations; according to Eq. 6.68, 6.80 and 6.76.

used in the derivation of these formulae, they become incorrect, giving zero or negative shear and Young’s moduli values for volume concentrations of pores over 50–55% instead of asymptotic approaching zero at c = 1. The results of the bulk creep calculation for porous media are shown in Fig. 6.34. It is interesting to compare it with similar calculations for the bulk creep of the medium with absolutely rigid inclusions (Fig. 6.9). With increase of void concentration the instantaneous bulk modulus is decreasing and the instantaneous strain is increasing (the opposite to the rigid inclusion case). However, the creep is increasing in both cases. It is happening due to the same reason – shear stress concentration near inclusions.

6.8

Viscoelasticity of fibrous composites with nano-filled matrices

A matrix filled by nanoparticles is not only stiffer and stronger but is also working slightly differently in its mechanism of load transmission from fiber to fiber in the composites compared with conventional reinforcing fibers. This effect is especially pronounced in the dynamic loading of composites. The example of an experimental study23, 24 of 3D fabric reinforced plastic filled by SiC nanoparticles is shown in Fig. 6.35. It shows that the dissipation of energy by material is significantly increasing due to the presence of nanoparticles in the

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6.34  Effective bulk creep function 〈JK〉 (volume creep curve divided by the hydrostatic stress) of material containing different volume percentages of spherical voids; τEcr is the strain retardance time in the creep at unidirectional tension/compression. Here EM∞ / EM0 = 0.6, 〈K0〉 is the instantaneous bulk modulus of composite and KM0 is the instantaneous bulk modulus of matrix.

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6.35  Dependencies of tangent of the angle of mechanical losses at glass transition temperature on weight concentrations of SiC nanoparticles in 3D fabric reinforced vinyl ester system in different fiber directions: (a) 90°, (b) 45°, (c) 0°.

polymer matrix. In Sun’s works (see for example reference 25) it was shown that the adding of nanoclay particles to polymeric matrix increases longitudinal compression strength by several tens of percent, probably due to additional support of fibers in matrix in buckling. Adding nanoparticles to the matrix of the syntactic foams also improves their properties,26 probably due to a different mechanism of the load transfer from one microballoon to another one. There are more investigation results described in the literature.27–40 However, due to complexity of the system it is necessary to accumulate more experimental data before making definite qualitative conclusions.

6.9

Concluding remarks

Despite the enthusiasm of scientists working on synthesis of new nanoparticles based on their success in achieving much higher stiffness and especially strength of nanoparticles, composites industry experts remain skeptical about the possibility of the realization of such improvements in composites. Particles and short fibers do not improve properties in comparison with matrix properties even by one decimal order despite the difference in properties of matrix and fillers which achieved two or three decimal orders. The behavior of the majority of conventional fillers, as it is shown in multiple examples in this chapter, is very close to the behavior of absolutely rigid particles. Further improvement of fillers’ mechanical properties going from micro- to nano-scale gives in many cases small gains in composites’ properties only. Nevertheless, it is not related to the physical and chemical properties. Significant effect is possible if continuous nanofibers can be produced on an industrial scale. In this case the increase in longitudinal mechanical properties of nanocomposites in comparison with conventional composites can be expected in the order of several tens of percent.

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Viscoelasticity of nanocomposites is determined first of all by the interaction of the nanoparticles and the viscoelasticity of the polymeric matrix. It can never be completely suppressed by using even ideally elastic or absolutely rigid inclusions if all sizes of these inclusions are finite ones.

6.10 Notation c

volumetric concentration of inclusions m volumetric concentration of solid phase in single hollow inclusion t time E Young’s modulus F functional G shear modulus H thickness I Abel operator J compliance K bulk modulus L, M, W viscoelastic operators N number of inclusions R radius S surface area V volume α, β, γ, λ, µ parameters of viscoelastic operators αij tensor of coefficients of linear thermal expansion εij strain tensor σij stress tensor ν Poisson’s ratio η viscosity φ diameter ξ form factor τ time before the moment t χ defined by Equation 6.53 κ inversed aspect ratio ω angular frequency Π,Υ,Λ viscoelastic operators Φ(t – τ), kernels of operators Ψ(t – τ)

Γ ∋ Θ Ξ

gamma-function Rabotnov operator Rzhanitsyn operator Koltunov operator

Indices Upper: × * ′ ″ 0 Lower: M I 0 ∞ E G K

cr

upp low

1,2,3

Brackets 〈 〉

operator viscoelasticity complex number real part imaginary part free strains (shrinkage, physical transitions, etc.) matrix inclusion instantaneous value in creep or relaxation long time value in creep or relaxation related to tension/ compression related to shear related to volumetric deformation related to creep upper estimation lowest estimation related to the directions x1, x2, x3 or order number of element effective characteristics

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6.11 Acknowledgement Part of the material contained in this work was obtained due to the support of ONR Grant N00014-05-1-0532 and Dr Yapa Rajapakse, Solid Mechanics Program.

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