Linear viscoelasticity material functions

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

For positive exponent values, the symbol m with m > 0 is used. The spectrum has the same format as in Eq. 8-1, H X) = H0(X/X0)m, however, the positive exponent results in a completely different behavior. One important difference is that the upper limit of the spectrum, 2U, has to be finite in order to avoid divergence of the linear viscoelastic material functions. This prevents the use of approximate solutions of the above type, Eqs. 8-2 to 8-4. 

Fig. 6. Chart showing the paths to interrelate the linear viscoelastic material functions. Equation numbers refer to Chapter 3 of Ferry s book (9) unless otherwise indicated. Determination of G (co) from G"(co) and J (o ) from J"(a>) and vice versa comes from the Kramer-Kronig relation and is discussed in Tschoegl (10).

S. W. Park and R. A. Schapery, Methods of Interconversion between Linear Viscoelastic material Functions. Part. I.-A Numerical Method Based on Prony Series Int. J. Solids and Struct. 36, 1653-1675 (1999). 

Marrucci G, Maffettone PL (1993) Liquid crystalline polymers. Pergamon, New York Mead DW (1994) Determination of molecular weight distributions of linear flexible polymers from linear viscoelastic material functions. J Rheol 38 1797-1827 Moldenaers P, Yanase H, Mewis J (1990) Effect of shear history on the theological behavior of lyotropic liquid crystals. Liq Cryst Polym 24(1990) 370-380 Muir MC, Porter RS (1989) Processing rheology of liquid taystal polymers a review. Mol Cryst Liq Cryst 169 83-95... 

D. W. Honerkamp, Numerical interconversion of linear viscoelastic material functions, J. Rheol (1994) 38, pp. 1769-1795... 

D. W. Mead, Determinations of Molecular Weight Distributions of Linear Flexible Polymers from Linear Viscoelastic Material Functions J. Rheol. 38, 1797-1827 (1994). 

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

Stresses in viscoelastic materials "remember" deformation prehistory and so are not an unambiguous function of instantaneous deformations however, they may be expressed by a functional. For a linear viscoelastic material, the relationship between stresses and deformations... 

As discussed earlier for a Hookean solid, stress is a linear function of strain, while for a Newtonian fluid, stress is a linear function of strain rate. The constants of proportionality in these cases are modulus and viscosity, respectively. However, for a viscoelastic material the modulus is not constant it varies with time and strain history at a given temperature. But for a linear viscoelastic material, modulus is a function of time only. This concept is embodied in the Boltzmann principle, which states that the effects of mechanical history of a sample are additive. In other words, the response of a linear viscoelastic material to a given load is independent of the response of the material to ary load previously on the material. Thus the Boltzmann principle has essentially two implications — stress is a linear function of strain, and the effects of different stresses are additive. 

The most commonly used model is the Boltzmann superposition principle, which proposes that for a linear viscoelastic material the entire loading history contributes to the strain response, and the latter is simply given by the algebraic sum of the strains due to each step in the load. The principle may be expressed as follows. If an equation for the strain is obtained as a function of time under a constant stress, then the modulus as a function of time may be expressed as... 

Buschmann DM. Numerical conversion of transient to harmonic response functions for linear viscoelastic materials. J Biomechanics 1997 30 197-202... 

This solution illustrates an important point for a linear viscoelastic material, it is possible to convert between the viscoelastic response functions, without Imowledge of the stress/strain/time differential equations to which th correspond. 

Generalization of Hooke s law shows that in the range of linear viscoelasticity, material parameters become a function of time ... 

According to CEN EN 12697-26 (2012), complex modulus, E, is defined as the relationship between stress and strain for a linear viscoelastic material submitted to a sinusoidal load wave form at time t, where applied stress o x sin (co x t) results in a strain e x sin (o) x (t - rf>)) that has a phase angle O, with respect to stress (co = angular speed, in radians per second). The amplitude of strain and the phase angle are functions of the loading frequency, f, and the test temperature, . 

The formulations of (47) and (55) have been criticized by Leonov [L9, Lll], among others, as not being tested for consistency with the second law of thermodynamics. For Newtonian fluids, such testing requires a positive shear viscosity. For a linear viscoelastic material, one may show that the relaxation modulus function must be always positive to satisfy the second law. The requirements for Eqs. (47) and (55) are not so clear. Leonov has sought to develop nonlinear viscoelastic rheological models based on thermodynamic arguments. 

The Viscoelastic Material Functions. In linear viscoelasticity, the moduli discussed for the elastic case can be recast as time- or fi equency-dependent functions. The same is true for the compliance functions that are discussed here. For simplicity, consider the shear modulus G which becomes G(t) or G (a>) in the case of the viscoelastic material. An important point here is that the viscoelastic modulus functions all exhibit time (frequency) dependence. Hence, one will have functions for K(t) and E(t) [or, eg, G t) and v i)] and these are required in the case of a three-dimensional strain or stress field. 

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

It can be shown that if a material is linear in the sense of Equation (10.1) then it is also linear in the sense of Equation (10.2), and vice versa J and G are mathematically related [1]. Thus, a linear material is one for which the creep compliance function or the stress relaxation modulus is a function of time only. When this is not the case, the material is non-linear. For example, the following simple forms are characteristic of non-linear viscoelastic materials ... 

The nonlinear constitutive law due to Schapery may be linearized by assuming that the nonlinearizing parameters 8 y d g2 have a value of unity. In addition, the stress-dependent part of the exponent in the definition of the shift function is set to zero. Consequently, the constitutive law reduces to the hereditary integral form commonly used to describe a linear viscoelastic material. 

If the material is subjected to a time-dependent strain, the situation becomes more complicated. However, in the case of a linear viscoelastic material (like many food products) the superposition principle can be applied the response of the stress to a strain increment is independent of the already existing strain. The effect of the strain as a function of time can therefore be integrated, and the generalized Hooke s law can be extended to describe the stress-strain behavior of linear viscoelastic materials relatively easily. 

Figure 2. Three-parameter model of linear viscoelastic material (a) Creep compliance function (t) (1 / oo) l - (1 - / p) exp(t / T) (b) Phase angle dbetween cyclic...

In a linear viscoelastic material, the moduli and compliances (t), G(t), D(t), and J t) (tensile and shear modulus and tensile and shear comphance respectively) are functions of t, although they remain independent of stress or strain. Development of constitutive equations for such materials is generally based on the idea that the effects of small increases in stress or strain are additive, which is known as the Boltzmann superposition principle [4, 10, 11]. A strain e(t) may be considered to result from a sum of step strains applied at time u and maintained for a time t-u[Eq.(31)j. 

Typical examples of tensile (isochronous) linear and nonlinear stress-strain diagrams for elastic and viscoelastic materials are shown in Fig, 10.1. For elastic materials, the response is time independent, so there is a single curve for multiple times and the nonlinearity is apparent as a deviation of the stress-strain response from linear. For linear viscoelastic materials, the isochronous response is linear, but the effective modulus decreases with time so that the stress-strain curves at different times are separated from one another. When a viscoelastic material behaves nonlinearly, the isochronous stress-strain curves begin to deviate from linearity at a certain stress level. Fig. 10.2 shows creep compliance data for an epoxy adhesive as a function of stress level for various time intervals after initial loading. 

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