Linear viscoelasticity creep compliance function

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

In Chapter 4 we introduced linear viscoelasticity. In this scheme, observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The definition of the creep compliance function J t), which is given as the ratio of creep strain e t) to the constant stress o, may be recalled as... 

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 ... 

In the course of extensive studies of the creep and recovery behaviour of textile fibres already referred to, Leaderman [13] became one of the first to appreciate that the simple assumptions of linear viscoelasticity might not hold even at small strains. For nylon and cellulosic fibres he discovered that although the creep and recovery curves may be coincident at a given level of stress - a phenomenon associated with linear viscoelasticity (Section 4.2.1) the creep compliance plots indicated a softening of the material as stress increased, except at the shortest times (Figure 10.4). Thus, the creep compliance function is non-linear and of the... 

Schapery [16, 17] has used the theory of the thermodynamics of irreversible processes to produce a model that may be viewed as a further extension of Leaderman s. Schapery continued Leaderman s technique of replacing the stress by a function of stress /(a) in the superposition integral, but also replaced time by a function of time, the reduced time ip. The material is assumed to be linear viscoelastic at small strains, with a creep compliance function of the form [17]... 

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...

The notation used to describe the contacts is shown in Figure 1. P t) is the time dependent applied load, S P,t) the deformation, a(P,t) the contact radius, and R and Ri the radii of curvature of the two bodies at the point of contact. We consider only flat substrates so that R R and R2 = >. Each elastic material is described by its Young modulus E, Poisson ratio v, and is assumed to be isotopic so that the shear modulus is G = Ejl + v). Viscoelastic materials are assumed to be linear with stress relaxation functions E t) and creep compliance functions J t), All properties are assumed to be independent of depth. 

In Chapter 5, we introduced linear viscoelasticity. In this scheme, the observed creep or stress relaxation behaviour can be viewed as the defining characteristic of the material. The creep compliance function - the ratio of creep strain e t) to the constant stress a - is a function of time only and is denoted as J t). Similarly and necessarily, the stress relaxation modulus, the ratio of stress to the constant strain, is the function G(r). Any system in which these two conditions do not apply is non-linear. Then, the many useful and elegant properties associated with the linear theory, notably the Boltzmann superposition principle, no longer apply and theories to predict stress or strain are approximations that must be supported by experiment. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

Since the linear viscoelasticity of a material is described with a material function G(t), any experiment which gives full information on G(t) is sufficient it is not necessary to give the stresses corresponding to various strain histories. We will restrict the discussion to incompressible isotropic materials. In this case, different types of deformation such as elongation and shear give equivalent information in the range of linear viscoelasticity. Several types of experiments measure relaxation modulus, creep compliance, complex modulus etc which are equivalent to the relaxation modulus (1). 

Measurement of C requires more sophisticated and expensive rheometers and more involved experimental procedures. It must be remembered that experiments have to he carried out below the critical strain value (see Sec II), or in [he region of linear viscoelastic behavior. This region is determined by measuring the complex modulus G as a function of the applied strain at a constant oscillation frequency (usually 1 Hz). Up to 7, G does not vary with the strain above Yr, G tends to drop. The evaluation of oscillatory parameters is more often restricted to product formulation studies and research. However, a controlled-fall penetrometer may be used to compare the degree of elasticity between different samples. Creep compliance and creep relaxation experiments may be obtained by means of this type of device. In fact, a penetrometer may be the only way to assess viscoeIa.sticity when the sample does not adhere to solid surfaces, or adheres too well, or cures to become a solid or semisolid. This is the case of many dental products such as fillings, impression putties, sealants, and cements. 

With constant stress, G t) = Gy, where creep strain y t) is constant [y(t) = Gq/G] for a Hookean solid. It would be directly proportional to time for a Newtonian liquid [(y(0 = Go/r])t]. Here t is the initial time at which recovery of the viscoelastic material begins. For a viscoelastic fluid, when stress is applied, there is a period of creep that is followed by a period of recovery. For such liquids, strains return back toward zero and finally reach an equilibrium at some smaller total strain. For the viscoelastic liquid in the creep phase, the strain starts at some small value, but builds up rapidly at a decreasing rate until a steady state is reached. After that the strain simply increases linearly with time. During this linear range, the ratio of shear strain to shear stress is a function of time alone. This is shear creep compliance, J t) The equation of shear creep compliance can be written as follows ... 

Two test cases are used to validate the linear viscoelastic analysis capability implemented in the present finite-element program named NOVA. In the first case, the tensile creep strain in a single eight-noded quadrilateral element was computed for both the plane-stress and plane-strain cases using the program NOVA. The results were then compared to the analytical solution for the plane-strain case presented in Reference 49. A uniform uniaxial tensile load of 13.79 MPa was applied on the test specimen. A three-parameter solid model was used to represent the tensile compliance of the adhesive. The Poisson s ratio was assumed to remain constant with time. The following time-dependent functions were used in Reference 49 to represent the tensile compliance for FM-73M at 72 °C ... 

The linear viscoelastic properties G(t)md J t) are closely related. Both the stress-relaxation modulus and the creep compliance are manifestations of the same dynamic processes at the molecular level in the liquid at equilibrium, and they are closely related. It is not the simple reciprocal relationship G t) = 1/J t) that applies to Newtonian liquids and Hookean solids. They are related through an integral equation obtained by means of the Boltzmann superposition principle [1], a link between such linear response functions. An example of such a relationship is given below. 

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. 

We can interrelate the relaxation and creep functions and the dynamic moduli and compliances via Boltzmann s superposition principle which states that all effects of past history can be considered independently in their contributions to the present state of the (linear) viscoelastic material. Thus, if one subjects the material to, say, incremental strains yo — 0), (y — y ),.. . , (y y -1) at times , ... 

A number of small strain experiments are used in rheology. Some of the more common techniques are stress relaxation, creep, and sinusoidal oscillations. In the linear viscoelastic region all small strain experiments must be related to one another through G(t), as indicated by the basic constitutive equation, eq. 3.2.7, or through M(t), eq. 3.2.4. Different experimental methods are used because they may be more convenient or better suited for a particular material or because they provide data over a particular time range. Furthermore, it is often not easy to transform results from one type of linear viscoelastic experiment to another. For example, transformation from the creep compliance J t) to the stress relaxation modulus G(t) is generally difficult. Thus both functions are often measured. 

A corner-stone of the theory of linear viscoelasticity is the Boltzmann superposition principle. It allows the state of stress or strain in a viscoelastic body to be determined from knowledge of its entire deformation history. The basic assumption is that during viscoelastic deformation in which the applied stress is varied, the overall deformation can be determined from the algebraic sum of strains due to each loading step. Before the use of the principle can be demonstrated it is necessary, first of all, to define a parameter known as the creep compliance J(t) which is a function only of time. It allows the strain after a given time e(t) to be related to the applied stress or for a linear viscoelastic material since... 

Thus, creep recovery leads to the same material function as creep (as long as we are in the linear viscoelastic regime). The ultimate recoil or recoverable shear is the recovered shear in the limit of long time when recoil has ended, and this quantity is directly related to the steady-state compliance. 

During the creep of PET and PpPTA fibres it has been observed that the sonic compliance decreases linearly with the creep strain, implying that the orientation distribution contracts [ 56,57]. Thus, the rotation of the chain axes during creep is caused by viscoelastic shear deformation. Hence, for a creep stress larger than the yield stress, Oy,the orientation angle is a decreasing function of the time. Consequently, we can write for the viscoelastic extension of the fibre... 

In this report we present a model for the primary creep of well-oriented aramid fibres. It has been shown for well-oriented fibres of PET, cellulose and poly-(p-phenylene terephthalamide), abbreviated here as PpPTA, that the dynamic compliance, S, is a linear function of the second moment of the orientation distribution of the chains / / /2/ /3/. By measuring S during creep and relaxation of a fibre, the changes in the orientation distribution can be followed. As shown here, such an experiment offers a valuable tool for the investigation of the viscoelasticity in polymer fibres. 

As the stress-strain linearity limit of most thermoplastics and their blends is very low, nonlinear viscoelastic behavior of heterogeneous blends needs to be considered in most cases. The nonlinearity is at least partly ascribed to the fact that the strain-induced expansion of materials with Poisson s ratio smaller than 0.5 markedly enhances the fractional free volume (240). Consequently, the retardation times are perpetually shortened in the course of a tensile creep in proportion to the achieved strain. Thus, knowledge of creep behavior over appropriate intervals of time and stress is of great practical importance. The handling and storage of the compliance curves D (t,a) in a graphical form is impractical, so numerous empirical functions have been proposed (241), eg. 

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