Linear Creep compliance

Figure 3.12 shows that, at long times, the creep compliance is directly proportional to time for the polymers shown. For polystyrene (M = 600,000) at 100°C, the following values describe the linear portion of the datat ... 

Plot the creep compliance (cm /dyn) as a function of time using a logarithmic time scale. Would the curve show the upward curvature on a linear time scale ... 

Studies have been conducted on creep compliance tests in which paint films were subjected to tensile loads of 4-7 psi (27.2-47.6 x 10 N/m ) and to 6% ozone for 505 h. A typical result for a high-quality emulsion-base paint is shown in Figure 13-1. Creep compliance is reduced by exposure to 6% ozone. If the effect is linearly related to ozone concentration, we might expect the same reduction in creep compliance at 0.1-ppm ozone in 3 x 10 h, or some 30,(XX) yr. Thus, reduction in creep compliance is not viewed as having a serious ozone contribution. 

Fig. 4. Reduced creep compliance (with and without plastic flow term) for the linear array as a function of molecular weight...

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

We will first consider the parameters we are trying to model. Let us start with stress relaxation, where it is usual to describe properties in terms of a relaxation modulus, defined in Table 13-5 for tensile [ (r)] and shear [G(r)] experiments. The parameter used to describe the equivalent creep experiments are the tensile creep compliance [D(r)] and shear creep compliance [7(0]. It is important to realize that the modulus and the compliance are inversely related to one another for linear, tune-independent behavior, but this relationship no longer holds if the parameters depend on time. 

Consider two experiments carried out with identical samples of a viscoelastic material. In experiment (a) the sample is subjected to a stress CT for a time t. The resulting strain at f is ei, and the creep compliance measured at that time is D t) = e la. ln experiment (b) a sample is stressed to a level CT2 such that strain i is achieved immediately. The stress is then gradually decreased so that the strain remains at f for time t (i.e., the sample does not creep further). The stress on the material at time t will be a-i, and the corresponding relaxation modulus will be y 2(t) = CT3/C1. In measurements of this type, it can be expected that az> 0 > ct and Y t) (D(r)) , as indicated in Eq. (11-14). G(t) and Y t) are obtained directly only from stress relaxation measurements, while D(t) and J(t) require creep experiments for their direct observation. Tliese various parameters can be related in the linear viscoelastic region described in Section 11.5.2. 

Creep-compliance studies conducted in the linear viscoelastic range also provide valuable information on the viscoelastic behavior of foods (Sherman, 1970 Rao, 1992). The existence of linear viscoelastic range may also be determined from torque-sweep dynamic rheological experiments. The creep-compliance curves obtained at all values of applied stresses in linear viscoelastic range should superimpose on each other. In a creep experiment, an undeformed sample is suddenly subjected to a constant shearing stress, Oc. As shown in Figure 3 1, the strain (y) will increase with time and approach a steady state where the strain rate is constant. The data are analyzed in terms of creep-compliance, defined by the relation ... 

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). 

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. 

The creep compliance of the Maxwell model is linear in time,... 

The strain of a viscoelastic liquid in creep is shown as the top curve in Fig. 7.24. The slope in Fig. 7.24 at long times is the shear rate 7 and the viscosity is therefore determined using Newton s law of viscosity [Eq. (7.100)]. For liquids, the long-time creep compliance is linear in time and its form is reminiscent of the Maxwell model [Eq. (7.125)] -----------... 

Two steady states are recognized for the long-time creep compliance of materials. Either the sample is a solid and the compliance becomes time independent or the sample is a liquid and the compliance becomes linear in -time. Once steady state has been achieved in creep, the stress can be removed (a = 0) and the elastic recoil, called creep recovery, can be measured. Recovery strain is defined as 7r(0 s 7(0) — 7(0 for t > 0, where t is defined to be zero at the start of recovery. The recoverable compliance is defined as the ratio of the time-dependent recovery strain 7r(0 and the initially applied stress a, where both 7r and t are now defined to be zero at the start of recovery ... 

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. 

Figure 2. Creep compliance vs. time for linear polyurethane encapsulants at constant load. See Figure 1 for nomenclature.

In Figure 5.8d an intermediate behavior, called viscoelastic, is depicted such a relation is often called a creep curve, and the time-dependent value of the strain over the stress applied is called creep compliance. On application of the stress, the material at first deforms elastically, i.e., instantaneously, but then it starts to deform with time. After some time the material thus exhibits flow for some materials, the strain can even linearly increase with time (as depicted). When the stress is released, the material instantaneously loses some of it deformation (which is called elastic recovery), and then the deformation decreases ever slower (delayed elasticity), until a constant value is obtained. Part of the deformation is thus permanent and viscous. The material has some memory of its original shape but tends to forget more of it as time passes. 

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). 

Equation (d) then gives an expression, for the strain at time t in a body whose creep compliance is given by equation (a) when it is subjected to a linear... 

Abstract Based on the theory of irreversible process thermodynamics, non-linear stress-strain-temperature equations are derived, together with an expression for time-temperature equivalence. In addition, an equation of shift factor for time-temperature equivalence is also obtained. The parameters in the equations are experimentally determined and the main curves for creep compliance and cohesion of TOP granite are obtained by a series of creep tests. As a result, it is proved that both deformation and strength of the TOP granite follow the time-temperature equivalent principle. 

There is linear viscoelastic behaviour in the stress region where the isochronous stress-strain curve is linear (to within 5%). The creep compliance /( ), defined by Eq. (7.4), is independent of stress. However, above this stress region (stresses >1 MPa for the data in Fig. 7.7 for a time of 1 year) there is non-linear viscoelastic behaviour and the creep compliance becomes stress dependent... 

The rule for linearly viscoelastic design is Replace 1/E in the elastic formula by the creep compliance J(t), to obtain the time-dependent deflection. Applying this rule, the cantilever arm deflection is given by... 

Assuming linearity, i.e. that each part of the strain is proportional to the applied stress, a time-dependent creep compliance /(t) can be defined as... 

A straight rod of polymer is 10 mm in diameter (2r) and 1 m long. The pol5aner behaves in a linear viscoelastic manner with a tensile creep compliance that can be well approximated by J t) = (2 — ) GPa ,... 

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. 

The most common technique employed to date has been that of creep in uniaxial tension. It was shown above that with the inclusion of lateral strain measurements this is a powerful technique giving access to up to 6 independent creep compliance functions. This is more than for any other known method. It further has the overwhelming advantage over many methods, such as say torsional or flexural creep, that the stress is sensibly uniform over the working volume of the specimen. This advantage is paramount in studies of materials displaying non-linear behaviour in creep since analysis of the non-uniform stress situation in non-linear systems is not well developed. Attempts to overcome the non-uniform stress situation in torsion, by recourse to, say, torsion of thin walled tubes, lead to severe difSculties in specimen preparation in oriented materials, when anisotropy of behaviour is to be studied. 

Sel Seltzer, R., Mai, Y.-W. Depth sensing indentation of linear viscoelastic-plastic solids A simple method to determine creep compliance. Eng. Fract. Mech. 75 (2008) 4852-4862. 

A spherical vessel is moulded from a pofymer whose one-month tensile creep compliance D(1 month) is 2 GPa . The vessel is of diameter 400 mm and wall thickness S mm. A constant internal pressure is applied to the vessel, giving rise to a tensile stress 1.6 MPa acting uniform in all directions in the plane of the vessel wall. Find (a) the change in diameter, and (b) the change in wall thickness, after 1 month of pressurization, assuming the polymer to be linearly viscoelastic with constant Poisson ratio v = 0.41. 

A straight rod of solid polymer is of length 1 m and diameter 10 mm. The polymer is linearly viscoelastic with a tensile creep compliance... 

Use the shear creep data in Figure 4.4, together with the method of time-temperature superposition, to estimate the shear creep compliance for linear polyethylene at 20°C and a creep time 10 s. Ust the assumptions that you make in this long extrapx>lation of the creep data. 

Show that, for a linear viscoelastic beam under a constant bending moment Mg, the curvature of the beam 1/R (R is the radius of curvature) increases with time t according to an equation similar to that for a linear elastic beam, except that the reciprocal tensile creep compliance (or creep modulus ) takes the place of Young s modulus ... 

Use the integral form of the Boltzmann superposition principle to show that the creep compliance and stress relaxation modulus of any linear viscoelastic material are related through... 

The flexural creep stiffness, or the flexural creep compliance, describes the low-temperature stress-strain time response of bituminous binder at the test temperature within the range of linear viscoelastic response. 

Linear expansion coefficient Glass transition temperature Softening temperature Melting temperature Creep, compliance... 

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