Time dependence of strain

Figure 11. Time dependence of strain at break silicone sealant at 72°F (22°C)...

Normally a logarithmic time scale is used to plot the creep curve, as shown in Figure 3.9b, so that the time dependence of strain after long periods can be included. If a material is linearly viscoelastic (Equation 3.17), then at any selected time each line in a family of creep curves (with equally spaced stress levels) should be offset along the strain axis by the same amount. Although this type of behavior maybe observed for plastics at low strains and short times, in most cases the behavior is nonlinear, as indicated in Figure 3.9c. 

The total time rate of change of extensional strain of the surface includes contributions due to the time dependence of strain at a material point, the time dependence of the orientation of the surface, and the motion of the surface with respect to the material particles instantaneously on it. If these effects are taken into account, then... 

The roles of stress and strain are reversed in a creep experiment stress is the disturbance and strain the response. In simple shear, a constant shear stress cxq is imposed and the time dependence of strain y(t) is recorded. In the creep recovery phase, the sample is unloaded (the shear stress is set to zero), and the strain at subsequent times is recorded. Because the stress is constant, the creep strain y t) would be a constant, y t) = Oq/G, for the Hookean solid and directly proportional to time, y t) = (oo/r))t, for the Newtonian liquid. In the recovery phase, the strain recoils immediately to zero for the solid and remains fixed at (cTo/v)ti for the liquid, ti being the time at which recovery began. 

In the course of tensile creep, the form of the time dependence of strain (as expressed by the stretch ratio X, for example) depends on the magnitude of tensile stress at high stresses." " Recovery is considerably more rapid than would be predicted from the Boltzmann superposition principle, as illustrated in Fig. 13-23 for polyisobutylene of high molecular weight. " The course of recovery is predicted successfully by the theory of Bernstein, Kearsley, and Zapas. 2 - 22 -pije stress-dependent recoverable steady-state compliance D = which is equal to Z) at low stresses, decreases with increasing Ot- This effect, moderate when the tensile strain e is defined as X — 1, is more pronounced when it is replaced by the Hencky strain, defined as In X. The stress dependence of steady-state compliance in shear will be discussed in Chapter 17. The reader is referred to the review by Petrie" for more details. 

In this section we consider a different experimental situation the case of creep. In a creep experiment a is maintained at a constant value and the time dependence of the strain is measured. Thus it is the exact inverse of the relaxation... 

The maximum temperature at which mild steel can be used is 550°C. Above this temperature the formation of iron oxides and rapid scaling makes the use of mild steels uneconomical. For equipment subjected to high loadings at elevated temperatures, it is not economical to use carbon steel in cases above 450°C because of its poor creep strength. (Creep strength is time-dependent, with strain occurring under stress.)... 

For a linear viscoelastic material in which the strain recovery may be regarded as the reversal of creep then the material behaviour may be represented by Fig. 2.49. Thus the time-dependent residual strain, Sr(t), may be expressed as... 

Time dependence Viscoelastic deformation is a transition type behavior that is characterized by the occurrence of both elastic strain and time-dependent flow. It is the time dependence of the mechanical properties of plastics that makes the behavior of these materials difficult to analyze by mathematical theory. 

So far we have employed in this discussion a critical shear stress as a criterion for fibre fracture. In Sect. 4 it will be shown that a critical shear strain or a maximum rotation of the chain axis is a more appropriate criterion when the time dependence of the strength is considered. 

The total contribution of the shear strain to the fibre strain is the sum of the purely or immediate elastic contribution involving the change in angle, A0e=0o- , occurring immediately upon loading of the fibre at f=0, and the time-dependent or viscoelastic and plastic contribution A0(f)=0(f)-0o [7-10]. According to the continuous chain model for the extension of polymer fibres, the time-dependent shear strain during creep can be written as... 

The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

The properties of a material must dictate the applications in which it will best perform its intended use. All materials made to date with polymerized sulphur show time-dependent stress-strain behaviour. The reversion to the brittle behaviour of orthorhombic sulphur is inevitable as the sulphur transforms from the metastable polymeric forms to the thermodynamically stable crystalline structure. The time-span involved of at most 15 months (to date) would indicate that no such materials should be used in applications dependent on the strain softening behaviour. Design should not be based on the stress-strain relationships observed at an age of a few days. Since the strength of these materials is maintained, however, uses based on strength as the only mechanical criterion would be reasonable. 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

Viscoelasticity is termed linear when the time-dependent compliance (strain/stress) of a material is independent of the magnitude of the applied stress. All materials have a linearity limit (see Table 9.2). 

According to results reported in the literature [1-13] if the shear stress is canceled out after steady-state conditions are reached, the time dependence of the recoverable deformation [er(t)—cre(t)/r ] is obtained where e(t) is the shear strain, a is the stress and r is the viscosity (2.6). The higher temperature, the greater the unrecoverable contribution to the shear deformation i.e. the viscous deformation. Figure 2.3 shows the effect of temperature on the strain. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

Models of mechanical behavior of tissues have been difficult to develop primarily because of the time dependence of the viscoelasticity. Analysis of viscoelastic behavior of even simple polymers at strains greater than a few percent is not accurate. In addition, most tissues undergo strains larger than a few percent, which makes the analysis require an understanding of the elongation behavior. In this chapter we focus on using modeling techniques to analyze the physical basis for determination of the tensile behavior of ECMs found in connective tissue. 

Time-dependent stress-strain behavior of the neat resins was studied using an Instron (Model 1122) tensile tester. Dog-boneshaped epoxy specimens were prepared in accordance to ASTM D1708-66. Strain-rate used was 5x 10 5 s 1. 

Very important phenomena in polymer behaviour, such as viscoelasticity, stress, strain, volume and enthalpy relaxation, ageing, etc., are characterised by time-dependence of the polymer properties. 

FIG. 13.17 Left time dependent reduced stress, Maxwell element after applying a constant strain vs. log reduced time, log (t/t). Right time dependent reduced strain, of a Voigt-Kelvin... 

Al-Saidi LF, Mortensen K, Almdal K (2003) Environmental stress cracking resistance behaviour of polycarbonate in different chemicals by determination of the time-dependence of stress at constant strains. Polym Degradat Stabil 82(3) 451—461... 

In discussing shear deformation, it is convenient to distinguish between the initial elastic and viscoelastic response of the polymer to the applied load and the subsequent time-dependent response. However, the distinction is somewhat arbitrary and is not as fundamental as that between elastic volume response and crazing. Viscoelastic shear deformation continues throughout the period under load. The observed time-dependence of lateral strain reflects both generalized viscoelastic relaxation and shear band formation. Since crazing consists simply of displacement in the tensile stress direction, it makes no contribution to lateral strain therefore —e specifically measures deformation by shear processes. 

In order to better understand the shear modulus (and the time dependence of the shear modulus) we shall describe it mathematically. Elastic behavior is the instantaneous response to a stress as shown in Figure 4.11. When a stress is imposed, the material deforms instantaneously. When the stress is removed the material relaxes completely. The amount of deformation (strain) is related to the stress (pressure) by the shear modulus according to ... 

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