Delayed elastic strain

The measurements reported by Bach (56) apparently were not under constant temperature conditions. Strain recovery after loading in the plasticized state is small. The longer the loading period the smaller the recoverable strain. This suggests plastic flow under load and a conversion of delayed elastic strain into an irreversible deformation. 

Experimentally, as indicated in Fig. 12.13, we find that D/Dq depends on the shear stress at the wall xw (a flow variable) and the molecular weight distribution (MWD) (a structural variable) (22). The length-to-diameter ratio of the capillary (a geometric variable) also influences D/Dq. The swelling ratio at constant xw decreases exponentially with increasing L/Dq and becomes constant for L/Dq > 30. The reason for this decrease can be explained qualitatively as follows. Extrudate swelling is related to the ability of polymer melts and solutions to undergo delayed elastic strain recovery, as discussed in... 

The usual way in which the deformation changes with time, has been dealt with in 6.1. The best representation appeared to be a Maxwell element with a Kelvin-Voigt element in series the deformation is then composed of three components an immediate elastic strain, which recovers spontaneously after removal of the load, a delayed elastic strain which gradually recovers, and a permanent strain. Moreover, we noticed that a single retardation time (a single Kelvin-Voigt element) is not sufficient we need to introduce a spectrum ... 

A) Schematic illustration of strain when a viscoelastic material is subjected to a constant stress, a, at time /g. Total strain, a, is the sum of the instantaneous elastic (e ), delayed elastic (fip), and viscous (sy) strain. When the stress is removed at time /, the elastic (thick bar) and delayed elastic strains recover. 

Figure 6.21 Burger solid and its response to a constant applied stress. On loading under tq an instantaneous elastic strain yi, a delayed elastic strain ya and a viscous strain ya appear. On unloading only elastic strain y-, and delayed elastic strain ya recover.

Equation (6.58) produces an instantaneous elastic strain (first term, j i = -ro/Gi), a delayed elastic strain (second term, Y t) = (tq/G2)[1 - exp(-t/r )]) and a viscous strain (third term, y (t) =-rot/GiT ). The first two terms recover on unloading, whereas viscous flow is irreversible. Often used in the literature, anelasticity refers to the instantaneous plus delayed recoverable deformation while viscous flow, in contrast, is not recoverable. 

The creep response according to the Burgers model in Fig. 34.4 covers all elementary aspects of time-dependant viscoelastic behavior including instantaneous elastic strain, secondary steady state creep in the long-term area, and a delayed elastic strain transition behavior that can be, for example, fitted to experimental data according to the choice of the t/i, Ei, ijz, and Ez parameters. 

O Figure 34.10 represents the typical course of a creep experiment using single lap shear specimen bonded with a viscoelastic acrylic adhesive under tensile load. The curve progression can be divided into three phases of creep. At the end of phase I, instantaneous elastic strain and delayed elastic strain are completed and the creep progress in phase II is dominated by secondary creep. At the end of phase II, creep accelerates leading to failure of the specimen at the end of phase III. 

Beyond collecting empirical creep data for specific adhesives or adhesive joints, it is in some cases desirable to derive viscoelastic parameters in relation to the previously discussed mechanical substitute models of viscoelasticity from the test results. The challenge hereby is to distinctively separate instantaneous elastic and delayed elastic strain (primary creep) from non-recoverable strain (secondary creep) and to determine, whether the principles of linear viscoelasticity may or may not be applied. 

Figure 16 (145). For an elastic material (Fig. 16a), the resulting strain is instantaneous and constant until the stress is removed, at which time the material recovers and the strain immediately drops back to 2ero. In the case of the viscous fluid (Fig. 16b), the strain increases linearly with time. When the load is removed, the strain does not recover but remains constant. Deformation is permanent. The response of the viscoelastic material (Fig. 16c) draws from both kinds of behavior. An initial instantaneous (elastic) strain is followed by a time-dependent strain. When the stress is removed, the initial strain recovery is elastic, but full recovery is delayed to longer times by the viscous component.

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

The same parameters can also be determined by applying a constant shear stress to the interface and measuring the resulting shear strain as a function of time (see fig. 3.40), so-called interfacial creep tests. At t = 0, a shear stress is suddenly applied, and kept constant thereafter. For ideally viscous monolayers a steady increase of the shear strain with t will be observed, while for an elastic material the observed strain will be instantaneous and constcmt in time. For a viscoelastic material, as in fig. 3.40, there is first am Instantaneous increase AB in the strain, the elastic response followed by a delayed elastic response BC and a viscous... 

In a Voigt (or Kelvin) element tlie spring and dashpot are parallel. If a stress is suddenly applied the spring cannot respond immediately because of the resistance caused by the viscous flow (delayed elasticity). Monolayers with a two-dimensional network and viscous material between the cross-links will display such behaviour. So, the increase of the strain is retarded. Eventually the maximum strain / K° is attained, see fig. 3.52a. After cessation of the strain the energy stored in the spring relaxes, again with a rate determined by the parallel viscosity, till AA— 0. Behaviour like this is semi-solid. In the limit of r] - 0 the block diagram of fig. 3.49b is retrieved. 

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. 

The early work on viscoelasticity was performed on silk, mbber, and glass, and it was concluded that these materials exhibited a delayed elasticity manifest in the observation that the imposition of a stress resulted in an instantaneous strain, which continued to increase more slowly with time. It is this delay betweai cause and effect that is fundamental to the observed viscoelastic response, and the three major examples of this hysteresis effect are (1) creep, where there is a delayed strain response afto the rapid application of a stress, (2) stress-relaxation (Section 13.15), in which the material is quickly subjected to a strain and a subsequent decay of stress is observed, and (3) dynamic response (Section 13.17) of a body to the imposition of a steady sinusoidal stress. This produces a strain oscillating with the same frequeney as, but out of phase with, the stress. For maximum usefulness, these measurements must be carried out over a wide range of temperature. 

In many materials, the mechanical response can show both elastic and viscous types of behavior the combination is known as viscoelasticity. In elastic solids, the strain and stress are considered to occur simultaneously, whereas viscosity leads to time-dependent strain effects. Viscoelastic effects are exhibited in many different forms and for a variety of structural reasons. For example, the thermoelastic effect was shown earlier to give rise to a delayed strain, though recovery of the strain was complete on unloading. This delayed elasticity is termed anelastic-ity and can result from various time-dependent mechanisms (internal friction). Figure 5.9 shows an example of the behavior that occurs for a material that has a combination of elastic and anelastic behavior. The material is subjected to a constant stress for a time, t. The elastic strain occurs instantaneously but, then, an additional time-dependent strain appears. On unloading, the elastic strain is recovered immediately but the anelastic strain takes some time before it disappears. Viscoelasticity is also important in creep but, in this case, the time-dependent strain becomes permanent (Fig. 5.10). In other cases, a strain can be applied to a material and a viscous flow process allows stress relaxation (Fig. 5.11). 

Reference Initial strain at 5N Immediate Elasticitv Delayed Elasticity Permanent set... 

Creep describes time-dependent permanent deformation of materials resulting from constant structural stress. The creep of polymers can be divided into two main stages primary creep and steady-state creep. The creep strain rate decreases with time in the primary creep and is constant in the steady-state creep. Strain recovery occurs with the removal of external load after a creep time. Therefore, the total strain (e) consists of three separate parts el, e2, and e3. The el and e2 are the immediate elastic deformation and delayed elastic deformation, respectively. The e3 is the Newtonian flow. It was found that the el and e2 decreased with increasing clay contenf indicating lower creep recovery with the addition of C20A. The creep compHance J, the ratio of strain and applied load, can be expressed as... 

Instantaneous elastic strain and delayed viscous strain arise in the part only when it is released from the mould there is a change in the shape of the product, with equilibrium and relaxation of internal stresses. 

The effect of applying a similar loading programme to a linear viscoelastic solid has several similarities (Figure 5.2(b)). In the most general case, the total strain e is the sum of three separate parts e, and ez. e and are often termed the immediate elastic deformation and the delayed elastic deformation respectively, ez is the Newtonian flow, that is that part of the deformation, which is identical with the deformation of a viscous liquid obeying Newton s law of viscosity. 

Therefore, the Kelvin model exhibits delayed elasticity with the creep strain approaching its final value gradually (O Fig. 23.4). 

The initial stress upon application of a constant strain is governed by the Maxwell spring element Ei, and relaxation on a long-term time scale is dominated by the Maxwell dashpot tji, while the Kelvin-Voigt section with and r/2 governs the delayed elastic relaxation. Solving the differential equation by Laplace transformation is, for example, comprehensively described in (Betten 2008). 

Problem Determine material specifications for a design of a shock absorber for a design that requires 63% of the elastic strain to be delayed (retarded) 2.0 s. Assume a Kelvin solid-type material behavior and assume the maximum elastic strain is limited to 0.01 when a constant stress of 1000 psi is applied. 

The elongation of a stretched fiber is best described as a combination of instantaneous extension and a time-dependent extension or creep. This viscoelastic behavior is common to many textile fibers, including acetate. Conversely, recovery of viscoelastic fibers is typically described as a combination of immediate elastic recovery, delayed recovery, and permanent set or secondary creep. The permanent set is the residual extension that is not recoverable. These three components of recovery for acetate are given in Table 1 (4). The elastic recovery of acetate fibers alone and in blends has also been reported (5). In textile processing strains of more than 10% are avoided in order to produce a fabric of acceptable dimensional or shape stabiUty. 

---