Viscoelastic Mechanical Models PDF

21. Viscoelastic mechanical models 22.1. Viscoelastic mechanical models. It is clear that any theoretical explanations of the above phenomena should be able to account for the dependence of the stress and strain upon time. Ideally, it should be possible to predict for example, the stress relaxation...

21. Viscoelastic mechanical models 22.1. Viscoelastic mechanical models. It is clear that any theoretical explanations of the above phenomena should be able to account for the dependence of the stress and strain upon time. Ideally, it should be possible to predict for example, the stress relaxation behavior from knowledge of the creep curve, but in practice, with real polymers this is somewhat difficult to do. Hence, the situation is often simplified by assuming that the polymer behaves as a linear viscoelastic material, i.e., it is assumed that the deformation of the polymer comprises independent elastic and viscous components so that the deformation of the polymer can be described by a combination of Hooke’s law and Newton’s law. It should be noted that these equations only apply at small strains. A particularly useful method of formulating the combination of elastic and viscous behavior is through the use of mechanical models. The two basic components used in the models are an elastic spring of modulus E, which obeys Hooke’s law and a viscous dashpot containing an incompressible liquid of viscosity η, which obeys Newton’s law. The various models that have been proposed involve different combinations of these two basics elements. 22.2. Maxwell Model It consists of a spring and dashpot in series. Under the action of an overall stress σ, there will be an overall strain e in the system, which is given by e = e1+ e2 Where e2 is the strain in the spring; e2 is the strain in the dashpot. Since the elements are in series, the stress will be identical in each one and so 𝜎 = 𝜎1 = 𝜎2 From Hooke’s law and Newton’s law, 1  𝑑𝜎 𝑑𝑒1 𝑑𝑒2 =𝐸 𝑎𝑛𝑑 𝜎 = 𝜂 𝑑𝑡 𝑑𝑡 𝑑𝑡 From e = e1+ e2, we know 𝑑𝑒 𝑑𝑒1 𝑑𝑒2 = + 𝑑𝑡 𝑑𝑡 𝑑𝑡 So for the Maxwell model 𝑑𝑒 1 𝑑𝜎 𝜎 = + 𝑑𝑡 𝐸 𝑑𝑡 𝜂 At this stage, it is worth considering how closely the Maxwell model predicts the mechanical behaviour of a polymer. In the case of creep, the stress is held constant at σ=σ0 and so dσ/dt =0. Equation can be written as 𝑑𝑒 𝜎0 = 𝑑𝑡 𝜂 Thus, the Maxwell model predicts Newtonian flow. The strain is expected to increase linearly with time, which is clearly not the case for a viscoelastic polymer where de/dt decreases with time. The model is of more use in predicting the response of a polymer during stress relaxation. In this case, a constant strain e =e0 is imposed on the system and so de/dt =0. Then 1 𝑑𝜎 𝜎 0= + 𝐸 𝑑𝑡 𝜂 2  𝑑𝜎 𝐸 = − 𝑑𝑡 𝜎 𝜂 This can be readily integrated if at t =0, σ=σ0 to give 𝐸𝑡 𝜎 = 𝜎0 exp (− ) 𝜂 where σ0 is the initial stress. The term η/E will be constant for a given Maxwell model and it sometimes is referred to as a relaxation time τ0. The equation can then be written as 𝑡 𝜎 = 𝜎0 exp (− ) 𝜏0 which predicts an exponential decay of stress. This is a rather better representation of polymer behavior than the Maxwell model for creep. However, the equation predicts that the stress relaxes completely over a long period of time, which is not normally the case for a real polymer. 22.3. Voigt Model (Kelvin model) It consists of the same elements as are in the Maxwell model but in this case they are in parallel rather than in series. The parallel arrangement of the spring and the dashpot means that the strains are uniform, i.e. e = e1 = e2 and the stresses in each component will add to make an overall stress of σ such that σ = σ1 + σ 2 𝑑𝑒 𝜎1 = 𝐸𝑒 𝑎𝑛𝑑 𝜎2 = 𝜂 𝑑𝑡 σ 𝑑𝑒 𝐸𝑒 = + 𝜂 𝑑𝑡 𝜂 The Voigt or Kelvin model is particularly useful in describing the behavior during creep where the stress is held constant at σ=σ0. σ0 𝑑𝑒 𝐸𝑒 = + 𝜂 𝑑𝑡 𝜂 3 Thus, σ0 𝐸𝑡 𝑒= (1 − exp (− )) 𝐸 𝜂 The constant ratio η/E can again be replaced by τ0, the relaxation time and so the variation of strain with time for a Voigt model undergoing creep loading is given by σ0 𝑡 𝑒= (1 − exp (− )) 𝐸 𝜏0 it clearly represents the correct form of behavior for a polymer undergoing creep. The strain rate decreases with time and e → σ0/E as t → ∞. On the other hand, the Voigt model is unsuccessful in predicting the stress relaxation behavior of a polymer. In this case, the strain is held constant at e0 and since de/dt= 0, then the equation becomes σ = 𝐸𝑒0 Which is the linear elastic response. 22.4. Standard linear solid It has been demonstrated that the Maxwell model describes the stress relaxation of a polymer to a first approximation, whereas the Voigt model similarly describes creep. A logical step forward is, therefore, to find some combination of these two basic models that can account for both phenomena. A simple model that does this is known as the standard linear solid, one example of which is shown in Figure 20.2c. In this case, a Maxwell element and spring are in parallel. The presence of the second spring will stop the tendency of the Maxwell element to undergo simple viscous flow during creep loading, but will still allow the stress relaxation to occur. 4