BM Unit 3 Viscoelasticity PDF

Viscoelasticity  Viscoelasticity When a body is suddenly strained and then the strain is maintained constant afterward, the corresponding stresses induced in the body decrease with time. This phenomenon is called stress relaxation, or relaxation for short I...

Viscoelasticity  Viscoelasticity When a body is suddenly strained and then the strain is maintained constant afterward, the corresponding stresses induced in the body decrease with time. This phenomenon is called stress relaxation, or relaxation for short If the body is suddenly stressed and then the stress is maintained constant afterward, the body continues to deform, and the phenomenon is called creep. If the body is subjected to a cyclic loading, the stress- strain relationship in the loading process is usually somewhat different from that in the unloading process, and the phenomenon is called hysteresis 2 Examples of Creep, Relaxation and Hysteresis Creep Relaxation Hysteresis 3  The features of hysteresis, relaxation and creep are found in many materials. Collectively they are called features of viscoelasticity 4  Models Mechanical models are often to discuss the viscoelastic behavior of materials In this figure are shown three mechanical models of material behavior, namely the Maxwell model, the Voigt model, and the Kelvin model (also called standard linear solid), 5 6  All of these models are composed of combination of linear springs with spring constant μ and dashpots with coefficient of viscosity η. A linear spring is supposed to produce instantaneously a deformation proportional to the load. A dashpot is supposed to produce a velocity proportional to the load at any instant. Thus if F is the force acting in a spring and u is its extension, then F = μu. If the force F acts on a dashpot, it will produce a velocity of deflection , and F= η 7  The shock absorber on an aeroplane's landing gear is an example of a dashpot Now, in a Maxwell model as shown in this figure, the same force is transmitted from the spring to the dashpot This force produces a displacement F/μ in the spring and a velocity F/η in the dashpot 8  Maxwell Model The velocity of the spring extension is /μ if we denote a differentiation aspect with respect to time by a dot The total velocity is a sum of these two (Maxwell model).... Eq(1) 9  Initial Condition Furthermore, if the force is suddenly applied at the instant of time t=0, the spring will be suddenly deformed to u(0)=F(0)/μ, but the initial dashpot deflection would be zero, because there is no time to deform. Thus the initial condition of differential equation (1) is..... Eq (2) 10  Voigt model For the Voigt model, the spring and the dashpot have the same displacement If the displacement is u, the velocity , and the spring and dashpot will produce forces μu and η respectively. The total force F is therefore (Voigt model).........Eq(3) If F is suddenly applied, the appropriate initial condition is u(0)=0..........Eq (4) 11  Kelvin Model For the Kelvin model (or standard linear model), let us break down the displacement u into u1 and ú1 for the spring, whereas the total force F is the sum of the force F0 from the spring and F1 from the Maxwell element. Thus..... Eq (5) 12  From this we can verify by substitution that Hence Replacing the last term by μ1ú1 and using Eq(5a), we obtain........... Eq (6) 13  Equation (6) can be written in the form (Kelvin model)........ Eq (7) Where........ Eq (8) For a suddenly applied force F(0) and displacement u(0), the initial condition is........ Eq (9) The constant τε is called the relaxation time for constant strain, whereas τσ is called the relaxation time for constant stress. 14  If we solve Equations (1), (3) and (7) for u(t) when F(t) is a unit- step function 1(t), the results are called creep functions, which represent the elongation produced by a sudden application at t=0 of a constant force of magnitude unity. They are: Maxwell solid.......... Eq (10) Voigt solid.......... Eq (11) standard linear solid.......... Eq (12) 15  Where the unit-step function 1(t) is defined as.......... Eq (13) 16  A body that obeys a load-deflection relation like that given by Maxwell's model is said to be a Maxwell solid Since a dashpot behaves as a piston moving in a viscous fluid, these models are called models of viscoelasticity. 17  Interchanging the roles of F and u, we obtain the relaxation function as a response F(t)=k(t) corresponding to an elongation u(t)=1(t) The relaxation function k(t) is the force that must be applied in order to produce an elongation that changes at t=0 from zero to unity and remains unity thereafter. They are Maxwell solid........... Eq (14) Voigt solid........... Eq (15) standard linear solid........... Eq (16) 18  Here, we have used the symbol δ(t) to indicate the unit impulse function or Dirac-delta function, which is defined as a function with a singularity at the origin δ(t)=0 (for t0)................ Eq (17) Where f(t) is an arbitrary function, continuous at t=0 These functions c(t), k(t) are illustrated in next slide 19 20  Creep Relaxation For the Maxwell solid, the sudden application of a load induces an immediate deflection by the elastic spring, which is followed by "creep" of the dashpot. On the other hand, a sudden deformation produces an immediate reaction by the spring, which is followed by the stress relaxation according to an exponential law Eq(14) The factor η/μ,with the dimension of time, maybe called the 21 relaxation time: it characterises the rate of decay of the force  Creep Relaxation For the Voigt solid, a sudden application of force will produce no immediate deflection, because the dashpot, arranged in parallel with the spring, will not move instantaneously Instead, a deformation will gradually build up, while the spring takes a greater and greater share of the load as shown in Equation (11) The dashpot displacement relaxes exponentially Here the ratio η/μ is again a relaxation time: characterises the rate of 22 relaxation of the dashpot  Creep Relaxation For the standard linear solid, the constant τε is the time of relaxation of load under the condition of constant deflection as shown in Eq (16) whereas, the constant τσ is the time of relaxation of deflection under the condition of constant load as shown in Eq (12) As , the dashpot is completely relaxed and the load-deflection relation becomes that of the springs, as characterised by constants ER in equations (12) and (16) 23 Therefore ER is called relaxed elastic modulus  Maxwell introduced the model represented by Eq(1), with the idea that all fluids are elastic to some extent. Lord Kelvin showed the inadequacy of the Maxwell and Voigt models in accounting for the rate of dissipation of energy in various materials subjected to cyclic loading Kelvin's model is commonly called the standard linear model because it is the most general relationship that includes the load, the deflection and their first (commonly called "linear") derivatives More general models may be built by adding more and more elements to Kelvin model Equivalently we may add more and more exponential 24 terms to the creep function or to the relaxation function.  The most general formulation under the assumption of linearity between cause and effect is due to Boltzmann (1844-1906) In the one-dimensional case, we may consider a simple bar subjected to a force F(t) and elongation u(t). The elongation u(t) is caused by the total history of the loading up to the time t. If the function F(t) is continuous and differentiable, then in a small time interval dτ, at time τ, the increment of loading is (dF/dτ)dτ. This increment remains acting on the bar and contributes an element du(t) to the elongation at time t, with a proportionality constant c depending on the time 25 interval t-τ.  Hence we may write....... Eq (18) Let the origin of time be taken at the beginning of motion and loading. Then, on summing over the entire history, which is permitted under Boltzmann's hypothesis we obtain......... Eq (19) A similar argument with the role of F and u interchanged, gives......... Eq (20) 26  These laws are linear, since doubling the load doubles the elongation, and vice versa. The functions c(t-τ) and k(t-τ) are the creep and relaxation functions, respectively 27  Boltzmann Formulation The Maxwell, Voigt and Kelvin models are special examples of the Boltzmann formulation. More generally, we can write the relaxation function in the form............. Eq (21) which is generalisation of the Eq (16) 28  If we plot the amplitude αn associated with each characteristic frequency vn on a frequency axis, we obtain a series of lines that resembles an optical spectrum. Hence, αn(vn) is called a spectrum of relaxation function. The example shown in following figure is a discrete spectrum 29  A generalisation to a continuous spectrum may sometimes be desired. In the case of a living tissue such as mesentery, experimental results on relaxation, creep and hysteresis cannot be reconciled unless a continuous spectrum is assumed 30  Generalisation to tensor equations We will generalise Equations (19) and (20) into tensor equations relating stress and strain tensors of a viscoelastic solid Eq(19): Eq(20): 31  If F and u are considered to be stress and strain tensors, then equations (19) and (20) can be considered as constitutive equations of a viscoelastic solid if c and k are tensors defining the creep and relaxation characteristics of the material. We have to be careful about the time derivatives F. and u if the deformation is large, because the time derivatives of the components of stress and strain of any material particle (such as the small cubic element shown above) may depend not only on how fast the element stretches, but also on how fast it rotates 32  To account for the stress rate (i.e. the rate of change of stress tensor) in finite deformation turns out to be quite complex Hence, for the present discussion, we will limit our considerations to a small deformation, with infinitesimal displacements, strains and velocities. Under this restriction we can express our idea more explicitly 33  Let us denote the stress and strain tensors by σij and eij (which are functions of space x(i.e. x1, x2, x3) and time t) Then a material is said to be linear viscoelastic if σij(x, t) is related to eij(x, t) by the following convolution integral.......Eq (22) or its inverse as.......Eq (23) Gijkl is called the tensorial relaxation function Jijkl is called the tensorial creep function Note that the lower limit of integration is to be taken to be -∞, which is to mean that the integration is to be taken before the very 34 beginning of motion  If the motion starts at time t=0,and σij=eij=0 for t