2 Plastic deformation occurs by dislocation motions in crystallic structures. For noncrystallic structures plastic deformations are due to viscous flow.The characteristic property for viscous flow, viscosity, is a measure of a noncrystalline material’s resistance to deformation.LPlate AFVWhen a tangential force (F) acts on the plate, the plate moves with respect to the bottom.
3 The velocity of the liquid particles in each layer is a function of the distance L. Thus the rate at which the particles change their position is the measure of the rate of flow.dγdLdV=velocity gradientdtrate of flowAdLdVNewton expresses: F = ηSince τ =F / Aη : coefficient of viscositydtdγτ = ητ = ηdLdV&12
4 Unit of viscosity is Pa.s (Pascal-seconds) (N.s/m2) The liquids that follow equations (1) & (2) are termed as Newtonian Liquids.Newtonian liquidsη
5 Viscosity also varies with temperature. A: ConstantE: Energy of activationR: Gas constantT: Absolute temperatureη1= A . e-E/RTWhen solid particles are introduced into Newtonian liquids, viscosity increases.η0: Coefficient of viscosity of the parent liquid.Ø: Volume concentration of suspended particlesη = η0 (1+2.5 Ø)η = η0 (1+2.5 Ø Ø2)
6 NON-NEWTONIAN MATERIALS In certain materials, τ-dγ/dt does not obey the linearity described by Newton, i.e. Viscosity may vary with the rate of shear strain.Dilatant: η increases with increasing dγ/dt or τ (clay)Newtonian: (all liquids)Pseudoplastic: η decreases with dγ/dt or τ (most plastics)Newtonianη
7 The relationship between dγ/dt & τ can be described by the following general equation. η1= τn .dtdγIf n=1 → Newtoniann > 1 → Pseudoplasticn < 1 → Dilatant
8 Fresh cement pastes & mortars, have highly concentrated solid particles in the liquid medium. Such a behaviour is described by Bingham’s equation.dtdγτ = τy + ηdtdγττy(Upto τy there is no flow)
9 VISCOELASTICITY & RHEOLOGICAL CONCEPTS Viscoelastic behaviour, as the name implies, is a combination of elasticity & viscosity.Such a behaviour can be described by Rheological Models consisting of springs (for elasticity) & dashpots (for viscosity).
11 Models to explain the viscoelastic behavior: Maxwell ModelKelvin Model4-Element Model (Burger’s Model)
12 1. Maxwell Model: A spring & a dashpot connected in series. σk = Eβ = 1/ηA spring & a dashpot connected in series.The stress on each element is the same:σspring = σdashpotHowever, the deformations are not the same:εspring ≠ εdashpot
13 When the force (stress) is applied the spring responds immediately and shows a deformation εspring = σ/EAt the same time the dashpot piston starts to move at a rate βσ = σ/ηand the displacement of the piston at time t, is given by:Therefore the total displacement becomes:
15 Relaxation: An important mode of behavior for viscoelastic materials can be observed when a material is suddenly stroked to ε0 & this strain is kept constant.Instead of this loading pattern strain is kept constant.ε0εtσ0σ
16 Solving this differential equation Where : σ0 = Eε0 &If ε is constant →Solving this differential equationWhere :σ0 = Eε0
17 εRelaxation time is a parameter of a viscoelastic material.ε0tIf the body is left under a constant strain, the stress gradually disappears (relaxes). This pehenemenon can be observed in glasses & some ceramics.σσ0t=00.37σ0trelt
18 2. Kelvin Model:Consists of a spring & a dashpot connected in parallel.In this case the deformations are equal but the stresses are different.εspring = εdashpotσspring ≠ σdashpotσ = σspring + σdashpotE1/ησ
20 At each strain increment the spring will extend by σ/E so that a part of the load is taken over and the force on the piston decreases. Thus a final displacement is reached asymtotically and when the load is removed, there will be an asymtotic recovery until σ=0.When a viscoelastic Kelvin Body is subjected to a constant stress, σ0, the response could be obtained by solving the differential equation.
21 is reached at time t=∞ retardation time σtis reached at time t=∞retardation timeWhen stressed the elastic deformation in the spring is retarded by the viscous deformation of the dashpot.εtretε0Retarded elastic strain (delayed elastic strain)0.63ε0
22 3. Burger’s Model:The actual viscoelastic behavior of materials is very complex. The simplest models, Maxwell & Kelvin Models, explain the basic characteristics of viscoelastic behavior.The Maxwell Model, for example, has a viscous character and explains the relaxation behavior of viscoelastic materialsThe Kelvin Model on the other hand has a solid character and explains the retarded elasticity behavior.
23 However, none of the mentioned models completely explain the real behavior of viscoelastic materials.There are other models with different E and η constants but they are rather complex.One such model is given by BURGER, which consists of a Maxwell Model and Kelvin Model connected in series.
25 Kelvin (retarded elasticity) Spring (elastic)Dashpot (viscous)Kelvin (retarded elasticity)Most engineering materials show certain deviations from the behavior described by the 4-Element Model. Therefore the deformation equation is usually approximated as:Instantaneous elasticRetarded elasticViscous
26 Where “k, β & γ” are material constants & “α, n” are constants accounting for nonlinearity.
27 Example 1: For a certain oil, the experimentally determined shear stress, rate of flow data provided the following plot. Determine the viscosity of the oil.dγ/dt (1/sec)0.90.60.3302010τ (Pa)
28 Example 2: When a concrete specimen of 75 cm in length is subjected to a 150 kgf/cm2 of constant compressive stress, the following data were obtained.t (month)ε10.00060.0007Assumewhere B is constant.What will be the total deformation under 150 kgf/cm2 after 6 months?
29 Example 3: A glass rod of 2. 5 cm in diameter & 2 Example 3: A glass rod of 2.5 cm in diameter & 2.5 m in length, is subjected to a tansile load of °C.Calculate the deformation of the rod after 100 hrs.Determine trel (relaxation time)What is the time during which the stress in the material would decay to 5% of its initial volume?η=2x °C & E=1.55x105 kgf/cm2Assume that the behavior of glass at this temperature can be approximated by a Maxwell Model.
30 For Normal Stresses & Strains the viscous behavior is described by dε/dt=σ/λ where λ is called “the Coefficient of Viscous Traction” & equals to “3η”.η=2x1012 poise (1 poise = 1 dyne.sec/cm2) & (1 kgf = 106 dyne)After 100 hrs the total strain is 3.4x10-7x100x60x60 =Δ = x250 = 30.6 cm