Continuum Mechanics and Rheology: Rheology Part 1 - Ideal Material Behaviours

Chapter 4 Rheology part 1: Ideal material behaviours 4.1 Reading assignment Chapter 18 in Twiss and Moores, pages 361 – 385, deals with both ideal models for rock deformation as well as experimental investigation of ductile flow. Chapter 9, pages 165 – 190, deals with the mechanics of brittle fracture. This is good stuff, be­ lieve. 4.2 Ideal behaviours The next section of the course will review what we know about the relationship be­ tween stress and strain; that is, the laws that govern deformation. We start with "ideal" behaviours, and then compare these to experimental results. A material’s re­ sponse to an imposed stress is called the rheology. A mathematical representation of the rheology is called a constitutive law. stress } dx Figure 4.1: Simple, 1­D Hooke body (linear elasticity) rate stress Figure 4.2: In a Newtonian fluid, stress and strain­rate are linearly proportional to each other. The slope of the stress­strain rate line is the viscosity. 4.2.1 Elastic behaviour – Hooke’s law The characteristics of elasticity are: 1. strain is instantaneous upon application of stress; 2. stress and strain are linearly related; 3. strain is perfectly recoverable. σ∝� The constant of proportionality is called the elastic modulus; the exact modulus you use depends on whether the strain is volumetric, uniaxial or in shear. Respectively, these are: σh = K �v σn = E �n σs = 2µ�s Where K , E , µ are the bulk modulus, Young’s modulus and the shear modulus. Another important elastic constant is ν, Poisson’s ratio. Poisson’s ratio relates the elastic strain for orthogonal directions. 4.2.2 Viscous behaviour – Newtonian fluids Strain in a viscous fluid is time dependant and non­recoverable. Time­dependance is the fundamental difference with elastic behaviour: instead of a linear relationship between stress and strain, viscous materials exhibit a linear relationship between stress and strain­rate. τ = η�˙ strain softening: (”shear rate thinning”) strain hardening: (”shear rate thickening”) Figure 4.3: Strain­rate softening or hardening The constant of proportionality, η, is the viscosity, and has units of Pascal­seconds. A plot of strain against time is a line, provided that the viscosity remains constant. Viscosity is a measure of the strength of a material: higher viscosity materials are stronger. Typical geological viscosities are: Material Water Lava Glacier ice Salt Window glass Athenospheric mantle Viscosity (Pa­s) 10−3 0.1 ­ 10 1013 14 10 ­ 1020 1021 1021 The effective viscosity can change during the course of deformation. If the vis­ cosity increases, we call the process strain hardening or shear rate thickening; con­ versely, if the material becomes weaker and the viscosity decreases, the process is strain softening or shear rate thinning. 4.2.3 Visco­elasticity I: Maxwell bodies If we represent elastic behaviour with a spring, and viscous behaviour with a dash­ pot1 , we can also combine dashpots and springs in a number of different ways to simulate various possible ideal material responses or rheologies. 1 What the heck is a dashpot? They show up a lot in material science, geodynamics and differential equations, and are invariably introduced as though they are a familiar, intuitive part of one’s day to day experience. Except that they aren’t, really. Ask the random person on the street what a dashpot is and you’ll get a pretty blank look. Dashpots are simple pistons combined – usually – with a hydraulic fluid. Examples of objects that include dashpots are car shocks, bike shocks, and are found on some doors. Figure 4.4: Maxwell visco­elastic body The most basic combination is to put a spring and a dashpot in series. This combination is known as Maxwell viscoelasticity. The constitutive relationship for Maxwell viscoelasticity comes from the linear addition of the relationships for elas­ ticity and viscous fluids2 : σ̇ σ �˙ = + 2µM 2η M The subscript M just indicates that we’re dealing with Maxwell­rigidity and Maxwell­ viscosity. This equation can be manipulated by stipulating either constant stress (σ = σ0 ) or constant strain (�˙ = 0). Under constant stress: �= σ0 σ0 + t 2µM 2η M On a strain vs. time plot (fig. 4.5), this shows instantaneous elastic strain, fol­ lowed by steady state linear viscous strain. The case of constant strain is a little more difficult to intuit, but the equation, and plot of stress against time are clear enough. � � µM σ = σ0 exp − t ηM A plot of stress against time demonstrates the viscous stress relaxation of a Maxwell body (fig. 4.5). That is, a Maxwell body should relax to an isotropic (i.e. hydrostatic) stress state on a time scale that is captured by the ratio of η M : µM . Expressed as a fraction, this ratio has units of time, and is known as the Maxwell time. The Maxwell time of the asthenospheric mantle is on the order of a thousand years and is what sets the time scale of phenomena like post glacial rebound. 4.2.4 Visco­elasticity II: Kelvin bodies Another way to combine a spring and a dashpot (that is, an elastic response and a viscous response) are to put them in parallel, rather than in series. This material is known as a Kelvin body, and sometimes called fermoviscous behaviour. This is the idealization of a phenomenon (that is, something that actually happens in the real world) called elastic afterworking, which is just that, in the real world, most springs 2 The notation I use differs somewhat from what Clark used in class. Mine follows standard texts such as Ranalli, G. (1995) Rheology of the Earth, and Turcotte, D. and Schubert (2002) Geodynamics. Constant Strain Constant Stress strain stress viscous relaxation viscous } } recovery elastic time time Figure 4.5: Stress and strain versus time for Maxwell viscolelastic bodies and supposedly elastic materials don’t always respond instantaneously to the im­ posed stress. A Kelvin body shows time­dependant, recoverable strain. Kelvin be­ haviour is given by σ = 2µK � + 2η K �˙ Upon loading, the elastic response of the spring is damped by the dashpot, and goes asymptotically to σ0 /2µK : � � �� µK σ0 �= 1 − exp − t ηK 2µK When the load is removed, the strain is recovered, but not instantaneously. Sup­ pose, a strain �0 had been accummulated, then �(t ) is � � µK � = �0 exp − t ηK 4.2.5 Other ideal rheologies Four other ideal behaviours are worth mentioning. The first are two end­members of Newtonian viscosity (fig. 4.7): the Pascal liquid and the Euclid solid, which have viscosities 0 and ∞, respectively. The last two are types of plastic behaviour. A plas­ tic material is one that has a yield stress, but otherwise behaves as a Newtonian fluid (or a non­Newtonian, power­law fluid). The St. Venant body (also called an elastic­ plastic body) is idealized as a friction block being pulled by an elastic spring (fig. 4.8). Such a body is a material that exhibits linear elasticity up to a certain point, the yield point. Then, the material fails abruptly. This sort of rheology is relevant to understanding elastic strain accumulations that drive the earthquake cycle. A Bingham body, also called viscoplastic, is idealized as a spring in series with a friction block itself in parallel with a dashpot 3 . Its behaviour is governed by the equations: σ = 2µ� σ < σY σ = σY + 2η B �˙ σ ≥ σY 3 Note Twiss and Moores leave out the leading spring, which is a special case. constant strain remove stress viscous recovery time Figure 4.6: Kelvin visco­elasticity, strain versus time. no strain, regardless of stress infinite strain, for no stress Figure 4.7: Idealized end­members of Newtonian viscosity Friction Block Analog - St.Venant body (stick-slip) friction static friction threshold “Yield Point” Figure 4.8: Model and stress­strain plot for St. Venant (or "stick­slip") rheology The Bingham body behaves elastically at stresses lower than the yield stress, and flows as a linear fluid above the yield strength with a strain rate proportional to σ − σY . Examples of Bingham materials are certain clays, some kinds of submarine debris flows, oil paintings, drilling muds, toothpaste and bread dough. 4.3 Study and review questions You should be aware of the differences between: linear elastic, linear viscous, Maxwell viscoelastic, and Kelvin behaviour. St. Venant and Bingham bodies are also relevant since they introduce the concept of a yield stress. For each of the above, draw a spring and dashpot idealization, and draw curves that relate stress and strain (or strain­rate). For Maxwell and Kelvin bodies, also plot stress against time for constant strain, and strain (or strain rate) against time for constant stress. Think of a real world example or two for each behaviour. yield point Figure 4.9: Bingham viscoplastic rheology