Notes IV: Rheology Part 1 - Ideal Material Behaviours

Notes IV Rheology part 1: Ideal material behaviours Fall 2005 1 Reading assignment Chapter 18 in Twiss and Moores, pages 361  385, deal with both ideal models for rock deformation as well as experimental investigation of ductile ow. Chapter 9, pages 165  190 deal with the mechanics of brittle fracture. This is good stu, believe. 2 Ideal behaviours The next section of the course will review what we know about the relationship between 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 response to an imposed stress is called the rheology. A mathematical representation of the rheology is called a constitutive law. 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∝� There ought to be a constant of proportionality on there, 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. 1 2.2 Viscous behaviour  Newtonian uids Strain in a viscous uid is time dependant and non-recoverable. Time-dependance is the fundamental dierence with elastic behaviour: instead of a linear relationship between stress and strain, viscous materials exhibit a linear relationship between stress and strain-rate. τ = η�˙ The constant of proportionality, η , is the viscosity, and has units of Pascal-seconds. 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 (Pa-s) 10−3 0.1 - 10 1013 14 10 - 1020 1021 1021 The eective viscosity can change during the course of deformation. If the viscosity increases, we call the process strain hardening or shear rate thickening; conversely, if the material becomes weaker and the viscosity decreases, the process is strain softening or shear rate thinning. 2.3 Visco-elasticity I: Maxwell bodies If we represent elastic behaviour with a spring, and viscous behaviour with a dashpot1 , we can also combine dashpots and springs in a number of dierent ways to simulate various possible ideal material responses or rheologies. 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 elasticity and viscous uids2 : σ̇ σ �˙ = + 2µM 2ηM 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 2ηM 1 What the heck is a dashpot? They show up a lot in material science, geodynamics and dierential 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 uid. Examples of objects that include dashpots are car shocks, bike shocks, and are found on some doors. 2 The notation I use diers 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. 2 On a strain vs. time plot, this shows instantaneous elastic strain, followed by steady state linear viscous strain. The case of constant strain is a little more dicult 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. 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 ηM : 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. 2.4 Visco-elasticity 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 and supposedly elastic materials don't always respond instantaneously to the imposed stress. A Kelvin body shows time-dependant, recoverable strain. Kelvin behaviour is given by σ = 2µK � + 2ηK �˙ Upon loading, the elastic response of the spring is damped by the dashpot, and goes asymptotically to σ0 /2µK : � � �� µK σ0 1 − exp − t �= 2µK ηK When the load is removed, the strain is recovered, but not instantaneously. Suppose, a strain �0 had been accummulated, then �(t) is � � µK � = �0 exp − t ηK 2.5 Other ideal rheologies Four other ideal behaviours are worth mentioning. The rst are two end-members of Newtonian viscosity: the Pascal liquid and the Euclid solid, which have viscosities 0 and ∞, respectively. The last two are types of plastic behaviour. A plastic material is one that has a yield stress, but otherwise behaves as a Newtonian uid (or a non-Newtonian, power-law uid). The St. Venant body (also called an elastic-plastic body) is idealized as a friction block being pulled by an elastic. Such a body is a material that exhibits linear elasticity up to a certain point, the yield point. Then, the material fails abruptly. A Bingham body, also called viscoplastic, is idealized as a spring in parallel with a friction 3 block itself in parallel with a dashpot3 . Its behaviour is governed by the equations: σ = 2µ� σ < σY σ = σY + 2ηB �˙ σ ≥ σY The Bingham body behaves elastically at stresses lower than the yield stress, and ows as a linear uid above the yield strength with a strain rate proportional to σ − σY . Examples of Bingham materials are certain clays, some kinds of submarine debris ows, oil paintings, drilling muds, toothpaste and bread dough. 3 Study and review questions You should be aware of the dierences 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 strain-rate). For Maxwell and Kelvin bodies, also plot stress against time for constant strain, and strain (or strain rate) against time for constant stress. Think o a real world example or two for each behaviour. 3 Note that in class, this was shown, in error, as a dashpot and a friction block in series. Twiss and Moores leave out the leading spring, which is a special case. 4