Rheology, or the study of the flow of matter. Panta rei. (Panta rhei)

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1 Rheology, or the study of the flow of matter Panta rei (Panta rhei)

2 Overview Basics of rheology Linear elasticity Linear viscosity Linear viscoelasticity To infinity... and beyond! Coming back to Earth

3 Stress and strain σ= Force Surface ϵ= Elongation Initial length

4 Solids and fluids A solid is characterized by the capability of sustaining long-term stresses with a finite deformation. A fluid cannot sustain (tangential) stresses at all, and deforms continually that is, it flows. If the strain increases linearly with the applied stress, the material is purely elastic If the strain rate increases linearly with the applied stress, the material is purely newtonian

5 Constitutive equation The relation between stress and strain is expressed by means of a constitutive equation: σ= f (ϵ ; p,t,t,...) To test the response of a certain material, two simple (real or thought) experiments can be performed: creep and relaxation.

6 Elasticity First proposed by Robert Hooke. He described his discovery in the anagram ceiiinosssttuv, whose solution he published in 1678 as Ut tensio, sic vis meaning As the extension, so the force. σ=e ϵ Schematically represented by a spring:

7 Seismology More in general, for a linear isotropic elastic solid, we have the following constitutive equation S σ=2 G ϵ +3 K ϵ I

8 Viscosity Sir Isaac Newton, the famous English physicist, mathematician, astronomer, natural philosopher, alchemist, and thorough lunatic, proposed that in ideal fluids tangential stresses are proportional to deformation rates: σ=η ϵ Schematically represented by a dashpot:

9 The pitch drop experiment Started in 1927 by Professor Thomas Parnell of the University of Queensland in Brisbane, Australia. Large droplets form and fall over the period of about a decade. The 9th drop fell in April 2014 This pitch has a viscosity approximately 2.3 * 108 Pa s A similar experiment by Trinity College, Dublin

10 Mantle convection A geodynamicist contrary to a seismologist would argue that the whole Earth is indeed fluid! S σ=2 μ ϵ +3 ζ ϵ I

11 Viscoelasticity James Clerk Maxwell proposed in 1867 a constitutive equation for a material that displays short-term solid behavior but that is fluid in the long term. It can be represented by a viscous dashpot connected in series with an elastic spring. The spring and the dashpot are subject to the same stress (Newton's third law) σ η=σ E =σ ϵtot =ϵη +ϵe

12 Maxwell rheology ϵe = σ ; ϵ η = σ η E ϵ tot = σ + σ E η In a creep experiment the stress is constant, and solving for the strain yields: ϵ=ϵ0 + σ t ; ϵ0 = σ E E In a relaxation experiment the strain is constant, and solving for the stress yields: t t σ=σ 0 e m η ; σ 0 =E ϵ ; t m = E A Maxwell body relaxes on timescales of the order of the Maxwell time tm

13 Kelvin-Voigt rheology A Kelvin-Voigt material is characterized by a transient creep that leads to an elastic equilibrium. It can be represented by a dashpot and a spring connected in parallel: The spring and the dashpot are equally deformed, while the total stress is shared between them. σ tot =σ η +σ E ϵ η=ϵ E =ϵ σ E =E ϵ ; σ η =η ϵ σtot =E ϵ+η ϵ

14 Burgers model By connecting a Maxwell element in series with a Kelvin-Voigt element, a Burgers material is obtained: Karaman et al. 2012, Linear creep and recovery analysis of ketchup processed cheese mixtures using mechanical simulation models as a function of temperature and concentration.

15 Other non-linear behaviors Shear thickening fluids: viscosity increases with increasing shear strain rate. Shear thinning fluids: viscosity decreases with increasing shear strain rate. Thixotropic and rheopectic fluids: at constant strain rate, their viscosity changes with time. Bingham plastics deform like a solid at low stress,but at high enough stress they yield and flow viscously. Temperature-dependent viscosity.

16 Why bother? Most of the fluids we encounter in our everyday life are nonnewtonian. In some cases we just have to deal with it: food in cooking, organic fluids and tissues in medicine, metals, polycrystalline compounds in geophysics and geotechnics.

17 Why bother? In some other cases we specifically design them in order to exploit their non-newtonian properties, e.g.: inks and paints, lubricating oils, metal frames, asphalt mixtures.

18 OK, now I'm really into it, so... what about the Earth? The Earth as a whole has an extremely complex and diverse rheology, with effective viscosity spanning more than 20 orders of magnitude from the liquid outer core ( Pa s) to the solid lithosphere (>1025 Pa s). For the mantle, experimental and theoretical evidences point to a strongly temperature- and pressure-dependent viscosity; under certain p-t-σ conditions, mantle rocks are also shear thinning. While constraining the precise rheology of the Earth is difficult, general estimates of its effective viscosity are easily obtained by studying postglacial rebound, the geoid and mantle convection.

19 Boston Molasses Disaster! On the 15th of January 1919 (98 years ago!) a molasses storage tank 15 m high and 27 m in diameter burst, causing a 7 m high molasses wave traveling at 50 km/h, killing 21 people and injuring 150.