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Introduction to Continuum Mechanics and Geological Applications

This book provides an overview of continuum mechanics and its applications in the field of geoscience. It covers topics such as stress, strain, stretching, and constitutive relationships, using examples and exercises to enhance understanding. The text is written in a clear and concise manner, making it accessible to both students and professionals in the field.

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Introduction to Continuum Mechanics and Geological Applications

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  1. The Australian Computational Earth Systems Simulator (ACcESS) Multi-Scale Behaviour in the Geo-Science II: Introduction to Continuum Mechanics and Geological Applications by Hans Mühlhaus

  2. Overview Introduction 1D relationships Statics and kinematics Stress, Strain, Stretching, Spin, Objective Variables, Constitutive Relationships, stress equilibrium Level set method Outline, upwinding (Taylor Galerkin), two step methods, examples Exercises

  3. Displacement, Strain and Stretching Strain: f2 (Force) u2 (displacement) L=L0+u2 (length after application of force f2) Stretching: x2 Where L0 (initial length) x1

  4. 1D Force and Stress Equilibrium x2 x1 The sum of all vertical forces must vanish for force equilibrium: Constitutive Relationship (Hooke’s law) Thus

  5. 1D Constitutive Relations and Balance Equations More Constitutive Relationships: Newtonian Creep Insert into stress equilibrium: Considering the definition of the material time derivative:

  6. 1D Constitutive Relations and Balance Equations C is a Concentration, R is a Reaction Term (e.g. Mass Source), qi Flux of Concentration Change of concentration due to change of size x2 Assumption: x3 Thus: x1 Example 2: Heat Equation Thermal conductivity Heat source (radioactive decay) Heat capacity

  7. Governing Equations Stretching: Stress Equilibrium: Einstein’s summation convention: Heat Equation:

  8. Governing equations Stress Equilibrium Heat Equation Melt viscosity for magma with 0.65% water content Here we use: with

  9. Satellite Image of Volcanic Event

  10. 2nd talk: Volcano modelling Montserrat, West Indies

  11. Intraplate: Hotspots • Anomalous areas of volcanism • Mantle plumes • Ocean: low-viscosity basaltic magmas, Hawaiian Islands • Continental: high silica (high viscosity) rhyolites, Yellowstone • Little information on magma source

  12. VolcanoFacts • 1511 known eruptions in last 10000 years • 238000 deaths in last 400 years • Biggest eruption: Yellowstone, USA (2500km3) • Potential problem: Vesuvius, Italy Poorly understood natural phenomena with approximately 30 eruptions in any given year. Volcanoes also produce many natural resources such as important minerals and metals.

  13. Generic Volcano • Magma chamber at depth (5 – 60km) • Plumbing from chamber to surface not well constrained • 800 to 1200 degree C • Changes in regional stress, earthquakes, can cause the volcano to erupt • New eruption from exertion of magma forces, increased gas pressure or both • Long term activity governed by rate of supply of new magma • Different styles of volcanoes relate to different hazards

  14. Physical properties of magma • Magma =melt + crystals + gas. • Melt: Temperature 800-1300 оС, pressure 103 -10-1MPa • Crystals: size10-7-10-1 m, number densityup to1017 m-3, fraction up to95 % • Gas: H2O - 60-95%, CO2- 0-35%,mass fraction0.1-7 % • Melt viscosity 102 -1012Pa•s • Bulk viscosity depends upon: • Chemical composition • - more SiO2 → higher viscosity • Temperature • - higher temperature → lower viscosity • Water content • - higher content → lower viscosity • Crystal content • - higher content → higher viscosity

  15. Dome Growth Styles Ross Griffiths & Jonathan Fink Axisymmetrical lava dome Platy lava dome Lobate lava dome Spiny lava dome

  16. t=0 t>0 H D/2 z=0 z=0 r=0 r=0 r=R r=R D/2

  17. The Level Set Method: Presentation • Implicit representation of the interface by the zero level set of a smooth function φ • φis usually a “signed” distance function • At each time step, φ is updated solving the advection equation:

  18. The Level Set Method: Solving the advection equation (1/4) Test: A gaussian is advected in a constant 1D velocity field. • Explicit • Implicit • Taylor Galerkin

  19. The level set method …continued Advection dominated pde’s need to require special treatment…..upwinding etc Taylor-Galerkin:

  20. The level set method …continued 2-step alternative to Taylor-Galerkin upwinding (very effective in the presence of diffusion terms….):

  21. Formulation • Davies, M., Gross, L., Mühlhaus, H.–B., 2004, Scripting High Performance Earth Systems Simulations • on the SGI Altix 3700, Proc. 7th Intl Conf. on High Performance Computing and Grid in Asia Pacific Region, • 244-251. Finley PDE: Example : Momentum and Heat equation

  22. EScript • for i in range(numDim):\par • for j in range(numDim):\par • tau += stress[i,j] * stress[i,j]\par • tau = sqrt(0.5 * tau + small)\par • map["tau"] = tau\par • \par • # tau_Y\par • \par • # release memory\par • # power law\par • Xi_P1 = (tau / tau_0) ** (1 - n1)\par • Xi_P2 = (tau / tau_Ystep) ** (1 - n2)\par • Xi_P = Xi_P1 * Xi_P2 / (Xi_P1 + Xi_P2)\par • map["Xi_P"] = Xi_P\par • \par • # release memory\par • del tau_Ystep \par • \par • # melting temperature\par • T_M = T_M0 + gamma * p\par • map["T_M"] = T_M\par • \par

  23. The Level Set Method: Solving the advection equation (2/4) Taylor Galerkin: The gaussian keeps its shape. Implicit: The gaussian is deformed in the direction of the velocity field.

  24. Level set applied to a cantilever beam Presentation of the test case: Constitutive relationship: Stress equilibrium:

  25. Level set applied to a cantilever beam

  26. Level set applied to a cantilever beam Influence of contributions to stress rate: Oldroyd stress rate:

  27. Level set applied to a cantilever beam Accuracy of the method: Conservation of volume

  28. The Level Set Method: Solving the advection equation (3/4) Previous test: No topological change in the solution Need for a new test with: and New test: shearing flow • Mesh: 100x100 • Courant Number: 0.25 • 1000 steps forward • 1000 steps with -v

  29. The Level Set Method: Solving the advection equation (4/4) The shape gets “noisy”… Problem: φ looses its distance function property Reinitialisation needed!

  30. The Level Set Method: Reinitialisation (1/3) • Idea: • Rebuild a “signed” distance function ψ from the distorted function φ • Requirements: • The interface must not be changed • ψ must represent a distance function • Solution: • Solve to steady state the equation: • Rewritten as: with Interpretation: The “distance information” is carried by w, a unit vector pointing away from the interface.

  31. The Level Set Method: Reinitialisation (2/3) 1D 2D 3D

  32. The Level Set Method: Reinitialisation (3/3) Same test as before, with reinitialisation

  33. The Level Set Method: Benchmarks (1/2) Axisymmetrical case: A fluid is submitted to a centrifugal force Interest: The analytical steady state is known (grey line) Parameters: mesh:20x20, density of air: 0 kg/m3, density of fluid: 103 kg/m3 Results:

  34. The Level Set Method: Benchmarks (2/2) Rayleigh-Taylor instability

  35. Level set cont. : Merger of small and large bubbles Parameters: Surface tension: Calculation, includes inertia, implicit, Courant Number=0.5, msh:30 by 45 8 node quad’s

  36. Level set cont. : Merger of small and large bubbles

  37. Level set cont. : Merger of small and large bubbles

  38. Level set cont. : Merger of small and large bubbles

  39. t=0 t>0 H D/2 z=0 z=0 r=0 r=0 r=R r=R D/2

  40. Axisymmetric volcano

  41. Axisymmetric volcano (T= 1000-1001)

  42. Exercises z h0 r R A cylindrical container of radius R is filled initially to height h0 with an incompressible fluid of density r and viscosity h. The container is then rotated around his axis at a constant spin w. Determine the steady state position of the free surface of the fluid.

  43. Exercise5Infinite vent: Hagen-Poiseulle flow Stress Equilibrium Heat Equation D z Dimensionless form r Here

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