Geology 2112 – Principles and Applications of Geophysical Methods WEEK 1 Lecture Notes – Week 1 A Review of the basic properties and mechanics of materials Suggested Reading: Relevant sections from any basic physics or engineering text. Objectives: •
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Review some basic properties of materials relevant to the study of geophysics Introduce some new concepts relating to the mechanics of materials Highlight some common problems with using the basic and derived material properties in modeling behaviour Introduction: •
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Basic (intrinsic) and derived material properties are used to describe the mechanical behaviour of rocks and other natural earth materials Knowledge of properties of earth materials allows geophysicists to identify anomalies and model mechanical behaviour Models are formulations (equations) that approximately predict interactions and behaviours; they are never perfect, and often require major assumptions and simplifications that must be considered when applying them For example, the following equation describes the energy associated with the propagation of fractures in weak snowpack layers in simulations of avalanche release: 2
⎛ 3 ⎞ ⎛⎜ ( ρ slab⋅ g⋅cosψ ) ⎞⎟
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C. Sigrist, PhD Thesis, 2006 (Swiss Federal ⎟ ⋅ CrLψ Gf = ⎜ ⎟ ⎜
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⋅
E
Thn
slab
Institute) ⎝ 2 ⎠⎝
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Where: GF is fracture energy, ρSLAB is the density of the snow ‘slab’ of interest, g is the acceleration due to gravity, Ψ is the slope angle that the slab might slide down, E is the ‘Young’s Modulus’ of the slab, ThnSLAB is the thickness of the slab, CrL is the critical length that a small crack must attain before it propagates. For this course, it is not important exactly what this equation or model is telling us – we will focus on the material properties used as inputs, and look at what they actually mean. 1 Geology 2112 – Principles and Applications of Geophysical Methods WEEK 1 Length: •
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Used directly in the fracture energy formulation in slab thickness and critical length terms Used to derive density, acceleration, Young’s Modulus, even slope angle SI unit: metre (m) – note British/Canadian spelling vs. U.S. ‘meters’ A metre was originally standardized/defined by an arbitrary distance between two markings on an alloy bar The current standard metre is: the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second This is derived from the speed of light in a vacuum: 299 792 458 m/s. Some basic derived measures are area (m2) and volume (m3) Caution: Always consider the ‘route dependence’ of a length parameter. Sometimes length is simply the straight‐line distance between two points; in other cases, it might be referring to a distance along a surface – but that could depend on how smooth the surface is or how we measure it. Density: •
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Used directly in the fracture energy formulation in the density term Defined as mass per unit volume, or mass/volume SI unit: kilograms (kg)/cubic metre (m3) Mass: the standard kg is another arbitrary unit defined/standardized by the mass of a chunk of alloy Volume is often measured in litres (L = 1000 cm3), cubic centimeters (cc or cm3) etc. Caution: Always consider how the ‘density’ of something is measured, and what it means in a formulation. In proper ‘materials’ it doesn’t matter what volume of sample you take, the volume is the same. In other complex or porous ‘materials’ it depends on how big and what location the sample comes from. For example – what is the density of the Centennial Building? Or of a sample of snow with over 80% porosity? If the formulation or application is really using the mass part, you must use a large representative sample. But sometimes the density term has real physical significance, referring to how big the molecules making up a material are, and how far apart they are in the crystal structure. In these cases, the density of the porous ‘material’ may not apply. Acceleration due to gravity: •
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The ‘g’ term in the fracture energy formulation refers to ‘the free‐fall acceleration near the earth’s surface’ Usually use a value of approximately 9.81 m/s2 Caution: 9.81 m/s2 is a useful value for mast formulations and calculations, but it is not strictly correct everywhere. The actual value at a given location on the earth’s surface depends on many factors – such as distance from the centre of the earth, nearby massive bodies, ‘missing’ mass, or 2 Geology 2112 – Principles and Applications of Geophysical Methods WEEK 1 the equilibrium of the local crust, etc. Gravity surveys use this variation to say something meaningful about the causes and local conditions. In most cases the standard value is precise enough for formulations, but always consider what effect, if any, local variations might have on the output of a model or calculation. Also – don’t forget that the acceleration is a vector, and it doesn’t always point straight down! Young’s Modulus: •
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This experimentally determined material property describes its ‘stiffness’ Young’s Modulus (E in the fracture energy formulation) is simply the ratio between the stress applied to a material and the strain that it causes. This must be determined experimentally, but is often related by some empirical function or model to other properties such as density Young’s Modulus = STRESS/STRAIN •
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STRESS = FORCE/AREA, where: o Force is that which can cause a mass to accelerate o SI unit for force is the Newton (N). A Newton is the force required to accelerate 1 kg by 1 m/s2 o Area is simply the area (in m2) over which the force is applied o The SI unit for STRESS is the Pascal (Pa), and stress is equivalent to ‘pressure’ o Pa = N/m2 o Stress or pressure are often reported in multiples of the Pa (GPa, MPa, KPa) or others (mb, mm of Hg, psi, etc.) STRAIN refers to a change in length or some other ‘dimension’ of an object due to applied stress, in relation to the stress‐free length o For Young’s Modulus the strain is determined for tension: L0 L1 •
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Strain (elongation) = L1‐L0/L0 o Strain is dimensionless (m/m) Units for Young’s Modulus = Stress/Strain = Pa/(m/m) = Pa For ‘stiffer’ things, more stress is required for a given elongation, so stiffer things have a larger Young’s Modulus 3 Geology 2112 – Principles and Applications of Geophysical Methods WEEK 1 •
There are many other so‐called ‘elastic moduli’ , such as the Shear Modulus (shear strain is the change in angles within a material), or the Compressive or Bulk Modulus (strain is compressive rather than tensile) Caution: Physically, and in most applications of it, the elastic moduli of a material are referring to how firmly the molecules in its structure are held together. Consider porous materials or other structures, such as snow or the Centennial Building – can we measure their Young’s Modulus? If so, where is the elongation occurring? Do the values of the moduli mean the same thing as in a solid material like steel or ice? As always, think about what the formulation is using the modulus value for, and what assumptions or simplifications are required so that it makes sense. Elastic behaviour: All of the moduli assume that a sample or material behaves elastically, meaning that the strain returns to zero after the stress is removed: Stress Strain
Plastic behaviour is permanent, unrecoverable strain that remains after the stress returns to zero: Plastic yielding Stress Strain
4 Brittle failure, rupture – max stress = strength Geology 2112 – Principles and Applications of Geophysical Methods WEEK 1 Energy: •
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In the example formulation from Sigrist (2006), the ‘fracture energy’ is the ultimate outcome ‘Energy’ is a scalar quantity associated with the state of something – in the fracture energy example it refers to the relationship between strain energy and surface energy There are many different ‘states’ an object can be in, and there are many different energies used to describe them: kinetic energy, strain energy, thermal energy, etc. etc. SI Unit for energy is Joule = kg∙m2/s2 Work is the act of transferring energy via a force Note that ‘strain energy’ is equal to the area beneath the linear part of the stress‐strain curves on the previous page Summary: •
All models of our understanding of how the earth works use inputs of material properties to determine or predict some other value of interest – ALWAYS understand why a model or formulation is using that property, and that the value you are using is appropriate. Sometimes we do the best we can, but must always be aware about the assumptions and simplifications that are required to make it work. 5