What is stress (σ)?
Stress is the intensity of force acting on a body, measured as force per unit area (N/m²). It represents the internal forces within a material in response to external loads.
What is strain (ε)?
Strain is the deformation of a body caused by stress. It is dimensionless and represents relative change in length (ΔL/L) or volume (ΔV/V). It quantifies how much a material deforms under load.
What is linear elasticity (Hooke’s Law)?
For small deformations, strain is proportional to stress: σ_ij = c_ijkl ε_kl, where c_ijkl is the elastic stiffness tensor. This is the elastic region where deformations are fully recoverable when stress is removed.
What does ‘fully elastic’ mean?
Fully elastic means deformations disappear completely when stress is removed. In seismic exploration, stress and strain are well within the elastic regime, so Hooke’s Law applies.
How is Young’s modulus (E) measured?
Young’s modulus E = σ/ε = (F/A)/(ΔL/L) is measured using a uniaxial compression/tension test. Apply force F to a rod with cross-sectional area A and measure the length change ΔL relative to original length L.
What is Young’s modulus?
Young’s modulus E is the ratio of tensile stress to tensile strain in a uniaxial test. It measures material stiffness: how much a material resists stretching or compression along one axis.
How is Poisson’s ratio (ν) measured?
Poisson’s ratio ν = -ε_r/ε_l = -(Δw/w)/(ΔL/L) is measured in the same uniaxial test as Young’s modulus. Measure both longitudinal strain (ΔL/L) and transverse strain (Δw/w) when stretching a rod.
What is Poisson’s ratio?
Poisson’s ratio ν is the ratio of fractional transverse contraction to fractional longitudinal extension. It describes how much a material contracts laterally when stretched axially. Typical values: 0.1-0.35 for rocks.
How is bulk modulus (k) measured?
Bulk modulus k = -ΔP/(ΔV/V) is measured using a hydrostatic pressure test. Apply uniform pressure ΔP and measure volume change ΔV. Alternatively, calculate from k = E/[3(1-2ν)] using Young’s modulus and Poisson’s ratio.
What is bulk modulus (k)?
Bulk modulus k is the stress-strain ratio under hydrostatic pressure, also called incompressibility. It measures resistance to uniform compression: k = -ΔP/(ΔV/V). Higher k means more incompressible.
How is shear modulus (μ) measured?
Shear modulus μ = τ/tan(θ) is measured using a shear test. Apply tangential force (shear stress τ) and measure the angle of deformation θ. Alternatively, calculate from μ = E/[2(1+ν)] using Young’s modulus and Poisson’s ratio.
What is shear modulus (μ)?
Shear modulus μ (also called rigidity) is the ratio of shear stress to shear strain. It measures resistance to shape change without volume change. Fluids have μ = 0 (cannot support shear).
Write the generalized Hooke’s Law
σ_ij = c_ijkl ε_kl, where σ_ij is stress tensor, c_ijkl is elastic stiffness tensor, ε_kl is strain tensor. Inverse form: ε_ij = s_ijkl σ_kl, where s_ijkl is elastic compliance tensor.
How many independent elastic coefficients for a fully anisotropic material?
21 independent elastic coefficients. Starting from 81 components (3×3×3×3 tensor), symmetry of stress/strain tensors reduces to 36, and energy symmetry (c_ijkl = c_klij) reduces to 21.
How many elastic constants for isotropic material?
2 independent elastic constants. Commonly: λ and μ (Lamé parameters), or k and μ (bulk and shear modulus), or E and ν (Young’s modulus and Poisson’s ratio). All other moduli can be derived from any pair.
How many elastic constants for Transverse Isotropic (TI) material?
5 independent elastic constants. TI symmetry (cylindrical symmetry about one axis) reduces the 21 constants of fully anisotropic material to 5. TI is common in layered shales.
Why introduce material symmetries?
Material symmetry reduces the number of elastic constants needed: Fully anisotropic (21) → Transverse Isotropic/TI (5) → Isotropic (2). This simplifies modeling while capturing essential physics of layered rocks like shale.
What is Transverse Isotropic (TI) media?
TI media have symmetry about one axis normal to a plane of isotropy (cylindrical symmetry). Properties are uniform in the plane but different perpendicular to it. Example: horizontally layered shale (VTI).
Why is TI important in seismic exploration?
Shale is often TI due to horizontal layering. Fractured carbonates with parallel fractures can be TI. TI is more realistic than isotropic assumptions but simpler than fully anisotropic. Affects velocity, AVO, time-depth conversion, and NMO correction.
What is anisotropy?
Anisotropy means physical properties vary with direction. In seismic: wave velocity depends on propagation direction. Causes: layering (shale), aligned fractures, stress-induced alignment. Isotropic: properties same in all directions (e.g., well-sorted sandstone).
What are the three main types of anisotropy?
1) VTI (Vertical Transverse Isotropy): horizontal layering, vertical symmetry axis (shale). 2) TTI (Tilted Transverse Isotropy): dipping layers, tilted symmetry axis. 3) HTI (Horizontal Transverse Isotropy): vertical fractures, horizontal symmetry axis.
What physical principles form the basis of the wave equation?
1) Newton’s 2nd law (F = ma), 2) Hooke’s law (σ = cε). Combining these gives the elastodynamic wave equation: ∂²u/∂t² = (c/ρ)(∂²u/∂x²) for 1D case.
Write the 1D wave equation
∂²u/∂t² = (c/ρ)(∂²u/∂x²), where u is particle displacement, t is time, c is elastic constant, ρ is density, x is position. Wave velocity V = √(c/ρ).
Write the P-wave velocity equation
V_p = √[(λ + 2μ)/ρ] = √[(k + 4μ/3)/ρ], where V_p is P-wave velocity, λ is Lamé’s first parameter, μ is shear modulus, k is bulk modulus, ρ is density.
