STRESS & STRAIN

SYSTEMATIC JOINTS / FRACTURES

Strike-slip fault near Las Vegas, NV, Source: M. Miller, U. of Oregon Faults Strike-Slip Fault Strike-Slip Fault (left-lateral)

Syncline, Israel Folds

FORCES & VECTORS Force is any action which alters, or tends to alter Force is any action which alters, or tends to alter Newton II law of motion :F = M a Newton II law of motion :F = M a Unit force : kgm/s 2 = newton (N) or dyne = gram cm/s 2 ; N = 10 5 dynes Unit force : kgm/s 2 = newton (N) or dyne = gram cm/s 2 ; N = 10 5 dynes BASIC CONCEPTS (a). Force: vector quantity with magnitude and direction (b). Resolving by the parallelogram of forces Modified Price and Cosgrove (1990) Two Types of Force Body Forces (i.e. gravitational force) Body Forces (i.e. gravitational force) Contact Forces (i.e. loading) Contact Forces (i.e. loading)

SCALARS temperature speed volume time length VECTORS force and stress (on a surface) temperature gradient acceleration Earth’s gravity field Earth’s magnetic field velocity Mantle convection flow ocean currents Scalars vs. Vectors

VECTOR & COORDINATE SYSTEM

Term for Stress & Strain *) Important distinction between two quantities

STRESS Stress defined as force per unit area: σ = F/A A = area, Stress units = Psi, Newton (N), Pascal (Pa) or bar (10 5 Pa) A = area, Stress units = Psi, Newton (N), Pascal (Pa) or bar (10 5 Pa) (Davis and Reynolds, 1996) (Twiss and Moores, 1992)

Deformation Dilation: a change in volume Translation: a change in place Rotation: a change in orientation Distortion: a change in form

Basic Fundamental Structural Geology STRESS (  ) & STRAIN (  ) “As Geologist I don’t believe in stress (John Ramsay)”

GEOLOGY CARTESIAN COORDINATE SYSTEM

STRESS Stress at a point in 2D Stress at a point in 2D Types of stress Types of stress Stress (  ) Normal Stress (  n ) Shear Stress (  s ) Normal stress (  N ) (+) Compressive (-) Tensile Shear stress (  S ) (+)(-)

STRESS ON A PLANE AND AT A POINT Stress Tensor Notation  11  12  13  =  21  22  23  31  32  33

Geologic Sign Convention of Stress Tensor (Twiss and Moores, 1992)

Stress Ellipsoid FUNDAMENTAL STRESS EQUATIONS Principal Stress:  1     All stress axes are mutually perpendicular All stress axes are mutually perpendicular Shear stress are zero in the direction of principal stress Shear stress are zero in the direction of principal stress Stress Tensor Notation  11  12  13  11  12  13  =  21  22  23  31  32  33  31  32  33  12 =  21,  13 =  31,  23 =  32

Stress Ellipsoid a) Triaxial stress b) Principal planes of the ellipsoid the ellipsoid (Modified from Means, 1976)

σ2σ2 σ1σ1 σ3σ3 σ1σ1 σ1σ1 σ1σ1 σ1σ1 σ1σ1 σ2σ2 σ2σ2 σ2σ2 σ2σ2 σ3σ3 σ3σ3 σ3σ3 σ3σ3 σ2σ2 ELIPSOID TEGASAN σ 1 > σ 2 = σ 3 σ 1 = σ 2 > σ 3 σ 1 > σ 2 > σ 3

The State of Two-Dimensional Stress at Point (Twiss and Moores, 1992) Principal Stress:  1    x,  z = Surface Stress

The State of 3-Dimensional Stress at Point (Twiss and Moores, 1992) Principal Stress:  1    

SPECIAL STATE OF STRESS

Mohr Diagram 2-D A. Physical Diagram A. Mohr Diagram (Twiss and Moores, 1992)

 1 +  3 -  1 –  3  N = cos 2  22  s = Sin 2   1 –  3 2 Stress Equation:

B. Mohr Diagram A. Physical Diagram Planes of maximum shear stress Mohr Diagram 2-D (Twiss and Moores, 1992)

Mohr diagram is a graphical representative of state of stress Mohr diagram is a graphical representative of state of stress Mean stress is hydrostatic component which tends to produce dilation Mean stress is hydrostatic component which tends to produce dilation Deviatoric stress is non hydrostatic which tends to produce distortion Deviatoric stress is non hydrostatic which tends to produce distortion Differential stress, if greater is potential for distortion Differential stress, if greater is potential for distortion (Davis and Reynolds, 1996)

Image of Stress

Body force works from distance and depends on the amount of materials affected (i.e. gravitational force). Body force works from distance and depends on the amount of materials affected (i.e. gravitational force). Surface force are classes as compressive or tensile according to the distortion they produce. Surface force are classes as compressive or tensile according to the distortion they produce. Stress is defined as force per unit area. Stress is defined as force per unit area. Stress at the point can be divided as normal and shear component depending they direction relative to the plane. Stress at the point can be divided as normal and shear component depending they direction relative to the plane. Structural geology assumed that force at point are isotropic and homogenous Structural geology assumed that force at point are isotropic and homogenous Stress vector around a point in 3-D as stress ellipsoid which have three orthogonal principal directions of stress and three principal planes. Stress vector around a point in 3-D as stress ellipsoid which have three orthogonal principal directions of stress and three principal planes. Principal stress  1 >  2 >  3 Principal stress  1 >  2 >  3 The inequant shape of the ellipsoid has to do with forces in rock and has nothing directly to do with distortions. The inequant shape of the ellipsoid has to do with forces in rock and has nothing directly to do with distortions. Mohr diagram is a graphical representative of state of stress of rock Mohr diagram is a graphical representative of state of stress of rock STRESS

STRAIN UNDEFORMEDDEFORMED Strain is defined as the change (in size and shape) of a body resulting from the action of an applied stress field

TYPES OF STRAIN

Fundamental Strain Equations Extension (e) = (l f – l o )/l o Stretch (S) = l f /l o = 1 + e Lengthening e>0 and shortening e<0

SHEAR STRAIN

Strain Ellipsoid S 1 = Maximum Finite Stretch S 3 = Minimum Finite Stretch (Davis and Reynolds, 1996)

τ1τ1 τ3τ3 τ2τ2 ELIPSOID TERAKAN τ1τ1 τ3τ3 τ2τ2 τ1τ1 τ3τ3 τ2τ2 τ1τ1 τ3τ3 τ2τ2 τ1τ1 τ3τ3 τ2τ2 τ1τ1 τ3τ3 τ2τ2 τ 1 > τ 2 = τ 3 τ 1 = τ 2 > τ 3 τ 1 > τ 2 > τ 3

Progressive Deformation (Davis and Reynolds, 1996)

Strain Measurement Geological Map Geologic Cross-section Seismic Section Outcrop Thin Section Knowing the initial objects Shape Size Orientation

Strain Measurement from Outcrop

 = gap 

STRESS vs. STRAIN

Relationship Between Stress and Strain Evaluate Using Experiment of Rock Deformation Rheology of The Rocks Using Triaxial Deformation Apparatus Measuring Shortening Measuring Strain Rate Strength and Ductility

(Modified from Park, 1989) Deformation and Material A.Elastic strain B.Viscous strain C.Viscoelastic strain D.Elastoviscous E.Plastic strain Hooke’s Law: e =  /E, E = Modulus Young or elasticity Newtonian :  =  viscosity,  = strain-rate

Stress Ellipsoid Strain Ellipsoid

Relationship Between Stress and Strain Evaluate Using Experiment of Rock Deformation Rheology of The Rocks Using Triaxial Deformation Apparatus Measuring Shortening Measuring Strain Rate Strength and Ductility

STRESS – STRAIN RELATIONS

Stress – Strain Diagram A.Onset plastic deformation B.Removal axial load C.Permanently strained D.Plastic deformation E.Rupture

Effects of Temperature and Differential Stress

BRITTLE & DUCTILE

CATACLASTIC AND PLASTIC DEFORMATION MECHANISMS