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2 stress & strain
- 1. Deformation:Deformation: STRESS & STRAINSTRESS & STRAIN
- 2. DeformationDeformation Dilation:Dilation: a change in volumea change in volume Translation:Translation: a change in placea change in place Rotation:Rotation: a change in orientationa change in orientation Distortion:Distortion: a change in forma change in form
- 3. Term for Stress & Strain *) Important distinction between two quantities
- 4. SCALARSSCALARS temperaturetemperature speedspeed volumevolume timetime lengthlength VECTORSVECTORS force and stressforce and stress (on a surface)(on a surface) temperaturetemperature gradientgradient accelerationacceleration Earth’sEarth’s gravitygravity fieldfield Earth’sEarth’s magneticmagnetic fieldfield velocityvelocity MantleMantle convectionconvection flowflow oceanocean currentscurrents ScalarsScalars vs.vs. VectorsVectors
- 5. VECTOR & COORDINATE SYSTEMVECTOR & COORDINATE SYSTEM
- 6. FORCES & VECTORSFORCES & VECTORS • ForceForce is any action which alters, or tends to alteris any action which alters, or tends to alter • Newton II law of motion :Newton II law of motion : F = M aF = M a • Unit force : kgm/sUnit force : kgm/s22 = newton (N) or dyne = gram cm/s= newton (N) or dyne = gram cm/s22 ; N = 10; N = 1055 dynesdynes BASIC CONCEPTSBASIC CONCEPTS (a). Force: vector quantity with magnitude and direction(a). Force: vector quantity with magnitude and direction (b). Resolving by the parallelogram of forces(b). Resolving by the parallelogram of forces Modified Price and Cosgrove (1990)Modified Price and Cosgrove (1990) Two Types of ForceTwo 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)
- 7. STRESSSTRESS Stress defined as force per unit area:Stress defined as force per unit area: σ = F/Aσ = F/A A = area, Stress units = Psi, Newton (N),A = area, Stress units = Psi, Newton (N), Pascal (Pa) or bar (10Pascal (Pa) or bar (1055 Pa)Pa) (Davis and Reynolds, 1996)(Davis and Reynolds, 1996) (Twiss and Moores, 1992)(Twiss and Moores, 1992)
- 8. STRESSSTRESS • Stress at a point in 2DStress at a point in 2D • Types of stressTypes of stress Stress(Stress(σσ)) NormalStress( NormalStress(σσ nn)) Shear Stress ( Shear Stress (σσ ss )) Normal stress (Normal stress (σσNN)) (+) Compressive(+) Compressive (-) Tensile(-) Tensile Shear stress (Shear stress (σσSS)) (+)(+) (-)(-)
- 9. STRESS ON A PLANE AND AT A POINT Stress Tensor Notation σ11 σ12 σ13 σ = σ21 σ22 σ23 σ31 σ32 σ33
- 10. Stress EllipsoidStress Ellipsoid FUNDAMENTAL STRESS EQUATIONSFUNDAMENTAL STRESS EQUATIONS Principal Stress:Principal Stress: σσ11 >> σσ22 >> σσ33 • All stress axes are mutuallyAll stress axes are mutually perpendicularperpendicular • Shear stress are zero in theShear stress are zero in the direction of principal stressdirection of principal stress Stress Tensor NotationStress Tensor Notation σσ1111 σσ1212 σσ1313 σσ == σσ2121 σσ2222 σσ2323 σσ3131 σσ3232 σσ3333 σσ1212 == σσ2121,, σσ1313 == σσ3131,, σσ2323 == σσ3232
- 11. Stress EllipsoidStress Ellipsoid a) Triaxial stressa) Triaxial stress b) Principal planes ofb) Principal planes of the ellipsoidthe ellipsoid (Modified from Means, 1976)(Modified from Means, 1976)
- 12. σ2 σ1 σ3 σ1 σ1 σ1 σ1 σ1 σ2 σ2 σ2 σ2 σ3 σ3 σ3 σ3 σ2 ELIPSOID TEGASAN σ1 > σ2 = σ3 σ1 = σ2 > σ3 σ1 > σ2 > σ3
- 13. B. Principal stress components σ1 z x σ3 x1 x3 y y x2 x x y z σ2 x σzy σxy σyy σyz σyx σxx σzx σzz σxz z y Arbitrary coordinate planes A. Stress elipsoid C. General stress components z Principal coordinate planes The State ofThe State of 3-Dimensional3-Dimensional Stress at PointStress at Point (Twiss and Moores, 1992)(Twiss and Moores, 1992) Principal Stress:Principal Stress: σσ11 >> σσ22 >> σσ33
- 14. n - Planes of maximum shear stress Clockwise shear stress x3 x1 σs σs Counterclockwise shear stress θ' = +45º σ1 x3σ3 σ1 n + σs x1 θ = +45º σ1 σ3 2θ = +90º σn σs max Clockwise 2θ = −90' º σs max Counter clockwise σ3 B. Mohr DiagramB. Mohr DiagramA. Physical DiagramA. Physical Diagram Planes of maximum shear stressPlanes of maximum shear stress Mohr Diagram 2-DMohr Diagram 2-D (Twiss and Moores, 1992)(Twiss and Moores, 1992)
- 15. σσcc == σσoo + tan+ tan θθ ((σσnn)) The Coulomb Law of FailureThe Coulomb Law of Failure σσcc = critical shear stress= critical shear stress σσoo = cohesive strength= cohesive strength tantan θθ = coefficient= coefficient of internal frictionof internal friction σσnn = normal stress= normal stress (Modified from Davis and Reynolds, 1996)(Modified from Davis and Reynolds, 1996) Compressive FracturesCompressive Fractures
- 19. • Body force works from distance and depends on the amount of materialsBody force works from distance and depends on the amount of materials affected (i.e. gravitational force).affected (i.e. gravitational force). • Surface force are classes as compressive or tensile according to theSurface force are classes as compressive or tensile according to the distortion they produce.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 componentStress at the point can be divided as normal and shear component depending they direction relative to the plane.depending they direction relative to the plane. • Structural geology assumed that force at point are isotropic andStructural geology assumed that force at point are isotropic and homogenoushomogenous • Stress vector around a point in 3-D as stress ellipsoid which have threeStress vector around a point in 3-D as stress ellipsoid which have three orthogonal principal directions of stress and three principal planes.orthogonal principal directions of stress and three principal planes. • Principal stressPrincipal stress σσ11>>σσ22>>σσ33 • The inequant shape of the ellipsoid has to do with forces in rock and hasThe inequant shape of the ellipsoid has to do with forces in rock and has nothing directly to do with distortions.nothing directly to do with distortions. • Mohr diagram is a graphical representative of state of stress of rockMohr diagram is a graphical representative of state of stress of rock STRESSSTRESS
- 20. STRAIN UNDEFORMED DEFORMED Strain is defined as the change (in size and shape) of a body resulting from the action of an applied stress field
- 24. TYPES OF STRAIN B. Inhomogeneous strain A. Homogeneous strain H I H
- 28. L l = 5 cmo L' = 3 cm L l = 8 cmf L' = 4.8 cm Fundamental Strain Equations Extension (e) = (lf – lo)/lo Stretch (S) = lf/lo = 1 + e Lengthening e>0 and shortening e<0 Strain B. Shear strain Deformed State Strain R e= n Deformed State Undeformed State A. Extension and stretch Undeformed State R = 1 θ θ r θ θ r = Sn T R e tans = 1/2 ψt ψ γ ψ= tan ψ Shear Strain ( )γ
- 29. SHEAR STRAIN
- 31. S2 S2 S3 S3 S3 S1 S1 S1 Strain Ellipsoid S1 = Maximum Finite Stretch S3 = Minimum Finite Stretch (Davis and Reynolds, 1996)
- 32. τ1 τ3 τ2 ELIPSOID TERAKAN τ1 τ3 τ2 τ1 τ3 τ2 τ1 τ3 τ2 τ1 τ3 τ2 τ1 τ3 τ2 τ1 > τ2 = τ3 τ1 = τ2 > τ3 τ1 > τ2 > τ3
- 34. O N SimpleShear (NoncoaxialStrain) A B M S1 ML PureShear (CoaxialStrain) S3S3 S1 25%Flattering S3 S1 S3 S1+22º +31º S3 S1 S1 S3 30%Flattering +45º 40%Flattering Progressive Deformation (Davis and Reynolds, 1996)
- 35. Strain Measurement • Geological Map • Geologic Cross-section • Seismic Section • Outcrop • Thin Section Knowing the initial objects • Shape • Size • Orientation
- 36. Strain Measurement from Outcrop
- 37. ∆ = gap ∆
- 40. 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
- 41. (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 = σ/E, E = Modulus Young or elasticity Newtonian : σ = ηε, η = viscosity, ε = strain-rate
- 43. 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
- 44. STRESS – STRAIN RELATIONS
- 45. BRITTLE & DUCTILE DEFORMATIONS
- 50. THANK YOU
- 51. GEOLOGY CARTESIAN COORDINATE SYSTEMGEOLOGY CARTESIAN COORDINATE SYSTEM