Strain-Stress B PDF

Deformation Compiled By: Prof. Fathy Hassan Measures of Strain Strain may be recognized as a change in line length, angles between lines, or volume. A component of deformation dealing with shape and volume change. Distance between some particles changes. Angle between p...

Deformation Compiled By: Prof. Fathy Hassan Measures of Strain Strain may be recognized as a change in line length, angles between lines, or volume. A component of deformation dealing with shape and volume change. Distance between some particles changes. Angle between particle lines may change. The quantity or magnitude of the strain is given by several measure based on change in: ♣ Length (longitudinal strain) – ε ♣ Angle (angular or shear strain) -  ♣ Volume (volumetric strain) – εv Extension or Elongation, ε : change in length per length ε = (ℓ-ℓo) / ℓo = Δℓ/ℓo [dimensionless] Where ℓ and ℓo are the final and original lengths of a linear object Note: Shortening is negative extension (i.e., ε < 0) e.g., ε = - 0.2 represents a shortening of 20% Example: Iflongitudinal a belemnite of an original length (ℓo) of 10 cm is now 12 cm (i.e., ℓ=12 cm), the strain is positive, and ε = (12-10)/10 * 100% which gives an extension, ε = 20% Other Measures of Longitudinal Strain Stretch: s = ℓ/ℓo = 1+ ε =  [no dimension] X =   = s1 Y =   = s2 Z =   = s3 These principal stretches represent the semi-length of the principal axes of the strain ellipsoid. For Example: ♣ Given ℓo = 100 and ℓ = 200 ♣ Extension: ε = (ℓ-ℓo)/ ℓo = (200-100)/100 = 1 or 100% ♣ Stretch: s = 1+ ε = ℓ/ℓo = 200/100 = 2 ♣ i.e., The line is stretched twice its original length! Quadratic elongation:  = s2 =(1+ ε)2 Reciprocal quadratic elongation:  =   NOTE: Although ´ is used to construct the strain Mohr circle,  can be determined from the circle! Example: ♣ Given ℓo = 100 and ℓ = 200, then  = s2 = 4  Measures of Longitudinal Strain (ε) ℓo ℓ ε = (ℓ-ℓo)/ ℓo = Δℓ/ℓo s = ℓ/ℓo εx = 100% Sx = 2 εy = -50% S y = 0.5 Volumetric Strain (Dilation) Gives the change of volume compared with its original volume Given the original volume is vo, and the final volume is v, then the volumetric stain, εv is: εv =(v-vo)/vo = Δ v/vo [no dimension] Angular Shear (γ): change in angle between two perpendicular lines.  Lines of no finite elongation - lnfe ▪ If we draw a circle on a deck of card, and deform the deck, the strain ellipse will intersect the original circle (if we redraw it) along the two lines of lnfe ▪ If we draw a new circle in the deformed state (in a different color, say red) and restore the deck to its original unstrained configuration, the red circle becomes an ellipse ▪ This ellipse which has long axis perpendicular to the strain ellipse is called the reciprocal strain ellipse lnfe  Extension and Shortening Fields Fields in the strain ellipsoid, separated by the line of no incremental (infinitesimal) longitudinal strain (lnie) Boundaries between the shortening and extension are always at 45 degrees either side of incremental principal axes of shortening and extension during each incremental strain event As ellipse flattens, material lines migrate through the boundaries separating fields of shortening and extension Note that rock particle lines or planes can undergo transition from shortening (folding) to extension (boudinage) without change in orientation of the principal axes of strain.  Fields of Shortening and Extension  Boudins Sandstone beds in Ain Sokhna region along Gulf of Suez Road, Egypt Strain Ratio ▪ We can think of the strain ellipse as the product of strain acting on a unit circle ▪ A convenient representation of the shape of the strain ellipse is the strain ratio ▪ Rs = (1+ ε1)/(1+ ε3) = S1/S3 = X/Z ▪ It is equal to the length of the long axis over the length of the short axis. Types of Homogeneous Strain at Constant Volume ▪ 1. Axially symmetric extension ▪ Extension in one principal direction (1) and equal shortening in all directions at right angles (2 and 3) ♣ 1 > 2 = 3 < 1 ♣ X>Y=Z ♣ Produces prolate strain ▪ The strain ellipsoid is ellipsoid with extension in prolate spheroid or the X direction and shortening in Y and Z cigar shaped ♣ Hotdog to football shaped Types of Homogeneous Strain at Constant Volume ▪ 2. Axially symmetric shortening ▪ This involves shortening in one principal direction (3) and equal extension in all directions at right angles (1 and 2 ). ♣ X=Y>Z ♣ 1 = 2 > 1 > 3 ♣ Produces an oblate ellipsoid with equal amounts of extension in ▪ Strain ellipsoid is oblate the directions perpendicular to spheroid or pancake- the shortening direction ♣ The strain ellipsoid resembles a shaped hamburger Types of Homogeneous Strain at Constant Volume ▪ 3. Plane Strain ▪ The intermediate axis of the ellipsoid has the same length as the diameter of the initial sphere, i.e.: ε2= 0, or 2 =(1+ ε2)2 =1 ▪ Shortening (3) and extension (1), respectively, occur parallel to the ♣ X>1>Z other two principal directions ♣ One axis remains the same as ▪ The strain ellipsoid is a triaxial before deformation, and commonly this is Y ellipsoid (has different lengths) ♣ The description is often that of a ♣ 1 > 2 = 1 > 3 two-dimensional ellipse in the XZ plane, with extension along X and contraction along Z Types of Homogeneous Strain at Constant Volume ▪ 4. General Strain ▪ Involves extension or shortening in each of the principal directions of strain ♣1 > 2 > 3 ♣ X>Y>Z ♣ Also known as triaxial strain all  1 ♣ NOT the same as general shear  Types of Homogeneous Deformation of a cube Containing a Sphere  Simple Shear Simple shear is a three-dimensional constant-volume, plane strain A single family of parallel planes is undistorted in the deformed state, and parallel the same family of planes in the undeformed state Involves a change in orientation of material lines along two of the principal axes (1 and 3 ). Simple shear is analogous to the process that occurs when you place a deck of cards on a table and then gently press down on the top of the deck and move your hands to the right or left A cube subjected to a simple shear event is converted into a parallelogram resulting in a rotation of the finite strain axes The sides of the parallelogram will progressively lengthen as deformation proceeds but the top and bottom surfaces neither stretch nor shorten. Instead, they maintain their original length, which is the length of the edge of the original cube.   = tan  Shearing Note: that the area of the initial box and final parallelogram is the same.  = tan (45°) = 1 Every whole number (1,2,3…) Flattening represents shear of one shear zone width - i.e. g = 2 means that the shear zone has slipped 2 shear zone width units.  Pure Shear In contrast to simple shear, pure shear is a three-dimensional constant- volume, irrotational, homogeneous flattening, which involves either plane strain or general strain. Lines of particles that are parallel to the principal axes of the strain ellipsoid have the same orientation before and after deformation It does not mean that the principal axes coincided in all increments! During homogeneous flattening a sphere is transformed into a pancake-like shape and a box is changed into a tablet or book-like form. During pure shear the sides of the cube that are parallel to the z-axis are shortened, while the lengths of the sides that are parallel to the x-axis increase. In contrast, the lengths of the sides of the cube that are parallel to the y-axis remain unchanged. When such geometrical changes occur during the transformation of a rock body to a distorted state then the mechanism of distortion is termed plane strain.  Strain Markers Strain Markers - A deformed feature in the rock that can be measured to determine strain. Have to know the original shape for comparison. Should have the same mechanical characteristics as the original rock. Good strain Fossils Pebbles markers include: ♣ Pebbles ♣ Ooids Ooids ♣ Fossils ♣ Vesicles ♣ Pillow Pillows Basalts ♣ Burrows  Line changes when a circle becomes an ellipse Initially circular object will become ellpises when homogeneously deformed. The burrow can be used as a strain gauge. By determining the stretch (S) and extension (ε) of the long and short axes of the ellipse we can describe the amount of lengthening and shortening the rock containing the burrow experienced. How to get at ε and S? If we assume there has been no volume change: Aellipse = Acircle ( ab = r2) ab = r2 What’s ε&S a = 2.6/2 = 1.3 b = 2.2/2 = 1.1  Angular Shear: Change of angles between lines Angular shear:  (psi) - we need to find a line (L) that was originally perpendicular to the line in question. Angular shear strain describes the departure of this line from its initially perpendicular relationship with L. The sign convention is CW = (+), CCW = (-), magnitude is in degrees (°) this is also the classic right-hand rule. All lines have changed Original state length, 6 have changed orientation. Final state Calculate the angular shear in each photo  Shape and Intensity –Flinn Diagram The Flinn diagram, named for British geologist Derek Flinn, is a plot of axial ratios. In strain analysis, we typically use strain ratios, so this type of plot is very useful. The horizontal axis is the ratio Y/Z = b (intermediate stretch/minimum stretch) and the vertical axis is X/Y = a (maximum stretch divided by intermediate stretch). The parameters a and b may be written: ♣ a = X/Y = (1 + ε1)/(1 + ε2) ♣ b = Y/Z = (1 + ε2)/(1 + ε3)  Flinn Diagram  How Rocks Deform General characteristics of rock deformation. Elastic deformation – Temporary change in shape or size that is recovered when the deforming force is removed. Think “rubber band”. The rock returns to nearly its original size and shape when the stress is removed. Elastic limit – Point beyond which permanent deformation occurs. Once the elastic limit (strength) of a rock is surpassed, it either flows (ductile deformation) or fractures (brittle deformation). Plastic deformation – Permanent change in shape or size that is not recovered when the stress is removed. Occurs by the slippage of atoms or small groups of atoms past each other in the deforming material, without loss of cohesion, think “deck of cards” Brittle deformation (rupture) – Loss of cohesion of a body under the influence of deforming stress. Usually occurs along sub-planar surfaces that separate zones of coherent material. Brittle Deformation Plastic (Rupture) Deformation  Law of Elasticity - Hooke’s Law A linear equation, with no intercept, relating stress () to strain (ε) For longitudinal strain: =Eε The proportionality constant ‘E’ between stress and longitudinal strain is the Young’s modulus Typical values of E for crustal rocks are on the order of 10-11 Pa Typically, elastic strains are less than a few. percent of the total strain Neither the strain (ε) nor the strain rate (ε ) of a plastic solid is related to stress (). The terms elastic and plastic describe the nature of the material Brittle and ductile describe how rocks behave. Rocks are both elastic and plastic materials, depending on the rate of strain and the environmental conditions (stress, pressure, temperature), and we say that rocks are viscoelastic materials. Stress- Strain Relation Brittle vs. Ductile ♣ Brittle rocks fail by fracture at less than 3-5% strain ♣ Ductile rocks are able to sustain, under a given set of conditions, 5-10% strain before deformation by fracturing  Strain Rate, ε. Strain rate: The time interval it takes to accumulate a certain amount of strain ♣ Change of strain with time (change in length per length per time). ♣ Slow strain rate means that strain changes slowly with time ♣ How fast change in length occurs per unit time. ε = dε /dt = (dℓ/ℓo)/dt [T-1] e.g., s-1 ▪ Examples: 30% extension (i.e., dε = 0.3) in one hour (i.e., dt =3600 s) translates into:. ♣ ε = dε /dt = 0.3/3600 s. ♣ ε = 0.000083 s-1 = 8.3 x 10-5 s -1 30% extension (i.e., dε = 0.3) in 1 my (i.e., dt = 1000,000 yrs. ) translates into: ε =. dε /dt ♣ = 0.3/1000,000 yrs. ; 0.3/(1000000)(365 x 24 x 3600 s)= 9.5 x 10-15 s-1 If the rate of growth of your fingernail is about 1 cm/year, the strain rate of your fingernail is: ε = (ℓ-ℓo) / ℓo = (1-0)/0 = 1 (no units) ε. = d ε /dt = 1/yr = 1/(365 x 24 x 3600 s); = 3.1 x 10-8 s-1  Factors that affect deformation ▪Temperature; Pressure; Strain rate; Rock type. 1. Temperature ♣ Increasing T increases ductility by activating crystal-plastic processes ♣ Increasing T lowers the yield stress (maximum stress before plastic flow), reducing the elastic range ♣ Increasing T lowers the ultimate rock strength 2. Confining Pressure – as pressure increases (weight of overlying rocks), materials become more ductile. 3. Rate of deformation – if stress is applied rapidly, rocks will tend to be brittle. Example: chewing gum 4. Composition of rocks – at the same temperature and pressure, some rocks will be more brittle than others. Example: slate (brittle) and copper (ductile) Effects of Rock Type on Deformation Incompetent Rock ▪ Some rocks are stronger than others. ▪ Competent: rocks that deform only under great stresses. ▪ Incompetent: rocks that deform under moderate to low stresses. ▪ Rock competency (competence is a term that describes the resistance of rocks to Competent Rock flow. ▪ Examples of incompetent rocks (ductile) are slate, phyllite, schist, or salt and shale, whereas crystalline granite, basalt, gneiss, or quartz sandstone and metaquartzite are examples of competent rocks ( Brittle). ▪ In boudinage structures rupture of competence layer occurs, while the incompetence layer flows around the brittle competence rocks. Competent & Incompetent Competent Rock Incompetent Rock Than k You