Deformation and Strain Deformation occurs in response to forces (stress) No simple stress-deformation relations Deformation: collective displacements of points in a body. distortion translation rotation Strain: describes changes displacements of points in a body relative to one another. Components of deformation: The displacement of the corner of the square (rigid body) represent the deformation field with strain ellipse. Strain along axes x and y, which are the principal strain axes. The displacement field is subdivided into 3 components: Distortion Translation Rotation What is the missing element Change in shape Deviatoric stress: unequal shape changes Hydrostatic component: equal changes of volume (burial) Dilation: area or volume change positive = volume increase negative = volume decrease Review of deformation Rigid body translation Rigid body rotation Strain (distortion) Volume change (or dilation) Homogenous strain and the strain ellipsoid Homogeneous strain Original straight lines remain straight Origin parallel lines remain parallel Circles become ellipses, in 3-D, spheres becomes ellipsoids Homogenous strain (2-D): two material lines that do not rotate relative to one another Homogeneous strain describes the transformation of a square to a rectangle or a circle to an ellipse. Two material lines that remain perpendicular before and after strain are the principal axes of the strain ellipse The dashed lines are material lines that do not remain perpendicular after strain, the rotate towards the long axes of the strain ellipse. Homogeneous strain Note the two material lines that form the ellipse change length. In 3-D, the three material lines that remain perpendicular also change length from initial to final stages. The lines that are perpendicular before and after strain are called Principal Strain Axes. Finite strains Xf and Yf are the same in (a) and (b), but the strain path is different. Importance of incremental strain history of rocks and limitation of finite strain analysis. Coaxial and non-coaxial strain accumulation Non-coaxial strain accumulation All material lines except those that remain perpendicular before and after strain, rotate relative to one another. Here the principal incremental strain axes rotate relative to the finite strain axes. Coaxial and non-coaxial strain accumulation Coaxial strain accumulation The principal incremental strain axes remain perpendicular to the finite strain axes. The magnitude of strain axes change with each step in both types of strain Strain Quantities How much strain is necessary to produce these strains? How do we determine this? Quantifying Strain Line length change Volume change Angular change Small fold with axial plane cleavage. Longitudinal strain Longitudinal strain is expressed by elongation, e: e = (lf – lo)/lo Negative e = shortening Positive e = extension Maximum, e1 Minimum, e3 e x 100 = % extension or shortening Volumetric strain Volumetric strain, D D = Vf – Vo)/Vo or D = dV/Vo Negative D = V loss Positive D = V gain Angular strain Angular strain is the change in angle between 2 lines that were initially perpendicular. The change in the angle, angular shear, y. The tangent of y is shear strain, g g = tan y D = dV/Vo Other strain quantities Quadratic elongation l = (lf / lo)2 or (1 + e)2 l used in Mohr Circle for strain Stretch, root of l s = l ½ = lf / lo = 1 + e Convenient measures. They describe the lengths of the principal axes (X, Y, and Z) and the strain ellipsoid. change in the angle, angular shear, y. Mohr Circle for Strain Unit square is shortened 50% and lengthened 100% e1 = 1 and e3 = -.0.5 thus l1 = 4 and l3 = 0.25 Now get reciprocal values, thus l1 = 0.25 and l3 = 4 Plot on Mohr Strain Circle Mohr Circle for Strain For a deformed rock: we use the reciprocal values of the principal strain a)Plotted in gl - space, where l = 1/l and g = gl Where, l is quadratic elongation (l/lo)2 and g is shear strain (tangent of angular shear) Mohr Circle for Strain Plot on Mohr Circle with l1 = 0.25 and l3 = 4 Easy to find longitudinal strain (e value) and angular strain for any line oriented at angle °) to strain axes. Plot 2 ° and find corresponding strain values l = 1.1 and g = 1.5 Strain States (a) General strain (b) Axially symmetric extension (c) Axially symmetric shortening (d) Plane strain (e) Simple shortening Strain Representation XY sectional ellipse of the finite strain ellipsoid in Alps and Sectional ellipsoid on cross-section Long axis of strain ellipse lies in the direction of thrust transport Strain ratio increases with depth What do we learn? Measure strain magnitudes across a region, an outcrop or hand specimen Quantify strain history Sharp increase of strain may define a region of high strain = ductile shear zone How much strain is required to fold a rock? Shortening rate in metamorphic rocks or along brittle faults