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Finite strain II We have learned how to quantify the strain of each individual line. Often, there are several material lines, oriented at different directions. In that case we would like to describe the changes of the body as a whole. Finite strain II: The strain ellipse A simple geometric object that describes the orientation of lines at all directions is a circle. A circle undergoing homogeneous strain becomes an ellipse. Likewise, a sphere becomes ellipsoid following deformation. Finite strain II: The strain ellipse The principal axes of strain: Are the long and the short axes of the strain ellipse. Lines parallel to the principal axes undergo no shear strain. Principal strains: Are the stretch along the principal axes. In 2D : S1> S3 In 3D : S1 > S2 > S3 The principal axes and the principal strains are the Eigenvectors and the Eigenvalues (squared) of DTD, respectively. Finite strain II: The strain ellipse • Principal axes are perpendicular lines that were perpendicular prior to the deformation. • Lines parallel to the principal axes undergo no shear strain. Finite strain II: The strain ellipse Non-unique strain path: Different strain paths may produce identical strain ellipse. For example: Finite strain II: The strain ellipse Non-commutability of strain: In general, the order at which strains and rotations are superposed makes a difference in terms of the final product. For example: Finite strain II: Infinitesimal versus finite strain ellipses The finite strain may be thought as the sum of many infinitesimal strains. At any stage in the deformation, there are two strain ellipses that represent the strain of the rock. • The finite strain ellipse: represents the total deformation. • The infinitesimal strain ellipse: represents the instantaneous strain that the particle feels. (show a computer simulation) Finite strain II: Infinitesimal versus finite strain ellipses • Coaxial deformation: If the axes of the finite and infinitesimal strain ellipse remain parallel throughout the deformation. • Non-coaxial deformation: if the axes of the finite and infinitesimal strain ellipse are not parallel. We have seen that while progressive pure shear is coaxial, progressive simple shear is non-coaxial. Finite strain II: Infinitesimal versus finite strain ellipses Lines of no finite elongation (LNFE): Two lines for which the final and initial lengths are equal. • f+ : lines within that sector are longer in the final state. • f-: lines within that sector are shorter in the final state. Finite strain II: Infinitesimal versus finite strain ellipses Lines of no infinitesimal elongation (LNIE): Two lines for whose lengths remain unchanged during the next increment of infinitesimal strain. • i+ : lines within that sector will become infinitesimally longer during the next infinitesimal strain. • i-: lines within that sector will become infinitesimally shorter during the next infinitesimal strain. Finite strain II: Infinitesimal versus finite strain ellipses Note that: • The LNIE are perpendicular. • The angle between the LNIE and the principal axes of the infinitesimal strain ellipse is equal to 45 degrees. Finite strain II: Infinitesimal versus finite strain ellipses In the most general case, the superposition of the finite and the infinitesimal strain ellipse may form 4 fields: • (f-,i-) lines are shorter than they started, and they will continue to shorten in the next increment. • (f-,i+) lines are shorter than they started, but will begin to lengthen in the in the next increment. • (f+,i+) lines are longer than they started, and they will continue to lengthen in the next increment. • (f+,i-) lines are longer than they started, but will begin to shorten in the next increment. Finite strain II: Infinitesimal versus finite strain ellipses Finite strain II: Infinitesimal versus finite strain ellipses So far we have dealt with straight lines. Let's examine the case of curved lines and lines of finite thickness (i.e., beds). (show computer animation) Finite strain II: homogeneous versus heterogeneous strains The deformation is homogeneous if: • Straight lines remain straight. • Parallel lines remain parallel. Finite strain II: homogeneous versus heterogeneous strains Otherwise, the deformation is heterogeneous. Often it is a matter of scale, i.e. the deformation may be homogeneous on a small scale, but heterogeneous on a broader scale (e.g., a fold). Finite strain II: Continuous versus discontinuous Continuous: when strain vary smoothly throughout the volume. Discontinuous: abrupt changes in the strain distribution, i.e., faults, joints, etc. Mathematically, the treatment of discontinuous deformation is much more complicated.