Mathematical Geology

, Volume 29, Issue 1, pp 93–130 | Cite as

A general deformation matrix for three-dimensions



A deformation that is obtained by any simultaneous combination of two steady-state progressive deformations: simple shearing and a coaxial progressive deformation, involving or not a volume change, can be expressed by a single transformation, or deformation matrix. In the general situation of simple shearing in a direction non-orthogonal with the principal strains of the coaxial progressive deformation, this deformation matrix is a function of the strain components and the orientation of shearing. In this example, two coordinate systems are defined: one for the coaxial progressive deformation (xi system), where the principal and intermediate strains are two horizontal coordinate axes, and another for the simple shear (x i t’ system), with any orientation in space. For steady-state progressive deformations, from the direction cosines matrix that defines the orientation of shear strains in the xi coordinate system, an asymmetric finite-deformation matrix is derived. From this deformation matrix, the orientation and ellipticity of the strain ellipse, or the strain ellipsoid for three-dimensional deformations, can be determined. This deformation matrix also can be described as a combination of a rigid-body rotation and a stretching represented by a general coaxial progressive deformation. The kinematic vorticity number (W k is derived for the general deformation matrix to characterize the non-coaxiality of the three-dimensional deformation. An application of the deformation matrix concept is given as an example, analyzing the changes in orientation and stretching that variously-oriented passive linear markers undergo after a general two-dimensional deformation. The influence of the kinematic vorticity number, the simple and pure shear strains, and the obliquity between the two deformation components, on the linear marker distribution after deformation is discussed.

Key Words

simple shear coaxial progressive deformation velocity gradient tensor deformation tensor vorticity tensor strain ellipsoid passive linear marker 



Components of the velocity gradient tensor L


Constant coefficients of Equation (19) [see Eq. (B.10)]


Deformation tensor


Principal strain of the coaxial progressive deformation


Direction cosines of aij


Velocity gradient tensor (matrix a)


Rigid-body rotation matrix (direction cosines matrix) and axial ratio or ellipticity of the strain ellipse


Components of the simple shearing velocity gradient tensor


Stretching tensor


Principal stretch of the strain ellipsoid


Velocity field tensor


Magnitude of the vorticity vector


Vorticity or spin tensor


Kinematic vorticity number (varies between 1 for simple shearing, and 0 for pure shearing)


Coordinate system in the deformed state for the coaxial progressive deformation


Coordinate system in the deformed state for the simple shearing progressive deformation


Coordinate system in the undeformed state


Angle betweenx i andX j


Simple shear strain along the i -x j plane


Instantaneous simple shear strain rate along the i - x j plane


Dilation or anisotropic volume change


Kronecker delta (unit diagonal matrix)


Principal strain rate of the coaxial progressive deformation


Orientation of any line in the undeformed state with respect tox 1


Orientation of any line in the deformed state with respect to x1′ (e.g., principal stretch of the strain ellipse)


Angle between the flow direction of simple shearing and the principal straink 1 of pure shearing (between x1′ and x1)


Eigenvalue of the velocity gradient tensor


Second moment of the tensor S


Length of a unit line after deformation (i.e., square of the quadratic elongation or extension, λ)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Coward, M. P., 1976, Strain within ductile shear zones: Tectonophysics, v. 34, no. 3/4, p. 181–197.CrossRefGoogle Scholar
  2. Coward, M. P., and Kim, J. H., 1981, Strain within thrust sheets,in McKlay, K. R., and Price, N. J., eds., Thrust and nappe tectonics: Geol. Soc. London, Spec. Publ., v. 9, p. 275–292.Google Scholar
  3. Coward, M. P., and Potts, G. J., 1983, Complex strain pattern developed at the frontal and lateral tips to shear zones and thrust zones: Jour. Struct. Geology, v. 5, no. 3/4, p. 383–399.CrossRefGoogle Scholar
  4. Dias, R., and Ribeiro, A., 1994, Constriction in a transpressive regime: an example in the Iberian branch of the Iberian-Armorican arc: Jour. Struct. Geology, v. 16, no. 11, p. 1543–1554.CrossRefGoogle Scholar
  5. Elliot, D., 1972, Deformation path in structural geology: Geol. Soc. America Bull., v. 83, no. 9, p. 2621–2638.CrossRefGoogle Scholar
  6. Flinn, D., 1962, On folding during three dimensional progressive deformation: Geol. Soc. London Quart. Jour., v. 118, no. 4, p. 385–433.Google Scholar
  7. Flinn, D., 1979, The deformation history and the deformation ellipsoid: Jour. Struct. Geology, v. 1, no. 4, p. 299–307.CrossRefGoogle Scholar
  8. Fossen, H., and Tikoff, B., 1993, The deformation matrix for simultaneous simple shearing, pure shearing and volume change, and its application to transpression-transtension tectonics: Jour. Struct. Geology, v. 15, no. 3-5, p. 413–422.CrossRefGoogle Scholar
  9. Ghosh, S. K., 1987, Measure of non-coaxiality: Jour. Struct. Geology, v. 9, no. 1, p. 111–113.CrossRefGoogle Scholar
  10. Ghosh, S. K., and Ramberg, H., 1976, Reorientations of inclusions by combinations of pure and simple shear: Tectonophysics, v. 34, no. 1/2, p. 1–70.CrossRefGoogle Scholar
  11. Hobbs, B. E., Means, W. D., and Williams, P. F., 1976, An outline of structural geology: Wiley International, New York, 571 p.Google Scholar
  12. Jaeger, J. C., 1956, Elasticity, fracture and flow: Methuen & Co., Ltd., London, 208 p.Google Scholar
  13. Jones, R. R., and Tanner, P. W. G., 1995, Strain partitioning in transpression zones: Jour. Struct. Geology, v. 17, no. 6, p. 793–802.CrossRefGoogle Scholar
  14. Kligfield, R., Crespi, J., Naruk, S., and Davis, G. H., 1984, Displacement and strain patterns of extensional orogens: Tectonics, v. 3, no. 5, p. 577–609.Google Scholar
  15. Malvern, L. E., 1969, Introduction to the mechanics of a continuous medium: Prentice-Hall, Englewood Cliffs, New Jersey, 713 p.Google Scholar
  16. McKenzie, D., and Jackson, J., 1983, The relationship between strain rates, crustal thickening, paleomagnetism, finite strain and fault movements within a deformation zone: Earth Planet. Sci. Lett., v. 65, no. 1, p. 182–202.CrossRefGoogle Scholar
  17. Means, W. D., 1976, Stress and strain, Basic concepts of continuum mechanics for geologists: Springer-Verlag, New York, 339 p.Google Scholar
  18. Means, W. D., 1983, Application of the Mohr-circle construction to problems of inhomogeneous deformation: Jour. Struct. Geology, v. 5, no. 3-4, p. 279–286.CrossRefGoogle Scholar
  19. Means, W. D., 1990, Kinematics, stress, deformation and material behavior: Jour. Struct. Geology, v. 12, no. 8, p. 953–971.CrossRefGoogle Scholar
  20. Means, W. D., Hobbs, B. E., Lister, G. S., and Williams, P. F., 1980, Vorticity and non-coaxiality in progressive deformations: Jour. Struct. Geology, v. 2, no. 3, p. 371–378.CrossRefGoogle Scholar
  21. Merle, O., 1986, Pattern of stretch trajectories and strain rates within spreading-glidding nappes: Tectonophysics, v. 124, no. 3/4, p. 211–222.CrossRefGoogle Scholar
  22. Passchier, C. W., 1986, Flow in natural shear zone—the consequences of spinning flow regimes: Earth Planet. Sci. Lett., v. 77, no. 1, p. 70–80.CrossRefGoogle Scholar
  23. Passchier, C. W., 1987, Efficient use of the velocity gradients tensor in flow modelling: Tectonophysics, v. 136, no. 1/2, p. 159–163.CrossRefGoogle Scholar
  24. Passchier, C. W., 1988, Analysis of deformation paths in shear zones: Geol. Rdsch., v. 77, no. 1, p. 308–318.CrossRefGoogle Scholar
  25. Ramberg, H., 1975, Particle paths, displacement and progressive strain applicable to rocks: Tectonophysics, v. 28, no. 1/2, p. 1–37.CrossRefGoogle Scholar
  26. Ramsay, J. G., 1967, Folding and fracturing of rocks: McGraw-Hill Book Co., New York, 568 p.Google Scholar
  27. Ramsay, J. G., and Graham, R. H., 1970, Strain variation in shear belts: Can. Jour. Earth Science, v. 7, no. 3, p. 786–813.Google Scholar
  28. Ramsay, J. G., and Huber, M. I., 1983, The techniques of modem structural geology, v. 1, Strain analysis: Academic Press, London, 307 p.Google Scholar
  29. Sanderson, D. J., 1976, The superposition of compaction and plane strain: Tectonophysics, v. 30, no. 1/2, p. 35–54.CrossRefGoogle Scholar
  30. Sanderson, D. J., 1982, Models of strain variation in nappes and thrust sheets: a review: Tectonophysics, v. 88, no. 3/4, p. 201–233.CrossRefGoogle Scholar
  31. Sanderson, D. J., and Marchini, W. R. D., 1984, Transpression: Jour. Struct. Geology, v. 6, no. 5, p. 449–458.CrossRefGoogle Scholar
  32. Sanderson, D. J., Andrews, J. R., Phillips, W. E. A., and Hutton, D. H. W., 1980, Deformation studies in the Irish Caledonides: Jour. Geol. Soc. London, v. 137, no. 3, p. 289–302.Google Scholar
  33. Schwerdtner, W. M., and Gapais, D., 1983, Calculation of finite incremental deformations in ductile geological materials and structural models: Tectonophysics, v. 93, no. 1/2, p. T1-T7.CrossRefGoogle Scholar
  34. Simpson, C., and De Paor, D. G., 1993, Strain and kinematic analysis in general shear zones: Jour. Struct. Geology, v. 15, no. 1, p. 1–20.CrossRefGoogle Scholar
  35. Strang, G., 1980, Linear algebra and its applications: Academic Press, London, 414 p.Google Scholar
  36. Tikoff, B., and Fossen, H., 1993, Simultaneous pure and simple shear: the unifying deformation matrix: Tectonophysics, v. 217, no. 5, p. 267–283.CrossRefGoogle Scholar
  37. Tikoff, B., and Fossen, H., 1995, The limitations of three-dimensional kinematic vorticity analysis: Jour. Struct. Geology, v. 17, no. 12, p. 1771–1784.CrossRefGoogle Scholar
  38. Truesdell, C., 1953, Two measures of vorticity: Jour. Rational Mech. Analysis, v. 2, p. 173–217.Google Scholar
  39. Truesdell, C., and Toupin, R., 1960, The classical field theories,in Flugge, S., ed., Encyclopedia of Physics 3: Springer, Berlin, p. 226–793.Google Scholar
  40. Weijermars, R., 1991, The role of stress in ductile deformation: Jour. Struct. Geology, v. 13, no. 9, p. 1061–1078.CrossRefGoogle Scholar
  41. Weijermars, R., 1992, Progressive deformation in anisotropic rocks: Jour. Struct. Geology, v. 14, no. 6, p. 723–742.CrossRefGoogle Scholar
  42. Weijermars, R., 1993, Progressive deformation of single layers under constantly oriented boundary stresses: Jour. Struct. Geology, v. 15, no. 7, p. 911–922.CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geology 1997

Authors and Affiliations

  1. 1.Instituto Andaluz de Ciencias de la Tierra and Departamento de GeodinamicaC.S.I.C.-University of Granada, Faculty of SciencesGranadaSpain

Personalised recommendations