Skip to main content
SpringerLink
Log in

A general deformation matrix for three-dimensions

  • Published:
Mathematical Geology Aims and scope Submit manuscript

Abstract

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 t’i 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a ij :

Components of the velocity gradient tensor L

c ij :

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

D :

Deformation tensor

k i :

Principal strain of the coaxial progressive deformation

l ij :

Direction cosines of aij

L :

Velocity gradient tensor (matrix a)

R :

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

s ij :

Components of the simple shearing velocity gradient tensor

S :

Stretching tensor

S i :

Principal stretch of the strain ellipsoid

v:

Velocity field tensor

w :

Magnitude of the vorticity vector

W :

Vorticity or spin tensor

W k :

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

x i :

Coordinate system in the deformed state for the coaxial progressive deformation

x ′i :

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

X i :

Coordinate system in the undeformed state

α ij :

Angle betweenx ′i andX j

γ ij :

Simple shear strain along the ′i -x ′j plane

gg ij :

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

δ:

Dilation or anisotropic volume change

δ ij :

Kronecker delta (unit diagonal matrix)

ge ij :

Principal strain rate of the coaxial progressive deformation

θ :

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

θ′ :

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

Φ :

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

Xi :

Eigenvalue of the velocity gradient tensor

II:

Second moment of the tensor S

√λ:

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

References

  • Coward, M. P., 1976, Strain within ductile shear zones: Tectonophysics, v. 34, no. 3/4, p. 181–197.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Elliot, D., 1972, Deformation path in structural geology: Geol. Soc. America Bull., v. 83, no. 9, p. 2621–2638.

    Article  Google Scholar 

  • Flinn, D., 1962, On folding during three dimensional progressive deformation: Geol. Soc. London Quart. Jour., v. 118, no. 4, p. 385–433.

    Google Scholar 

  • Flinn, D., 1979, The deformation history and the deformation ellipsoid: Jour. Struct. Geology, v. 1, no. 4, p. 299–307.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ghosh, S. K., 1987, Measure of non-coaxiality: Jour. Struct. Geology, v. 9, no. 1, p. 111–113.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hobbs, B. E., Means, W. D., and Williams, P. F., 1976, An outline of structural geology: Wiley International, New York, 571 p.

    Google Scholar 

  • Jaeger, J. C., 1956, Elasticity, fracture and flow: Methuen & Co., Ltd., London, 208 p.

    Google Scholar 

  • Jones, R. R., and Tanner, P. W. G., 1995, Strain partitioning in transpression zones: Jour. Struct. Geology, v. 17, no. 6, p. 793–802.

    Article  Google Scholar 

  • 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 

  • Malvern, L. E., 1969, Introduction to the mechanics of a continuous medium: Prentice-Hall, Englewood Cliffs, New Jersey, 713 p.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Means, W. D., 1976, Stress and strain, Basic concepts of continuum mechanics for geologists: Springer-Verlag, New York, 339 p.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Means, W. D., 1990, Kinematics, stress, deformation and material behavior: Jour. Struct. Geology, v. 12, no. 8, p. 953–971.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Merle, O., 1986, Pattern of stretch trajectories and strain rates within spreading-glidding nappes: Tectonophysics, v. 124, no. 3/4, p. 211–222.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Passchier, C. W., 1987, Efficient use of the velocity gradients tensor in flow modelling: Tectonophysics, v. 136, no. 1/2, p. 159–163.

    Article  Google Scholar 

  • Passchier, C. W., 1988, Analysis of deformation paths in shear zones: Geol. Rdsch., v. 77, no. 1, p. 308–318.

    Article  Google Scholar 

  • Ramberg, H., 1975, Particle paths, displacement and progressive strain applicable to rocks: Tectonophysics, v. 28, no. 1/2, p. 1–37.

    Article  Google Scholar 

  • Ramsay, J. G., 1967, Folding and fracturing of rocks: McGraw-Hill Book Co., New York, 568 p.

    Google Scholar 

  • 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 

  • 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 

  • Sanderson, D. J., 1976, The superposition of compaction and plane strain: Tectonophysics, v. 30, no. 1/2, p. 35–54.

    Article  Google Scholar 

  • Sanderson, D. J., 1982, Models of strain variation in nappes and thrust sheets: a review: Tectonophysics, v. 88, no. 3/4, p. 201–233.

    Article  Google Scholar 

  • Sanderson, D. J., and Marchini, W. R. D., 1984, Transpression: Jour. Struct. Geology, v. 6, no. 5, p. 449–458.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Strang, G., 1980, Linear algebra and its applications: Academic Press, London, 414 p.

    Google Scholar 

  • Tikoff, B., and Fossen, H., 1993, Simultaneous pure and simple shear: the unifying deformation matrix: Tectonophysics, v. 217, no. 5, p. 267–283.

    Article  Google Scholar 

  • Tikoff, B., and Fossen, H., 1995, The limitations of three-dimensional kinematic vorticity analysis: Jour. Struct. Geology, v. 17, no. 12, p. 1771–1784.

    Article  Google Scholar 

  • Truesdell, C., 1953, Two measures of vorticity: Jour. Rational Mech. Analysis, v. 2, p. 173–217.

    Google Scholar 

  • 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 

  • Weijermars, R., 1991, The role of stress in ductile deformation: Jour. Struct. Geology, v. 13, no. 9, p. 1061–1078.

    Article  Google Scholar 

  • Weijermars, R., 1992, Progressive deformation in anisotropic rocks: Jour. Struct. Geology, v. 14, no. 6, p. 723–742.

    Article  Google Scholar 

  • Weijermars, R., 1993, Progressive deformation of single layers under constantly oriented boundary stresses: Jour. Struct. Geology, v. 15, no. 7, p. 911–922.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto Andaluz de Ciencias de la Tierra and Departamento de Geodinamica, C.S.I.C.-University of Granada, Faculty of Sciences, Campus Fuentenueva, 18008, Granada, Spain

    Juan Ignacio Soto

Authors
  1. Juan Ignacio Soto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Ignacio Soto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Soto, J.I. A general deformation matrix for three-dimensions. Math Geol 29, 93–130 (1997). https://doi.org/10.1007/BF02769621

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02769621

Key Words

Search

Navigation