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in which can be separately determined by solving the second equation. For undrained behavior, we can integrate the second equation when H = 0 and obtain an antisymmetric system which can be made symmetric by multiplying the second equation by minus unity (Zienkiewicz and Taylor 1985)

      (3.68)

      It is interesting to observe that in the steady state, we have a matrix which, in the absence of fluid compressibility, results in

      (3.69)

      which only can have a unique solution when the number of variables n_u is greater than the number of variables n_p. This is one of the requirements of the patch test of Zienkiewicz et al. (1986a, 1986b) and of the Babuska‐Brezzi (Babuska 1973 and Brezzi 1974) condition.

      3.2.6 Damping Matrices

      In general, when dynamic problems are encountered in soils (or other geomaterials), the damping introduced by the plastic behavior of the material and the viscous effects of the fluid flow are sufficient to damp out any nonphysical or numerical oscillation. However, if the solutions of the problems are in the low‐strain range when the plastic hysteresis is small or when, to simplify the procedures, purely elastic behavior is assumed, it may be necessary to add system damping matrices of the form to the dynamic equations of the solid phase, i.e. changing (3.23) to

      (3.70)

      Indeed, such damping matrices have a physical significance and are always introduced in earthquake analyses or similar problems of structural dynamics. With the lack of any special information about the nature of damping, it is usual to assume the so‐called “Rayleigh damping” in which

      (3.71)

where α and β are coefficients determined by experience (see, for instance, Clough and Penzien (1975) or (1993)). In the above, M is the same mass matrix as given in (3.24) and K is some representative stiffness matrix of the form given in (3.47).

3.3 Theory: Tensorial Form of the Equations

      The equation numbers given here correspond to the ones given earlier in the text.

      (3.8b)

      (3.9b)

Noting that the engineering shear strain dγ_xy is defined as:

Equation (3.10) is scalar

      (3.11b)

      (3.12b)

Equation (3.13) is scalar.

Equation (3.14) is scalar.

      (3.15b)

Equation (3.16) is scalar

      (3.17b)Скачать книгу