Computational Geomechanics. Manuel Pastor

Computational Geomechanics

target="_blank" rel="nofollow" href="#fb3_img_img_a80d478b-d82d-5332-90f9-2e846ed2ddfb.png" alt="Start 2 By 2 Matrix 1st Row 1st Column bold upper K 2nd Column minus bold upper Q overTilde 2nd Row 1st Column 0 2nd Column bold upper H EndMatrix StartBinomialOrMatrix bold u overbar Choose bold p Superscript w Baseline overbar EndBinomialOrMatrix equals StartBinomialOrMatrix bold f Superscript left-parenthesis 1 right-parenthesis Baseline Choose bold f Superscript left-parenthesis 2 right-parenthesis EndBinomialOrMatrix"/>

in which bold p Superscript w Baseline overbar 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 bold u overbar variables n_u is greater than the number of bold p Superscript w Baseline overbar 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 bold upper C ModifyingAbove Above bold u overbar With ampersand c period dotab semicolon 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) bold upper C equals alpha bold upper M plus beta bold upper K

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) sigma Subscript italic i j comma j Baseline minus rho ModifyingAbove u With two-dots Subscript i Baseline plus rho b Subscript i Baseline equals 0

(3.9b) d epsilon Subscript italic i j Baseline equals one half left-parenthesis italic d u Subscript i comma j Baseline plus italic d u Subscript j comma i Baseline right-parenthesis

Noting that the engineering shear strain dγ_xy is defined as:

d gamma Subscript italic x y Baseline equals 2 d epsilon Subscript italic x y

Equation (3.10) is scalar

(3.11b) sigma identical-to sigma Subscript italic i j

(3.12b) sigma double-prime Subscript italic i j Baseline equals sigma Subscript italic i j Baseline plus italic alpha delta Subscript italic i j Baseline p

Equation (3.13) is scalar.

Equation (3.14) is scalar.

(3.15b) normal d sigma prime Subscript italic i j Superscript prime Baseline equals upper D Subscript italic ijkl Baseline left-parenthesis normal d epsilon Subscript italic k l Baseline minus normal d epsilon Subscript italic k l Superscript 0 Baseline right-parenthesis

Equation (3.16) is scalar

(3.17b)Скачать книгу