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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)![Start 2 By 2 Matrix 1st Row 1st Column bold upper M 2nd Column 0 2nd Row 1st Column 0 2nd Column 0 EndMatrix StartBinomialOrMatrix ModifyingAbove Above bold u overbar With two-dots Choose ModifyingAbove Above bold p Superscript w Baseline overbar With two-dots EndBinomialOrMatrix plus Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column 0 2nd Row 1st Column 0 2nd Column 0 EndMatrix StartBinomialOrMatrix ModifyingAbove Above bold u overbar With ampersand c period dotab semicolon Choose ModifyingAbove Above bold p Superscript w Baseline overbar With ampersand c period dotab semicolon EndBinomialOrMatrix plus Start 2 By 2 Matrix 1st Row 1st Column bold upper K 2nd Column minus bold upper Q overTilde 2nd Row 1st Column bold upper Q overTilde Superscript normal upper T Baseline 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 0 EndBinomialOrMatrix]()
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)![Start 2 By 2 Matrix 1st Row 1st Column bold upper K 2nd Column minus bold upper Q overTilde 2nd Row 1st Column bold upper Q overTilde Superscript normal upper T Baseline 2nd Column 0 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 0 EndBinomialOrMatrix]()
which only can have a unique solution when the number of ![bold u overbar]()
variables nu is greater than the number of ![bold p Superscript w Baseline overbar]()
variables np. 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)![bold upper M ModifyingAbove Above bold u overbar With two-dots plus bold upper C ModifyingAbove Above bold u overbar With ampersand c period dotab semicolon plus integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline bold sigma Superscript double-prime Baseline d upper Omega minus bold upper Q bold p overbar Superscript w Baseline minus bold f Superscript left-parenthesis 1 right-parenthesis Baseline equals 0]()
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 dγxy 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)![StartLayout 1st Row left-parenthesis k Subscript italic i j Baseline left-parenthesis minus p Subscript w comma j Baseline plus upper S Subscript w Baseline rho Subscript f Baseline b Subscript j Baseline right-parenthesis right-parenthesis Subscript comma i Baseline plus alpha ModifyingAbove u With ampersand c period dotab semicolon Subscript i comma i Baseline plus StartFraction ModifyingAbove p With ampersand c
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