Computational Geomechanics. Manuel Pastor

Computational Geomechanics

o n normal upper Gamma equals normal upper Gamma Subscript w"/>

It is of interest to note, as shown by Zienkiewicz (1982), that some typical soil constants are implied in the formulation. For instance, we note from (2.16) that for undrained behavior, when w_i,i = 0, i.e. with no net outflow, we have (neglecting the last two terms which are of the second order).

normal d p equals minus italic upper Q alpha normal d bold epsilon Subscript italic i i

normal d p equals minus italic upper Q alpha bold m Superscript normal upper T Baseline normal d bold epsilon

and

StartLayout 1st Row normal d sigma Subscript italic i j Baseline equals normal d sigma Subscript italic i j Superscript double-prime Baseline minus italic alpha delta Subscript italic i j Baseline normal d p equals upper D Subscript italic ijkl Baseline normal d bold epsilon Subscript italic k l Baseline plus alpha squared upper Q normal d bold epsilon Subscript italic i j Baseline 2nd Row equals left-parenthesis upper D Subscript italic ijkl Baseline plus alpha squared upper Q delta Subscript italic i k Baseline delta Subscript italic j l Baseline right-parenthesis normal d bold epsilon Subscript italic k l EndLayout

normal d bold sigma equals left-parenthesis bold upper D plus alpha squared bold m upper Q bold m Superscript normal upper T Baseline right-parenthesis normal d bold epsilon

If the pressure change dp is considered as a fraction of the mean total stress change m^T dσ/3 or dσ_ii/3, we obtain the so‐called B soil parameter (Skempton 1954) as

StartLayout 1st Row upper B identical-to StartFraction minus 3 normal d p Over normal d sigma Subscript italic i i Baseline EndFraction equals StartFraction 3 italic alpha upper Q normal d bold epsilon Subscript italic k k Baseline Over upper D Subscript italic iikl Baseline normal d bold epsilon Subscript italic k l Baseline plus 3 alpha squared upper Q normal d bold epsilon Subscript italic k k Baseline EndFraction 2nd Row upper B identical-to StartFraction minus 3 normal d p Over bold m Superscript normal upper T Baseline normal d bold sigma EndFraction equals StartFraction 3 italic alpha upper Q bold m Superscript normal upper T Baseline normal d bold epsilon Over bold m Superscript normal upper T Baseline left-parenthesis bold upper D plus alpha squared bold m upper Q bold m Superscript normal upper T Baseline right-parenthesis normal d bold epsilon EndFraction EndLayout

Using the assumption that the material is isotropic so that

StartLayout 1st Row upper D Subscript italic ijkl Baseline equals left-parenthesis upper K Subscript upper T Baseline minus two thirds mu right-parenthesis delta Subscript italic i j Baseline delta Subscript italic k l Baseline plus mu left-parenthesis delta Subscript italic i k Baseline delta Subscript italic j l Baseline plus delta Subscript italic i l Baseline delta Subscript italic j k Baseline right-parenthesis 2nd Row upper B equals StartFraction 3 italic alpha upper Q normal d epsilon Subscript italic k k Baseline Over left-parenthesis left-parenthesis 3 upper K Subscript upper T Baseline minus 2 mu right-parenthesis delta Subscript italic k l Baseline plus 2 italic mu delta Subscript italic k l Baseline right-parenthesis normal d epsilon Subscript italic k l Baseline plus 3 alpha squared upper Q normal d epsilon Subscript italic k k Baseline EndFraction 3rd Row equals StartFraction italic alpha upper Q Over upper K Subscript upper T Baseline plus alpha squared upper Q EndFraction equals StartFraction alpha Over left-parenthesis upper K Subscript upper T Baseline slash upper Q right-parenthesis plus alpha squared EndFraction 4th Row upper B equals StartFraction 3 italic alpha upper Q bold m Superscript upper T Baseline bold m Over bold m Superscript upper T Baseline left-parenthesis bold upper D plus alpha squared bold m upper Q bold m Superscript upper T Baseline right-parenthesis bold m EndFraction equals StartFraction 9 italic alpha upper Q Over bold m Superscript upper T Baseline bold upper D m plus 9 alpha squared upper Q EndFraction 5th Row equals StartFraction 9 italic alpha upper Q Over 9 upper K Subscript upper T Baseline plus 9 alpha squared upper Q EndFraction equals StartFraction italic alpha upper Q Over upper K Subscript upper T Baseline plus alpha squared upper Q EndFraction equals StartFraction alpha Over left-parenthesis upper K Subscript upper T Baseline slash upper Q right-parenthesis plus alpha squared EndFraction EndLayout

where K_T is (as defined in equation (1.10), the bulk modulus of the solid phase and μ is once again Lamé’s constant. B has, of course, a value approaching unity for soil but can be considerably lower for concrete or rock. Further, for unsaturated soils, the value will be much lower (Terzaghi 1925; Lambe and Whitman 1969; Craig 1992).

2.2.2 Simplified Equation Sets (u–p Form)

The governing equation set (2.11), (2.13), and (2.16) together with the auxiliary definition system can, of course, be used directly in numerical solution as shown by Zienkiewicz and Shiomi (1984). This system is suitable for explicit time‐stepping computation as shown by Sandhu and Wilson (1969) and Ghaboussi and Wilson (1972) and later by Chan et al. (1991). However, in implicit computation, where large algebraic equation systems arise, it is convenient to reduce the number of variables by neglecting the apparently small (underlined) terms of equations (2.11) and (2.13). These contain the variable w_i(w) which now can be eliminated from the system.

The first equation of the reduced system becomes (from (2.11))

(2.20a) 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

(2.20b) bold upper S Superscript normal upper T Baseline bold sigma minus rho ModifyingAbove bold u With two-dots plus rho bold b equals 0

The second equation is obtained by coupling (2.13) and (2.16) using the definition (2.14) and thus eliminating the variable w_i(w). We now have, omitting density changes

(2.21a)

(2.21b)

This reduced equation system is precisely the same as that used conventionally in the study of consolidation if the dynamic terms are omitted or even of static problems if the steady state is reached and all the time derivatives are reduced to zero. Thus, the formulation conveniently merges with procedures used for such analyses. However,

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