Computational Geomechanics - Manuel Pastor

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to obtain the unique undrained condition.

      The existence of the two steady states is well known and what we have indicated here is a process by which various matrices given in the original computer program can be used to obtain either of the steady state solutions. However, this does require an alternative to the original computer program. Though, it is possible to obtain such steady states by the code, using the previous time‐stepping procedure. Two types of undrained conditions exist: (a) when k = 0 throughout; (b) k ≠ 0 but the complete boundary is impermeable. Both cases can be computed with no difficulties.

      Provided that the boundary conditions are consistent with the existence of drained and undrained steady state conditions, the time‐stepping process will, in due course, converge with

StartLayout 1st Row normal upper Delta ModifyingAbove Above bold u overbar With two-dots Subscript n plus 1 Baseline right-arrow 0 2nd Row normal upper Delta ModifyingAbove Above bold u overbar With ampersand c period dotab semicolon Subscript n plus 1 Baseline right-arrow 0 3rd Row normal upper Delta bold u overbar Subscript n plus 1 Baseline right-arrow 0 4th Row normal upper Delta ModifyingAbove Above bold p overbar With ampersand c period dotab semicolon Subscript n plus 1 Superscript w Baseline right-arrow 0 5th Row normal upper Delta bold p overbar Subscript n plus 1 Superscript w Baseline right-arrow 0 EndLayout

      However, this process may be time‐consuming even if large time steps, Δt are used. A simpler procedure is to use the GN00 scheme with

bold u overbar Subscript n plus 1 Baseline equals bold u overbar Subscript n Baseline plus normal upper Delta bold u overbar Subscript n bold p overbar Subscript n plus 1 Superscript w Baseline equals bold p overbar Subscript n Superscript w Baseline plus normal upper Delta bold p overbar Subscript n Superscript w integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline left-parenthesis bold sigma double-prime Subscript n plus 1 Baseline minus bold sigma double-prime Subscript n right-parenthesis d upper Omega minus bold upper Q bold p overbar Subscript n plus 1 Superscript w Baseline equals normal f Subscript n plus 1 Superscript left-parenthesis 1 right-parenthesis Baseline minus bold upper Q bold p overbar Subscript n Superscript w Baseline minus bold f Subscript n Superscript left-parenthesis 1 right-parenthesis bold upper Q overTilde Superscript normal upper T Baseline bold u overbar Subscript n plus 1 Baseline plus bold upper S bold p overbar Subscript n plus 1 Superscript w Baseline equals bold upper Q overTilde Superscript normal upper T Baseline bold u overbar Subscript n Baseline plus bold upper S bold p overbar Subscript n Superscript w

      If the material behavior is linearly elastic, then the equation can be solved directly yielding the two unknowns bold u overbar n+1 and bold p overbar Subscript n plus 1 Superscript w and if the material is nonlinear, an iteration scheme such as the Newton–Raphson, Quasi‐Newton, Tangential Matrix or the Initial Matrix method can be adopted. With a systematic change of the external loading, problems such as the load–displacement curve of a nonlinear soil and pore–fluid system can be traced.

      

      3.2.5 The Structure of the Numerical Equations Illustrated by their Linear Equivalent

      If complete saturation is assumed together with a linear form of the constitutive law, we can write the effective stress simply as

      (3.62)bold sigma double-prime equals bold upper D upper B bold u overbar

      We can now reduce the governing u–p Equations (3.23) and (3.28) to the form given below

      (3.63)bold upper M ModifyingAbove Above bold u overbar With two-dots plus bold upper K bold u overbar minus bold upper Q overTilde bold p Superscript w Baseline overbar minus bold f Superscript left-parenthesis 1 right-parenthesis Baseline equals 0

      and

      (3.64)bold upper Q overTilde Superscript normal upper T Baseline ModifyingAbove Above bold u overbar With ampersand c period dotab semicolon plus upper H bold p Superscript w Baseline overbar plus normal upper S ModifyingAbove Above normal p overbar With ampersand c period dotab semicolon minus bold f Superscript left-parenthesis 2 right-parenthesis Baseline equals 0

      where bold p Superscript w Baseline overbar = bold p Superscript w Baseline overbar

      (3.65)and bold upper K equals integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline bold upper D upper B d normal upper Omega

      The overall system can be written in the terms of the variable set [bold u overbar, bold p Superscript w Baseline overbar]T as

      (3.66)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 bold upper Q overTilde Superscript normal upper T Baseline 2nd Column upper S 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 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 Baseline EndBinomialOrMatrix equals StartBinomialOrMatrix 0 Choose 0 EndBinomialOrMatrix

      Once again the uncoupled nature of the problem under drained condition is evident (by dropping the time derivatives) giving

      (3.67) Скачать книгу