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Computational Geomechanics. Manuel Pastor

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Computational Geomechanics - Manuel Pastor

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equals sigma Subscript italic i j Baseline plus italic alpha delta Subscript italic i j Baseline p"/>

      (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

      (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 period dotab semicolon Subscript w Baseline Over upper Q Superscript asterisk Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0 2nd Row t Subscript i Baseline equals sigma Subscript italic i j Baseline n Subscript j Baseline equals t overbar Subscript i Baseline o n normal upper Gamma equals normal upper Gamma Subscript t Baseline 3rd Row u Subscript i Baseline equals u overbar Subscript i Baseline o n normal upper Gamma equals normal upper Gamma Subscript u EndLayout

      and

      (3.18b)n Subscript i Baseline w Subscript i Baseline equals n Subscript i Baseline 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 equals w overbar Subscript n Baseline o n normal upper Gamma equals normal upper Gamma Subscript w

      and assuming isotropic permeability

      (3.19b)StartLayout 1st Row n Subscript i Baseline w Subscript i Baseline equals k left-parenthesis minus p Subscript w comma n Baseline plus upper S Subscript w Baseline normal rho Subscript normal f Baseline n Subscript i Baseline b Subscript i Baseline right-parenthesis equals w overbar Subscript n Baseline equals minus q overbar o n normal upper Gamma equals normal upper Gamma Subscript w Baseline 2nd Row u Subscript i Baseline approximately-equals ModifyingAbove u With ampersand c period circ semicolon Subscript i Baseline equals upper N Subscript k Superscript u Baseline u overbar Subscript italic upper K i Baseline 3rd Row p Subscript w Baseline approximately-equals ModifyingAbove p With ampersand c period circ semicolon Subscript w Baseline equals upper N Subscript upper L Superscript p Baseline p overbar Subscript upper L Superscript w EndLayout

      The summation range for the upper‐case indices will depend on the number of nodes with solid displacement and pore water pressure degrees‐of‐freedom (dofs), respectively.

integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript u Baseline left-parenthesis sigma Subscript italic i j comma j Baseline minus rho ModifyingAbove u With two-dots Subscript i Baseline plus rho b Subscript i Baseline right-parenthesis d upper Omega equals 0 integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript u Baseline left-parenthesis sigma Subscript italic i j comma j Baseline minus rho upper N Subscript upper L Superscript u Baseline ModifyingAbove Above u overbar With two-dots Subscript italic upper L i Baseline plus rho b Subscript i Baseline right-parenthesis d upper Omega equals 0

      Applying Green’s identity to the internal force term (first term on the left‐hand side)

StartLayout 1st Row integral Underscript normal upper Omega Endscripts left-parenthesis minus upper N Subscript upper K comma j Superscript u Baseline sigma Subscript italic i j Baseline minus upper N Subscript upper K Superscript u Baseline rho upper N Subscript upper L Superscript u Baseline ModifyingAbove Above u overbar With two-dots Subscript italic upper L i Baseline plus upper N Subscript upper K Superscript u Baseline rho b Subscript i Baseline right-parenthesis d upper Omega plus integral Underscript normal upper Gamma Subscript t Baseline Endscripts upper N Subscript upper K Superscript u Baseline n Subscript j Baseline sigma Subscript italic i j Baseline d upper Gamma equals 0 2nd Row integral Underscript normal upper Omega Endscripts left-parenthesis minus upper N Subscript upper K comma j Superscript u Baseline sigma Subscript italic i j Baseline minus upper N Subscript upper K Superscript u Baseline rho upper N Subscript upper L Superscript u Baseline ModifyingAbove Above u overbar With two-dots Subscript italic upper L i Baseline plus upper N Subscript upper K Superscript u Baseline rho b Subscript i Baseline right-parenthesis d upper Omega plus integral Underscript normal upper Gamma Subscript t Baseline Endscripts upper N Subscript upper K Superscript u Baseline t overbar Subscript i Baseline d upper Gamma equals 0 EndLayout

      Rearranging

      (3.20)integral Underscript normal upper Omega Endscripts upper N Subscript upper K comma j Superscript u Baseline sigma Subscript italic i j Baseline d upper Omega plus left-bracket integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript u Baseline rho upper N Subscript upper L Superscript u Baseline d upper Omega right-bracket ModifyingAbove Above u overbar With two-dots Subscript italic upper L i Baseline equals integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript u Baseline rho b Subscript i Baseline d upper Omega plus integral Underscript normal upper Gamma Subscript t Baseline Endscripts upper N Subscript upper K Superscript u Baseline t overbar Subscript i Baseline d upper Gamma

sigma Subscript italic i j Baseline equals sigma double-prime Subscript italic i j Baseline minus alpha chi delta Subscript italic i j Baseline p

      (3.23b)upper M Subscript italic upper K upper L Baseline ModifyingAbove Above u overbar With two-dots Subscript italic upper L i Baseline plus integral Underscript normal upper Omega Endscripts upper N Subscript upper K comma j Superscript u Baseline sigma Subscript italic i j Superscript double-prime Baseline d upper Omega minus upper Q Subscript italic upper K i upper L Baseline p overbar Subscript upper L Superscript upper W Baseline minus f Subscript italic upper K i Superscript left-parenthesis 1 right-parenthesis Baseline equals 0

      (3.24b)upper M Subscript italic upper K upper L Baseline equals integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript u Baseline normal rho upper N Subscript upper L Superscript u Baseline d upper Omega

      (3.25b)upper Q Subscript italic upper K i upper L Baseline equals integral Underscript normal upper Omega Endscripts upper N Subscript upper K comma i Superscript u Baseline normal alpha chi upper N Subscript upper L Superscript p Baseline d upper Omega

      (3.26b)

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