Manuel Pastor

Computational Geomechanics

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d upper Omega equals 0"/>

      Neglecting source term and integrating by part the first part of the first term

minus integral Underscript normal upper Gamma Subscript w Baseline Endscripts upper N Subscript upper K Superscript p Baseline n Subscript i Baseline k Subscript italic i j Baseline p Subscript w comma j Baseline d upper Gamma plus integral Underscript normal upper Omega Endscripts left-bracket upper N Subscript upper K comma i Superscript p Baseline k Subscript italic i j Baseline p Subscript w comma j Baseline plus upper N Subscript upper K Superscript p Baseline left-parenthesis k Subscript italic i j Baseline upper S Subscript w Baseline rho Subscript f Baseline b Subscript j Baseline right-parenthesis Subscript comma i Baseline plus upper N Subscript upper K Superscript p Baseline alpha ModifyingAbove u With ampersand c period dotab semicolon Subscript i comma i Baseline plus upper N Subscript upper K Superscript p Baseline StartFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w Baseline Over upper Q Superscript asterisk Baseline EndFraction right-bracket d upper Omega equals 0

      Inserting the shape functions

      (3.28b)StartLayout 1st Row minus integral Underscript normal upper Gamma Subscript w Baseline Endscripts upper N Subscript upper K Superscript p Baseline n Subscript i Baseline q Subscript i Baseline d upper Gamma plus integral Underscript normal upper Omega Endscripts left-bracket upper N Subscript upper K comma i Superscript p Baseline k Subscript italic i j Baseline upper N Subscript upper L comma j Superscript p Baseline p overbar Subscript upper L Superscript w Baseline plus upper N Subscript upper K Superscript p Baseline left-parenthesis k Subscript italic i j Baseline upper S Subscript w Baseline rho Subscript f Baseline b Subscript j Baseline right-parenthesis Subscript comma i Baseline plus upper N Subscript upper K Superscript p Baseline alpha upper N Subscript upper L comma i Superscript u Baseline ModifyingAbove Above u overbar With ampersand c period dotab semicolon Subscript upper L Baseline plus upper N Subscript upper K Superscript p Baseline StartFraction 1 Over upper Q Superscript asterisk Baseline EndFraction upper N Subscript upper L Superscript p Baseline ModifyingAbove Above p overbar With ampersand c period dotab semicolon Subscript upper L Superscript w Baseline right-bracket d upper Omega equals 0 2nd Row minus integral Underscript normal upper Gamma Subscript w Endscripts upper N Subscript upper K Superscript p Baseline n Subscript i Baseline q Subscript i d upper Gamma plus integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript p Baseline left-parenthesis k Subscript italic i j Baseline upper S Subscript w Baseline normal rho Subscript f Baseline b Subscript j Baseline right-parenthesis Subscript comma i d upper Omega plus integral Underscript normal upper Omega Endscripts upper N Subscript upper K comma i Superscript p Baseline k Subscript italic i j Baseline upper N Subscript upper L comma j Superscript p d upper Omega p overbar 3rd Row plus integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript p Baseline alpha upper N Subscript upper L comma i Superscript u Baseline d upper Omega ModifyingAbove Above u overbar With ampersand c period dotab semicolon Subscript upper L Baseline plus integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript p Baseline StartFraction 1 Over upper Q asterisk EndFraction upper N Subscript upper L Superscript p Baseline d upper Omega ModifyingAbove Above p overbar With ampersand c period dotab semicolon Subscript upper L Superscript w Baseline equals 0 4th Row upper Q overTilde Subscript italic upper L i upper K Baseline ModifyingAbove Above u overbar With ampersand c period dotab semicolon Subscript italic upper L i Baseline plus upper H Subscript italic upper K upper L Baseline p overbar Subscript upper L Superscript w Baseline plus upper S Subscript italic upper K upper L Baseline ModifyingAbove Above p overbar With ampersand c period dotab semicolon Subscript upper L Superscript upper W Baseline minus f Subscript upper K Superscript left-parenthesis 2 right-parenthesis Baseline equals 0 EndLayout

      (3.29b)upper Q overTilde 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 alpha upper N Subscript upper L Superscript p Baseline d upper Omega

      (3.30b)upper H Subscript italic upper K upper L Baseline equals integral Underscript normal upper Omega Endscripts upper N Subscript upper K comma i Superscript p Baseline k Subscript italic i j Baseline upper N Subscript upper L comma j Superscript p Baseline d upper Omega

      (3.31b)upper S Subscript italic upper K upper L Baseline equals integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript upper P Baseline StartFraction 1 Over upper Q asterisk EndFraction upper N Subscript upper L Superscript p Baseline d upper Omega

      (3.32b)f Subscript upper K Superscript left-parenthesis 2 right-parenthesis Baseline equals minus integral Underscript normal upper Omega Endscripts upper N Subscript upper K comma i Superscript p Baseline k Subscript italic i j Baseline upper S Subscript w Baseline rho Subscript f Baseline b Subscript j Baseline d upper Omega plus integral Underscript normal upper Gamma Subscript w Baseline Endscripts upper N Subscript upper K Superscript p Baseline q overbar d upper Gamma

      In this chapter, the governing equations introduced in Chapter 2 are discretized in space and time using various implicit and explicit algorithms. They are now ready for implementation into computer codes. In Chapter 5, we shall address some special modeling aspects and in Chapters 6–8, we shall show some applications for static, quasi‐static, and dynamic examples to illustrate the practical applications of the method and to validate and verify the schemes and constitutive models used.

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      12 Katona,

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