Computational Geomechanics. Manuel Pastor

Computational Geomechanics

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)

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

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)

The definition of the B matrix in Equation (3.21) is not needed in tensorial form.

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)

(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)

(3.27b)

integral Underscript normal upper Omega Endscripts upper N Subscript upper K Superscript p Baseline 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 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 <hr><noindex><a href=

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