Computational Geomechanics. Manuel Pastor

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Название Computational Geomechanics
Автор произведения Manuel Pastor
Жанр Физика
Серия
Издательство Физика
Год выпуска 0
isbn 9781118535301
Computational Geomechanics - Manuel Pastor
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double-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 plus sigma double-prime Subscript italic i k Baseline normal d omega Subscript italic k j Baseline plus sigma double-prime Subscript italic j k Baseline normal d omega Subscript italic k i"/>

      The large strain rotation components are small for small displacement computation and can be frequently neglected. Thus, in the derivations that follow, we shall do so – though their inclusion presents no additional computational difficulties.

      (2.6a)normal d epsilon Subscript italic i j Baseline equals one half left-bracket normal d u Subscript i comma j Baseline plus normal d u Subscript j comma i Baseline right-bracket

      and

      The comma in the suffix denotes differentiation with respect to the appropriate coordinate specified. Thus

u Subscript i comma j Baseline identical-to StartFraction partial-differential u Subscript i Baseline Over partial-differential x Subscript j Baseline EndFraction e t c period

      If the vectorial notation is used, as is often the case in the finite element analysis, the so‐called engineering strains are used in which (with the repeated index of ∂ui,i not summed)

      (2.7a)normal d epsilon Subscript i Baseline equals normal d u Subscript i comma i

      or

normal d epsilon Subscript x Baseline equals StartFraction normal d Over partial-differential x EndFraction normal d u Subscript x Baseline e t c period

      However, the shear strain increments will be written as

      (2.7b)normal d gamma Subscript italic i j Baseline equals 2 normal d epsilon Subscript italic i j Baseline equals normal d u Subscript i comma j Baseline plus normal d u Subscript j comma i

      or

normal d gamma Subscript italic x y Baseline equals StartFraction partial-differential normal d u Subscript y Baseline Over partial-differential x EndFraction plus StartFraction partial-differential normal d u Subscript x Baseline Over partial-differential y EndFraction

      We shall usually write the process of strain computation using matrix notation as

      (2.8)normal d bold epsilon equals bold upper S normal d bold u

      where

      (2.9)bold u equals left-bracket u Subscript x Baseline u Subscript y Baseline u Subscript z Baseline right-bracket Superscript normal upper T

      And for two dimensions, the strain matrix is defined as:

      We can now write the overall equilibrium or momentum balance relation for the soil–fluid “mixture” as

      (2.11a)sigma Subscript italic i j comma j Baseline minus rho ModifyingAbove u With two-dots Subscript i Baseline minus ModifyingBelow rho Subscript f Baseline left-bracket ModifyingAbove w With ampersand c period dotab semicolon Subscript i Baseline plus w Subscript j Baseline w Subscript i comma j Baseline right-bracket With bar plus rho b Subscript i Baseline equals 0

      or

      (2.11b)StartLayout 1st Row bold upper S Superscript upper T Baseline bold sigma minus rho ModifyingAbove bold u With two-dots minus ModifyingBelow rho Subscript f Baseline left-bracket ModifyingAbove bold w With ampersand c period dotab semicolon plus bold w italic nabla Superscript upper T Baseline bold w right-bracket With bar plus bold rho b equals 0 2nd Row where ModifyingAbove w With ampersand c period dotab semicolon Subscript i Baseline identical-to StartFraction normal d w Subscript i Baseline Over normal d t EndFraction e t c period EndLayout

      In the above, wi (or w) is the average (Darcy) velocity of the percolating water.

      The underlined terms in the above equation represent the fluid acceleration relative to the solid and the convective terms of this acceleration. This acceleration is generally small and we shall frequently omit it. In derivations of the above equation, we consider the solid skeleton and the fluid embraced by the usual control volume: dx ⋅ dy ⋅ dz.

      Further, ρf is the density of the fluid, b is the body force per unit mass (generally gravity) vector, and ρ is the density of the total composite, i.e.

      where ρS is the density of the solid particles and n is the porosity (i.e. the volume of pores in a unit volume of the soil).

      The second equilibrium equation ensures the momentum balance of the fluid. If again we consider the same unit control volume as that assumed in deriving (2.11) (and we further assume that this moves with the solid phase), we can write

      (2.13a)minus p Subscript i Baseline minus upper R Subscript i Baseline minus rho Subscript f Baseline ModifyingAbove u With two-dots Subscript i Baseline minus ModifyingBelow rho Subscript f Baseline left-bracket ModifyingAbove w With ampersand c period dotab semicolon Subscript i Baseline plus w Subscript j Baseline w Subscript i comma j Baseline right-bracket slash n With bar plus rho Subscript f Baseline b Subscript i Baseline equals 0

      or

      (2.13b)minus italic nabla p minus bold upper R minus rho Subscript f Baseline ModifyingAbove </div> </div> </div> <hr> <div class=

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