Computational Geomechanics. Manuel Pastor
Читать онлайн книгу.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"/>
where the last two terms account for simple rotation (via the definition in 2.6b) of the existing stress components and are known as the Zaremba (1903a, 1903b)–Jaumann (1905) stress changes. We omit here the corresponding vectorial notation as this is not easy to implement.
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.
The strain and rotation increments of the soil matrix can be determined in terms of displacement increments dui as
(2.6a)
and
The comma in the suffix denotes differentiation with respect to the appropriate coordinate specified. Thus
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)
or
However, the shear strain increments will be written as
(2.7b)
or
We shall usually write the process of strain computation using matrix notation as
(2.8)
where
(2.9)
And for two dimensions, the strain matrix is defined as:
(2.10)
with corresponding changes for three dimensions (as shown in Zienkiewicz et al. 2005).
We can now write the overall equilibrium or momentum balance relation for the soil–fluid “mixture” as
(2.11a)
or
(2.11b)
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)
or
(2.13b)