Читать книгу Computational Geomechanics - Manuel Pastor - Страница 48

The equation numbers given here correspond to the ones given earlier in the text.

(3.8b)

(3.9b)

Noting that the engineering shear strain dγ_xy is defined as:

Equation (3.10) is scalar

(3.11b)

(3.12b)

Equation (3.13) is scalar.

Equation (3.14) is scalar.

(3.15b)

Equation (3.16) is scalar

(3.17b)

and

(3.18b)

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.

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

Rearranging

(3.20)

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

(3.23b)

(3.24b)

(3.25b)

(3.26b)

(3.27b)

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

Inserting the shape functions

(3.28b)

(3.29b)

(3.30b)

(3.31b)

(3.32b)

Equation (3.33) is scalar.