6.7.2 Geostatic stress state

**Products: **ABAQUS/Standard ABAQUS/CAE

A geostatic stress field procedure:

is used to verify that the initial geostatic stress field is in equilibrium with applied loads and boundary conditions and to iterate, if necessary, to obtain equilibrium;

accounts for pore pressure degrees of freedom when pore pressure elements are used;

is usually the first step of a geotechnical analysis, followed by a coupled pore fluid diffusion/stress or static analysis procedure; and

can be linear or nonlinear.

Establishing geostatic equilibrium

The geostatic procedure is normally used as the first step of a geotechnical analysis; in such cases gravity loads are applied during this step. Ideally, the loads and initial stresses should exactly equilibrate and produce zero deformations. However, in complex problems it may be difficult to specify initial stresses and loads that equilibrate exactly. ABAQUS/Standard will check for equilibrium during the geostatic procedure and iterate, if needed, to obtain a stress state that equilibrates the prescribed boundary conditions and loads. This stress state, which is a modification of the stress field defined by the initial conditions (“Initial conditions,” Section 27.2.1), will then be used as the initial stress field in a subsequent coupled pore fluid diffusion/stress or static analysis.

If the stresses given as initial conditions are far from equilibrium under the geostatic loading and there is some nonlinearity in the problem definition, this iteration process may fail. Therefore, you should ensure that the initial stresses are reasonably close to equilibrium.

If the deformations produced during the geostatic step are significant compared to the deformations caused by subsequent loading, the definition of the initial state should be reexamined.

Input File Usage: | *GEOSTATIC |

ABAQUS/CAE Usage: | Step module: |

Most geotechnical problems begin from a geostatic state, which is a steady-state equilibrium configuration of the undisturbed soil or rock body under geostatic loading. The equilibrium state usually includes both horizontal and vertical stress components. It is important to establish these initial conditions correctly so that the problem begins from an equilibrium state. Since such problems often involve fully or partially saturated flow, the initial void ratio of the porous medium, , the initial pore pressure, , and the initial effective stress must all be defined.

If the magnitude and direction of the gravitational loading are defined by using the gravity distributed load type, a total, rather than excess, pore pressure solution is used (see “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1). This discussion is based on the total pore pressure formulation.

The *z*-axis points vertically in this discussion, and atmospheric pressure is neglected. We assume that the pore fluid is in hydrostatic equilibrium during the geostatic procedure so that

where is the user-defined specific weight of the pore fluid (see “Permeability,” Section 20.7.2). (The pore fluid is not in hydrostatic equilibrium if there is significant steady-state flow of pore fluid through the porous medium: in that case a steady-state coupled pore fluid diffusion/stress analysis must be performed to establish the initial conditions for any subsequent transient calculations—see “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1.) If we also take to be independent of

where is the height of the

We usually assume that there are no significant shear stresses , . Then, equilibrium in the vertical direction is

where is the dry density of the porous solid material (the dry mass per unit volume),

ABAQUS/Standard requires that the initial value of the effective stress, , be given as an initial condition (“Initial conditions,” Section 27.2.1). Effective stress is defined from the total stress, , by

where is a unit matrix. Combining this definition with the equilibrium statement in the

again using the assumption that is independent of

In many cases *s* is constant. For example, in fully saturated flow everywhere below the phreatic surface. If we further assume that the initial porosity, , and the dry density of the porous medium, , are also constant, the above equation is readily integrated to give

In more complicated cases where *s*, , and/or vary with height, the equation must be integrated in the vertical direction to define the initial values of .

In many geotechnical applications there is also horizontal stress, typically caused by tectonic action. If the pore fluid is under hydrostatic equilibrium and , equilibrium in the horizontal directions requires that the horizontal components of effective stress do not vary with horizontal position: only, where is any horizontal component of effective stress.

The initial effective geostatic stress field, , is given by defining initial stress conditions. The initial state of stress must be close to being in equilibrium with the applied loads and boundary conditions. See “Initial conditions,” Section 27.2.1.

You can specify that the initial stresses vary only with elevation, as described in “Initial conditions,” Section 27.2.1. In this case the horizontal stress is typically assumed to be a fraction of the vertical stress: those fractions are defined in the *x*- and *y*-directions.

In problems involving partially or fully saturated porous media, initial pore fluid pressures, , void ratios, , and saturation values, *s*, must be given (see “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1).

In partially saturated cases the initial pore pressure and saturation values must lie on or between the absorption and exsorption curves (see “Sorption,” Section 20.7.4). A partially saturated problem is illustrated in “Wicking in a partially saturated porous medium,” Section 1.8.3 of the ABAQUS Benchmarks Manual.

Boundary conditions can be applied to displacement degrees of freedom 1–6 and to pore pressure degree of freedom 8 (“Boundary conditions,” Section 27.3.1).

The boundary conditions should be in equilibrium with the initial stresses and applied loads. If the horizontal stress is nonzero, horizontal equilibrium must be maintained by fixing the boundary conditions on any nonhorizontal edges of the finite element model in the horizontal direction or by using infinite elements (“Infinite elements,” Section 22.2.1).

The following loading types can be prescribed in a geostatic stress field procedure:

Concentrated nodal forces can be applied to the displacement degrees of freedom (1–6); see “Concentrated loads,” Section 27.4.2.

Distributed pressure forces or body forces can also be applied; see “Distributed loads,” Section 27.4.3. The distributed load types available with particular elements are described in Part VI, “Elements.” The magnitude and direction of gravitational loading are defined by using the gravity or body force distributed load types.

Pore fluid flow is controlled as described in “Pore fluid flow,” Section 27.4.6.

The following predefined fields can be specified in a geostatic stress field procedure, as described in “Predefined fields,” Section 27.6.1:

Although temperature is not a degree of freedom in coupled pore fluid diffusion/stress analysis, nodal temperatures can be specified.

The values of user-defined field variables can be specified; these values affect only field-variable-dependent material properties, if any.

Any of the mechanical constitutive models available in ABAQUS/Standard can be used to model the porous solid material.

If a porous medium will be analyzed subsequent to the geostatic procedure, pore fluid flow quantities such as permeability and sorption should be defined (see “Pore fluid flow properties,” Section 20.7.1).

Any of the stress/displacement elements in ABAQUS/Standard can be used in a geostatic procedure. Continuum pore pressure elements can also be used for modeling fluid in a deforming porous medium. These elements have pore pressure degree of freedom 8 in addition to displacement degrees of freedom 1–3. See “Choosing the appropriate element for an analysis type,” Section 21.1.3, for more information.

The element output available for a coupled pore fluid diffusion/stress analysis includes the usual mechanical quantities such as (effective) stress; strain; energies; and the values of state, field, and user-defined variables. In addition, the following quantities associated with pore fluid flow are available:

VOIDR | Void ratio, |

POR | Pore pressure, . |

SAT | Saturation, |

GELVR | Gel volume ratio, . |

FLUVR | Total fluid volume ratio, . |

FLVEL | Magnitude and components of the pore fluid effective velocity vector, . |

FLVELM | Magnitude, , of the pore fluid effective velocity vector. |

FLVEL n | Component |

The nodal output available includes the usual mechanical quantities such as displacements, reaction forces, and coordinates. In addition, the following quantities associated with pore fluid flow are available:

POR | Pore pressure at a node. |

RVF | Reaction fluid volume flux due to prescribed pressure. This flux is the rate at which fluid volume is entering or leaving the model through the node to maintain the prescribed pressure boundary condition. A positive value of RVF indicates fluid is entering the model. |

All of the output variable identifiers are outlined in “ABAQUS/Standard output variable identifiers,” Section 4.2.1.

*HEADING … *MATERIAL, NAME=mat1Data lines to define mechanical properties of the solid material… *DENSITYData lines to define the density of the dry material*PERMEABILITY, SPECIFIC=Data lines to define permeability, , as a function of the void ratio,… *INITIAL CONDITIONS, TYPE=STRESS, GEOSTATICeData lines to define the initial stress state*INITIAL CONDITIONS, TYPE=PORE PRESSUREData lines to define initial values of pore fluid pressures*INITIAL CONDITIONS, TYPE=RATIOData lines to define initial values of the void ratio*INITIAL CONDITIONS, TYPE=SATURATIONData lines to define initial saturation*BOUNDARYData lines to define zero-valued boundary conditions** *STEP *GEOSTATIC *CLOAD and/or *DLOAD and/or *DSLOADData lines to specify mechanical loading*FLOW and/or *SFLOW and/or *DFLOW and/or *DSFLOWData lines to specify pore fluid flow*BOUNDARYData lines to specify displacements or pore pressures*END STEP