Geostatic Stress State

The geostatic procedure requires that the initial stresses are close to the equilibrium state; otherwise, the displacements corresponding to the equilibrium state might be large. Abaqus/Standard checks for equilibrium during the geostatic procedure and iterates, 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), is then used as the initial stress field in a subsequent static or coupled pore fluid diffusion/stress (with or without heat transfer) 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.

If heat transfer is modeled during the geostatic step through the use of coupled temperature–pore pressure elements, the initial temperature field and thermal loads, if specified, must be such that the system is relatively close to a state of thermal equilibrium. Steady-state heat transfer is assumed during a geostatic step.

To obtain equilibrium in cases when the initial stress state is unknown or is known only approximately, you can invoke an enhanced procedure. Abaqus automatically computes the equilibrium corresponding to the initial loads and the initial configuration, allowing only small displacements within user-specified tolerances. (The default tolerance is ${10}^{-5}$.) The procedure is available with a limited number of elements and materials and is intended to be used in analyses in which the material response is primarily elastic; that is, inelastic deformations are small.

The procedure is supported for both geometrically linear and geometrically nonlinear analyses. However, in general, the performance in the geometrically linear case will be better. Therefore, it might be advantageous to obtain the initial equilibrium in a geometrically linear step, even though a geometrically nonlinear analysis is performed in subsequent steps.

When coupled temperature–pore pressure elements are used, heat transfer is modeled in these elements by default. However, you may optionally choose to switch off heat transfer within these elements during a geostatic step. This feature may be helpful in reducing computation time if temperature and associated heat flow effects are not important.

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, $e^0$, the initial pore pressure, $u_w$, 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). 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 ${\gamma }_w$ is the user-defined specific weight of the pore fluid (see Permeability). (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.) If we also take ${\gamma }_w$ to be independent of z (which is usually the case, since the fluid is almost incompressible), this equation can be integrated to define

where $z_w^0$ is the height of the phreatic surface, at which $u_w=0$ and above which $u_w<0$ and the pore fluid is only partially saturated.

We usually assume that there are no significant shear stresses ${\tau }_{xz}$, ${\tau }_{yz}$. Then, equilibrium in the vertical direction is

where $\rho$ is the dry density of the porous solid material (the dry mass per unit volume), g is the gravitational acceleration, $n^0$ is the initial porosity of the material, and s is the saturation, $0\le s\le 1.0$ (see Permeability). Since porosity is the ratio of pore volume to total volume and the void ratio is the ratio of pore volume to solids volume, $n^0$ is defined from the initial void ratio by

Abaqus/Standard requires that the initial value of the effective stress, $\overline{\mathit{\sigma }}$, be given as an initial condition (Initial Conditions). Effective stress is defined from the total stress, $\mathit{\sigma }$, by

where $\mathbf{I}$ is a unit matrix. Combining this definition with the equilibrium statement in the z-direction and hydrostatic equilibrium in the pore fluid gives

again using the assumption that ${\gamma }_w$ is independent of z. $z_1^0$ is the position of the surface that separates the dry soil from the partially saturated soil. The soil is assumed to be dry ($s=0$) for $z_1^0<z$, and it is assumed to be partially saturated for $z_w^0<z<z_1^0$ and fully saturated for $z\le z_w^0$.

In many cases s is constant. For example, in fully saturated flow $s=1.0$ everywhere below the phreatic surface. If we further assume that the initial porosity, $n^0$, and the dry density of the porous medium, $\rho$, are also constant, the above equation is readily integrated to give

where $z^0$ is the position of the surface of the porous medium, $z_w^0<z^0$.

In more complicated cases where s, $n^0$, and/or $\rho$ vary with height, the equation must be integrated in the vertical direction to define the initial values of ${\overline{\sigma }}_{zz}\left(z\right)$.

In many geotechnical applications there is also horizontal stress, typically caused by tectonic action. If the pore fluid is under hydrostatic equilibrium and ${\tau }_{xz}={\tau }_{yz}=0$, equilibrium in the horizontal directions requires that the horizontal components of effective stress do not vary with horizontal position: ${\overline{\sigma }}_h\left(z\right)$ only, where ${\overline{\sigma }}_h$ is any horizontal component of effective stress.

There are two different conventions to define the effective stress. The effective stress, $\overline{\mathit{\sigma }}$, defined above, is often referred to as the soils mechanics effective stress. Another form of effective stress that takes into account the effect of the bulk modulus of the solid grains is often referred to as the rock mechanics effective stress, ${\overline{\sigma }}^{**}$. The rock mechanics effective stress is used to evaluate the damage state of the material if a material damage model is present or the element is enriched. The material plasticity constitutive behavior is always computed based on the soils mechanics effective stress regardless of the material damage state. The rock mechanics effective stress, ${\overline{\sigma }}^{**}$, is related to the soils mechanics effective stress, $\overline{\mathit{\sigma }}$, by

where $\alpha$ is the so-called Biot's coefficient. Biot's coefficient is defined as $\alpha =1-\frac{K}{K_s}$, where $K$ is the bulk elastic modulus of the porous media and $K_s$ is the bulk elastic modulus of the solid grains.