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10. Soil analysis

10.1 Plane strain consolidation

10.1.1 Plane strain consolidation

Product: Abaqus/Standard  

Most consolidation problems of practical interest are two- or three-dimensional, so that the one-dimensional solutions providedby Terzaghi consolidation theory (see “ The Terzaghi consolidation problem, ”   Section 1.15.1 of the Abaqus Benchmarks Guide ) are useful only as indicators of settlement magnitudes and rates. This problem examines a linear, two-dimensional consolidation case: the settlement history of a partially loaded strip of soil.This particular case is chosen to illustrate two-dimensional consolidation because an exact solution is available (Gibson et al., 1970), thus providing verification of this capability in Abaqus.

Geometry and model

The discretization of the semi-infinite, partially loaded strip of soil is shown in Figure 10.1.1–1. The loaded region is half aswide as the depth of the sample. The reduced-integration plane strain element with pore pressure, CPE8RP, is used in this analysis. Reduced integration is almost always recommended when second-order elements are used because it usually gives more accurate results and is less expensive than full integration. No

mesh convergence studies have been done, although the reasonable agreement between the numerical results provided by this model and the solution of Gibson et al. (1970) suggests that the model used is adequate—at least for the overall displacement response examined. In an effort to reduce analysis cost while at the same time preserve accuracy, the mesh is graded from six elements through the height, under the load, to one element through the height at the outer boundary of the model, where a single infinite element (type CINPE5R) is used to model the infinite domain. This requires the use of two kinematic constraint features provided by Abaqus. Consider first the displacement degrees of freedom along line   in Figure 10.1.1–1. The 8-node isoparametric elements used for the analysis allow quadratic variation of displacement along their sides, so the displacementsof nodes a and b in elements x and y may be incompatible with thedisplacement variation along side   of element z. To avoid this, nodes a and b must be constrained to lie on the parabola defined by the displacements of nodes A, B, and   The QUADRATIC MPC (“multi-point constraint”) is used to enforce this kinematic constraint: it must be used at each node where this constraint is required (see planestrainconsolidation.inp). Pore pressure values are obtained by linear interpolation of values at the corner nodes of an element. When mesh gradation is used, as along line   in this example, an incompatibility in pore pressure values may result for the same reason given for thedisplacement incompatibility discussed above. To avoid this, the pore pressure at node B must be constrained to be interpolated linearly from the pore pressure values at A and   This is done by using the P LINEAR MPC.

The material properties assumed for this analysis are as follows:the Young's modulus is chosen as 690 GPa (108 lb/in2); the Poisson's ratio is 0; the material's permeability is 5.08 × 10–

7 m/day (2.0 × 10–5 in/day); and the specific weight of pore fluidis chosen as 272.9 kN/m3 (1.0 lb/in3).

The applied load has a magnitude of 3.45 MPa (500 lb/in2). The strip of soil is assumed to lie on a smooth, impervious base, so the vertical component of displacement is prescribed to be zero

on that surface. The left-hand side of the mesh is a symmetry line (no horizontal displacement). The infinite element models the other boundary.

Time stepping

As in the one-dimensional Terzaghi consolidation solution (see “ The Terzaghi consolidation problem, ”   Section 1.15.1 of the Abaqus Benchmarks Guide), the problem is run in two steps. In thefirst transient soils consolidation step, the load is applied andno drainage is allowed across the top surface of the mesh. This one increment step establishes the initial distribution of pore pressures which will be dissipated during the second transient soils consolidation step.

During the second step drainage is allowed to occur through the entire surface of the strip. This is specified by prescribing thepore pressure (degree of freedom 8) at all nodes on this surface (node set TOP) to be zero. By default, in a transient soils consolidation step such boundary conditions are applied immediately at the start of the step and then held fixed. Thus, the pore pressures at the surface change suddenly at the start ofthe second step from their values with no drainage (defined by the first step) to 0.0.

Consolidation is a typical diffusion process: initially the solution variables change rapidly with time, while at the later times more gradual changes in stress and pore pressure are seen. Therefore, an automatic time stepping scheme is needed for any practical analysis, since the total time of interest in consolidation is typically orders of magnitude larger than the time increments that must be used to obtain reasonable solutions during the early part of the transient. Abaqus uses a tolerance on the maximum change in pore pressure allowed in an increment tocontrol the time stepping. When the maximum change of pore pressure in the soil is consistently less than this tolerance, the time increment is allowed to increase. If the pore pressure changes exceed this tolerance, the time increment is reduced and the increment is repeated. In this way the early part of the

consolidation can be captured accurately and the later stages areanalyzed with much larger time steps, thereby permitting efficient solution of the problem. For this case the tolerance ischosen as 0.344 MPa (50 lb/in2), which is 10% of the applied load. This is a fairly coarse tolerance but results in an economical and reasonable solution.

The choice of initial time step is important in consolidation analysis. As discussed in “ The Terzaghi consolidation problem, ”   Section 1.15.1 of the Abaqus Benchmarks Guide , the initial solution (immediately following a change in boundary conditions) is a local, “skin effect” solution. Due to the coupling of spatial and temporal scales, it follows that no useful information is provided by solutions generated with time steps smaller than the mesh and material-dependent characteristictime. Time steps very much smaller than this characteristic time provide spurious oscillatory results (see Figure 3.1.5–2). This issue is discussed by Vermeer and Verruijt (1981), who propose the criterion

where   is the distance between nodes of the finite element mesh near the boundary condition change, E is the elastic modulusof the soil skeleton, k is the soil permeability, and   is the specific weight of the pore fluid. In this problem   is 8.5 mm (0.33 in). Using the material properties shown in Figure 10.1.1–1,

We actually use an initial time step of 2 × 10–5 days, since the immediate transient just after drainage begins is not considered important in the solution.

Results and discussion

The prediction of the time history of the vertical deflection of the central point under the load (point P in Figure 10.1.1–1) is plotted in Figure 10.1.1–2, where it is compared with the exact

solution of Gibson et al. (1970). There is generally good agreement between the theoretical and finite element solutions, even though the mesh used in this analysis is rather coarse.

Figure 10.1.1–2 also shows the time increments selected by the automatic scheme, based on the tolerance discussed above. The figure shows the effectiveness of the scheme: the time increment changes by two orders of magnitude over the analysis.

Input file

planestrainconsolidation.inp

Input data for this example.

References

Gibson, R. E., R. L. Schiffman, and S. L. Pu, “Plane Strain and Axially Symmetric Consolidation of a Clay Layer on a Smooth Impervious Base,” Quarterly Journal of Mechanics and Applied Mathematics, vol. 23, pt. 4, pp. 505–520, 1970.

Vermeer, P. A., and A. Verruijt, “An Accuracy Condition for Consolidation by Finite Elements,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 5, pp. 1–14, 1981.

Figures

Figure 10.1.1–1 Plane strain consolidation example: geometry and properties.

Figure 10.1.1–2 Consolidation history and time step variation history.

1.15.1 The Terzaghi consolidation problem

Product: Abaqus/Standard  

This one-dimensional problem has a well-known linear solution (see Terzaghi and Peck, 1948) and, thus, provides a simple verification of the consolidation capability in Abaqus. The analysis of saturated soils requires solution of coupled stress-diffusion equations, and the formulation used in Abaqus is described in detail in “ Analysis of porous media, ”   Section 2.8 of the Abaqus Theory Guide, and “ Plasticity for non-metals, ”   Section 4.4 of the Abaqus Theory Guide. The coupling is approximated by the effective stress principle, which treats the saturated soil as a continuum, assuming that the total stress at each point is the sum of an “effective stress” carried by the soil skeleton anda pore pressure in the fluid permeating the soil. This fluid porepressure can change with time (if external conditions change, such as the addition of a load to the soil), and the gradient of the pressure through the soil that is not balanced by the weight of fluid between the points in question will cause the fluid to flow: the flow velocity is proportional to the pressure gradient in the fluid according to Darcy's law. A typical case is a consolidation problem. Here the addition of a load (usually an overburden) to a body of soil causes pore pressure to rise initially; then, as the soil skeleton takes up the extra stress, the pore pressures decay as the soil consolidates. The Terzaghi problem is the simplest example of such a process. For illustration purposes, the problem is treated with and without finite-strain effects. The small-strain version is the classical case discussed by Terzaghi and Peck (1948), and the finite-strainversion has been analyzed numerically by a number of authors, including Carter et al. (1979).

Problem description

The problem is shown in Figure 1.15.1–1. A body of soil 2.54 m (100 in) high is confined by impermeable, smooth, rigid walls on all but the top surface. On that surface perfect drainage is

possible, and a load is applied suddenly. Gravity is neglected. Because of the boundary conditions, the problem is one-dimensional, the only gradient being in the vertical direction. The purpose of the analysis is to predict the evolution of displacement, effective stress, and pore pressure throughout the soil mass as a function of time following the load application.

Geometry and models

Abaqus contains no one-dimensional elements for effective stress calculations. Therefore, we use a two-dimensional plane strain mesh, with one element only in the x-direction. Element type CPE4P is chosen to perform the finite-strain analysis, and element type CPE8P is chosen for the small-strain analysis. We recommend the use of linear elements for applications involving finite strain, impact, or complex contact conditions and second-order elements for problems where stress concentrations must be captured accurately or where geometric features such as curved surfaces must be modeled. In this particular example the linear and second-order elements yield almost identical results.

The soil is assumed to be linear elastic, with a Young's modulus of 689.5 GPa (108 lb/in2) and Poisson's ratio of 0.3. The specific weight of the pore fluid is assumed to be 276.8 × 103 N/m3 (1 lb/in3). The permeability is assumed to vary linearly with the void ratio, with a value of 8.47 × 10–8 m/sec (2.0 × 10–

4 in/min) at a void ratio of 1.5 and a value of 8.47 × 10–9 m/sec (2.0 × 10–5 in/min) at a void ratio of 1.0. The void ratio is assumed to be 1.5 initially throughout the sample. Abaqus uses effective permeability, which is permeability divided by the specific weight of the pore fluid. Therefore, the fluid in this problem is assigned the value 276.8 × 103 N/m3 (1 lb/in3) for the specific weight (water, for example, has a specific weight of 9965 N/m3, 0.036 lb/in3) and the permeability is scaled accordingly.

The boundary conditions are as follows. On the bottom and two vertical sides, the normal component of displacement is fixed (

0 on the bottom and  0 on the sides), and no flow of pore

fluid through the walls is permitted. This latter is the natural boundary condition in the fluid mass conservation equation, so noexplicit specifications need to be made (as with zero tractions in the equilibrium equation). On the top surface a uniform downward load (an overburden) is applied suddenly. The magnitude of this load is taken to be 689.5 GPa (108 lb/in2). This large load will cause considerable deformation, thus illustrating the difference between the small- and large-strain solutions. This surface allows perfect drainage so that the excess pore pressure is always zero on this surface.

Time stepping

The problem is run in two steps. The first step is a single increment of a transient soils consolidation analysis with an arbitrary time step, with no drainage allowed across the top surface (the natural boundary condition in the mass conservation equation governing the pore fluid flow). This establishes the initial solution: uniform pore pressure equal to the load throughout the body, with no stress carried by the soil skeleton (zero effective stress). The actual consolidation is then done with a second soils consolidation step, using automatic time stepping.

The accuracy of the time integration for the second soils consolidation procedure, during which drainage is occurring, is controlled by specifying the maximum allowable pore pressure change per time step,  . Even in a linear problem this valuecontrols the accuracy of the solution, because the time integration operator is not exact (the backward difference rule is used). In this case   is chosen as 344.8 GPa (5.0 × 107 lb/in2), which is a relatively large value and, so, should only give moderate accuracy: this is considered to be adequate for the purposes of the example.

An important issue in such consolidation problems is the choice of initial time step. As the governing equations are parabolic, the initial solution (immediately after the sudden change in load) is a local, “skin effect,” solution. In this one-

dimensional case the form of the initial solution is sketched in Figure 1.15.1–2 for illustration purposes. With a finite element mesh of reasonable size for modeling the solution at a later time (when the changes in pore pressure have diffused into the bulk of the body soil), this initial solution will be modeledpoorly. With smaller initial time steps the difficulty becomes more pronounced, as sketched inFigure 1.15.1–2. As in any transient problem, the spatial element size and the time step arerelated to the extent that time steps smaller than a certain sizegive no useful information. This coupling of the spatial and temporal approximations is always most obvious at the start of diffusion problems, immediately after prescribed changes in the boundary values. For this particular case the issue has been discussed in detail by Vermeer and Verruijt (1981), who suggest the simple criterion

where   is a characteristic element size near the disturbance (that is, near the draining surface in our case), E is the elastic modulus of the soil skeleton, k is the soil permeability,and   is the specific weight of the permeating fluid. For our model we choose   254 mm (10 in); and we have   689.5  GPa (108 lb/in2),  8.47 × 10–8 m/s (2.0 × 10–4 in/min),  2.768 × 105 N/m3 (1.0 lb/in3), which gives  .05 s (0.833 × 10–3 min). Based on this calculation, an initial time step of .06 sec (0.001 min) is used. This gives an initial solution with no “overshoot” at all, as expected.

In this case we wish to continue the analysis to steady-state conditions. This is defined by asking Abaqus to stop when all pore pressure change rates fall below 11.5 KN/m2/s (100 lb/in2/min).

Results and discussion

In the small-strain analysis the “steady-state” condition (rate of change of pore pressure with time below the prescribed value)

is reached after 20 increments, the last time increment taken being 491 seconds (8.19 min)—about 8000 times the initial time increment. This very large change in time increment size is typical of such diffusion systems and points out the value of using automatic time stepping with an unconditionally stable integration operator for such problems.

The results of the small-strain analysis are summarized in Figure1.15.1–3 to Figure 1.15.1–5. Figure 1.15.1–3 shows pore pressure profiles (pore pressure as a function of elevation) at various times in the solution. As we would expect, the solution begins byrapid drainage at the top of the sample and loss of pore pressurein that region. This effect propagates down the sample until the entire sample is steadily losing pore pressure throughout its length. At steady state the solution has zero pore pressure everywhere, with the load being carried as a uniform effective vertical stress. Figure 1.15.1–4 shows this transfer of load fromthe fluid to the skeleton at the 1.905 m (75 in) elevation as a function of time. Figure 1.15.1–5 compares these numerical results with the solution quoted in Terzaghi and Peck (1948). Here the downward displacement of the top surface of the soil, asa fraction of its steady-state value (the “degree of consolidation”), is plotted as a function of normalized time, defined as

where k is the permeability of the soil, E is the Young's modulusof the soil,   is the specific weight of the pore fluid, H is the height of the soil sample, and t is time.

Figure 1.15.1–5 shows that the numerical solution agrees reasonably well with the analytical solution, with some loss of accuracy at later times. This latter effect is attributable to the coarse time stepping tolerance chosen. Higher accuracy could be obtained with a tighter tolerance on the allowable pore pressure stress change parameter ( ). However, the solution is clearly adequate for design use.

In the finite-strain analysis of soils, changes in the void ratiocan lead to large changes in permeability, therefore affecting the transient response in a consolidation analysis. Typical soilsshow a strong dependence of permeability on the void ratio (as soil compacts, it becomes increasingly harder for fluid to pass through it), with the consequence that “plugging” may result. This means that a soil that was relatively permeable in its original state becomes less permeable as it consolidates.

In this example the permeability of the soil is assumed to decrease by an order of magnitude as the void ratio decreases from its initial value of 1.5 to a value of 1.0. Such logarithmicdependence of permeability on the void ratio is not uncommon in fully saturated clays. Two finite-strain analyses are run, one with permeability treated as a constant and a second with this variation in permeability. The results are shown in Figure 1.15.1–6, together with the results of the small-strain analysis under similar load. The “plugging” effect of void ratio dependence of permeability is clearly seen in this figure. Since the permeability decreases with the consolidation of the soil, the time required for all excess pore pressure to dissipate increases. The final value of displacement under the applied loadis not a function of permeability and is correctly predicted by both large-strain analyses. (The exact solution for this displacement is very easily calculated.) It is interesting to observe that, if the permeability is not dependent on the void ratio, the finite-strain results show more rapid initial consolidation than the corresponding small-strain analysis.

A separate suite of files (terzaghi_cpe8p_rigid.inp, terzaghi_cpe4p_rigid.inp, and terzaghi_cpe8p_ss_rigid.inp) is provided to illustrate the use of a contact pair in problems involving pore pressure elements. Three rigid surfaces are used to model the three impermeable sides of the specimen shown in Figure 1.15.1–1, thus replacing the boundary conditions used in terzaghi_cpe8p.inp,terzaghi_cpe4p.inp, and terzaghi_cpe8p_ss.inp.

Input files

terzaghi_cpe8p.inp

Small-strain analysis (element type CPE8P).

terzaghi_cpe4p.inp

Finite-strain case with permeability depending on the void ratio (element type CPE4P).

terzaghi_cpe8p_ss.inp

Small-strain steady-state solution (element type CPE8P).

terzaghi_cpe8p_perm.inp

Small-strain case with velocity-dependent permeability (Forchheimer flow) and velocity coefficient depending on the voidratio.

terzaghi_postoutput1.inp

*POST OUTPUT postprocessing of terzaghi_cpe8p.inp.

terzaghi_postoutput2.inp

*POST OUTPUT postprocessing of terzaghi_cpe8p_perm.inp.

terzaghi_cpe8p_rigid.inp

Identical to terzaghi_cpe8p.inp except that rigid surfaces are used to impose the boundary conditions.

terzaghi_cpe4p_rigid.inp

Identical to terzaghi_cpe4p.inp except that rigid surfaces are used to impose the boundary conditions.

terzaghi_cpe8p_ss_rigid.inp

Identical to terzaghi_cpe8p_ss.inp except that rigid surfaces areused to impose the boundary conditions.

References

Carter,  J. P., J. R. Booker, and J. C. Small, “The Analysis of Finite Elasto-Plastic Consolidation,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 3, pp.107–129, 1979.

Terzaghi,  K., and R. B. Peck, Soil Mechanics in Engineering Practice, John Wiley and Sons, New York, 2nd, 1948.

Vermeer,  P. A., and A. Verruijt, “An Accuracy Condition for Consolidation by Finite Elements,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 5, pp. 1–14, 1981.

Figures

Figure 1.15.1–1 Terzaghi consolidation problem definition.

Figure 1.15.1–2 Solutions at very early times.

Figure 1.15.1–3 Pore pressure at various times.

Figure 1.15.1–4 Pore pressure and effective stress at elevation 1.905 m (75 in).

Figure 1.15.1–5 Degree of consolidation versus time factor.

Figure 1.15.1–6 Comparisons of finite- and small-strain solutionsto the Terzaghi consolidation problems.

1.15.2 Consolidation of a triaxial test specimen

Product: Abaqus/Standard  

This example illustrates inelastic deformation of a soil specimenwhose constitutive behavior is modeled with modified Cam-clay plasticity. The elastic part of the behavior is modeled with boththe linear elastic and porous elastic models. The Cam-clay plasticity theory, which is one of the critical state plasticity theories developed by Roscoe and his colleagues at Cambridge, is described in “ Plasticity for non-metals, ”   Section 4.4 of the Abaqus Theory Guide. Verification of the model is provided by “ Triaxial tests on a saturated clay, ”   Section 3.2.4 .

The geometric configuration is one of the most common soils tests: a triaxial specimen, confined by an enclosing membrane, being squeezed axially between platens (see Figure 1.15.2–1). Perfectly smooth and perfectly rough platens are both considered.The platen motion is assumed to be very slow compared to characteristic diffusion times in the soil, and the platen is assumed to provide perfect drainage, so that the pore pressures throughout the soil specimen are always essentially zero. Pore fluid diffusion is, thus, not a significant effect in this case. See “ The Terzaghi consolidation problem, ”   Section 1.15.1 , and “ Plane strain consolidation, ”   Section 10.1.1 of the Abaqus Example Problems Guide, for cases where transient effects in the pore fluid diffusion are an important aspect of the overall response.

As the specimen is compressed, the elastic-plastic response of the specimen consists of two competing effects. Elastically, the increased compressive hydrostatic effective stress on the soil skeleton causes a stiffening of the response. When the soil yields, inelastic deformation results in softer behavior. Eventually the stress state in some region of the specimen reaches critical state, where the soil skeleton response is perfectly plastic. When this region is sufficiently developed, a limit state is attained, and the specimen's resistance to further

compression no longer increases. The analysis is intended to track the response from the initial loading to this limit state.

Problem description

The soil sample is an axisymmetric cylinder, as shown in Figure 1.15.2–1. The model takes advantage of symmetry about the midplane, as well as the axisymmetry of the configuration. The specimen has a length to diameter ratio of 3. Two cases are considered: one in which the platen is assumed to be perfectly smooth, so that the stress state in the specimen will be homogeneous, and one in which the platen is assumed to be completely rough, so the soil in contact with the platen cannot move with respect to the platen. This latter case results in an nonhomogeneous stress state, as the specimen bulges during compression. The eight element mesh shown inFigure 1.15.2–1 is not expected to capture this nonhomogeneous state accurately but should suffice for the present demonstration purposes.

The material properties of the soil are based on the example usedby Zienkiewicz and Naylor (1972). The properties for the Cam-claymodel with porous elasticity are shown in Figure 1.15.2–1. The Cam-clay model with linear elasticity uses a Young's modulus of 15 GPa. This value is based on the elastic stiffness (at the end of the loading step) of the examples that use porous elasticity.

Initial conditions

The Cam-clay model assumes that the soil has no stiffness at zerostress, so that some initial (compressive) stress state must be defined for the material. In this case we assume that the soil sample is under an initial hydrostatic pressure of 0.1 MPa (14.5 lb/in2), and this confining pressure remains constant throughout the test. Since the soil may drain through the platen,this pressure is carried as an effective stress in the soil skeleton. This initial stress state is defined using initial conditions. In this particular example it is trivial to see that this initial stress state is in equilibrium with the external

distributed pressure of the same magnitude. In more complex casesit may not be so simple to ensure that the discrete, finite element model is in equilibrium with the geostatic loading. Accordingly, the first step of any analysis involving an initial stress state should be a geostatic step. In that step the geostatic external loads (in this case the pressure on the specimen) should be specified. Abaqus will then check whether theinitial stress state is in equilibrium with these loads. If it isnot, Abaqus will iterate and attempt to establish an equilibrium stress field that balances the prescribed tractions. Such iteration does not occur in this case since the prescribed initial stress is in equilibrium with the applied tractions.

Loading

The specimen is compressed to 40% of its initial height over 34.56 × 106 sec (400 days). Although this represents a large strain of the specimen, geometric nonlinearity is ignored in thisexample because we wish to examine the effects of the material nonlinearity, and we only report the stress-strain response at points, rather than overall load-deflection response that will bepredicted quite inaccurately unless geometric nonlinearity is included. The loading is applied in a transient soils consolidation step specifying the time period, with an associatedboundary condition prescribing the travel of the platen during that time. The platen is assumed to drain freely throughout the analysis. This is specified by a boundary condition, fixing the pore pressure at zero on the top edge of the mesh. The loading isintended to represent very slow compression, sufficiently slow that the pore pressures never rise to any significant values. We can obtain a rough idea of this time scale by noting that a characteristic time for pore pressure dissipation is , where H is a typical dimension from the draining surface (60 mm, 2.362 in, in this case);   is the specific weight of thepore fluid (1.0 × 104 N/m3, 0.0369 lb/in3); k is the permeability of the soil (0.1728 mm/day, 6.803 × 10–3 in/day); and   is a typical soil modulus, which we compute as  , where   is thelogarithmic bulk modulus and p is a typical mean normal effective

stress. T is, thus, estimated as 0.05 days. This is about the time it takes for pore pressures to drop to 5% of their initial values, following sudden application of a load (see Terzaghi and Peck, 1967). Since the time scale chosen for the loading of the test specimen in this example is very long compared to this value, no significant pore pressures should ever arise in the analysis.

The same analysis could be performed by using a static procedure,in which case the coupled, effective stress formulation element type could be replaced with an element that models soil deformation only. We choose to use the coupled element type and the soils consolidation procedure to exercise these features.

The accuracy of the equilibrium solution within a time increment is controlled by iterating until the out-of-balance forces reduceto a small fraction of an average force magnitude calculated internally by Abaqus. The rough platen causes an nonhomogeneous stress state, which tends to cause an underestimation of this average force magnitude since stresses are locally higher in the region of the mesh near the platen and the reference force magnitude is averaged over the entire mesh. To avoid iterating toexcessive accuracy, we have overridden the default calculation ofthe average force magnitude and have defined that typical actual nodal forces will be of the order 100 N (22.52 lb). This is done using solution controls. The increment size choice is automatic, determined by allowing a maximum pore pressure change ( ) of0.16 KPa (.023 lb/in2) per increment, which should give sufficient definition of the solution.

Results and discussion

Figure 1.15.2–2 shows results for the rough platen case, when thestress field is nonhomogeneous, and shows results corresponding to point A in Figure 1.15.2–1: this is the stress output point atthe centroid of the element shown. Figure 1.15.2–3 is for the smooth platen case, when the stress field is homogeneous. The topsection of each figure shows the  –q plane. Here p is the equivalent effective pressure stress, defined by

and q is the equivalent deviatoric stress (the Mises equivalent stress) defined by

where

is the deviatoric stress (here   is a unit matrix).

The   plot in each case shows the critical state line, the initial yield surface, and the stress trajectory followed in the solution. The bottom section of each figure is a plot of the equivalent deviatoric stress, q, versus the vertical deflection of the platen. The behavior in both cases is as we would expect: a gradual softening of the specimen after yield, until critical state is reached, when the behavior becomes perfectly plastic. Inthe rough platen case, the response at the point plotted moves some way up the critical state line after it reaches that line: presumably this is because other points in the model have not yetreached the limit state.

Similar results are obtained for the Cam-clay model with linear elasticity.

Input files

triaxconsolid_cax8rp_por.inp

Rough platen case using the porous elasticity model with CAX8RP elements.

triaxconsolid_caxa8rp1_por.inp

Rough platen case using the porous elasticity model with CAXA8RP1elements. This analysis is done as basic verification of this element type.

triaxconsolid_cax8rp_lin.inp

Rough platen case using the linear elasticity model with CAX8RP elements. This analysis is done as basic verification of the Cam-clay model with linear elasticity.

The only change needed for the smooth platen case is to remove the boundary conditions in the radial direction at the top of themesh.

References

Terzaghi,  K., and R. B. Peck, Soil Mechanics in Engineering Practice, John Wiley and Sons, New York, 2nd, 1967.

Zienkiewicz,  O. C., and D. J. Naylor, “The Adaptation of Critical State Soil Mechanics Theory for Use in Finite Elements,” Stress-Strain Behavior of Soils, edited by R. H. G. Parry, G. T. Foulis and Co., Ltd., London, 1972.

Figures

Figure 1.15.2–1 Triaxial consolidation: geometry, properties, andloading.

Figure 1.15.2–2 Shear stress versus mean normal stress and axial strain. Rough platen case.

Figure 1.15.2–3 Shear stress versus mean normal stress and axial strain. Smooth platen case.

1.15.3 Finite-strain consolidation of a two-dimensional solid

Product: Abaqus/Standard  

This example involves the large-scale consolidation of a two-dimensional solid. Nonlinearities caused by the large geometry changes are considered, as well as the effects of the change in the void ratio on the permeability of the material. The material is assumed to be linear elastic. The example exhibits many features in common with the one-dimensional Terzaghi consolidation problem discussed in “ The Terzaghi consolidation problem, ”   Section 1.15.1 , notably the reduced settlement magnitudes predicted by finite-strain analysis in comparison withthe results provided by small-strain theory.

Problem description

The example considers a finite strip of soil, loaded over its central portion. Symmetry permits modeling half the strip, as shown in Figure 1.15.3–1: the half-width of the strip is equal toits height, and the ratio of the loaded portion of the strip to the width of the strip is 1:5. The finite element discretization used is also shown in Figure 1.15.3–1: 35 CPE8RP elements are used, and the mesh is graded in the vertical direction in length ratios 1:2:3:4:5 and horizontally in ratios 1:1:1:2:2:4:4. This is a coarse mesh, but it is expected to provide representative results. A similar mesh using CPE4P elements is included for verification purposes.

Figure 1.15.3–1 also summarizes the material properties and boundary conditions used. The ratio of the pressure load to the Young's modulus is 1:2, and the Poisson's ratio is specified as 0. The soil is assumed to have an initial void ratio of 1.5, and the permeability at this value of the void ratio is 0.508  m/sec(2.0 × 10–5 in/sec). The permeability is assumed to be one order of magnitude smaller at a void ratio of 1.0. These low permeability values are representative of clays.

The strip of soil is assumed to lie on a rigid, impermeable, smooth base. No horizontal displacement or pore fluid flow is permitted along the vertical sides of the model. Free drainage isassumed along the top surface of the model.

Loading and time stepping

The analysis is performed using two transient soils consolidationsteps. In the preliminary step the full load is applied over two equal fixed time increments. The load remains constant in the subsequent step during which the soil undergoes consolidation.

In the analysis accounting for finite-strain effects, the preliminary step requires six iterations for convergence of the first increment and seven iterations for convergence at full load. These relatively large numbers of iterations are due to thelarge geometry changes experienced by the soil. As shown in Figure 1.15.3–2, at full load the midpoint vertical deflectionin this case is about 0.49 times the width of the strip that is loaded. The geometrically linear analysis predicts the midpoint vertical deflection to be approximately 0.52 times the width of the strip that is loaded.

Practical consolidation analyses require solutions across severalorders of magnitude of time (see Figure 1.15.3–2, for example), and the automatic time stepping scheme is designed to generate cost-effective solutions for such cases. The algorithm is based on the user supplying a tolerance on the pore pressure change permitted in any increment,  . Abaqus uses this value in thefollowing manner: if the maximum change in pore pressure at any node is greater than  , the increment is repeated with a proportionally reduced time step. If the maximum change in pore pressure at any node is consistently less than  , the time step is increased. In this case   is set to 0.103 MPa (15 lb/in2). This represents about 3% of the maximum pore pressure in the model following application of the load. With this value the first time increment is 7.2 seconds, and the finaltime increment is 1853 seconds. This is quite typical of diffusion processes: at early times the time rates of pore

pressure are significant, and at later times these time rates arevery low.

Results and discussion

The first analysis considers finite-strain effects, and the soil permeability varies with the void ratio. This change of permeability with the void ratio is physically realistic—as soil is compressed, it becomes harder for pore fluid to flow through it. A small-strain analysis is also run with constant permeability. The midpoint settlement versus time for the finite-and small-strain analyses are shown in Figure 1.15.3–2. The two analyses predict large differences in the final consolidation: the small-strain result shows about 40% more deformation than thefinite-strain case. This is consistent with results from the one-dimensional Terzaghi consolidation solutions—(see “ The Terzaghi consolidation problem, ”   Section 1.15.1 ). Quite clearly, in cases where settlement magnitudes are significant, finite-strain effects are important.

The time scale in Figure 1.15.3–2 spans five orders of magnitude,pointing to the importance of automatic time incrementation for cost-effective solutions.

Figure 1.15.3–3 shows time histories of pore pressure at two points in the model, points a and b in Figure 1.15.3–1. The pore pressure results are normalized by the value of pore pressure at these points at the end of the preliminary step and are taken from the finite-strain analysis. The increase in pore pressure shown at point   is evidence of the Mandel-Cryer effect (see Prevost, 1981 and Lambe and Whitman, 1969) and is typical of two-and three-dimensional consolidation analysis.

Input files

finstrainconsolid2d_node.f

Generates the nodal coordinates required by finstrainconsolid2d_cpe8rp.inp.

finstrainconsolid2d_cpe8rp.inp

Finite-strain analysis (element type CPE8RP).

finstrainconsolid2d_cpe4p.inp

Element type CPE4P.

finstrainconsolid2d_autostab.inp

Input data with automatic stabilization added.

finstrainconsolid2d_autostab_adap.inp

Input data with adaptive automatic stabilization added.

The small-strain case is obtained by removing the NLGEOM parameter on the *STEP option.

References

Lambe,  T. W., and R. V. Whitman, Soil Mechanics, John Wiley and Sons, New York, 1969.

Prevost,  J. H., “Consolidation of an Elastic Porous Media,” Journal of the Engineering Mechanics Division, ASCE, vol.107, pp. 169–186, February 1981.

Figures

Figure 1.15.3–1 Two-dimensional elastic consolidation problem description.

Figure 1.15.3–2 Midpoint settlement time history.

Figure 1.15.3–3 Pore pressure time history.

1.15.4 Limit load calculations with granular materials

Products: Abaqus/Standard  Abaqus/Explicit  

This example presents solutions to limit load calculations for a strip of sand loaded by a rigid, perfectly rough footing. The foundation material is defined as a Mohr-Coulomb material; therefore, we show results obtained with the Mohr-Coulomb plasticity model available in Abaqus. The example also shows results obtained with different parameters used in the modified Drucker-Prager model in Abaqus, with and without a cap, matched to the classical Mohr-Coulomb yield model.

The classical failure model for granular materials is the Mohr-Coulomb model, which can be written as

where   and   are the maximum and minimum principal stresses (positive in tension),   is the friction angle, andc is the cohesion. The intermediate principal stress has no effect on yield in this model. Experimental evidence suggests that the intermediate principal stress does have an effect on yield; nonetheless, laboratory data characterizing granular materials are often presented as values of   and 

Abaqus offers a Mohr-Coulomb model for modeling this class of material behavior. This model uses the classical Mohr-Coulomb yield criterion: a straight line in the meridional plane and a six-sided polygon in the deviatoric plane. However, the Abaqus Mohr-Coulomb model has a completely smooth flow potential insteadof the classical hexagonal pyramid: the flow potential is a hyperbola in the meridional plane, and it uses the smooth deviatoric section proposed by Menetrey and Willam (1995). The Abaqus Mohr-Coulomb model is described in “ Mohr-Coulomb plasticity, ”   Section 23.3.3 of the Abaqus Analysis User's Guide .

Abaqus also offers two Drucker-Prager models, with and without a compression cap, to model this class of material behavior. The

Abaqus Drucker-Prager model without a cap provides a choice of three yield criteria. The differences are based on the shape of the yield surface in the meridional plane, which can be a linear form, a hyperbolic form, or a general exponent form (as describedin “ Extended Drucker-Prager models, ”   Section 23.3.1 of the Abaqus Analysis User's Guide). The linear form is used here to make direct comparisons with the classical linear Mohr-Coulomb model. In addition, the hyperbolic and exponential forms are also verified in this example by using parameters that reduce them to equivalent linear forms.

This section also illustrates how to match the parameters of a corresponding linear Drucker-Prager model,   andd, to the Mohr-Coulomb parameters,   and c, under plane strain conditions.

The Abaqus Drucker-Prager and Mohr-Coulomb models restrict possible flow patterns when the stress point is at a vertex of the Mohr-Coulomb yield surface. Thus, the models will not reproduce some localization effects exhibited by real materials, which are assumed to behave more in accordance with a vertex model than with a smooth model when the plastic flow direction wants to change rapidly with load. Either model must be used withnonassociated flow to avoid excessive dilatation in modeling realmaterials.

The Drucker-Prager/Cap model adds a cap yield surface to the modified Drucker-Prager model. The cap surface serves two main purposes: it bounds the yield surface in hydrostatic compression,thus providing an inelastic hardening mechanism to represent plastic compaction; and it helps control volume dilatancy when the material yields in shear by providing softening as a functionof the inelastic volume increase created as the material yields on the Drucker-Prager shear failure and transition yield surfaces. The model uses associated flow in the cap region and a particular choice of nonassociated flow in the shear failure and transition regions.

Problem description

The plane strain model analyzed is shown in Figure 1.15.4–1. The strip of sand is 3.66 m (12 ft) deep and of infinite horizontal extent. The footing is rigid and perfectly rough and spans a central portion 3.05 m (10 ft) wide. The model assumes symmetry about a center plane, and the region modeled with finite elementsextends 8.84 m (29 ft) to the right of the center plane. In Abaqus/Standard reduced-integration, second-order, plane strain quadrilaterals (element type CPE8R) are used for the finite region, and infinite elements (element type CINPE5R) are used beyond this line to simulate the rest of the strip. In Abaqus/Explicit these elements are substituted with their linear counterparts (element type CPE4R and element type CINPE4, respectively). In Abaqus the infinite elements are always assumedto have linear elastic behavior; therefore, they are used beyond the region where plastic deformation takes place. The base of thestrip is fixed in both the horizontal and vertical directions. The mesh is shown in Figure 1.15.4–1. No mesh convergence studieshave been performed.

Material

The material's elastic response is assumed to be linear and isotropic, with a Young's modulus 207 MPa (30 × 103 lb/in2) and aPoisson's ratio 0.3. Yield is assumed to be governed by the Mohr-Coulomb surface, with a friction angle   20° and cohesion, c, of 0.069 MPa (10 lb/in2). These constants can be used directly inthe Abaqus Mohr-Coulomb model.

“ Extended Drucker-Prager models, ”   Section 23.3.1 of the Abaqus Analysis User's Guide, describes the method for converting these Mohr-Coulomb parameters to Drucker-Prager parameters in plane strain. Applying the formulae given in the Abaqus Analysis User'sGuide provides   0.581 (  30.16°) and   0.137 MPa (19.8 lb/in2) for associated flow and   0.592 (  30.64°) and   0.140 MPa (20.2 lb/in2) for nondilatant flow. The example is run using the associated flow parameters together with   and using the nondilatant flow parameters together with  0.

The Drucker-Prager/Cap model is run using the same plane strain matching of the Mohr-Coulomb parameters. The cap eccentricity parameter is chosen as   0.1. The initial cap position (which measures the initial consolidation of the specimen) is taken as   0.00041, and the cap hardening curve is as shown in Figure 1.15.4–2. The transition surface parameter   0.01 isused.

For verification of the hyperbolic and exponent forms of the yield criteria, input files have been included that correspond tothe dilatant linear Drucker-Prager model. Reducing the hyperbolicyield function into a linear form requires that  . Reducing the exponent yield function into a linear form requires that   1.0 and that   ( )–1.

Loading and controls

We are mainly interested in obtaining the limit footing pressure and in estimating the vertical displacement under the footing as a function of load.

A convenient way of defining a rigid and perfectly rough footing is to use an equation constraint to constrain all of the nodes under the footing to have the same displacement, which is done byretaining the central top node (node 801) to represent the footing. A vertical displacement is then applied to this node while its horizontal displacement is constrained to be zero. The total footing load is obtained as the vertical reaction force at node 801. The average footing pressure is this vertical load divided by the width of the footing.

For the nonassociated flow cases unsymmetric matrix storage and solution is used: this is essential to obtain an acceptable convergence rate since nonassociated flow plasticity results in unsymmetric stiffness matrices.

Results and discussion

The load-displacement responses are shown in Figure 1.15.4–3, where we also show the limit analysis (slip line) Prandtl and Terzaghi solutions, as given by Chen (1975). The nondilatant Drucker-Prager and Mohr-Coulomb models give a softer response anda lower limit load than the corresponding dilatant versions. The cap model provides a response that is comparable to the corresponding Drucker-Prager nondilatant response. This response is due to the addition of the cap and the nonassociated flow in the failure region, which combine to reduce the dilation in the model and—therefore—approximate the Drucker-Prager nondilatant flow model.

The results obtained for the nondilatant Drucker-Prager model andthe corresponding cap model match closely those obtained with thenondilatant Mohr-Coulomb model. They provide almost identical limit loads, which lie between the Prandtl and Terzaghi solutions. This conclusion can be extended to general geotechnical problems that are analyzed under plane strain or axisymmetric assumptions.

Input files

Abaqus/Standard input files

granularlimitload_mc_nondilat.inp

Mohr-Coulomb nondilatant flow case.

granularlimitload_mc_dilat.inp

Mohr-Coulomb dilatant flow case.

granularlimitload_dp_nondilat.inp

Drucker-Prager nondilatant flow case.

granularlimitload_dp_dilat.inp

Drucker-Prager dilatant flow case.

granularlimitload_cap2.inp

Case with cap model.

granularlimitload_hyper_dilat.inp

Hyperbolic yield criterion, dilatant case.

granularlimitload_expo_dilat.inp

Exponential yield criterion, dilatant case.

Abaqus/Explicit input files

granularlimitload_mc_nondilat_xpl.inp

Mohr-Coulomb nondilatant flow case.

granularlimitload_mc_dilat_xpl.inp

Mohr-Coulomb dilatant flow case.

References

Chen,  W. F., Limit Analysis and Soil Plasticity, Elsevier, Amsterdam, 1975.

Menétrey,  Ph., and K. J. Willam, “Triaxial Failure Criterion for Concrete and its Generalization,” ACI Structural Journal, vol. 92, pp. 311–318, May/June 1995.

Figures

Figure 1.15.4–1 Model for limit load calculations on centrally loaded sand strip.

Figure 1.15.4–2 Cap hardening curve.

Figure 1.15.4–3 Limit load results.

1.15.5 Finite deformation of an elastic-plastic granular material

Products: Abaqus/Standard  Abaqus/Explicit  

Problem description

This is a simple verification study in which we develop the homogeneous, finite-strain inelastic response of a granular material subject to uniform extension or compression in plane strain. Results given by Carter et al. (1977) for these cases areused for comparison.

The specimen is initially stress-free and is made of an elastic, perfectly plastic material. The elasticity is linear, with a Young's modulus of 30 MPa and a Poisson's ratio of 0.3. Carter etal. assume that the inelastic response is governed by a Mohr-Coulomb failure surface, defined by the friction angle of the Coulomb line (  30°) and the material's cohesion (c). They alsoassume that the cohesion is twice the Young's modulus for the extension test and 10% of the Young's modulus in the compression test. The above problem is solved using the Mohr-Coulomb plasticity model in Abaqus with the friction angle and the dilation angle equal to 30°. However, note that this Abaqus Mohr-Coulomb model is not identical to the classical Mohr-Coulomb model used by Carter because it uses a smooth flow potential.

An alternative solution is to use the associated linear Drucker-Prager surface in place of the Mohr-Coulomb surface. In this caseit is necessary to relate   and c to the material constants   and   that are used in the Drucker-Prager model. Matching procedures are discussed in “ Extended Drucker-Prager models, ” Section 23.3.1 of the Abaqus Analysis User's Guide . In this case we select a match appropriate for plane strain conditions:

The first equation gives   40°. Using the assumptions of Carter et al., the second equation gives d as 86.47 MPa (  = 120 MPa) for the extension case and d as 4.323 MPa (  = 6 MPa) for the compression case.

Uniform extension or compression of the soil sample is specified by displacement boundary conditions since the load-displacement response will be unstable for the extension case.

Results and discussion

The results are shown in Figure 1.15.5–1 for extension and in Figure 1.15.5–2 for compression. The solutions for Abaqus/Standard and Abaqus/Explicit are the same. The Drucker-Prager solutions agree well with the results given by Carter et al.; this is to be expected since the Drucker-Prager parameters are matched to the classical Mohr-Coulomb parameters under plane strain conditions. The differences between the Abaqus Mohr-Coulomb solutions and Carter's solutions are due to the fact thatthe Abaqus Mohr-Coulomb model uses a different flow potential. The Abaqus Mohr-Coulomb model uses a smooth flow potential that matches the classical Mohr-Coulomb surface only at the triaxial extension and compression meridians (not in plane strain).

However, one can also obtain Abaqus Mohr-Coulomb solutions that match Carter's plane strain solutions exactly. As discussed earlier, the classical Mohr-Coloumb model can be matched under plane strain conditions to an associated linear Drucker-Prager model with the flow potential

This match implies that under plane strain conditions the flow direction of the classical Mohr-Coulomb model can be alternatively calculated by the corresponding flow direction of the Drucker-Prager model with the dilation angle  as computed

before. Therefore, we can match the flow potential of the Abaqus Mohr-Coulomb model to that of the Drucker-Prager model. Matching between these two forms of flow potential assumes   1 and results in

which gives   22° in the Abaqus Mohr-Coulomb model. These Abaqus Mohr-Coulomb solutions are shown inFigure 1.15.5–1 and Figure 1.15.5–2 and match Carter's solutions exactly.

Input files

Abaqus/Standard input files

deformgranularmat_mc3030.inp

Extension case with the Mohr-Coulomb plasticity model (  30° and   30°) and CPE4 elements.

deformgranularmat_dp.inp

Extension and compression cases with the linear Drucker-Prager plasticity model and CPE4 elements.

deformgranularmat_cpe4i_dp.inp

Extension case with the linear Drucker-Prager plasticity model and CPE4I incompatible mode elements.

deformgranularmat_mc3030_comp.inp

Compression case with the Mohr-Coulomb plasticity model (  30°and   30°) and CPE4 elements.

deformgranularmat_dp_comp.inp

Compression case with the linear Drucker-Prager plasticity model and CPE4 elements.

deformgranularmat_mc3022.inp

Extension case with the Mohr-Coulomb plasticity model (  30° and   22°) and CPE4 elements.

deformgranularmat_mc3022_comp.inp

Compression case with the Mohr-Coulomb plasticity model (  30°and   22°) and CPE4 elements.

Abaqus/Explicit input files

granular.inp

Extension and compression cases with the linear Drucker-Prager plasticity model and CPE4R elements.

deformgranularmat_mc3030_xpl.inp

Extension case with the Mohr-Coulomb plasticity model (  30° and   30°) and CPE4R elements.

deformgranularmat_mc3030_comp_xpl.inp

Compression case with the Mohr-Coulomb plasticity model (  30°and   30°) and CPE4R elements.

deformgranularmat_mc3022_xpl.inp

Extension case with the Mohr-Coulomb plasticity model (  30° and   22°) and CPE4R elements.

deformgranularmat_mc3022_comp_xpl.inp

Compression case with the Mohr-Coulomb plasticity model (  30°and   22°) and CPE4R elements.

Reference

Carter,  J. P., J. R. Booker, and E. H. Davis, “Finite Deformation of an Elasto-Plastic Soil,” International Journal forNumerical and Analytical Methods in Geomechanics, vol. 1, pp. 25–43, 1977.

Figures

Figure 1.15.5–1 Load-displacement results for uniform extension.

Figure 1.15.5–2 Load-displacement results for uniform compression.

1.15.6 The one-dimensional thermal consolidation problem

Product: Abaqus/Standard  

When soil is subjected to loads and temperature variation, a coupled system of equations that describe the deformation, pore fluid flow, and heat transfer through the soil must be solved to accurately predict the consolidation behavior. In this problem the ability of Abaqus/Standard to model one-dimensional thermal consolidation is illustrated. Consolidation behavior of a one-dimensional column of fully saturated soil subjected to constant surface loads and constant surface temperature is studied, and the results obtained are compared to those obtained by Aboustit et al. (1985).

Problem description

This problem can be considered as the thermal counterpart to “ The Terzaghi consolidation problem, ”   Section 1.15.1 . The discussion presented in that section is equally applicable to this problem and is not repeated here.Figure 1.15.6–1 shows one-dimensional thermo-elastic consolidation of a linear elastic soil column under constant surface pressure and constant surface temperature.The column is 7 units high and 2 units wide. The bottom of the column is restrained, and all sides of the column are impermeableexcept for the top surface where free flow is allowed. The top surface is subjected to a constant pressure of 1 unit and a constant temperature of 50 units. The soil is assumed to be fullysaturated. Gravity is neglected. The material properties reportedby Aboustit et al. (1985) are used. The soil is elastic with a modulus of 6000 units and Poisson's ratio of 0.4. The permeability of the soil is 4 × 10–6 units with a specific weightof 1 unit. Since Aboustit et al. (1985) used only one set of thermal properties, identical thermal properties for the solid and the pore fluid are used. The specific heat is 40 units, and the density is 1 unit. The conductivity of the soil as well as the pore fluid is 0.2 units, and the coefficient of thermal expansion is 0.3 × 10–6.

All displacements perpendicular to the sides are restrained to enforce one-dimensional behavior. The consolidation analysis is performed using a transient soils consolidation step with automatic time stepping. The time stepping for this problem is controlled by two parameters: one that controls the accuracy of time integration for the temperature field, and one that controlsthe accuracy of time integration for the pore fluid flow. The stability limit for the pore fluid solution is given by

which dictates the minimum time increment. Variables used in thisequation are defined in “ Coupled pore fluid diffusion and stress analysis, ”   Section 6.8.1 of the Abaqus Analysis User's Guide . Themesh used is identical to the one used by Aboustit et al. (1985),which led to a minimum time increment of 0.1. Because of the applied surface load, the elements near the surface immediately acquire a pore pressure equal to the applied load; hence, a maximum pore pressure change per increment of 1.1 with an initialtime increment of 0.1 is used. This ensures that time steps smaller than 0.1 are not used in the analysis to satisfy time integration accuracy for pore fluid flow. The value for the maximum allowable temperature change in an increment was chosen as 3 to avoid having to use time increments that were smaller than what the stability limit for pore fluid required. The value for the maximum allowable temperature change was obtained by first running the problem using only the value for the maximum pore pressure change and determining the incremental temperature change. The parameter values listed above result in a moderately accurate solution. If a more accurate solution is desired, a morerefined mesh should be used.

Nonlinear geometric effects are not important in this problem dueto small load magnitudes. Similarly due to very small fluid velocities, heat convection effects due to pore fluid flow are not dominant enough to necessitate the inclusion of unsymmetric stiffness. However, for completeness, we have chosen to activate geometric nonlinear analysis as well as unsymmetric stiffness. Results of small-strain analysis with symmetric stiffness are

indistinguishable from the results presented. The time period forthe step is 21.1, corresponding to the time at which the Abaqus/Standard results are compared to the reference solution.

Results and discussion

At the start of the analysis there is zero temperature throughoutthe domain except at the top surface, and the pore pressure equals the applied surface load as all of the load is carried by the pore fluid. As time progresses, the temperature front progresses from the top to the bottom and the applied surface load is transferred from the pore fluid to the soil skeleton as pore fluid exits at the top, reducing the pore pressure in the domain. In the steady-state limit, all of the domain has zero pore pressure and constant temperature equal to the applied surface temperature. The Abaqus/Standard solution for pore pressure and temperature are compared at time 21.1, when the temperature front has progressed some distance from the top surface and there is partial reduction of the pore pressure. The results are shown in Figure 1.15.6–2. The temperature and pore pressure values are normalized using the applied temperature and the applied surface pressure. The ordinate is normalized using the height of the soil column. Abaqus/Standard results compare well with those presented by Aboustit et al. (1985).

Input files

unidircon_c3d8pt.inp

Mesh with C3D8PTelements.

unidircon_c3d8rpt.inp

Mesh with C3D8RPT elements.

unidircon_c3d8pht.inp

Mesh with C3D8PHT elements.

unidircon_c3d8rpht.inp

Mesh with C3D8RPHT elements.

unidircon_c3d10mpt.inp

Mesh with C3D10MPT elements.

unidircon_cax4pt.inp

Mesh with CAX4PT elements.

unidircon_cax4rpt.inp

Mesh with CAX4RPT elements.

unidircon_cax4rpht.inp

Mesh with CAX4RPHT elements.

References

Aboustit,  B. L., S. H. Advani, and J. K. Lee, “Variational Principles and Finite Element Simulations for Thermo-Elastic Consolidation,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 9, pp. 49–69, 1985.

Figures

Figure 1.15.6–1 One-dimensional thermal consolidation model.

Figure 1.15.6–2 Normalized temperature and pore pressure along the z-direction at time 21.1.

1.15.7 Consolidation around a cylindrical heat source

Product: Abaqus/Standard  

This problem presents the solution of consolidation in saturated soil around a cylindrical heat source. The problem has been studied by Booker and Savvidou (1985) and represents an idealization of the problem of a canister of radioactive waste buried in saturated soil. The temperature changes that occur due to radiation of heat from the canister cause the pore water to expand a greater amount than the pores in the soil, resulting in an increase in the pore pressure around the heat source. The resulting pore pressure gradient drives pore fluid away from the heat source, resulting in the dissipation of the pore pressure with time. Booker and Savvidou developed an analytical solution for the fundamental problem of a point heat source buried deep insaturated soil. They subsequently used this analytical solution to derive an approximate solution to the problem of consolidationaround a cylindrical heat source. This problem provides a verification for the coupled thermal consolidation capability in Abaqus. The analysis of saturated soils requires solution of coupled stress-diffusion equations, and the formulation used in Abaqus is described in detail in “ Analysis of porous media, ”   Section 2.8 of the Abaqus Theory Guide . The thermal consolidation capability also solves the heat transfer equation (considering both conductive and convective effects) in a fully coupled manner with the stress-diffusion equations and, thereby, models the influence of the pore pressure on the temperature field in the pore fluid and the soil and vice versa.

Numerical values for the parameters that define the geometry and the material properties are based on the details presented in a parametric study of this problem by Lewis and Schrefler (2000).

Problem description

The problem setup is shown in Figure 1.15.7–1. A cylindrical heatsource of radius 0.1604 m and height 2.5 m is embedded within a

cylindrical volume of soil with radius and height both equal to 10 m. The cylindrical volume of soil, in effect, represents an infinite medium surrounding the heat source. Gravity is neglected. Because of the boundary conditions (discussed below indetail), the problem is essentially one-dimensional, the only gradient being in the radial direction. The purpose of the analysis is to predict the evolution of pore pressure and temperature throughout the soil mass, especially in the neighborhood of the heat source, as a function of time.

Geometry and models

Only half of the problem is modeled, taking advantage of the symmetry in the vertical direction. This problem is solved using both three-dimensional and axisymmetric coupled temperature–pore pressure elements. For the purpose of presenting the results, three-dimensional element type C3D8RPT is chosen. Both the three-dimensional and axisymmetric analyses are carried out using different variants (such as reduced-integration and hybrid) of the basic three-dimensional 8-node or axisymmetric 4-node elements, as well as the modified tetrahedral elements.

The response of the soil is assumed to be linear elastic, with a Young's modulus of 60.0 MPa and a Poisson's ratio of 0.4. The specific weight of the pore fluid is assumed to be 9800.0 N/m3 (1 lb/in3). The permeability is assumed to be constant, with a value of 4.63 × 10–8 m/sec. The thermal expansion coefficients of the soil and pore fluid are assumed to be 0.3 × 10–6 per °C and 0.21 × 10–5 per °C, respectively. The density, specific heat, and conductivity of the soil and the porefluid are assumed to be the same, with values of 1000 kg/m3, 40.0cal/(kg°C), and 11.9 W/cal/(m°C), respectively. The void ratio isassumed to be 1.0 initially throughout the soil volume. The initial temperature and pore pressure are assumed to be zero everywhere. It is also assumed that the pores are fully saturatedwith pore fluid.

Boundary conditions

The normal (vertical) component of displacement is constrained atthe base of the soil, while rigid body motion in the two lateral directions is prevented by constraining a set of points on the outer boundary. The pore pressure and the temperature are assumedto be zero at all points on the outer boundary of the soil volume. Thus, the outer boundary is assumed to be connected to a heat and fluid reservoir (as represented by the soil that surrounds the volume considered for this model) that allows transfer of heat and pore fluid such that the boundary temperature and pore pressure are maintained at the specified values. The heat source is specified as heat body flux per unit volume with a magnitude of 11.58.

Analytical solution

There are two distinct time scales associated with this problem: one for each of the two diffusive mechanisms that are operational. The first time scale is associated with diffusive heat transfer and is given by  , where   is the radius of the cylindrical heat source, while   represents the thermal diffusivity of the surrounding medium and is given by  .In the preceding expression,  ,  , and   represent the density,specific heat capacity, and conductivity, respectively, of the surrounding medium. In general, the thermal properties used in this expression need to be weighted average quantities based on the volume fraction of the pores. However, such averaging is not necessary in this example as the thermal properties of the soil and the pore fluid are assumed to be the same. The second time scale is associated with diffusive flow of the pore fluid and is given by  . In the preceding expression, the quantity   represents the consolidation coefficient that is defined as  , where   is the permeability of the porous medium,   and   are elastic constants, and   is the specific weight of the pore fluid. The choice of the different parameters for the problem is such that the ratio   is approximately equal to 2.

Booker and Savvidou obtained an analytical solution for consolidation around a point heat source in an otherwise infinitemedium and utilized this analytical solution to approximate the solution to the problem of consolidation around a cylindrical heat source. The latter was accomplished by simply integrating the point source solution throughout the cylindrical volume. The expressions for the temperature and pore pressure fields, as given in the above reference, are reproduced below. These expressions are used to obtain the analytical solutions for comparison with the numerical results. The value of the temperature at the point ( ,  ,  ) and at time   is given by

where   is the strength of the heat source (heat energy radiatedfrom the source per unit volume per unit time),  , and ( ,  ,  ) represent the coordinates of a point source within the cylindrical volume  . The function   is expressed in terms of the complementary error function  as

Likewise, under the assumption that  , the pore pressure field can be expressed as

where the quantity   depends upon the elastic properties of the soil skeleton and the (volumetric) thermal expansion coefficientsof both the soil skeleton and the pore fluid and is given by  . Booker and Savvidou note that the temperature reaches a maximum value of   at the midpoint of the cylindrical source. If the soil were to be impermeable ( ), the pore pressure would reach a maximum value of   at the same point.

The expression for pore pressure given in the above paragraph clearly suggests that the effect of pore fluid flow is to reduce (with time) the pore pressure at a given point. For the special case of an impermeable fluid, the pore pressure will simply buildup over time and never reduce. On the other hand, if  , the fluid diffuses at the same rate as the heat and, hence, the pore pressure never builds up.

Time stepping

The accuracy of the time integration for the transient soils consolidation procedure that also models heat transfer is controlled by the maximum allowable pore pressure and temperaturechanges per time step. Even in a linear problem these values control the accuracy of the solution because the time integrationoperators for the consolidation and heat transfer problems are not exact (the backward difference rule is used in both cases). In this case the allowable pore pressure change per time step is chosen as 0.5 Pa, which is a relatively large value compared to  . Simulations with smaller values, such as 0.1 Pa and 0.075Pa, produce essentially the same results, although the analysis takes progressively more increments to complete with lower values. The allowable temperature change per time step is chosen as 0.003°C, which is approximately 0.1 . A value of 0.0003°C (along with a value of 0.075 Pa) produced essentially the same results, although the analysis took a significantly larger numberof increments to complete.

The analysis is continued for a time period of approximately 1000  .

The simulations use solution controls to specify a nondefault initial value of the time average pore fluid flux. The default choice may not be appropriate in situations such as those encountered in the present problem, where the fluid velocities are, in relative terms, lower compared to typical flux values encountered for other fields (such as displacements or rotations). Without the above specification, the increments wouldbe treated as linear from the viewpoint of the continuity

equation. In other words, without using solution controls to specify a nondefault initial value of the time average pore fluidflux, the pore fluid part of the incrementation will be treated as linear. Consequently, the continuity equation would be assumedto have been satisfied at the first iteration itself, without performing any further iterations to compute corrections to pore pressure.

For the pore fluid flow equations, the simulations also use a nondefault and a relatively large value of the ratio of the largest residual flux to the average flux, which sets the convergence criterion for an increment. This setting is helpful as the fluid velocity in this problem is very small and ensures that the pore pressure increment is not considered converged without at least one correction (iteration). There is not much advantage in reducing the tolerance further as the flow residualsare already sufficiently small, and any further reduction in the residual does not make any difference to the overall solution.

Results and discussion

The automatic time incrementation capabilities of Abaqus/Standardwere used for all the simulations. As discussed earlier, the total number of increments to complete the analysis depends strongly on the choices of the maximum allowable pore pressure and temperature changes per time step for the transient soils consolidation procedure. For relatively loose tolerances, the time increment size increases by a factor of approximately 25,000from the beginning to the end of the analysis, while for relatively tight tolerances the factor reduces to approximately 20,000. These very large changes in the time increment size are typical of problems that are diffusion dominated and highlight the value of using automatic time stepping with an unconditionally stable integration operators for such problems.

Figure 1.15.7–2 and Figure 1.15.7–3 show the variations of temperature and pore pressure, respectively, with time at three different radii from the cylindrical heat source. These locationscorrespond to the outer surface of the heat source ( ), and

distances of 2 ( ) and 5 ( ) times, respectively, of the radius of the heat source, along  . The axes for temperature, pore pressure, and time (the latter shown on a logarithmic scale) are normalized by the quantities  ,  , and  , respectively.

The analytical solutions to the problem, based on the equations presented earlier, are also shown in the figures using data points (as opposed to continuous curves) and are labeled Analytical-1, Analytical-2, andAnalytical-5, respectively, and correspond to the three different radii discussed above. The analytical solutions were obtained by evaluating the integrals numerically.

The temperature results suggest that at the surface of the heat source, the temperature approaches   with time, while at distances further away from the heat source the temperature increases with time at a slower rate. The results agree well withthe analytical results, especially relatively early in the analysis. The mesh used for the simulations is relatively coarse with 2718 elements. The agreement between the finite element predictions and the analytical solutions is much better at all times if the analysis is carried out with a more refined mesh consisting of 9816 elements.

There is an initial elevation, followed by a reduction in the pore pressure with time. The initial increase in pore pressure isdue to the relatively higher volumetric expansion of the pore fluid compared to the pores. A gradient in the pore pressure field is necessary to drive pore fluid flow. The results suggest that at relatively early times, the diffusion of the pore fluid away from a material point is not strong enough to offset the increase in volume associated with an increase in temperature. Hence, the pore pressure increases with time. However, with the passage of time the rate of increase of temperature at a materialpoint slows down, and the diffusion of pore fluid picks up such that any further increase in temperature (and the associated volume change) does not result in additional increase in pore pressure, and the pore pressure decays with time.

Figure 1.15.7–4 and Figure 1.15.7–5 show the contour plot of porepressure and a vector plot of the magnitude of the fluid velocity, respectively, at some intermediate time (approximately 5700 seconds) during the analysis. The distribution of pore pressure is approximately axisymmetric, with higher pore pressures closer to the central heat source. The radial gradient in the pore pressure drives the pore fluid flow, resulting in pore fluid velocity vectors that point approximately in the radial direction. The mesh itself is not axisymmetric, which results in small variations in the solution from a purely axisymmetric state.

While this problem illustrates the coupled nature of the physicalproblem of a heat source embedded in soil, the coupling is of a relatively weak nature. Thus, while the pore fluid flow field is mainly driven by the relative thermal volumetric expansions of the pore fluid and the pores and, hence, depends directly on the temperature field, the heat transfer problem is insensitive to the pore fluid flow. A stronger coupling could be included, for example, by considering convective heat transfer where the rate of transfer of heat is directly influenced by the pore fluid velocities. Additional potential sources of coupling include the dependence of permeability on the void ratio, which can depend onthe level of straining (including thermal expansion) in the material. Although such effects are accounted for in the formulation in Abaqus/Standard, they are neglected in the presentproblem.

Input files

pointheatsrcconsl_c3d8pt.inp

Consolidation analysis with heat transfer using element type C3D8PT.

pointheatsrcconsl_c3d8pht.inp

Consolidation analysis with heat transfer using element type C3D8PHT.

pointheatsrcconsl_c3d8rpt.inp

Consolidation analysis with heat transfer using element type C3D8RPT.

pointheatsrcconsl_c3d8rpht.inp

Consolidation analysis with heat transfer using element type C3D8RPHT.

pointheatsrcconsl_cax4pt.inp

Consolidation analysis with heat transfer using element type CAX4PT.

pointheatsrcconsl_cax4rpt.inp

Consolidation analysis with heat transfer using element type CAX4RPT.

pointheatsrcconsl_cax4rpht.inp

Consolidation analysis with heat transfer using element type CAX4RPHT.

pointheatsrcconsl_c3d10mpt.inp

Consolidation analysis with heat transfer using element type C3D10MPT.

References

Booker,  J. R., and C. Savvidou, “Consolidation Around a PointHeat Source,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 9, pp. 173–184, 1985.

Lewis,  R. W., and B. A. Schrefler, The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, John Wiley & Sons Ltd., 1998.

Figures

Figure 1.15.7–1 Geometry of the heat source that is embedded in an infinite soil medium (modeled as a cylindrical domain with finite dimensions).

Figure 1.15.7–2 Variation of normalized temperature with normalized time at three different radii.

Figure 1.15.7–3 Variation of normalized pore pressure with normalized time at three different radii.

Figure 1.15.7–4 Contour plot of pore pressure at an intermediate time.

Figure 1.15.7–5 Vector plot of pore fluid velocity at an intermediate time.