Rate Dependent Elasto Plastic Deformation Of Shear Bands In Sensitive Clays
ICG and Geotechnical Division, NTNU, Trondheim, Norway
Vikas.thakur@ntnu.no
Professor, ICG and Geotechnical Division, NTNU, Trondheim, Norway
Steinar.nordal@ntnu.no
|
|
Rate Dependent Elasto Plastic Deformation Of Shear Bands In Sensitive Clays
ICG and Geotechnical Division, NTNU, Trondheim, Norway Professor, ICG and Geotechnical Division, NTNU, Trondheim, Norway |
ABSTRACT
Failure in sensitive clays (with a post-peak strain softening behavior) is often associated with the development of shear bands, i.e. narrow zones of localized deformation, and the failure loads depend on the thickness of these shear bands. This feature is both a conceptual and numerical challenge and the real physical thickness of the shear band becomes an important issue. Contracting behavior of sensitive clay is responsible for generating excess pore pressure within shear band, while a high hydraulic gradient between elastically unloaded body and shear band will cause dissipation of pore water pressure. Consequently, a reduced rate of strain softening will act as a counterbalancing mechanism and may contribute to the thickness of shear band.
In this paper, rate dependent one dimensional finite element model is simulated and coupled pore water strain localization phenomenon is studied.
Keywords: sensitive clays, shear band, strain softening, finite element
INTRODUCTION
A negative second order work, strain softening, is often noticed in contracting material like sensitive clays. Failure in such clays will lead to the formation of localized deformation zone of intense inelastic strain, known as shear band. Conditions, emergence of shear band and inclination of localized zone has been very well demonstrated in past couple of decades, but a definite thickness of shear band is difficult to obtain due to lack of internal characteristic length parameter within the framework of classical continuum theory. Finite element analyses in elasto-plastic clay suffer from mesh dependency, ill-possed boundary and loss of ellipticity in static problems (Høeg, 1972; Rudnicki and Rice, 1975; Vardoulakis, 1980; Pietrusczak and Mroz, 1981; Muhlhaus and Vardoulakis, 1987; Nordal, 1983; Jostad, 1993; Andresen and Jostad, 2002). The maximum carrying capacity of soil depends on thickness of shear band and rate of shearing. A laboratory evidence for detection of shear band thickness in highly sensitive clay like Norwegian quick clay (sensitivity>35) is reported by the present authors together with Grimstad et al. (2005).
Importance of local drainage in formation of shear bands has been very well accepted in literature. However, there are very few literatures available where partial drainage is actually studied within the framework of localization. To the best of knowledge of the authors only a single study reported by Atkinson and Richardson (1987) where effect of partial drainage on undrained shear strength is demonstrated. Thakur et al. (2005) is presented a study on generation - dissipation of pore water in a shear band and the surrounding soil has been studied for soft, sensitive clay. A one dimensional model was considered both numerically and analytically with a constant thickness of shear band. Results show how the degree of local drainage of the shear band increases when the rate of shear strain is lowered. The results further show how the softening in the shear band is less pronounced with reduced excess pore pressures in the shear band.
Another study made by Thakur et al. (2005) propagation and thickness of shear band is simulated in a direct simple shear (DSS) model. It is found that propagating shear band results a non uniform shear strain and stress field within the localized zone and thus mesh dependency is observed.
In this paper, a rate dependent simple soil column model is simulated for partial drainage condition. Coupled pore water and strain localization theory is analogous to Visco-plastic modeling i.e. rate dependent modeling, due to the fact that hydraulic gradient will act as a default regularization parameter, and thus ill possedness can be avoided up to certain extent using classical continuum based finite element.
ELASTO PLASTICITY AND STRAIN SOFTENING
Elastic material models and solutions based on theory of elasticity may be used in geotechnical engineering for some problem where the deformation and strains are limited. Wave propagation is one example. When the deformation involved is larger such as strain localization, some of the deformation will be inelastic and remain after unloading, in such condition we apply theory of elasto-plasticity. (Nordal, 1983).
Total strain due to elastic and plastic contribution at any stress state can be written as equation 1;
 |
(1) |
Equation 1 can also be written in rate form as;
 |
(2) |
Elastic constitutive behavior can be written as;
 |
(3) |
 |
(4) |
Where E=C-1 is young modulus. Young’s modulus can be defined using two lames constants.
 |
(5) |
It is important to know whether plastic deformation has occurred during loading and for that a yield criterion must be defined. Normally the yield criteria are denoted as an equation in stress tensor components and state variables.
 |
(6) |
And equation (6) must be satisfied if any plastic strain shall develop.
Flow rule of plasticity describe the direction of plastic flow
 |
(7) |
In case of associated flow rule yield surface and plastic potential, Q, coincides. Thus
 |
(8) |
Internal state variable, k, can be expressed as
 |
(9) |
By combining equation 3, 4, 7, 8 and 9
 |
(10) |
 |
(11) |
Where
 |
(12) |
On set of loading from one plastic state to another plastic state, consistency condition must be satisfied. Thus
 |
(13) |
Where
 |
(14) |
The consistency condition allows defining L, which in turn represents the plastic strain increment
 |
(15) |
Where, A is the plastic modulus. Solve for L, gives
 |
(16) |
or
 |
(17) |
By inserting equation 17 in to equation 11, tangent stiffness matrix D can be derived as
 |
(18) |
Strain softening implies development of plastic strains for a decreasing yield stress or a contracting yield surface. The softening curve will have a negative tangent stiffness D. At the peak D = 0 and A= 0, which is also known as bifurcation point.
 |
(19) |
In softening branch, both D and A will have a negative values.
Incremental internal energy based Hill’s stability criteria (1962) is widely recognized and accepted to analyses the body undergoing through deformation (Jostad, 1993).
Incremental internal energy  |
(20) |
Strain softening is having negative second order work which means
 |
(21) |
Also, during the softening we must know the loading and unloading criteria. A more general and suiTable criteria will be based on testing the sign of L
L > 0 implies plastic loading
L = 0 implies neutral
L < 0 implies elastic unloading
A simple one dimensional example may illustrate the concepts, as shown in Figure 1
Figure 1. Strain Softening in One Dimensional Case (Nordal, 2004)
For one dimensional case;
 |
(22) |
or same can also be written as
 |
(23) |
Where c is plastic multiplier
Since dc>0, we must have A > -E. Setting A=E makes the tangent stiffness, D, of the resulting stress strain curve goes to “minus infinity,” refer to Figure 1 (Nordal, 2004).
STRAIN LOCALIZATION IN SENSITIVE CLAYS
According to bifurcation theory, for perfectly homogeneous materials localization should theoretically not occur unless it is triggered by geometric constrains. In practice weak elements in the soil may trigger shear banding. The consequence of the strain-softening behavior of sensitive clays is the possibility of progressive development of shear bands (localized zones with large shear deformation). The thickness of these shear bands may under rapid shear deformation be very thin Therefore, even before global failure, the shear strain in these zones may be extremely large (Jostad and Andresen, 2004). Another consequence of strain softening is generation of excess pore pressure inside the shear bands. Due to high hydraulic gradient between shear band and outside elastically unloaded media, dissipation of excess pore pressure can occur.
Figure 2 demonstrates the shear banding in a soil body intersected by a slip surface in a failing natural slope and the pore pressure profile across the shear band.
Figure 2. Strain Localization in Sensitive Clays
It is well know, generation of excess pore pressure in shear band is directly related to the magnitude of shear strain, shear strain rate and angle of dilatancy.
Thakur et al. (2005) have proposed an analytical solution, refer equation 24, which can be use to calculate excess pore pressure developed within shear band. This equation is valid for locally undrained situation, which means a case when there is zero volumetric strain at each and every material point within undrained media.
 |
(24) |
Where and are excess pore pressure, incremental shear strain, shear modulus, friction angle, Poisson’s ratio and angle of dilatancy respectively.
To verify the above equation, a simple shear box is simulated in Plaxis, finite element code. Equation 24 is tested for different dilatancy angle and compared with finite element solution. Figure 3, represents the comparison between FE and analytical solution. It can be noted that both are showing good agreements.
Figure 3. Validation of Equation
It can be noted that negative angle of dilatancy, which represents the contractant behavior of soil, will be responsible for generation of excess pore pressure however in case of positive dilatancy, which defines a dilating behavior of soil will be responsible for negative pore pressure i.e. suction.
SHEAR BAND SIMULATION AND DISCUSSIONS
There are many factor which are responsible for formation of shear band in sensitive clays, like non- homogeneity in the materials, initial weakness, non uniform loading and boundary constraints. In the present study, a weak element is considered to trigger localization. A simple analogy can be presented as an embedded weak spring within the series of spring of uniform stiffness. Within the elastic domain, all springs will be having uniform deformation and hence, uniform stresses. However, weak spring will starts yielding before it reaches to global failure load and thus adjacent springs will have to carry additional load. This additional loading to adjacent spring will cause a development of progressive failure with a propagating shear band. In the paper, purpose of shear band simulation is to see the effect of pore pressure generation dissipation on strain localization and thus propagation of band in not taken in account.
To simulate a one dimensional shear band a perturbation is embedded within soil body. A one dimensional model “soil column”, which is infinite in length compare to the thickness of shear band (in reality) and subjected to shear, refer Figure 2 a-d is modeled in plaxis.
Figure 4. One Dimensional Soil Column Model
A purely frictional (f = 30o) weightless clay is used in the model having dilatancy angle ?= -1o, Poisson’s ratio, ?, is 0.25 and shear modulus,G, is 5000 kPa. The model is simply supported and initially subjected to isotropic stress equal to 100kPa. Prescribed displacement at boundaries triggers the localization in a predefined shear band. Different strain rate is obtained by varying the application time of the prescribed displacement. Perturbation is having less frictional angle (f = 25o) compare to rest of the body and thus first yield will occur in it. The sensitive clay develops a considerably high excess pore pressure in shear band for large shear strain rates. Pore water dissipation occurs due to a high hydraulic gradient between the shear band and resting body which is a rigid mass. A closed consolidation boundary around the soil body should in theory be used to describe the globally undrained condition.
A coupled analysis of pore water and strain localization is performed, which means Biot equilibrium is coupled with strain localization, and thus pore water pressure governs a vital role in formation of shear band. Excess pore pressure will only generate inside shear band and rest of the body will be elastically unloaded. At any time t, under the shearing stage
 |
(25) |
And also at any depth z, during the coupled shearing
 |
(26) |
Mohr Coulomb elastic perfectly plastic model with negative dilatancy is considered. Soil column has an embedded weak layer, refer Figure 4 (a,b), which will start localizing first i.e. shear band and generating excess pore pressure and rest of the body will be elastically unloaded as show in Figure 4 (c). However neighboring body will receive dissipated pore water pressure from shear band, which depends upon the rate of shearing. On set of localization, high plasticshear strain develops within shear band. To define shear band, incremental shear strain must be used as shown in Figure 4 (d).
To check the rate dependency of study, model is sheared in 4 different strain rates i.e. from locally undrained rate to locally drained but globally undrained rate. Model is tested with 4 different 0.00005%, 0.005%, 0.05% and 0.5 % h-1, shear strain rate. It is very well understood that at high strain rate, there will not be simultaneous dissipation of pore water from shear band; therefore effective stress will decrease rapidly. Faster softening is observed at high strain rate, which leads to a very thin localized zone. But at slow rate of strain, sufficient pore water will be dissipating from shear band and thus effective stresses will not reduce significantly and ultimately mesh independency can be obtained, as presented in Figure 5 and details are depicted in Table 1.
Table 1. Details of Strain Rates and Normalized Pore Pressure
| ||
| Figure 5 | strain rates (h-1) |
normalized pore pressure |
| ||
| a | 0.00005% | 0.01 (1.0%) |
| b | 0.005% | 0.11 (11.0%) |
| c | 0.05% | 0.74 (74.0%) |
| d | 0.5% | 1.00 (100%) |
| ||
Figure 5. Simultaneous Generation and Dissipation of Pore Water Presure from Shear Band
Simultaneous dissipation of pore water at low strain rate will be high and thus a very low pore pressure will be retaining inside shear band. In other words, as consequences of low strain rate, hydraulic gradient will also be low due to longer drainage path and thus pore pressure will also be low and vise versa.
Normalized pore pressure is used to show the variation of pore pressure at different strain rate and can be defined as:
 |
(27) |
Incremental shear strain at different strain rates are presented in Figure 6. it can be noted from Figure that at very low rate of strain (0.00005%h-1), degree of drainge will be high and thus strain softening will be very slow. Such condition can be assumed as nearly drained condition and thus shear band will dilute, as we can in Figure 6 (a). However, at partially drained condition (0.005% h-1 and 0.05% h-1) there shear band will form without having mesh sensitivity. At 0.005% and 0.5% h-1, strain rate, thickness of shear band is equal to the size of 4 elements and 2 elements respectively. However, at a very high strain rate (0.5% h-1) of mesh dependency is observed i.e. 1 element size shear band thickness, refer Figure 6(d). It is interesting to mention that Figure 5 and Figure 6 are analogous to each other for defining shear band thickness, due to the fact that in coupled analysis incremental shear strain and excess pore pressure will develop only within shear band.
Figure 6. Incremental Shear Strain within Shear Band
A stress strain response curve for different strain rates is plotted in Figure 7. At lower strain rate a slow softening rate can be observed and same is true other way around. Atkinson and Richardson (1985) have reported the influence of partial drainage in peak undrained shear strength and a validated similar trend can be found in Figure 7. Which means higher is the local drainge; higher will be the peak undrained shear strength.
Figure 7. Stress Strain Plot
CONCLUSIONS
In order to understand the contracting behavior of shear band in sensitive material, rate dependent finite element simulation is presented, It is found that finite element can be used to simulate strain softening problem without any mesh sensitivity provided a coupled analysis is made with appropriate strain rate. And only plastic analysis will not be able to capture well possed localization due to the fact that there will not be any internal length available in the calculation. However, using hydraulic gradient as default internal characteristics length, successful modeling can be done in finite element method. In the present study, still mesh sensitivity can be observed for higher strain rates, which is acceptable because of local undrained condition and ill possedness of boundary. This study is quit similar, in sense of principle, with visco plastic regularization technique, where one can observed the mesh dependency if proper time step is not chosen. It is also necessary to mentioned that various regularization techniques available in literatures do not guarantee to get rid of ill posed boundary condition, but provides a better iterative procedure and controlled mesh sensitive analysis. In the respect, this study is quit similar. However, it is believed that this paper will give idea to the readers to using existing finite element method to successful modeling of strain softening problems.
This study also indicates one of the reasons for not being able to simulate actual laboratory results with finite element method without a physical thickness of shear band. Andresen (2001) reported a comparative between laboratory result for a biaxial testing and finite element simulation using Program called BIFURC. It was found that rate of softening obtained after finite element analysis, independent of mesh sensitivity, is quit higher than actual laboratory stress strain response, due to the lack of real shear band thickness.
In reality, thickness of shear band is very important to analyses progressive type failure mechanics where propagation of shear band depends upon accumulated plastic shear strain, critical tip angle and its thickness.
ACKNOWLEDGEMENT
International Centre for Geohazards (ICG) and Norwegian Geotechnical Institute (NGI) is gratefully acknowledged for their support and supervision. The Geotechnical division, Norwegian University of Science and Technology, NTNU, is acknowledging with utmost respect for financial support.
REFERENCES
 | |
| © 2005 ejge | |