A range of behaviour appears from the examples, from very little convergence of specific volumes during compression or shear to a slow but gradual convergence. Even slow convergence, however, may still preclude the definition of normal compression and critical state lines that are independent of initial density within a useful stress range.

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Today Updates. September Popular Files. September 6. Grewal Book Free Download April Check Anna University Internal Marks coe1. Working in terI? Matrices are introduced here simply as a convenient shorthand means of writing sets of equations. Thus, implies the foIlowing pair of equations:.

Writing equations in terms of matrix products is extremely helpful when trying to write compact, efficient computer programs. HoweveL no aptitude for manipulation of combinations of matrices is either expected or demanded in this book. The preferred strain increment and effective stress variables that were introduced in Section 1.

Tbe off-diagonal zeroes in 2. Cbange in mean stress p' produces no distortion beq , and change in the distortional deviator stress q produces no-change in volume. Tbe initial gradient of the stress:strain curve in Fig. The initial gradient of the volume change curve in Fig. EvidentIy, values ofYoung's mdulus and Poisson's ratio could be deduced using 2.

## SOIL BEHAVIOUR AND CRITICAL STATE SOIL MECHANICS

If, altematively, drainage from the triaxiar sample is prevented, then undrained constant volume response is observed. The initial response of the soil specimen may still be elastic, but now, with volume change prevented, pore pressures develop. The response of the soil can be depicted. The imposition of a condition ofconstant volume on 2. There Is no reason why the bulk modulus of the soil skeleton should be infinite; certainly the act of cIosing the drainage tap on the triaxiaI apparatus can have no influence on the elastic properties of the soil skeleton.

Consequently, the only reasonable solution to 2. The pore pressure changes reflect directIy the imposed changes in total mean stress 2. This result implies that for this isotropic elastic material the pore pressure parameter a in 1. The constant volume condition imposes no constraint on the change in shape of the soiI sampIe, and 2. For a conventionaI undrained triaxial compression test in which the cell pressure is held constant while the axial stress is increased, q.

AIthough the behaviour of soil elements is controlIed by changes in effective stresses, it is often useful to describe the eIastic response of soil in terms of changes in total stresses. The observed response of a soil element must, however, be identical whether it is treated in terms oftotal stresses oreffective stresses. The total stress equivalent of 2. The distinction between elastic properties in terms of total or effective stresses is only helpful for constant volume undrained conditions; this is the reason for the subscript u on the blk modulus and shear modulus in 2.

There can now be no constraint on the total stress path that is imposed. The condition of no volume change must emerge whatever the externally applied changes in total stress. The shearing, or change of shaPe, of the soil q calculated from 2. Given the link between shear modulus, Young's modulus, and Poisson's ratio implied by 24 , the undrained and drained values of Young's modulus, Eu and E', respectively, are not independent. For from 2. Young's modulus describes the slope of "the axial stress:axial strain relationship.

In conventional triaxial compression tests bu. Different slopes, in the ratio given by 2. The effect of allowing drainage to occur after sorne increments of undrained loading effective stress path Be after AB in Fig. In summary, for isotropic elastic soil there are only two independent elastic soil constants. Elastic constants to describe the behaviour of soil under special condtions e. Anisotropic elasticity The discussion in previous sections has been restricted to the ideal case of isotropic elasticity. Real soil may not tit into this simple picture. Deviations from this picture may result from inelasticity, but tbey can also occur if the soil is elastic but anisotropic.

A completely general description of an anisotropic elastic material req. Besides, the depositionaI history of many soils introduces symmetries which may reduce considerably the number of indepen. Many soils have been deposited over areas of large lateral extent, and the deformations they have experienced during and after deposition ha ve been essentially one-dimensional.

Soil particles have moved verticalIy downwards and possibly also upwards with, from symmetry, no tendency to move laterally Fig. The anisotropic elastic properties of the soil renect this history. The soil may respond differently if it is pushed in. For example, cylindrical sample A " in Fig. Most of the routine soiltests that are performed in practice are triaxial compression tests on samples such as A in Fig. Graham and Houlsby show that it is not possible from such tests to recover more than three elastic constants for the soil; since two constants are needed for the description of isotropic elastic response, that leaves only one constant through which sorne anisotropy can be incorporated.

They propose a particular form of one-parameter anisotropy which allows. Equation 2. Expressions 2. Still following Graharn and Houlsby , one finds the stiffness equation to be. The coupling between volumetric and distortional effects implies that constant volume effective stress paths are no longer vertical constant p' paths in the p':q pIane [ 2. The direction of the path wilI depend on the value of a.

The pore pressure that is observed in an undrained test will be different from the change in total mean stress [see 2. The direction of the effective stress path is, from 2. Comparison with 1. Sorne typical data for a natural cIay from Winnipeg, Manitoba are shown in Fig. For such a path, from 2. These limiting values correspond to compression at constant radial. Typical data for the Winnipeg cIay are shown in Fig. Tbe role of elasticity in soil mechanics' Because many subsequent sections of this book are devoted to discussion of the inelastic or plastic behaviour of soils, it may be wondered what real role elasticity' playsin soil mechanics.

Soil mechanics has traditionally been concerned firstly with ensuring that geotechnical structures do not actually collapse and secondly with ensuring that the working deformations of these structures are acceptable. If it can be assumed that the soil will respond elastically to applied loads, then the whole body of elastic theory becomes available to analyse the deformations of any particular problem. Settlements of foundations and deformations of piles are frequently estimated by using charts computed from elastic analyses of more or less standard situations: Much site investigation, whether with triaxial tests in the laboratory or with pressuremeter tests or pi ate loading tests in the field, is devoted to determining accurate values of moduli for soils for subsequent use in deformation analyses.

The results of such analyses will of course be as good as the ql! The behaviour of an isotropic elastic soil is encapsulated in just two elastic constants, which can be obtained from a simple programme f soil testing. Most soils cannot satisfactorily be described as isotropic and elastic, so a more elaborate model will be required to describe soil response.

Such models are used to extrapolate from available experimental data typicalIy obtained under th rather restrictive stress conditions imposed in conventionallaboratory tests to the complex states of stress and strain which develop around a prototype structure. The quality of the prediction of soil response will depend on the extent of this extrapolation.

If the soil response is very stress path dependent, then it is helpful if the laboratory testing can bear sorne relation to the stress paths to which soil elements in the ground around a geotechnical structure may be subjected. This circ1e has to be. Elastic stress distributions are available for many loading situations e. Stress paths for certain simple geotechnical problems will be considered in more detail in Chapter Paradoxica1ly, then.

A conventional undrained triaxial compression test, with the cell pressure CTr held constant, is carried out on a sample of stiff overconsolidated cIay. The stress:strain relationship is found to be linear up to failure, so it is deduced that the cIay behaves as an isotropic perfectly elastic material. For this stage of the test: calculate the values of L1u, L1p, L1p', L1o-;. Find an expression for the pore pressure change L1u in the soil if qrainage is not pennitted. Show that the cross section of the samples does not remain circular as they are compressed.

Determine the slope of the effective stress path in the p':q plane for the undrained test in terms of appropriate elastic constants. Assuming that measurements are made only ofaxial and volumetric strain in the usual way, determine the slope of the strain path in the ep:eq plane that would be deduced for the drained test. Check that these expressions reduce to the corrt;ct values for isotropic elastic soil. Show thatif the shear modulus G' is also depel. Introduction The behaviour of an elastic material can be described by generalisations of Hooke's original statement, ut tensio sic vis: the stresses are uniquely determined by the strains; that is, there is a one-to-one relationship between stress and strain.

Such a relationship may be linear or non-linear Fig. For many material s the overall stress:strain response cannot be condensed into such a unique relationship; many sta tes of strain can correspond to one state of stress and vice versa. If the wire is reloaded to loads less than the previous maximum load, then anessentially el as tic response is observed B 1 e l in Fig.

As soon as the previous maximum load is exceeded, the elastic description of the response ceases to apply and unloading from a higher load leaves the wire with a further permanen t extension B 1 e l A 2B 2 in Fig. Such a description would be of extremely " limited application, however, beca use it would not be able to cope with the observation that subsequent unloading does not retrace the same path. Such a pattern of behaviour can, however, be described using an.

The irrecoverable, permanent extensions that remain under zero load are plastic deformations and can be regarded as defining new reference states from which subsequent elastic response can be measured, provided the past maximum load is not exceeded. The departure from stifT elastic response that occurs as reloading proceeds beyond the past maximum load may be called yielding, and the past maximum load becomes a current yield point for the copper wire being loaded in simple tension. In general, in this chapter, yielding is associated with a transition from stifT to less stifT response, and the more or less well-defined kinks in stress:strain curves that can be established as marking such transitions are termed yield points.

This is a convenient, though not particularIy rigorous, definition of yield. Yielding oC metal tubes in combined tension and torsion Figure 3. We expect a thin-walled tube ofthe same material to show the same sort of response undero pure tensile loading and - unloading. Essentially similar behaviour would also be seen if the thin-walled tube were subjected to increasing cycles of loading and unloading in pure torsion instead of pure tension. It is oC interest to investigate the etTect of applying combinations of tension and torsion on the yielding and plastic deformation of the tube.

The schematic arrangement for their tests is shown in Fig. Copper tu bes ofexternal diameter 6. The path of a typical test is plotted in a load plane P: Q in Fig. Taylor and Quinney performed tests with eight ditTerent values of m, from 0. Through the eight yield points thus established, a yield curve could be drawn defining the combinations of tension P and torsion Q for which plastic defoIlJlations would begin to occur.

This is the current 3. If there were no interaction between the effects of tension and torsion, then the onset of plastic deformations would be associated with combinations of loads lying on the rectangle ACB, implying that the torque required to produce yield would be Ql' irrespective of thevalue of the tension. In fact, the experimental data lie on a curve fitting inside this rectangle. Yield criteria are more satisfactorily discussed in terms of stress components rather than components of load P and Q. The stresses acting on an element of metal.

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In the absence of internal and external pressures 'the radial normal stress through the tube which is a principal stress and the tangential normal stre. Part ofthe object ofthe experiments ofTaylor and Quinney was to discover which of the yield criteria due to Tresca and to von Mises best fitted the data.

These two yield criteria have found widespread acceptance and use in the theory orplasticity Hill, According to Tresca , yielding occurs when the maximum shear stress in the material reaches a critical value.

### RELATED BOOKS

This can be written in terms of principal stresses 11' 12' and 13 as 3. The three principal stress 1, 12, 13 can be used as threeorthogonal cartesian coordinate axes to define a 'principal stress space'. The surface Fig. For the thin-walled tubes, Tresca's criterion implies 2. According to von Mises , yidding occurs when the second invariant of the stress tensor reaches a critical value.

The statement of von Mises' yield criterion in this. The distance from the space diagonal to the principal stress state gives an indication of the magnitude of the distortional stress which is tending to change the shape of the material. Therefore, an alternative interpretation of von Mises' yield criterion is that yielding occurs when the el as tic strain energy of distortion reaches a critical value.

For the thin-walled tubes under combined tension and torsion, von Mises'yield criterion becomes. B which is again an ellipse in the a::! In this deviatoric view, the Tresca yield criterion becomes a regular hexagon and the von Mises criterion a circ1e. Because expressions 3. It is a familiar observation of metal plasticity that the presence of isotropic pressures has a negligibl effect on the occurrence of yielding.

The loading paths that Taylor and Quinney used to probe the yield curve consisted of changes of torque with no change of tension Fig. They were careful to ensure that the specimens that they tested behaved closely isotropically. For a material behaving isotropicalIy and elastically, the application of torque to a thin tube should produce twist but no change in length of the tube beca use this is a purely distortional process Section 2.

As the torque is increased there is initially only twist and no change in Fig.

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These deformation data can be presented on the stress diagram inwhich the yield data were plotted, provided' they are converted to appropriate work-conjugate strain quantities. The work! The plastic deformation data of Fig. The direction of each vector indicates the relative amounts of plastic twist and extension that occur when the yield curve is reached. It is apparent that. In principal stress space the correct strain parameters to associate with the principal stresses are the principal strain increments Taylor and. Quinney observed that, as expected for an isotropic material, the principal axes of strain increment and of stress were coincident after yield, within their experimental accuracy.

The plastic strain increment vectors are plotted in the deviatoric view of principal stress space in Fig. The maximum deviation from the radial direction, the direction of the normal to the von Mises circle, is about 3. The link between mechanismsof plastic deforrnation and the yield curve that appears to have emerged in Figs. For elastic materials the mechanism of elastic deforrnation depends on the stress increments; for plastic materials which are yielding, the mechanism ofplastic deforrnation depends on the stresses. Yielding oC clays Although a single qualitative picture of the yieIding of soils is being created, it is convenient to separate discussion of the yielding of cIays from the yielding of sands because of the differentexperimental procedures that have beenused to probe the yielding of these different soil types.

The discussion of the yielding of annealed copper in Section 3. A simple, one-degree-of-freedom test that is familiar in geotechnical engineering is the oedometer test used to study the one-dimensional compression characteristics of soils. By convention, the results of oedometer tests are usually plotted with the height of the sample or a volumetric parameter as ordinate and the appliedefTective stress as abscissa, orten plotted on a logarithmic scale Figs. However, ir the axes are interchanged Figs: 3.

A preconsolidation pressure is often sought in such tests; that is, the pressure at which the stiffness of -lhe soil in the oedometer falls rapidly, and the slope of. It is cIear that this can be thought of as a yield point for the soil. For stresses below the preconsolidation pressure O';c' the response in the oedometer test is stiff and essentially 'elastic'. If, after the preconsolidation pressure has been exceeded, the stresses are again reduced, then stiff elastic response can again be found.

The preconsolidatin pressure observed in oedometer tests is the most familiar example of yielding of soils, but a similar pattem can be found in isotropic compression tests Fig. In each case a stiff response is observed when the load is reduced below a previous maximum value, and the stiffness falls again when the load is increased beyond this past maximum value, which acts as a current yield point for the soil.

Again, sorne hysteresis is observed in cycles of unloading and reloading. This may actually be insi. Note that plotting the results of oedometer tests or isotro'pic compression tests with a logarithmic stress axis Figs. The oedometer sample is confined laterally by a rigid ring, and the lateral stress builds up as one-dimensional compression proceeds. Though shear stresses are being imposed, thedorrunant effect is one of increasing general mean stress level and hence of increasing stiffness. This is, of course, the only effect in the isotropic compression test, where no shear stresses at all are applied.

A pattem essentially identical to that. The tests illustrated in Figs. The combined tension and torsion tests on copper tubes studied the response of a number oftubes, which had all been given the same preloading history, to different combinations of tension and torsion. The soil mecnanics equivalent of this is a series of tests in which soil samples with the same preloading history are subjected to different modes of loading, such as. The ground provides a convenient so urce of soil samples with a single history of preloading. Many soil deposits have.

Iarge lateral extent, so thata series of samples taken from the same depth can be considered essentially identical. Suppose that three such samples [ 1 , 2 , 3 ] ha ve been set up in triaxial apparatus and are in equilibrium under the same cell pressure, so that they have effective stress state A in Fig. Sample 1 is subjected to isotropic. Sample 2 is subjected to one-dimensional compression the sort of loading that could be imposed in an oedometer, though lateral stress is not usually measured in oedometer tests by controUing the ceIl pressure as the axial stress is increased in such a way that lateral strain of the sample does not occur.

In this way the efTective stress path for one-dimensional loading can be folIowed and plotted in the p':q efTective stress plane Fig. Sample 3 is 5ubjected to a conventional undrained compression test with pore pressure measurement. The effective stress path is shown in Fig.

Yielding is observed at Y3 , where the stiffness of the sample in a plot of deviator stress against triaxial shear strain changes sharply Figs. Already a yield curve could be sketched, linking the yield points observed in these tests Fig. The yield surface can be regarded as a generalisedpreconsolidation pressure; the preconsolidation pressure observed in an oedmeter test corresponds to just one point on this yield surface. The test data that have been used in Figs. The phenomenon of yielding in insensitive soils such as this is often considerably less marked than the change in stifTness observed in the reloading of the annealed copper wire Fig.

For such soils it is cIear that considerable subjectivity may be involved in selecting precise yield points. Probing these clays can disturb this structure: for very sensj;ive cIays, the drop in stifTness associated with 'destructuration' may be extreme. For both the examples shown here, undisturbed samples of cIay have been recompressed to a common initial effective stress state and then subjected to rosettes of stress probes. Louis, Canada is shown in Fig.

The initial effective stress state used in their tests corresponded to their estimate of the ion si tu effective stress state. On each radial path in Fig. Yielding of the copper wire or tu bes, and of clay samples just discussed, has been deduced from an in crease in the rate at which a strain parameter increases with continuing increase in stress. Since different loading paths generate difTerent modes of deformation or straining, different strain variables may provide a more sensitive indication of the occurrence of yield on particular stress paths.

Tavenas et al. A further estimate is possible from consideration of the energy required to deform a sample. A simple uniaxial unconfined loading test leads to the axial stress:axial strain curve shown in Fig. The work done in straining the sample can be calculated at any stage from the area undemeath the stress:strain curve,. For such a one-dimensional system, the substitution' of work for strain may not appear to provide much benefit For more general states of stress.

A third estimate of the position of the yield point was obtained by Tavenas et al. The yield points deduced from these three different procedures were very similar, and the points plotted in Fig. In sorne of the probing tests performed by Tavenas et al. No matter what strain or energy variable is used, if the stress axis shows an unchanging quantity, then there is no possibility of detecting a yield point as a kink in the curve. No single plot is likely to be suitable for detection Fig.

Indeed, since yield points in tests on soils tend to be rather les s marked than those seen in repeated tension, or combined tension and torsion of annealed copper, the best approach may be to use as many different plots as possible so that a number of independent estimates ,of yield points can be made. There is, however, one derived plot which Graham, Noonan, and Lew have shown can be used in general: this usesthe cumulative work input 3. In general, even this plot should not be used in isolation, but in combination with other possibilities, as shown in Fig.

The yield points obtained from a series of plots such as these are similar but by no means identical. Graham and his co-workers have performed a large number of probing triaxial tests on WiilOipeg day in order to discover the shapes of the yield loci that are appropriate to samples at different depths in the soil deposit. Samples of soil taken from different depths Fig. Huwever, tpe past history of loading of soil elements at various depths can be expected to be similar - for example, one-dimensional compression and unloading and possible secondary effects such as cementation or ageing.

So it is reasonable to suppose that the general shape of the yield surface is the same for all depths and that. A simple indicator of the size of the yield surface at any particular depth is provided by the preconsolidation pressure u: c which. The vertical efTective stress measured in a conventionaI oedometer test does not provide sufficient inforrnation to plot an efTective stress state in the p':q plane because: the value of the horizontal stress is not known.

Knowledge of a preconsolidation pressureu: c thus only defines Fig. A single non-dimensionalised yield curve can be sketched through the data points oLFig. The examples shown here are concerned w1th Canadian clays. However, there is nothing exclusively transatIantic about the yielding of clays, and other studies of the yielding of natural clays include Larsson , clay from Backebol, Sweden; Bell , clay from Belfast, Northern Ireland; Berre and Ramanatha Iyer , clay from Drammen, Norway; as well as Wong and Mitchell , clay from Ottawa, Canada.

YieJding of sands Samples of clay can be taken from the ground with relatively minor disturbance to the samples. A yield locus deduced from probing tests in the triaxial apparatus on a series of field samples should represent correctIy the current yield locus for the clay in its in situ condition at a particular depth in the ground.

Sampling sand, unless it is strongly cemented, inevitably leads to serious disturbance of the particle structure. It is not usually feasible to reestablish field conditions of particle arrangements in the triaxial apparatus, anda more fundamental route has to be taken to study the yielding of sands. The de termina ti o n of the entire shape of a current yield sU,rface for a given soil sample is not feasible.

The detection bf yielding requires that the stress path that is being psed to probe the yield surface be taken well. As soon as the yield point has been passed, the yield surface starts to change. It is axioma tic that a stress state can lie on or inside but never outside a current yield surface. The passing of the yield point requires the current yield surface to change size and possibly shape to accommodate the new current stress state.

Any subsequent probing investigates the shape, not of the original yield surface but of this. The yieId point might be detected in a pIot of deviator stress q against triaxiaI shear strain eq! Tatsuoka subjected individual samples of Fuji River sand to. His procedure is illustrated in Fig.

A typical path might consist of isotropic compression from O to A followed by conventional, constant celI pressure compression [rom A to B. The yieIding of the sand is now governed by a yield locus passing through B, and its local form is investigatetl with the stress path BCDE. On this path. The stress:strain response on section DE is studied to establish the position of a yield point Y and thus deduce the local shape BY of a segment of the yield locus through B. A typical complete test path applied to an initially dense sample is shown in Fig.

A series of segments of yield curves are marked on Fig. This last figure contains sufiicient experimental observations so that the shape of the developing yield curves can be sketched and a mathematical description of a general yield curve can be generated. Such a mathematical description contains a size parameter, equivalent to preconsolidation pressure, which can be used to normalise the segments in Fig. Plastic deformations are Fig. A particular example is provided by isotropic compression: isotropc compression and unloading of Fuji River sand is shown in Fig.

The response is qualitatively the same as that seen earlier in isotropic compression and unloading of c1ays Fig. It is to be expected, then, that the yield loci will in fact be c10sed in the direction of increasing p', as suggested by the dotted curves in Fig. In an earlier set of tests reported by Poorooshasb, Holubec, and Sherbourne , a.

Whereas the yield segments found by Tatsuoka Fig. Here, too, sorne extra statement is needed to describe the yielding that occurs under in crease of mean efTective stress. The sand data seem to be supporting a slightly difTerent picture of yielding from that seen for c1ays. For sands, the dominant efTect leading to irrecoverable changes in partic1e arrangement is the stress ratio or mobilised friction. High mean stress levels are required to produce significant irrecoverable deformations in purely isotropic compression: this can be related to the hard, somewhat rotund shape of typical sand particles.

The quite difTerent character of cIay partic1es and their interactions leads to much more significant. Just as combinations of tension and torsion were applied by Taylor and Quinney to thin metal tubes to obtain information about the shape of the current yield locus or yield surface for difTerent metals, so combinations of mean normal stress and deviator stress can be applied to samples of soil in the triaxial apparatus to obtain information about the shape of the current yield locus or yield surface for the soil.

There is no difTerence in principIe between these results: in either case, changes of stress which remain inside the current yield surface are associated with stiff response and essentially reco-verable deformations, whereas changes of stress that push through the-current yield surface areassociated withless stifTresponse and the development of irrecoverable deformations. Of eourse, important difTerenees exist between the yielding of soils and the yielding of metals.

One has already been noted: yield points seen in tests on soils are in general less marked than yield points that are eommonly observed in tests on metals. As a result, yield surfaces for soils are rather less precisely defined than yield surfaees for metals. The tests of Taylor and Quinney were used to investigate whether the Tresca or the von Mises yield criterion was more relevant for metals.

The yield surfaces that emerge from the applieation of these two yield eriteria difTer only in their shape in the deviatorie plane Figs. It is a well-known experimental observation that the presence ofmean stress p' has no effect on the yielding of metals. In a diagram such as Fig. F or soils, on the other hand, the nature of the deviatoric sections of the yield surface has yet to be explored, but the mean normal efTeetive -stress p' is of primary importance.

Both for sands nd for clays the deviatorie size of the yield surface, that is, the range of values of q for stifT elastic response, is markedly dependent on p'. Indeed, particularly for clays, yielding occurs with increase of p' even in the complete absence of 3. For comparison with Figs. This marked difference of shape of yield surface should, however, be regarded as a difference in detail, and not a difference in principIe, between the plastic behaviour of metals and soils.

Though soils are in many ways more complex materials than metals to describe with numerical. Exercises E3.!. The stresses are then changed steadily, with 'fixed ratios of stress increments. Calculate the stress states at yield if the stress increment Fig. Repeat E3. The block is subjected to a simple shear test, with shear stresses. Use the dataof yielding of Winnipeg el ay shown in Fig. Introduction In this chapter we build a general but simple elastic-plastic model of soil behaviour, starting with the experimental observation of the existence of yield loci that was discussed in Chapter 3.

## Critical State Soil Mechanics

Other features are added as necessary, and their selection is aided sometimes by our knowledge of well-known characteristics of soil response and at other times by knowledge of the elastic-plastic behaviour of metals. Broadly, having established that yield surfaces exist for soils, it follows that, for stress changes inside a current yield surface, the response is elastic. As soon as a stress change engages a current yield surface, a combination of elastic and plastic responses occurs. It must be emphasised again that we are attempting to produce a simple broad-brush description of soil modelling which cannot hope to match all aspects of soil behaviour.

Sorne of the shortcomings of such models are discussed in Chapter For convenience of presentation, the discussion is largely restricted to combinations of stress and strain that can be applied in the triaxial apparatus, and the model is de. The possibility that changes in size of yield loci are related to distortional as well as volumetric efTects is included in Section 4.

Elastic volumetric straios A yield surface marks the boundary of the region of elasticaIIy attainable states of stress. Changes of stress within the yield surface are accompanied by purely elastic or recoverable deformations. The relationship hetween strain increments and stressincrements can be written if the elastic properties of the soil are known. O and recoverable changes in volume are associated only with changes in mean effective stress p'.

There would be no difficulty in incorporating an anisotropic elastic description of soil response within the yield surface. The specific volume v of this sample could be determined for sorne effective stress state such as A within the yield locus, and a point corresponding to A could be plotted in the compression plane p': v Fig. A change in stress which involves a change in mean stress p', such as from A to B in Fig. A new paint B can then be plotted in the compression plane Fig. Because the response is elastic, the route taken in the stress plane from A to B is immaterial.

## Soil behaviour and critical state soil mechanics.

As all stress sta tes within the yield locus are visited, a series of points in the compression plane is obtained, forming a single unloading-reloading line url Fig. The position, shape, and size of the yield locus for the soil shown in Fig. A likely historycould be one-dimensional compression and unloading.

The stress path associated with one-dimensional or other anisotropic normal - compression is a straight line such as OC in the p':q stress plane Fig. The combinations of specific volume v and mean effective stress p' at various stages of normal compression form a normal compression line nel to point e in the compression plane Fig. The two statements concerning the elastic behaviour within the yield loeus and the history of norma] compression which created the yield locus are combined in Fig.

The compression plane diagram of Fig. It is often found that the linearity of normal compression lines and unloading-reloading lines in. Evidently but unfortunately , the values of v. VI depend on the units chosen for the measurement of stress. Throughout this book it is assumed that the unit of stress is 1 kilopascal kPa. Equations 4. The principal difference between equations 4. The reason for the approximate relation between e; and K will emerge in Section