OBSERVATION AND PREDICTION

In a world where demands for improving life and work environment are ever increasing, particularly for densely populated areas and weak soil conditions, construction in and on the underground is complex. Proper insight in the short and long-term behaviour of soils and their interaction with structural elements is inevitable for achieving reliable and sustainable works in an efficient manner. Not always straightforward design tools are sufficient and sophisticated methods are required. In the process of design and construction, two aspects are highlighted in this chapter: physical simulation and numerical simulation. For both, understanding of the concepts of adopted material behaviour (constitutive models) in a multi-dimensional fashion is essential.

A PHYSICAL SIMULATION

Besides field observation, either by monitoring a pilot test or at an unexpected failure case, the use of physical simulation is inevitable for improvement of our understanding of soil behaviour. It provides valuable information on failures (liquefaction, landslides, dams), deformations (roads, foundations, tunnels), soil-structure interface behaviour (piles, anchors), and transport (contaminants, heat).

Physical simulation by means of tests and monitoring provides the means to check constitutive concepts, which form the fundaments of prediction models. In this respect physical simulation and numerical simulation are complementary. In fact, the objective of physical simulation is validation, while the objective of numerical simulation is prediction. A comprehensive evaluation of the application of physical testing methods is outlined in Chapter 3C.

A striking example of the need for physical simulation can be found from ‘odd’

behaviour of horizontal soil deformations due to a nearby excavation or landfill.

Elastic numerical simulation will result in predicting the horizontal deformation quite well in magnitude but completely wrong in direction: the reality will show a deformation in the opposite direction! Here, the soil behaviour must be simulated with suitable calibrated constitutive models that include plasticity and/or viscosity.

Physical simulation requires special skills and experience in scaling laws (centrifuge), fabrication (sample preparation), sensors and transducers, and data elaboration (visualisation).

Scaling

Before a large, expensive object is constructed, experimentation on models (modelling) is used to determine the best properties for construction and for its functioning. One must know how to scale up the results of a model test to a full-scale prototype. Here, the concept of physical similarity is central. This concept is satisfied when values of dimensionless parameters that characterise the phenomenon of interest are equal in model and prototype.

When developing models, it is not possible to take into account all factors that influence the phenomenon of interest; some should be disregarded and those of

crucial importance should be retained. A model is in fact an idealisation of reality.

With modelling, one looks for an appropriate idealisation.

Consider the consolidation process as the phenomenon of interest. The field equation is, according to (6.4)

cv w²u/wz² = wu/wt with cv = k/(JwD^{) [m2/s] (9.1)} By introducing scaling in space, time and the field variables, this equation is

rendered dimensionless, as follows

Cw²F/wZ ² = wF/wY with F = u/u0, Z = z/L, Y = t/T and C = cvT/L² (9.2) Here, L, T and u0 are arbitrary scaling factors, and C is a dimensionless consolidation number. Note that the field variable scale u0 does not appear in C, because (9.1) is a linear in u. For a non-linear process the field variable scale will appear in the dimensionless number. The general solution of (9.2) in terms of consolidation degree U as function of Y, according to (6.6) with L = h, becomes

U = 1 – (8/S ^{2) exp(–CY}S^{2/4) (9.3)}

Now, physical similarity for a model (m) and the prototype (p) is satisfied, when the number Cis similar for both, i.e. Cm = Cp. If the same material is tested, i.e. cv

is identical, then obviously Tm/Tp = (hp/hm)² must hold. The time in the prototype to reach the same consolidation degree as in the model will be (hm/hp)² faster. So, if 50% consolidation in an oedometer test of 2 cm high is reached in 0.5 day, then it takes 125 days for a layer of 5 metres in the field to reach 50% consolidation. In conclusion, it is sufficient to consider dimensionless numbers in order to achieve physical similarity.

Model tests are usually performed for situations where the solutions are not known. In most cases, the physical phenomena of interest can be described in terms of differential equations based on conservation principles (conservation of mass, momentum and/or energy) and constitutive laws. Making these dimensionless results in the physical numbers related to the different phenomena involved.

Imposing similarity for all of these numbers, i.e. making them identical for model and prototype, provides the necessary scaling rules.

In centrifuge testing the gravity is significantly increased. As an example, the physical similarity for two phenomena, equilibrium and consolidation, can be elaborated. Consider the fundamental equations for both processes (for simplicity one spatial dimension is considered), as follows

x g u m x

equilibriu

w

 w w

wV^' U _(9.4a)

t c

u x ion u

consolidat

vw w w w

2 2

(9.4b)

Introducing scaling factors L for length, V⁰ for pressure, g0 for gravity, and T for

Two dimensionless numbers occur: an acceleration number U^g0L/V^{0 and a} consolidation number c_vT/L². A convenient requirement is similarity of stress, V0m =

V0p and u_m = u_p. Similarity of mass, internal friction, cohesion, and compressibility by using the same materials provides identical deformation and failure states in model and prototype. The gravity is different: g_0m/g_0p = N, which may vary up to 100 and more, depending on the rotation speed of the centrifuge. Similarity of equilibrium is then satisfied by adopting L_m/L_p = 1/N. Physical similarity of consolidation involves beside length also time and the consolidation coefficient cv. Since c_v = k/U^gD ⁼ N ^/PD and one may adopt similarity of viscosity P ^and compressibility D⁴⁰, and the fact that the intrinsic permeability N is proportional to D² (D is the representative grain size), it shows that the consolidation number cvT/L² satisfies similarity, if also the grain size is scaled according to Dm/Dp = 1/N.

Then, time is equal in model and prototype. However, then the same material cannot be used in the model and in the prototype, and similarity of intrinsic parameters friction, cohesion, and compressibility may not hold. Usually the grain size is not scaled in centrifuge tests, and as a consequence permeability is N² larger and consolidation proceeds equally faster, which implies that transient boundary conditions must be faster by N² as well.

Implying scaling principles in modelling physical phenomena requires special expertise.

B CONSTITUTIVE MODELS

Soil is a complicated material that behaves non-linearly, anisotropic and time-dependently, when subjected to loading. It behaves differently for primary loading, unloading and reloading and it shows plastic and brittle failure behaviour. Since it is not practically possible to describe the true mechanical behaviour of saturated porous granular materials, approximate constitutive models are adopted at larger scale where soils are considered as continuous media. Though such models are commonly based on fundamental physics (continuum mechanics), the parameters involved are empirical in nature and should be verified by lab and field-testing, and in some cases monitored during construction and use. Observed phenomena, e.g.

deformations, stresses, pressures and rates offer indirect information about these

40 Note, that in reality D = K/V' and therefore consolidation in (9.4) is a non-linear partial differential equation.

constitutive parameters. Thus the underlying adopted model is crucial, in this respect. The same holds for predictions of structural behaviour performed by numerical simulations, as the outcome in terms of deformations and stresses depend on model chosen, corresponding parameter values adopted and the robustness of the computational algorithm. In general five basic aspects of soil behaviour can be recognised:

- the influence of pore water and pore pressures (permeability, consolidation) - the influences the soil stiffness (stress and strain level, stress path, anisotropy) - irreversible deformation (plasticity, creep, preconsolidation)

- strength (loading rate, density, drained-undrained) - compaction (contractancy, dilatancy)

p’, Hvp

q, J^p

H^p G

elastic

F p’, Hvp

q, J^p

H^p G

elastic

F

1 - sin\

2 sin\

cut off volume strain

vertical strain

(a) flow rule F and plastic potential G (b) dilatancy (triaxial test) Figure 9.1 Failure, plasticity and dilatancy concepts

Plasticity

It is assumed that plastic deformations can be represented by deformation rates wH^{ij /}wt and that these rates are zero in the elastic range and non-zero at the edge of the elastic range. A flow potential or yield surface, fully defined by effective stresses, describes the elastic range, e.g. F(V^ij') < 0. At the edge of the elastic range F = 0. It is further assumed that the strain rates that develop at F = 0 are related to a plastic potential G fully defined by effective stresses, e.g. wHij /wt = OwG/wVij'. If the flow potential F is identical to the plastic potential G the model behaves associated.

If not, the behaviour is non-associated (Fig 9.1a). In soil mechanics, an associated flow rule is commonly used to model the behaviour of normally consolidated clay.

A non-associated flow rule is frequently used to describe the behaviour of sands, particularly for dilation and contraction.

In geotechnical engineering many constitutive models have been suggested.

Here, commonly accepted constitutive soil models are shortly outlined: for sand the Mohr-Coulomb model and for clay the Cam-Clay model. Some remarks are given for peat behaviour and a few other constitutive models are mentioned briefly.

Mohr-Coulomb model for sand

The Mohr-Coulomb model (see Chapter 7) is one of the most popular constitutive models for soil behaviour (failure). The model adopts the maximum and minimum stress in the plane of failure, such that the stress in the orthogonal direction falls somewhere in between both. It is a (linear-)elastic perfectly-plastic

model with five intrinsic parameters. The stress-strain behaves linearly in the elastic range, involving Young’s modulus E and Poisson’s ratio Q. Two parameters define the failure criterion (flow potential F): friction angle M and cohesion c, by F

= |W^|  ^c V^'tanM, and G = |W^|  ^c  V^{' tan}\ is the plastic potential. The dilation angle \ applies to cover irreversible changes in volume due to shearing. In the case of associated behaviour M ⁼ \ (Drucker’s postulate), it is found for frictional materials that plastic deformations evolve with an increase of volume (dilatancy), which is not realistic. In the standard Mohr-Coulomb model the non-linear elasticity of soils is covered by an average linear elasticity E50, representing the stress-strain rate after 50% of the failure stress. The dilation angle \^{, which} describes the volume strain at failure, contributes substantially to the strength, particularly in sands. The actual friction angle and the dilation angle are interrelated, and M ⁼ Mcv + 0.82\ is used (Bolton). Here, Mcv is the friction angle at constant shear-induced volume change (critical state, critical density), which is roughly equivalent to the angle of repose of a loose dry material (ranges from 30º to 37º), see Fig 3.4b. It depends on mineralogy, grading and shape, but not on the test type. A common value for sands is Mcv = 33º. In the perfectly-plastic approach the dilation angle \ is usually kept constant. In reality, the value changes with the shear strain level, i.e. finally it tends to zero (Fig 9.1b).

The modified Mohr-Coulomb model adopts a non-linear shear modulus, according to

G = G0(p'/p'0)^1-E (9.6)

Here, G0 is a reference value at isotropic effective stress p'0. The value of E ^is usually around 0.4. For the plastic strain dependency of the actual dilation angle \ and the actual friction angle M in the modified Mohr-Coulomb model, the following formulas are adopted

sin\ ^{= sin}\^{0 exp(}H^pl/[^{) (9.7a)}

sinM ^{= (sin}M^cv+sin\^)/(1+sin\ ^sinM^{cv) (9.7b)} Here, \⁰ is the initial value, M^cv the value at constant shear-induced volume

change, H^pl the equivalent plastic strain (changing during the numerical calculation) and [ a parameter to be determined by long-duration triaxial testing ([ is in the order of 0.07). Adopting a kinematic yield cap in the compression domain is another adjustment (see literature).

For coarse soils, Steenfelt emphasised the importance of taking the strength and dilation parameters not as an easy matter. He showed practical examples and suggested that Bolton’s simplified expressions are appropriate.

plane: M_plM_cv 0.8\_max 5^oI_R ^(9.8a)

triaxial: M_trM_cv 0.8\_max 3^oI_R ^(9.8b)

with 1I_R D_r(Qlnp') . \max is the dilation angle at failure, D_r the relative density, p' the mean normal effective stress at failure, Q a parameter accounting for mineralogy (quartz: 10, limestone: 8, anthracite: 7, chalk: 5.5), and IR =

dH^v/(0.3dH1). For the projection of triaxial test results to plane symmetrical situations other empirical formulas are suggested, such as Mpl = Mtr(1+0.163 D_r).

Cam-Clay model for clay

Soil behaviour reveals irreversible strain and strain hardening, and plastic yielding is not synonymous with the maximum stress, as is assumed in the Mohr-Coulomb model. To cover these aspects, the soil mechanics group at Cambridge developed the Cam-Clay model (Schofield and Wroth). The Cam-Clay model is a linear-elastic plastic strain-hardening/softening model. It is based on Critical State theory, characterised by the critical state line (CSL), where soil elements can experience unlimited deformations without any changes in stress or volume. The model is able to describe deformation and failure especially for normally consolidated soft soils in applications involving loading conditions such as embankment or foundation. The adopted associated flow rule limits the simulation of undrained behaviour of sand and clay.

The state of a soil sample is characterised by three parameters: effective mean stress p', deviatoric shear stress q, and specific volume v (note: v = 1 + e). The change of state can be described by the isotropic compression line (loading) and swelling lines (unloading), both straight lines in the v-lnp space, characterised by the compression index O, the swelling index N, respectively.⁴¹ The critical state line corresponds to the failure stress and is parallel to isotropic compression line in the v-lnp space. The critical-state parameter M is related to the internal friction angle M^.42 Since the original Cam-Clay model led to too high K0 values, the Modified Cam-Clay model adopts an ellipse for the yield cap.⁴³

The cap supports a constant volume constraint at the critical state. The parameter

* defines the critical state line at unit pressure. In Fig 9.2 two cases are worked out, one for hardening (the preconsolidation pressure increases during loading) and one for softening (the preconsolidation pressure decreases). These cases are also referred to as the wet and dry side, because the wet side (hardening), in the higher-pressure zone, causes the clay to compact resulting is wet clay (pore water expelled), while the dry side (softening), in the lower pressure zone (overconsolidated) the clay swells resulting in dry clay (water sucked in). The latter case is more difficult to model in numerical codes. Originally, cohesion is not included in the Cam-Clay model; new versions do.

41O and N show similarity to a amd b, mentioned in Chapter 6

42 M = 6sinM / (3sinI) for triaxial tests

43 The original Cam-Clay model has a logarithmic yield cap.

q

Figure 9.2a Principle of the Modified Cam-Clay model; (a) the correspondence of the p-q space and the v-p space for a hardening stress path A-B-C, (b) the related v-lnp space, (c)

hardening behaviour

Figure 9.2b Principle of the Modified Cam-Clay model; (a) the correspondence of the p-q space and the v-p space for a softening stress path A-B-C, (b) the related v-lnp space, (c)

softening behaviour

Fig 9.2a shows a hardening stress path A-B-C of the Modified Cam-Clay model.

When loading starts (point A), the original yield surface is the smaller ellipse, characterised by the preconsolidation pressure p0. The corresponding over-consolidation ratio is OCR = p0/p'A. At point B the material starts yielding and the corresponding yield surface grows, until the stress path reaches the critical state (failure), point C. Fig 9.2a.b shows the v-lnp space, where characteristic lines becomes straight. Fig 9.2a.c shows the corresponding shear stress-strain hardening during yielding.

Fig 9.2b shows a softening stress path A-B-C of the Modified Cam-Clay model.

When loading starts (point A), the original yield surface is the larger ellipse, characterised by the preconsolidation pressure p0. The corresponding over-consolidation ratio is OCR = p0/p'A. The first part of the stress path A-B is elastic.

At point B the material starts yielding and the corresponding yield surface diminishes, until the stress path reaches the critical state (failure), point C. Fig 9.2b.b shows the v-lnp space, where characteristic lines becomes straight. Fig 9.2b.c shows the corresponding shear stress-strain softening during yielding.

Constitutive model for peat

Peat in particular behaves anisotropic related to the occurrence of fibres (organic remains) and depending on the rate of humification. Fibres are expected to mainly align horizontally, and therefore cause structural anisotropy. Since peat is a soft and very compressible, material loading may cause also induced anisotropy in stiffness. The compressibility of organic material, the main ‘solid particles’ in peat, is another aspect that differs from sand and clay where the solid particles are relatively incompressible. This will cause a decrease of the inter-particle stress and since equilibrium must hold, it may give rise to an additional pore pressure increase. It also affects the principle of effective stress, which becomes V ⁼ V^{' +} (1Z^{)u, with} Z the ratio between particle compressibility and bulk compressibility (Biot’s coefficient). Furthermore, the large flexibility of peat may have a pronounced effect on pore pressure development during consolidation, because the permeability will change significantly with the pronounced compression.

Therefore, in triaxial tests, horizontally retrieved samples may reveal differences compared to vertically retrieved samples (Zwanenburg). Conventional treatment in laboratory experiments of peat is therefore not sufficient. The constitutive behaviour of peat requires a different approach than provided in current constitutive models (Den Haan and Kruse).

Other constitutive models

A familiar elastic-plastic model is the hyperbolic model or the Duncan-Chang model, which adopts an incremental nonlinear stress-dependent concept, based on stress-strain curve in drained triaxial compression test of both cohesive and cohesion-less soil, approximated by a hyperbola (power law). The failure criterion is based on Mohr-Coulomb. The Duncan-Chang model is widely used as its soil parameters can be easily obtained directly from standard triaxial test. It is a simple enhancement of the Mohr-Coulomb model. It does not cover dilatancy, unloading behaviour and full plasticity.

For sand the Matsuoka-Nakai model is a modification between the Drucker-Prager criterion and the Mohr-Coulomb rule, and it is convenient for other stress paths than triaxial.

The Hardening Soil model (Vermeer c.s.) involves friction hardening to model the plastic shear strain in deviatoric loading and cap hardening to model the plastic volumetric strain in primary compression (see equations 9.7). Failure is defined according to Mohr-Coulomb. The model covers reduction of mean effective stress and mobilisation of shear strength. Soil dilation is covered and a power law formulation for stiffness is applied (Duncan-Chang model). This model can be used to accurately predict displacement and failure for general types of soils in various geotechnical applications. The model does not include anisotropy or creep.

V V2

Figure 9.3 Concept of double sliding along Coulomb shear planes

The concept that the flow of granular materials is governed by shear on critical surfaces can be used to formulate an elastic-plastic model. When the elastic strains are neglected, i.e. for large strains, the model becomes identical to the rigid-plastic model of de Josselin de Jong, which he named the Double Sliding model. For the critical surfaces, he adopted the conjugate shear planes of the Mohr-Coulomb model. A typical deformation according to rigid-plastic double sliding is shown in Fig 9.3.

Figure 9.4 Difference of Mohr-Coulomb (left) and Double Sliding (right)

When compared to the non-associative Mohr-Coulomb model (Fig 9.4) it is shown that the rotations and displacements are somewhat different (Teunissen).

The Double Shearing model predicts in general lower limit loads, because, for a

given state of stress, it allows for several possible directions of plastic flow rather than a unique direction, which coincides with one plastic potential in the Mohr-Coulomb approach. Here, a special remark is made about the use of model parameters. Since, as an example, the Mohr-Coulomb model is essentially different from the Double Sliding model, one may not use parameters obtained from laboratory tests by using Mohr-Coulomb for prediction with the Double Sliding model. This is inconsistent and may lead to erroneous results. Model parameters must be compatible with the constitutive model used for the calculation or prediction.

Hyper-elastic or Green model is suitable when the current state of stress depends on the current state of deformation and not on the history of strain. Hence, hyper-elastic materials can be characterised by a strain-energy function, assuming isotropic deformation (constant volume). Generally, it is suitable for materials that respond elastically when subjected to very large strains and under proportional

Hyper-elastic or Green model is suitable when the current state of stress depends on the current state of deformation and not on the history of strain. Hence, hyper-elastic materials can be characterised by a strain-energy function, assuming isotropic deformation (constant volume). Generally, it is suitable for materials that respond elastically when subjected to very large strains and under proportional