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Muir Wood, D., Diambra, A., & Ibraim, E. (2016). Fibres and soils: Aroute towards modelling of root-soil systems. Soils and Foundations,56(5), 765-778. https://doi.org/10.1016/j.sandf.2016.08.003
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Fibres and soils: a route towards modelling of root-soil systems
D. Muir Wooda,b,c, A. Diambrac, E. Ibraimc
aInstitutionen for Bygg- och miljoteknik, Chalmers tekniska hogskola, Goteborg, SwedenbDivision of Civil Engineering, Fulton Building, University of Dundee, United Kingdom
cDepartment of Civil Engineering, University of Bristol, United Kingdom
Abstract
Addition of flexible fibres to granular, cohesionless soils, has a marked influence on the stress:strain and volumetric response. Ex-perimental observations provide inspiration for the development of continuum models for the mechanical, pre-failure behaviour ofsuch fibre/soil mixtures. Such generic models and the deduced mechanisms of response should be applicable to other combinationsof soils and flexible fibres such as plant roots. Two features are particularly important: the distribution of orientations of fibres (nomethod of preparation produces an isotropic distribution) and the allowance for the volume of void space not only occupied butalso influenced by the presence of the fibres.
A simple shear element is used as a quasi-one-dimensional demonstrator platform for the presentation of the continuum consti-tutive model. Such an element represents a familiar configuration in which phenomena such as dilation and friction can be directlyobserved. A basic constitutive model for sand is adapted to this simple shear element; the fibres are added as a separate componentable to withstand tension but without flexural stiffness. As the soil-fibre mixture deforms, the straining of the soil generates stressesin favourably oriented fibres. The model is used to clarify some aspects of the response of fibre-soil mixtures: the influence of fibreson the volumetric behaviour; the existence and nature of asymptotic states; and the stress-dilatancy relationship.
Keywords: fibres; cohesionless soils; ground improvement; constitutive model
Introduction
It has been known, qualitatively, for many centuries thatthe presence of vegetation has beneficial effects on the stabil-ity and deformations of slopes through the reinforcing effect ofthe roots on the soil through which they are growing (Wu et al.,1988; Reubens et al., 2007). Roots, subject to the vagaries ofnature, present challenges for testing and modelling. The lab-oratory observations presented here relate to the behaviour ofcohesionless soil (sand) mixed with flexible polypropylene fi-bres which will have some similarity with the behaviour of soilscontaining actual plant roots. We are concerned only with me-chanical and not with hydrological effects. However, provideda model is available to describe the behaviour of the soil (sat-urated or unsaturated) in the absence of fibres/roots, then theeffect of the fibres can be added in a systematic way.
There have been several studies of the influence of flexiblefibres on the strength of soils. Failure criteria have been devel-oped using force equilibrium considerations in a localised shearband (Jewell and Wroth, 1987; Maher and Gray, 1990; Ranjanand Charan, 1996); energy based homogenisation approaches(Michałowski and Cermak, 2002); or the discrete superposi-tion of the sand and fibre effects (Zornberg, 2002). Quantitativemodelling of the pre-failure behaviour of fibre-soil mixtures hasreceived less attention and proposed models have dealt with
Email addresses: [email protected] (D. Muir Wood ),[email protected] (A. Diambra ),[email protected] (E. Ibraim )
elastic behaviour of the material (Ding and Hargrove, 2006) orhave been applied to soils reinforced with continuous thread(Texsol) (Villard et al., 1990; di Prisco and Nova, 1993). Thetwo dimensional DEM (Distinct Element Method) has been usedto investigate the micromechanical aspects of interaction be-tween grains and fibres and the distribution of the tensile stressesmobilised in the fibres (Ibraim et al., 2006; Ibraim and Maeda,2007).
Our modelling environment takes the form of an infinitesi-mal simple shear element (like an element at the centre of a di-rect shear box) (Fig 1). There are several reasons for taking thiselemental approach (Muir Wood, 2009): the direct shear boxis a particularly simple pedagogic device which shows studentsor other users exactly what is happening in terms of linked vol-umetric and shearing deformations; the simple shear elementis directly applicable to the deformation and sliding of a longslope and also to the propagation of shear waves in an earth-quake; and there have been a number of developments in con-stitutive modelling over the past few decades which have en-deavoured to include influences of fabric anisotropy and historyof loading or deformation by considering the overall responseto be the summation of responses of a series of shear elementsdistributed over all possible orientations. The microstructuralmodel of Calladine (1971) applied to soils a framework sug-gested by Batdorf and Budiansky (1949) for metals, and this ap-proach has been rediscovered in multilaminate modelling (Pandeand Sharma, 1983) and in the models of Chang and Hicher(2005).
Preprint submitted to Elsevier January 24, 2016
Q
P
z
x
τ
τ
σ'z
σ'z
cross-sectional area A
loading
platen
upper box
lower box
τ
σ'z
ε γ
a. b.
Fig. 1: (a) Simple shear element of soil with fibres; (b) correspondingto central region of shear box
The modelling framework has been described by Diambraet al. (2013) and Muir Wood et al. (2014); it will be summarisedbriefly here and used to illustrate some aspects of response offibre-soil mixtures: the influence of fibres on the volumetricbehaviour; the existence and nature of asymptotic states; andthe stress-dilatancy relationship for the mixture.
Experimental observations
Inspiration for the modelling has come from an extensiveexperimental study of the behaviour of mixtures of Hostun sand(d50 = 0.38mm, Cu = 1.9) with short flexible polypropylene fi-bres (length 35mm, diameter 0.1mm) (Ibraim and Fourmont,2007; Diambra et al., 2010). It is hypothesised that the be-haviour of soil containing flexible plant roots will be broadlysubject to the same characteristics of mechanical interaction.Fibres can be mixed with the soil in carefully monitored pro-portions: attention to detail of sample preparation techniquesencourages the formation of somewhat repeatable samples. Onthe other hand roots grow through the soil, feeling their way be-tween the soil particles or packets of particles, and developingbonding by a process of cavity expansion as the root expandswithin its chosen tortuous void space and develops restrainingconfinement stresses as it grows. The detailed fabric of soil-root mixtures is expected to be more variable, whether in thelaboratory or in the field, so the tests on polypropylene fibremixtures are consequently more useful for the initial develop-ment of constitutive models.
Direct shear tests with constant vertical stress σz = 55.3kPa(Fig 2) and with values of specific volume between 1.8 and2.0 (corresponding to relative densities of approximately 60%and 0%) show increased shear stress and increased dilatancyas a result of addition of flexible fibres (Ibraim and Fourmont,2007). Figures 2a, b, d, e show the variation of shear stressand vertical displacement or volume change (uz) with horizon-tal displacement ux. The rate of change of vertical displacementwith horizontal displacement, equivalent to an angle of dilationtanψ = −δuz/δux, is plotted against externally measured valuesof mobilised friction τ/σz in Fig 2c, f. The effect of fibres on di-
0 5 10 150
20
40
60
5 10 15
horizontal displacement ux (mm) horizontal displacement ux (mm)
horizontal displacement ux (mm)horizontal displacement ux (mm)
vertical
displacement
uz (mm)
vertical
displacement
uz (mm)
shear stress
(kPa)shear stress
(kPa)
0% 0%
0%
0%
0.3%0.3%
0.3%
0.3%
0.4%0.5%
0.5%
0.4%
1%
1%
0 5 10 150
20
40
60
0
5 10 15
0
0.5
1
1.5
a.
b.
d.
e.
0
0.5
1
1.5
-0.1 0.0 0.1 0.2 0.30.0
0.2
0.4
0.6
0.8
1.0
-0.1 0.0 0.1 0.2 0.30.0
0.2
0.4
0.6
0.8
1.0
−δuz/δux −δuz/δux
τ/σz τ/σz
dilationdilation
compression
compressionc.
f.
0%0%
0.3%
0.3%
0.4%
0.5%
1%
Fig. 2: Direct shear box tests on fibre/soil mixtures: sand void ratio:a, b, c. vo = 0.8; d, e, f. vo = 1.0; a, d: mobilised friction and sheardisplacement; b, e: vertical displacement and shear displacement; c,f: dilatancy δuy/δux and externally measured mobilised friction τ/σz;σz = 55.3kPa (Ibraim and Fourmont, 2007).
latancy is confirmed in undrained triaxial compression tests onloose fibre-sand mixtures which show reduced and even neg-ative pore pressures; the presence of fibres produces a signif-icant reduction in liquefaction potential (Ibraim et al., 2010a;Diambra et al., 2011).
However, not all published papers report increased dila-tancy (Heineck et al., 2005). This apparent contradiction can belinked to the different modelling assumptions implicit in the ex-perimental approach employed for comparison of unreinforcedand reinforced sand samples. It will be discussed in a later sec-tion of this paper.
Model for fibre/sand mixtures
The soil is seen as the active component and the fibres asthe reactive component. A series of hypotheses are introducedto describe the interactive behaviour (Diambra et al., 2013):
• The sand matrix in the presence of fibres can be describedby the same model as the unreinforced soil.
• Tensile strains in the soil try to stretch the fibres. Inter-action between fibres and soil requires some mechanicalbond or anchorage.
2
• Fibres are treated as forces with orientation and not asa continuous superimposed material. They have tensilestiffness and strength but negligible compression or flex-ural stiffness or strength.
• Stretched fibres try to resist extension and thus tend toincrease the normal stress on the soil but also contributedirectly to the shear stress.
• In their interaction with the soil, as the strains increase,the fibres may pull out of the soil or may reach their ten-sile strength and snap.
• Allowance must be made for the presence of the fibres incalculating the operational specific volume or void ratioof the soil.
The first three hypotheses relate to the two separate materials:soil and fibres. The last three hypotheses relate to the interac-tion between the fibres and the surrounding soil. This approachto modelling is described as the ‘Discrete framework’ by Li andZornberg (2013).
1. Severn-Trent sand
The description of the fibre-soil interaction can be com-bined with any model for the soil itself (Diambra and Ibraim,2014). Severn-Trent sand is an extended Mohr-Coulomb fric-tional model in which the strength and dilatancy vary as a func-tion of the distance of the current state of the soil from asymp-totic critical states (Gajo and Muir Wood, 1999a,b). This un-derpinning model for the sand is built round the interaction offour components (Fig 3). We will need to refer to elements ofthis model in subsequent discussion.
When subjected to monotonic shearing the sand reacheseventual asymptotic critical states in which shearing can con-tinue with no further change in effective stress or density orfabric (on average) (Fig 3a). In order to ensure that the criticalstate line does not suggest unreasonable values of void ratio eor specific volume v = 1+ e at very low or very high stresses, aform proposed by Gudehus (1997) has been used:
vc = vmin + ∆v exp[−(σz/σre f )β] (1)
where ∆v = vmax − vmin defines the range of values of specificvolume v = 1 + e; σre f is a reference stress; and β is a soilparameter. A‘state parameter’, ψ, (Wroth and Bassett, 1965;Been and Jefferies, 1985) can be defined which encapsulatesthe volumetric distance of the current state of the sand (σz andv) from the critical state condition for the same effective stress.The sand has a rather clear feeling for the change in volumetricpacking required to bring it to this asymptotic state.
The current strength of this frictional soil is not a constantbut depends on density and stress through the current value ofstate parameter (Fig 3b). Loose sands, with current specificvolume greater than the critical state specific volume (ψ > 0),show low current strength; dense sands, with current specificvolume below the critical state specific volume (ψ < 0), showhigh current strength.
critical
state
line
stress level
specific
volume
dilatancy
contractionexpansion
critical state
critical state
shear strain
mobilised strength
available strength
available
strength 1
state parameter ψ
ψ
0
0
mobilised
strength
1. shear strain increment
2. volumetric strain
increment (dilatancy)
3. change in
specific volume
(state parameter)
4. change in
available strength
a.
b.
c.
d.
Fig. 3: Four elements of Severn-Trent sand model: (a) critical stateline and state parameter; (b) strength dependent on state parameter; (c)monotonic hardening relationship; (d) stress:dilatancy relationship.
The plastic hardening of the soil is purely distortional, re-sulting from rearrangement of soil particles; the plastic stiff-ness falls steadily as the mobilised friction increases towardsthe currently available strength (Fig 3c). The plastic hardeningis described by a monotonic relationship. The flow rule link-ing plastic volumetric dilation with plastic distortion (Fig 3d)provides a feedback link.
The operation of the model can be simply described. Anincrement of (plastic) distortional strain leads to an increase inthe mobilisation of currently available strength, (Fig 3c). Theflow rule requires there to be plastic volumetric strains accom-panying the distortional strains (Fig 3d). The resulting changein volume moves the state of the sand closer to the critical state(from above or below) (Fig 3a). The resulting change in stateparameter leads to a change in the available strength (Fig 3b) sothat the distortional hardening is moving the state of the soil to-wards a moving target. The close interlocking of the elementsof the model (Fig 3) ensures that, with continuing monotonicshearing, the state of the sand heads for an asymptotic criti-cal state. The combination of these four components producesa satisfyingly rich range of simulated responses with a rathersmall number of soil parameters.
2. Contribution of fibres
Tensile strains in the soil try to stretch any fibres whose ori-entation engages with the tensile sector of the Mohr circle ofstrain increment (Fig 4). The simple shear element has two de-grees of strain increment freedom: vertical or volumetric strainand shear strain. The horizontal direction is always inexten-sional so that the Mohr circle of strain increment must intersect(or, in the limit, touch) the shear strain axis δϵ = 0. This Mohrcircle defines the range of orientations within the sample forwhich the strain increment has a tensile component (Figs 4a, b,d). There will always be some such orientations except whenthe sample is being subjected to pure (one-dimensional) com-pression (Fig 4c). For shearing at constant volume, fibres with
3
δε δε
δε
δγ/2
δγ/2
δγ/2
δγ/2
δεz δε
2χ
2χ
z z
z
x
x
x
compression
extension
extension
extension
z
a.
b.δεs/2 δεs/2
compression
δε
z
c. d.
x
compression
2χ = π
Fig. 4: Mohr’s circle of strain increment for simple shear sample: (a)shearing with volumetric compression δϵz > 0; (b) shearing with vol-umetric expansion δϵz < 0; (c) one-dimensional compression; (d) con-stant volume shearing δϵz = 0.
orientations between 0 and π/2 to the horizontal (in the direc-tion opposite to the shearing) will develop tensile strains (Fig4d).
It is obviously necessary to know the actual distribution oforientation of fibres in the sample that is to be simulated. Typi-cal techniques for preparation of fibre/soil mixtures do not pro-duce random distributions of fibre orientation (Diambra et al.,2007): moist tamping inevitably leaves the fibres in a somewhatsub-horizontal orientation (Michałowski and Cermak, 2003).Information is needed concerning both spatial distribution offibres and distribution of fibre orientations. A homogeneousspatial distribution is a reasonable experimental goal, whereasthe distribution of orientations is an outcome which must beknown even if it cannot be precisely controlled.
The same information is required for plant roots: distribu-tion and orientation of the flexible elements which may wellhave different diameters. Plants can be divided into two groups:‘oligorhizoid’ dicotyledons have a few rather substantial roots(such as mustard, Fig 5b); monocotyledons such as grasses tendto be more ‘polyrhizoid’ in character (Fig 5a) having more, finerroots which are much more randomly distributed. Such polyrhi-zoid species are more obviously suited to a continuum approachto modelling. Polyrhizoid plant species forming an interlock-ing cluster of reinforcement will provide an apparent cohesionin near surface soils for which the frictional strength is verylow. They are obvious candidates for improving slope stabilitythrough enhancement of the mechanical properties. The topol-ogy or architecture of plant roots is more complicated than thatof uniform identical flexible fibres. However, there exist com-pendia of immaculate drawings of roots for different species(Kutschera et al., 1960-2009) which can provide some initialguidance.
The outcome of these direct measurements, or estimates, is
0 0.1 0.20
1
2
3axial
stress
(MPa)
axial strain
a.
b. c.
diameter 0.49mm
diameter 0.78mm
Fig. 5: Root architecture for (a) rye grass Lolium mul. Wester-woldicum; (b) mustard Brassica nigra; (c) tensile tests on plant rootsof vetch (Vicia sativa) (Liang, 2016).
a probability density function N pθδθ describing the proportionof the total number N of fibres (of different diameters for roots)within the angular sector δθwith orientation θ crossing unit areaof the simple shear sample (Fig 6a).
A tensile test on a polypropylene fibre is shown in Fig 6cand tensile tests on roots of vetch are shown in Fig 5c. As a firstassumption we will assume that the response of the polypropy-lene fibres is linear elastic until plastic ductile failure is reachedat a yield strain ϵ f y and subsequent breakage strain ϵ f b. We as-sume a Young’s modulus E f to convert fibre strain to an axialforce in the elastic region δP = δσa f a f = E f a f δϵ f along thefibre of cross-sectional area a f . If ϵ f > ϵ f y then δP = 0.
Stressed fibres contribute vertical and horizontal compo-nents of force to the stress state on the horizontal plane of thesimple shear soil element (Fig 6b). Fibres try to resist stretch-ing because they are anchored in the soil by the clamping forcesof the soil particles along the length of the fibres. Consequentlyfibres will increase the normal stress δσz f ; fibres with orienta-tion θ < π/2 will also contribute to the shearing resistance ofthe composite element δτ f .
δσz f = N pθE f a f δϵ f sin θ; δτ f = N pθE f a f δϵ f cos θ (2)
Stretched fibres having orientations θ > π/2 (with tensile strainincrement range 2χ > π (Fig 4b)) will reduce the shearing re-sistance slightly while still boosting the normal stress.
Strains develop in the fibres because of the strains that occurin the soil around the fibres. However, the fibre-grain interac-tion is rather complex. The fibres take an erratic route betweenthe soil grains (Lirer et al., 2011; Heineck et al., 2005; Consoliet al., 2005) and the axial strains usually vary along the lengthof the fibres. Shear distortions at the interface between the twomaterials and end-effects occur in fibre reinforced composites(Hull and Clyne, 1996). These phenomena can be included ina continuum modelling approach by simply introducing a mis-match between the strains in the fibre and the soil (Diambra and
4
θ
fibre in
tension
additional
normal stress
additional
shear stress
θ
δθ
fibres within angle δθ at
angle θ to horizontal
unit width of
cross-section
P
a.
b.
c.
E = 900 MPa
σ (MPa)
ε (%)
0
100
200
300
0 40 80
εfy εfb
Fig. 6: (a) Orientation and distribution of fibres and (b) contribution tonormal stress and shear stress; (c) tensile test on polyproplene fibre.
Ibraim, 2015):δϵ f = fmδϵm (3)
where δϵ f and δϵm are the strain increments in the fibre and thesoil matrix respectively and fm < 1 is a strain ‘mismatch’ factor.Using appropriate modification of the shear lag theory for com-posite materials (Cox, 1952), Diambra and Ibraim (2015) de-rived a complete expression for fm which explicitly consideredthe geometry of fibre and grains, fibre stiffness, global stresslevel, soil density and the non-linearity of soil behaviour. Herewe have used a simpler expression for fm which accounts forthe fundamental effect of the stress level in the soil σzs whichwill be greater than the externally applied stress because of theextra stress generated by the stretched fibres:
fm = 1 − λ exp[−(σzs/σr) (4)
where σr is a reference stress, and λ controls the degree of mis-match for a given stress level. The mismatch between the soiland the matrix reduces as the surrounding stress level increases(Diambra and Ibraim, 2015). The incremental force in the fibreis then δP = E f a f δϵ f : the strain mismatch reduces the apparentfibre stiffness.
Simulation and discussion
A set of comparisons of simulated and laboratory directshear tests on fibre-sand mixtures, using a single set of soilparameters, (and with fibre orientations uniformly distributed)is shown in Fig 7. The simulations are described in terms ofstrains, the direct shear tests are reported in terms of displace-ments, but the general concordance between the observationsand simulations is good.
Volumetric interaction and fibrespaceThere are various ways in which the volumetric packing of
the sand can be described in the presence of the fibres; and the
increased
dilatancy
increased
strength
0 5 10 150.0
0.2
0.4
0.6
0.8
0 5 10 15
-1.0
-0.5
0.0
0.5
fibres: 1%
fibres: 1%
fibres: 1%
0.5%
0.5%0.5%
0.3%
0.3% 0.3%
0%
fibres: 1%
0.5%
0.3%
0%
0% 0%
stress
ratio
shear box displacement: mm
shear box displacement: mm
vertical
movement
mm
0 0.2 0.4
0 0.2 0.4
0.01
0
-0.02
-0.04
0
0.2
0.4
0.6
0.8
stress
ratio
shear strain
shear strain
vertical
strain
a.
b.
c.
d.
Fig. 7: Direct shear tests on fibre-sand mixtures: (a, b) observation; (c,d) simulation.
fibres fibres with stolen voids
voids
fibres
soil
volumes
fibre voids
fibres
soil voids
soil
volumes
Fig. 8: Fibres stealing void from sand to create fibrespace.
preparation technique, intending to prepare comparable sam-ples with different fibre contents, will itself make some assump-tion about what constitutes an appropriate measure of packing.
The fibres have volume V f , the soil particles have volumeVs and there are voids with volume Vv. Let us suppose thatthe fibres themselves require some volume of surrounding voids(Diambra et al., 2010) - in other words that they steal some voidratio from the soil in order to create their own fibrespace (Fig8). The volume of fibrespace might be somehow linked to thesurface area of the fibres (Muir Wood, 2012). The total volumeof voids is then divided into Vv f associated with the fibres andVvs associated with the soil. The specific volume of the fibresin fibrespace is:
v f =Vv f + V f
V f(5)
The volume proportion for the fibres ρ = V f /(V f + Vs) orvolume ratio V f /Vs = ρ/(1 − ρ). Samples will usually be pre-pared by mass: the proportion of masses f = M f /(M f + Ms).
5
Vvf
Vvs
Vf Vf
VfVf
Vs VsVs Vs
Vvf
VvsVvf
Vvs
Vvf
Vvs
e esf esv es
a. b. c. d.
Fig. 9: Alternative definitions of void ratio.
With specific gravity G f and Gs = G f /kG for fibres and soilparticles:
f =kGρ
1 − ρ(1 − kG)(6)
so that, if kG ∼ 1/3, and ρ ≪ 1, f ∼ ρ/3.The void ratio is e, the ratio of volume of all voids to the
volume of all solids (particles and fibres) (Fig 9a):
e =Vv
V f + Vs=
Vvs + Vv f
V f + Vs→ v = 1 + e (7)
The volume of fibres is small and they hardly provide a con-tinuous load bearing phase. A generous void ratio can then bedefined, treating everything apart from the soil particles them-selves as void space, es f (Fig 9b):
es f =Vv + V f
Vs=
e + ρ1 − ρ → vs f =
v1 − ρ (8)
If we associate all the voids with the soil particles but leavethe volume of fibres with no attached voids, there is an interme-diate void ratio esv (Fig 9c):
esv =Vv
Vs=
e1 − ρ → vsv =
v − ρ1 − ρ (9)
If we regard the voids contained in fibrespace as inalienablethen we can define a soil void ratio, es (Fig 9d):
es =Vvs
Vs=
e + (1 − v f )ρ1 − ρ → vs =
v − v f ρ
1 − ρ (10)
These various definitions of void ratio and specific volume arecompared in Fig 10a for v = 1.6 and v f = 3.
An immediate illustration of the effect of this stolen voidratio or fibrespace is provided by the results of the procedureadopted for preparation of the fibre-sand mixtures (Fig 10b)(Ibraim et al., 2012; Ibraim and Fourmont, 2007). For a givenamount of tamping effort, the final density of packing reducesas the fibre content increases. One-dimensional compressionlinked with tamping produces only compression direct strainincrements overall (Fig 4c) so that there is no obvious possi-bility at the ‘system’ level of fibres being stretched by tensilestrain increments in order to influence the compaction. How-ever, at the particle level there may be some mechanical inter-action with the fibres because of local fabric changes (Consoli
ρ
vs
vsv
vsf
v
0.00 0.02 0.04 0.06 0.08
1.6
1.8
2.0
2.2
volume proportion of fibres
specific volume
resulting from
moist tamping
a.
b.
0 0.04 0.081.3
1.4
1.5
1.6
1.7
1.8
Fig. 10: (a) Alternative definitions of specific volume (v = 1.6, v f =
3); (b) effect of fibre content on specific volume obtained by moisttamping sample preparation.
Vvf
Vvs
Vf
Vf
Vs
VsVso
Vvo Vvf
Vvs
a. b. c.
Fig. 11: Alternative strategies for preparation of soil/fibre mixtures:(a) unreinforced soil; (b) fibres replace soil (overall volume constant);(c) fibres replace void (overall volume constant).
et al., 2005; Ibraim et al., 2006; Diambra and Ibraim, 2015).If we suppose that the soil always reaches the same densityat the conclusion of tamping then we can ascribe the loweroverall density to the need to include the fibrespace. Analy-sis of the compaction produces fibrespace specific volumes ofv f ∼ 5 − 10. These may seem a little high but with this magni-tude the simulations become reasonable.
Is it possible to choose the initial density of the soil-fibrespecimens to guarantee direct comparability of response? Twostrategies have been adopted for preparation of fibre-soil mix-tures (Fig 11) (vo and ρ are the specific volume of the plain sandand the proportion of fibres by volume of fibres plus soil):
1. Some of the volume of soil particles is replaced by fi-bres so that the specific volume of the mixture matchesthe specific volume of the plain sand v = vo (Fig 11b)(Silva dos Santos et al., 2010; Michałowski and Cermak,2003; Heineck et al., 2005) [v = vo, vs f = vo/(1 − ρ),vsv = (vo − ρ)/(1 − ρ); vs = (vo − v fρ)/(1 − ρ)] .
2. The volume of sand is kept constant and the addition offibres replaces some of the voids (Fig 11c) so that thespecific volume vs f = vo (Fig 9b) (Diambra et al., 2010;Ibraim et al., 2010a) [v = vo(1 − ρ), vs f = vo, vsv =
6
shear strain εs
shear strain εs
vertical
strain εz
no fibres
no fibres
τ/σzo
σzs
specific
volume v critical
state line
no fibres
a.
b.
c.
0 2 4 60
1
2
3
0 2 4 6
0
fibres replace solid
fibres replace solid
fibres replace solid
fibres replace void
fibres replace void
fibres replace void
-0.04
0.04
0.08
0 100 200 300
1.46
1.5
1.54
1.58
1.62
remove
fibrespace
remove
fibrespace
remove
fibrespace
Fig. 12: Simulations of shearing with constant vertical stress σz = σzo;initial densities chosen according to different preparation strategies:fibres replace soil particles or fibres replace voids; fibrespace removedor not removed before calculating effective density; (a) shear stressτ/σzo and shear strain ϵs; (b) vertical strain ϵz and shear strain ϵs; (c)specific volume vs of soil and vertical stress experienced by soil σzs
(v f = 3, ρ = 0.03).
vo − ρ/(1− ρ), vs = vo − v f ρ/(1− ρ)]. This is the strategyadopted for the tests shown in Fig 7a, b.
Evidently vo > vo(1−ρ)−ρ and the second strategy will producedenser samples which will show greater dilation, even beforethe fibrespace of fibres and voids is removed from the calcula-tion of effective densities. This is confirmed in the simulationsin Fig 12.
A simple conclusion is that it is meaningless to say that ‘ad-dition of fibres increases (or decreases) dilatancy’, because sucha statement can only be made in the context of a complete de-scription of the procedure for preparing and testing and mod-elling the soil-fibre mixtures. Properly contextualised the ob-servation becomes another element of the dataset to incorporateinto the modelling.
Asymptotic states and stress-dilatancyWhen sheared continuously soils reach an asymptotic state
in which all aspects of the definition of state reach stationary
Fig. 13: Asymptotic states for fibre-sand mixtures: (a) fibres continu-ously pulling through soil; (b) fibres breaking.
values. The classical asymptotic critical state was concernedonly with stationary values of stresses and density (void ratio)(Roscoe et al., 1958). However, a properly asymptotic staterequires the fabric, particle grading and particle shape also tohave reached steady conditions. For the fibre-soil mixtures boththe soil and the fibres (in their interaction with the soil) in ourinfinitesimal simple shear element must have reached a steadystate. For the soil the critical state will be the same as that of thesoil tested on its own. For the fibre-sand mixtures we can en-visage two possible interactive asymptotic states (Fig 13). Thelimiting tensile force that can be transmitted by the fibre is de-pendent on the strength of the fibre and on the effectiveness ofthe anchorage of the ends of the fibre. A perfectly plastic lim-iting value of fibre stress may be reached either permanentlybecause the fibre is pulling out at constant stress (Fig 13a) ortemporarily because the fibre itself has an extended ductile re-gion of extension from a yield strain ϵ f y to a breakage strainϵ f b = κϵ f y (Fig 6b).
The second asymptotic possibility is one in which all thefibres have broken to a length (of the order of typical parti-cle size) at which they have no residual bond length (Fig 13b).Once broken, the fibre force in the infinitesimal element is zerofor all subsequent strain increments. However, the fragments offibre still occupy space in the fibre-soil mixture and thus con-tinue to influence the values of specific volume which recognisethe presence of the fibres, with or without their attendant voids,vsv (9) and vs (10).
In principle, infinite strain is needed to reach asymptoticstates in which all aspects of fabric and state have stopped chang-ing (Ibraim et al., 2010b). The concept of small strain asymp-totic or limiting response is slightly oxymoronic. Typical testapparatus are not capable of applying infinite strains (apart fromring shear (Consoli et al., 2005)): we seek tendencies towards,rather than arrivals at, asymptotic destinations.
Shearing at constant volume implies that the Mohr circle ofstrain increment is centred on the origin so that all fibres withorientation lying within one sector of π/2 from the horizontalwill be subject to extension stretching strains. The mechanicalcontribution of the fibres results from the interaction of the fibreorientations with the Mohr’s circles of strain increment (Fig 4).As a simple illustration suppose that the fibres are uniformlydistributed across all orientations so that pθ = 1/π, and that thefibre/soil sample is being sheared at constant volume. Succes-sive Mohr circles (centred on the origin) are shown in Fig 14a.Where the tensile strain is less than the yield strain ϵy the fibresare stretched elastically; for tensile strains in the range ϵy − ϵb
the fibres generate a constant yield or slipping force (Fig 13a).
7
ε
γ/2
εfyεfb
yieldbreakage
AB
C
D
E F
εs/εfy
εfb/εfy = 10
εfb/εfy >> 10
elastic
elastic +
slip/yield +
breakage
fibre contribution
to normal and
shear stress
(normalised)
a. b.
0 10 20 300
0.5
1
elastic +
slip/yield
Fig. 14: a. Mohr circles of strain for increasing shearing at constantvolume: yield/slip of fibres at tensile strain ϵy; breakage of fibres attensile strain ϵb (Mohr circle sectors AB and EF: elastic; BC and DE:ductile fibre response, constant P; CD: broken fibres); b. fibre con-tribution to normal stress and shear stress with increasing shear strainϵs.
50 60 70 80 90 100 110 1200
40
80
120
160
00
0.4
0.8
1.2
1.6
0.02 0.04 0.06 0.08 0.1 0.12shear strain
stress
ratio
vertical stress
shear
stresssoil without
fibres soil without
fibres
soil densified for
fibrespacesoil densified for
fibrespace
soil with fibres -
eventually slipping soil with fibres -
eventually slipping
soil with fibres -
eventually breaking
soil with fibres -
eventually breaking
a. b.
Fig. 15: Simulations of constant volume tests on sand-fibre mixture.
Where the strain exceeds ϵb the fibres break (Fig 13b). The re-sulting development of fibre contribution with strain is shownin Fig 14b - it is entirely determined by the Mohr circle of strain(for monotonic shearing).
We can build up the expected constant volume response ofan initially loose soil (state parameter ψ > 0) to which fibreshave been added (Fig 15). The loose soil on its own wishes tocontract as it is sheared in order to approach the critical state.The externally imposed normal stress decreases in order to per-mit elastic expansion to counter the tendency to contraction.The effective stress path heads towards the origin (Fig 15b).
Adding in the fibres, the theft of the voids to form fibrespaceleaves the soil feeling denser so that the tendency to contract isreplaced by a tendency to expand and the vertical stress has toincrease to counter this tendency. The stress:strain relationshipfor the sand alone shows higher stiffness and strength becauseof the higher perceived density and the effective stress path alsoshows dilative tendencies. The vertical stress initially falls butthen rises rapidly in order to counter the desire of the soil todilate.
However, the fibres being stretched by the shearing wantto compress the soil and the vertical stress therefore has todecrease in order to keep the volume constant. The eventualstress path for the mixture thus lies to the left of the path forthe pseudo-densified soil. The simulations shown have been
performed using a division of orientations into 36 sectors of5◦. The orientations of fibres have been uniformly distributedacross these orientations. Consequently the integrated contri-butions of the fibres to increase in shear stress and to decreasein normal stress are of equal magnitude.
With soil-fibre interaction chosen to lead to eventual per-fectly plastic pull-through ϵ f b ≫ ϵ f y (Fig 14b) the stress:strainresponse and stress path show a sustained benefit from the fi-bres. With alternative parameters which lead to fibre breakagethe benefit is steadily lost - fibres around π/4 to the horizontalexperience the largest strains and break first (Fig 14). Figure14b indicates the asymptotic responses corresponding to eitherof these limits. Breakage proceeds round the fibre orientations:the step-wise nature of the curves shown in Fig 15 correspondsto this sequential breakage. With complete breakage the fibre-soil mixture reverts to the response of the sand densified bythe removal of fibrespace: even with complete breakage thereis some residual benefit compared with the original loose sand(Fig 15).
Shearing in an asymptotic state must be occurring at con-stant volume - it would otherwise be unsustainable. In thestress-dilatancy plot of Severn-Trent sand (Fig 3d) the soil willreach its critical state as usual with mobilised friction corre-sponding to the critical state stress ratio M. However, the mo-bilised friction determined externally Rext = τext/σext is not thefriction mobilised in the soil Rs = τs/σs. The fibres beingstretched provide extra normal stress (σ f ) in addition to that ex-ternally applied: σs = σext + σ f . The fibres also provide someincreased shearing resistance beyond that generated in the soilτ f . Thus, for the soil the mobilised friction is:
Rs =τs
σs=
τs
σext + σ f=
τs
σzs(11)
reaching the value M at the critical state. The externally deter-mined mobilised friction is:
Rext =τext
σext=τs + τ f
σext=
τs + τ f
σs − σ f(12)
More to the point, the routes by which the soil resistanceand the contributions of the fibres are generated are quite dif-ferent. The stress-dilatancy plot (Fig 3d) forms part of the de-scription of the soil based on our experience of elastic-plasticconstitutive modelling of soils. The presence of the fibres ap-pears to push the stress-dilatancy plot away from the criticalstate for the soil alone. But we have two contributions - fibreand soil - which are responding mechanically in quite differentways and the pattern appropriate for one is not relevant for theother.
Roots
Much of our discussion has been generic so far as the na-ture of the flexible elements within the soil are concerned. Forour polypropylene fibres of uniform cross-section and length itis essential to know the distribution and orientation of the fi-bres. That necessity remains with roots but the variability in di-mensions and mechanical properties must be added. Fibre bun-dle or root bundle models (Pollen and Simon, 2005; Mickovski
8
et al., 2009; Schwarz et al., 2010a) provide a structured meansof describing such variability. In most models, roots have beenconsidered as very flexible elements, like our fibres, appropri-ate for the finest roots. ‘Structural roots’ with significant flex-ural resistance require a different sort of modelling (Reubenset al., 2007) - but also reflect different plant species which maybe less appropriate for general soil improvement. Roots mayhave different failure mechanisms, can break or pull-out, whilethe length, apparent Young’s modulus, and maximum tensileforce are functions of root diameter and age (Schwarz et al.,2010b). Root tortuosity can affect the root stiffness (Schwarzet al., 2011); root topology, branching angle and branching den-sity can significantly change the distribution of stresses andplastic strains within the soil (Stokes et al., 1996; Mickovskiet al., 2007; Dupuy et al., 2005; Loades et al., 2010; Danjon andReubens, 2008; Mickovski and van Beek, 2009). But some ofthese effects relate to the soil-root system - moving up a scalefrom the ‘infinitesimal’ continuum element that has been ourfocus.
Conclusion
We have developed a framework for modelling the inter-action of soil with flexible fibres. The mechanics of the indi-vidual components - soil and fibres - are not changed in theircombination but it is their interaction which provides a greaterchallenge. It is obvious that, whatever the nature of the flexibleinclusions, it will be necessary to know their distribution, orien-tation, dimensions, and mechanical properties if we are to havesome hope of being able to produce successful simulations.
Part of the description of the interaction between soil and fi-bres relates to the appropriate choice of description of packingof the mixture. The concept of stolen void ratio or fibrespacehas been invoked in order to be able to describe the significantchanges in dilatancy, which imply a reduction in state parame-ter, in the presence of the fibres. The consequences of fibres-pace require further exploration concerning both the physicaljustification and the potential for evolution with shearing or in-creased stress.
There are several different ways in which the volumetricproportions of different constituents in a mixture can be de-scribed. Basing an assessment of comparative response on onedescription rather than another is hardly conclusive. The gath-ering of completely defined experimental observations can mostusefully be fed into the parallel process of model development.
The interaction of soils with flexible fibres - or roots - canbe simulated rather satisfactorily with appropriate allowance forthe volumes occupied or demanded by the several phases. Thetest observations and the elements of the modelling for the soiland for the soil-fibre mixtures demonstrate once again the im-portance of considering volume and density change in soils inparallel with changes in effective stress - reinforcing the under-pinning message of critical state soil mechanics.
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