Revisión de similitud de cadenas

The term intergranular stress has become synonymous with effective stress. Whether or not the intergranular stressi is indeed equal to ⫺u cannot be ascertained

without more detailed examination of all the interpar-

1_{The terms}  and  are the principal total and effective stresses.

For general stress conditions, there are six stress components (11,

22,33,12,23, and31), where the first three are the normal stresses

and the latter three are the shear stresses. In this case, the effective stresses are defined as11⫽11⫺ u,22⫽22⫺ u,33⫽33⫺

u,12⫽ 12, 23⫽ ,23 and31⫽ .31

ticle forces in a soil mass. Interparticle forces at the microscale can be separated into the following three categories (Santamarina, 2003):

1. Skeletal Forces Due to External Loading These

forces are transmitted through particles from the forces applied externally [e.g., foundation load- ing) (Fig. 7.1a)].

2. Particle Level Forces These include particle

weight force, buoyancy force when a particle is submerged under fluid, and hydrodynamic forces or seepage forces due to pore fluid moving through the interconnected pore network (Fig. 7.1b).

3. Contact Level Forces These include electrical

forces, capillary forces when the soil becomes unsaturated, and cementation-reactive forces (Fig. 7.1c).

When external forces are applied, both normal and tangential forces develop at particle contacts. All par- ticles do not share the forces or stresses applied at the boundaries in equal manner. Each particle has different skeletal forces depending on the position relative to the neighboring particles in contact. The transfer of forces through particle contacts from external stresses was shown in Fig. 5.15 using a photoelastic model. Strong particle force chains form in the direction of major principal stress. The evolution and distribution of in- terparticle skeletal forces in soils govern the macro- scopic stress–strain behavior, volume change, and strength. As the soil approaches failure, buckling of particle force chains occurs and shear bands develop due to localization of deformation. Further discussion of microbehavior in relation to deformation and strength is given in Chapter 11.

Particle weights act as body forces in dry soil and contribute to skeletal forces, observed in the photo- elastic model shown in Fig. 5.15. When the pores are filled with fluids, the weight of the fluids adds to the body force of the soil–fluids mixture. However, hydro- static pressure results from the fluid weight, and the uplift force due to buoyancy reduces the effective weight of a fluid-filled soil. This leads to smaller skel- etal forces for submerged soil compared to dry soil. Seepage forces that result from additional fluid pres- sures applied externally produce hydrodynamic forces on particles and alter the skeletal forces.

7.4 INTERPARTICLE FORCES

Long-range particle interactions associated with elec- trical double layers and van der Waals forces are dis-

INTERPARTICLE FORCES 175 External Load Interparticle Forces Interparticle Forces (a) Body Force Buoyancy Force if Saturated Seepage Viscous Drag by Seepage Flow (b) Capillary Force or Cementation-reactive Force Electrical Forces (c) Figure 7.1 Interparticle forces at the particle level: (a) skeletal forces by external loading, (b) particle level forces, and (c) contact level forces (after Santamarina, 2003).

cussed in Chapter 6. These interactions control the flocculation–deflocculation behavior of clay particles in suspension, and they are important in swelling soils that contain expanding lattice clay minerals. In denser soil masses, other forces of interaction become impor- tant as well since they may influence the intergranular stresses and control the strength at interparticle con- tacts, which in turn controls resistance to compression and strength. In a soil mass at equilibrium, there must be a balance among all interparticle forces, the pres- sure in the water, and the applied boundary stresses. Interparticle Repulsive Forces

Electrostatic Forces Very high repulsion, the Born

repulsion, develops at contact points between particles.

It results from the overlap between electron clouds, and it is sufficiently great to prevent the interpenetra- tion of matter.

At separation distances beyond the region of direct physical interference between adsorbed ions and hy- dration water molecules, double-layer interactions pro- vide the major source of interparticle repulsion. The theory of these forces is given in Chapter 6. As noted there, this repulsion is very sensitive to cation valence, electrolyte concentration, and the dielectric properties of the pore fluid.

Surface and Ion Hydration The hydration energy of particle surfaces and interlayer cations causes large repulsive forces at small separation distances between unit layers (clear distance between surfaces up to about 2 nm). The net energy required to remove the last few

layers of water when clay plates are pressed together may be 0.05 to 0.1 J / m2_{. The corresponding pressure}

required to squeeze out one molecular layer of water may be as much as 400 MPa (4000 atm) (van Olphen, 1977).

Thus, pressure alone is not likely to be sufficient to squeeze out all the water between parallel particle sur- faces in naturally occurring clays. Heat and / or high vacuum are needed to remove all the water from a fine- grained soil. This does not mean, however, that all the water may not be squeezed from between interparticle contacts. In the case of interacting particle corners, edges, and faces of interacting asperities, the contact stress may be several thousand atmospheres because the interparticle contact area is only a very small pro- portion (⬍⬍1%) of the total soil cross-sectional area in most cases. The exact nature of an interparticle con- tact remains largely a matter for speculation; however, there is evidence (Chapter 12) that it is effectively solid to solid.

Hydration repulsions decay rapidly with separation distance, varying inversely as the square of the dis- tance.

Interparticle Attractive Forces

Electrostatic Attractions When particle edges and surfaces are oppositely charged, there is attraction due to interactions between double layers of opposite sign. Fine soil particles are often observed to adhere when dry. Electrostatic attraction between surfaces at differ- ent potentials has been suggested as a cause. When the

gap between parallel particle surfaces separated by dis- tance d at potentials V1 and V2 is conductive, there is

an attractive force per unit area, or tensile strength, given by (Ingles, 1962)

⫺6 2

4.4⫻ 10 (V1⫺ V )2 2

F⫽ ₂ N / m (7.1)

d

where F is the tensile strength, d is in micrometers, and V1 and V2are in millivolts. This force is indepen-

dent of particle size and becomes significant (greater than 7 kN / m2 _{or 1 psi) for separation distances less}

than 2.5 nm.

Electromagnetic Attractions Electromagnetic at- tractions caused by frequency-dependent dipole inter- actions (van der Waals forces) are described in Section 6.12. Anandarajah and Chen (1997) proposed a method to quantify the van der Waals force between particles specifically for fine-grained soils with various geomet- ric parameters such as particle length, thickness, ori- entation, and spacing.

Primary Valence Bonding Chemical interactions between particles and between the particles and adja- cent liquid phase can only develop at short range. Co- valent and ionic bonds occur at spacings less than 0.3 nm. Cementation involves chemical bonding and can be considered as a short-range attraction.

Whether primary valence bonds, or possibly hydro- gen bonds, can develop at interparticle contacts with- out the presence of cementing agents is largely a matter of speculation. Very high contact stresses be- tween particles could squeeze out adsorbed water and cations and cause mineral surfaces to come close to- gether, perhaps providing opportunity for cold weld- ing. The activation energy for soil deformation is high, in the range characteristic for rupture of chemical bonds, and strength behavior appears in reasonable conformity with the adhesion theory of friction (Chap- ter 11). Thus, interatomic bonding between particles seems possible. On the other hand, the absence of co- hesion in overconsolidated silts and sands argues against such pressure-induced bonding.

Cementation Cementation may develop naturally from precipitation of calcite, silica, alumina, iron ox- ides, and possibly other inorganic or organic com- pounds. The addition of stabilizers such as cement and lime to a soil also leads to interparticle cementation. If two particles are not cemented, the interparticle force cannot become tensile; they loose contact. However, if a particle contact is cemented, it is possible for some interparticle forces to become negative due to the ten- sile resistance (or strength) of the cemented bonds.

There is also an increase in resistance to tangential force at particle contacts. However, when the bond breaks, the shear capacity at a contact reduces to that of the uncemented contacts.

An analysis of the strength of cemented bonds should consider three cases: (i) failure in the cement, (ii) failure in the particle and (iii) failure at the ce- ment–particle interface. The following equation can be derived (Ingles, 1962) for the tensile strength T per

unit area of soil cross section:

1 n  ⫽T Pk

冉 冊

n (7.2) 1 ⫹e A

冘

i 1

where P is the bond strength per contact zone, k is the mean coordination number of a grain, e is the void ratio, n is the number of grains in an ideal breakage plane at right angles to the direction of T, and Ai is

the total surface area of the ith grain.

For a random and isotropic assembly of spheres of diameter d, Eq. (7.2) becomes

Pk

 ⫽T ₂ (7.3)

d (1⫹ e)

For a random and isotropic assembly of rods of length

l and diameter d

Pk

 ⫽T (7.4)

d(l⫹d / 2)(1 ⫹e)

Bond strength P is evaluated in the following way (Fig. 7.2) for two cemented spheres of radius R. It may be shown that

(R⫺ cos)

cosh ⫽ R sin (7.5)



so for known,can be computed. Then, for cement failure,

2

P⫽  ⫻ c (7.6)

where c is the tensile strength of the cement; for

sphere failure,

2

P ⫽ ⫻ s () (7.7) where  ⫽R sin , and s is the tensile strength of

the sphere, and for failure at the interface

INTERPARTICLE FORCES 177

Figure 7.2 Contact zone failures for cemented spheres.

sin ₂

Pⴖ ⫽ ⫻1 ⫻ 2R (1 ⫺ cos) (7.8)



where 1 is the tensile strength of the interface bond.

In principle, Eq. (7.6), (7.7), or (7.8) can be used to obtain a value for P in Eq. (7.2) enabling computation of the tensile strength Tof a cemented soil.

The behavior of cemented soils can depend on the timing of cementation development. Artificially ce- mented soils are often loaded after cementation has developed, whereas cementation develops during or af- ter overburden loading in natural soils. In the former case, the particles and cementation bonding are loaded together and contact forces can become negative de- pending on the tensile resistance of cementation bond- ing. The distribution and magnitude of skeletal forces are therefore influenced by both geometric arrange- ment of particles and the cementation bonding at the particle contacts. In the latter case, on the other hand, the contact forces induced by external loading are de- veloped before cementation coats the already loaded particles. In this case, it is possible that cementation creates extra forces at particle contacts. In some ce-

mented natural materials, if the soil is unloaded from high overburden stress, elastic rebound may disrupt ce- mented bonds.

Cementation allows interparticle normal forces to become negative, and, therefore, the distribution and evolution of skeletal forces may be different than in uncemented soils, even though the applied external stresses are the same. Thus, the stiffness and strength properties of a soil are likely to differ according to when and how cementation was developed. How to account for this in terms of effective stress is not yet clear.

Capillary Stresses Because water is attracted to soil particles and because water can develop surface tension, suction develops inside the pore fluid when a saturated soil mass begins to dry. This suction acts like a vacuum and will directly contribute to the effective stress or skeletal forces. The negative pore pressure is usually considered responsible for apparent and tem- porary cohesion in soils, whereas the other attractive forces produce true cohesion.

When the soil continues to dry, air starts to invade into the pores. The air entry pressure is related to the pore size and can be estimate using the following equa- tion, assuming a capillary tube as shown in Fig. 7.3a:

2awcos  ˆ

Pc⫽ (7.9)

rp

where Pcˆ is the capillary pressure at air entry, aw is

the air–water interfacial tension,is contact angle de- fined in Fig. 7.3, and rp is the tube radius. For pure

water and air, aw depends on temperature, for exam-

ple, it is 0.0756 N / m at 0C, 0.0728 N / m at 20C, and 0.0589 N / m at 100C. If the capillary pressure Pc

(⫽ ua ⫺ uw, where ua and uw are the air and water

pressures, respectively) is larger than P ,ˆc then air in- vades the pore.2_{Since soil has pores with various sizes,}

the water in the largest pores is displaced first followed by smaller pores. This leads to a macroscopic model of the soil–water characteristic curve (or the capillary pressure–saturation relationship), as discussed in Sec- tion 7.11.

If the water surrounding the soil particles remains continuous [termed the ‘‘funicular’’ regime by Bear (1972)], the interparticle force acting on a particle with radius r can be estimated from

2_{It is often assumed that u}

a⫽ 0 (for gauge pressure) or 1 atm (for

absolute pressure). However, this may not be true in cases such as rapid water infiltration when air in the pores cannot escape or the air boundary is completely blocked.

Capillary Tube Representing a Pore 2rp dc θ (a) (b) ua u_w Pc=ρwgdc= _r p ^ 2σ_awcosθ

Figure 7.3 Capillary tube concept for air entry estimation: (a) capillary tube and (b) bundle of capillary tubes to represent soil pores with different sizes.

2

2r aw cos

2 _ˆ

Fc⫽ r Pc⫽ (7.10)

rp

where rp is the size of the pore into which the air has

entered. Since the fluid acts like a membrane with neg- ative pressure, this force contributes directly to the skeletal forces like the water pressure as shown in Fig. 7.4a.

As the soil continues to dry, the water phase be- comes disconnected and remains in the form of me- nisci or liquid bridges at the interparticle contacts [termed the ‘‘pendular’’ regime by Bear (1972)]. The curved air–water interface produces a pore water ten- sion, which, in turn, generates interparticle compres- sive forces. The force only acts at particle contacts in contrast to the funicular regime, as shown in Fig. 7.4b. The interparticle force generally depends on the sep- aration between the two particles, the radius of the liq- uid bridge, interfacial tension, and contact angle (Lian et al., 1993). Once the water phase becomes discontin- uous, evaporation and condensation are the primary mechanisms of water transfer. Hence, the humidity of the gas phase and the temperature affect the water va- por pressure at the surface of water menisci, which in turn influences the air pressure ua.

7.5 INTERGRANULAR PRESSURE

Several different interparticle forces were described in the previous section. Quantitative expression of the in-

teractions of all these forces in a soil is beyond the present state of knowledge. Nonetheless, their exis- tence bears directly on the magnitude of intergranular pressure and the relationship between intergranular

pressure and effective stress as defined by  ⫽

 ⫺u.

A simplified equation for the intergranular stress in a soil may be developed in the following way. Figure 7.5 shows a horizontal surface through a soil at some depth. Since the stress conditions at contact points, rather than within particles, are of primary concern, a wavy surface that passes through contact points (Fig. 7.5a) is of interest. The proportion of the total wavy surface area that is comprised of intergrain contact area is very small (Fig. 7.5c).

The two particles in Fig. 7.5 that contact at point A are shown in Fig. 7.6, along with the forces that act in a vertical direction. Complete saturation is assumed. Vertical equilibrium across wavy surface x–x is con- sidered.3 _{The effective area of interparticle contact is} ac; its average value along the wavy surface equals the total mineral contact area along the surface divided by the number of interparticle contacts. Define area a as

3_{Note that only vertical forces at the contact are considered in this}

simplified analysis. It is evident, however, that applied boundary nor- mal and shear stresses each induce both normal and shear forces at interparticle contacts. These forces contribute both to the develop- ment of soil strength and resistance to compression and to the slip- ping and sliding of particles relative to each other. These interparticle movements are central to compression, shear deformations, and creep as discussed in Chapters 10, 11, and 12.

INTERGRANULAR PRESSURE 179

(a) Soil Particles

Continuous Water Film

Negative pore pressure acts all around the particles

(b)

Suction forces act only at particle contacts and the magnitude of the forces depends on the size of liquid bridges. Liquid Bridges Soil Particles Interparticle Forces Pores of Radius r_pFilled with Air

Air

Figure 7.4 Microscopic water–soil interaction in unsaturated soils: (a) funicular regime and (b) pendular regime.

Figure 7.6 Forces acting on interparticle contact A.

Figure 7.5 Surfaces through a soil mass.

the average total cross-sectional area along a horizontal plane served by the contact. It equals the total hori- zontal area divided by the number of interparticle con- tacts along the wavy surface. The forces acting on area

a in Fig. 7.6 are:

1. a, the force transmitted by the applied stress, which includes externally applied forces and body weight from the soil above.

2. u(a ⫺ ac), the force carried by the hydrostatic pressure u. Because a⬎⬎acand acis very small,

the force may be taken as ua. Long-range, double-layer repulsions are included in ua. 3. A(a ⫺ ac) ⬇ Aa, the force caused by the long-

range attractive stress A, that is, van der Waals and electrostatic attractions.

4. Aac, the force developed by the short-range at- tractive stress A, resulting from primary valence (chemical) bonding and cementation.

5. Cac, the intergranular contact reaction that is gen-

erated by hydration and Born repulsion. Vertical equilibrium of forces requires that

a ⫹Aa⫹ Aac⫽ua ⫹Cac (7.11)

Division of all terms by a converts the forces to stresses per unit area of cross section,

ac

 ⫽ (C⫺ A) ⫹ u⫺ A (7.12)

a

The term (C⫺ A)ac/a represents the net force across

the contact divided by the total cross-sectional area (soil plus water) that is served by the contact. In other words, it is the intergrain force divided by the gross area or the intergranular pressure in common soil me- chanics usage. Designation of this term by i gives

 ⫽  ⫹i A⫺ u (7.13)

Equations analogous to Eqs. (7.11), (7.12), and (7.13) can be developed for the case of a partly saturated soil. To do so requires consideration of the pressures in the water uw and in the air ua and the proportions of area a contributed by water awand by air aa with the con-

dition that

aw⫹aa ⫽ a i.e., ac→ 0

The resulting equation is

aw

 ⫽  ⫹i A⫺ ua⫺ (uw⫺ u )a (7.14)

a

In the absence of significant long-range attractions, this equation is similar to that proposed by Bishop (1960) for partially saturated soils

 ⫽  ⫺i ua⫹ (ua ⫺ u )w (7.15)

where ⫽aw/ a. Although it is clear that for a dry soil

 ⫽0, and for a saturated soil ⫽1.0, the usefulness of Eq. (7.15) has been limited in practice because of uncertainties about  for intermediate degrees of sat- uration. Further discussion of the effective stress con- cept for unsaturated soils is given in Section 7.12.

Limiting the discussion to saturated soils, two ques- tions arise:

1. How does the intergranular pressure i relate to the effective stress as defined for most analyses, that is, ⫽  ⫺ u?

2. How does the intergranular pressure i relate to the measured quantity,m⫽ ⫺u0, that is taken

as the effective stress, recalling (Section 7.2) that pore pressure can only be measured at points out- side the true interparticle zone?

Answers to these questions require a more detailed consideration of the meaning of fluid pressures in soils.

7.6 WATER PRESSURES AND POTENTIALS

Pressures in the pore fluid of a soil can be expressed

In document Guía para el solicitante de gtld. Version (página 70-75)