# Mechanics of Unsaturated Soils for Agricultural Applications

Authored by: Rainer Horn , Stephan Peth

# Handbook of Soil Sciences Properties and Processes

Print publication date:  November  2011
Online publication date:  November  2011

Print ISBN: 9781439803059
eBook ISBN: 9781439803066

10.1201/b11267-4

#### Abstract

Soils undergo intensive changes in their physical, chemical, and biological properties during natural soil development and as a result of anthropogenic processes such as plowing, sealing, erosion by wind and water, amelioration, excavation, and reclamation of devastated land. In agriculture, soil compaction as well as soil erosion by wind and water are classified as the most harmful processes. Irrespective of land-use systems, soil deformation as the sum of soil compaction and shear processes leads to numerous environmental changes affecting soil quality for crop production, soil biodiversity, filtering and buffering functions of soils, soil-water household components, trace gas emissions, soil erosion and nutrient export, and related off-site effects (Figure 3.1). Figure 3.1 Summary of the effects of stress application on soil properties and functions. (Translated and slightly modified from van der Ploeg, R.R., W. Ehlers, and R. Horn. <a href="#ref3_142">2006. Schwerlast auf dem Acker</a>. Spektrum der Wissenschaft August, 80–88. ISSN: 0170-2971.)

#### 3.1  Introduction

Soils undergo intensive changes in their physical, chemical, and biological properties during natural soil development and as a result of anthropogenic processes such as plowing, sealing, erosion by wind and water, amelioration, excavation, and reclamation of devastated land. In agriculture, soil compaction as well as soil erosion by wind and water are classified as the most harmful processes. Irrespective of land-use systems, soil deformation as the sum of soil compaction and shear processes leads to numerous environmental changes affecting soil quality for crop production, soil biodiversity, filtering and buffering functions of soils, soil-water household components, trace gas emissions, soil erosion and nutrient export, and related off-site effects (Figure 3.1).

Figure 3.1   Summary of the effects of stress application on soil properties and functions. (Translated and slightly modified from van der Ploeg, R.R., W. Ehlers, and R. Horn. 2006. Schwerlast auf dem Acker. Spektrum der Wissenschaft August, 80–88. ISSN: 0170-2971.)

An additional threat concerning soil degradation is caused by tillage erosion as the actual soil loss can exceed by far 20 Mg ha−1 and per tillage operation. This effect is the more pronounced, the drier the soil, the weaker the aggregates, the higher the dynamic energy input by agricultural machinery, and the greater the field size (Karlen, 2004; Reicosky et al., 2005; da Silva et al., 2006; Van Oost et al., 2006). The kinetic energy applied during seedbed preparation under dry conditions alone already results in an increased mass transport by wind. Additionally, the unproductive water loss due to evaporation from an increased accessible pore and particle surface and the enhanced organic-matter decomposition due to tillage both contribute to global change problems (Lal, 2005). Soil creep, mudflow after rainstorms, and intensified surface-water runoff are due to reduced/prevented vertical water infiltration in soils generating pronounced lateral fluxes of water, solids, and nutrients. The loss of shear strength, defined by the angle of internal friction and cohesion, of near-saturated soils subject to buoyancy forces that cause such lateral fluxes produces tremendous damage to the landscape and human beings. Related economic impacts are estimated to exceed billions of euros worldwide (van den Akker et al., 1998). In addition to productivity losses of soils in farming regions, the loss in biodiversity and the effects on global changes by modifying physicochemical processes result in the emission of greenhouse gases and the mobilization of heavy metals in soils due to redox reactions, which in turn cause groundwater pollution threats.

If, on the other hand, the tilled soil dries out, transport by wind may occur, resulting in severe reductions in potential site productivity. Furthermore, the preparation of a seedbed leads to abrupt changes in the transport of gas, water, ions, and heat between the tilled and deeper soil layers (Boone and Veen, 1994; Ball et al., 2000; Lipiec and Hatano, 2003). This is especially true concerning preferential flow through structured soils (Hendrickx and Flury, 2001). When it comes to the quantification of soil deformation with respect to the induced changes of physical functions, again the dynamic nature of soil loading processes during field traffic must be considered, where loads are applied mostly short term and where multiple loading events may cumulatively add to the total compression, although the statically determined precompression stresses are not exceeded and should result in no further compression.

Such anthropogenic changes make clear that the discussion of soil process dynamics within the soil profile is most relevant. Some of the interactions between soil structure, water status, and aeration of structured soils in relation to root growth and compressibility of arable land have been described by Emerson et al. (1978). In current discussions on sustainability, soil compaction is repeatedly mentioned as one of the main threats to agriculture, which should be avoided (Soane and van Ouwerkerk, 1994; Pagliai and Jones, 2002; Horn et al., 2006; Toth et al., 2008).

The aim of this chapter is to analyze the relationships between stress, strain, and strength as well as the consequences of soil deformation for physical and physicochemical properties. It also includes a description of well-established and new methods for measuring mechanical properties of soils and introduces possibilities for modeling soil strength and stress–strain relationships from the micro- to the macroscale.

#### 3.2.1.1  Definitions

Before discussing the methods of field and laboratory stress measurement as well as factors influencing compaction, one needs to differentiate between several terms used to define compressive properties. These definitions have been taken from Fredlund and Rahardjo (1993), Hartge and Horn (1999), Parry (2004), and McCarthy (2007).

#### 3.2.1.2  Mechanical Stresses

Stress is defined as force per area within a solid body. Stress can be induced by external or internal forces, which lead, if the body is nonrigid, to a change in the body’s volume and/or shape expressed as deformations or strains. The mechanical behavior of a soil can therefore be characterized by its stress–strain relationships.

Strength is typically referred to as the maximum amount of stress a solid material can withstand before it fails; thus, exceeding soil strength results in soil failure or yield. Strength depends on internal parameters such as particle-size distribution, type of clay minerals, nature and amount of adsorbed cations, content and type of organic matter, aggregation induced by swelling and shrinkage, stabilization by roots and humic substances, bulk density, pore-size distribution and pore continuity of the bulk soil and single aggregates, water content, and/or matric potential (Horn, 1981).

#### 3.2.1.2.1  Stress State

The number of stress state variables required to define the stress state depends primarily upon the number of phases involved. The effective stress σ’ can be defined as a stress variable for saturated conditions (Terzaghi, 1936) and is the difference between the total stress (σ) and the neutral stress (u w), which is equal to the pore water pressure:

3.1 $σ ′ = σ − u w ,$

where

• σ′ is transmitted by the solid phase
• u w is transmitted by the liquid phase, respectively

In unsaturated soil, stresses are transmitted by the solid, liquid, and gaseous phases. Thus, Equation 3.1 becomes (Bishop, 1959)

3.2 $σ ′ = ( σ − u a ) + χ ( u a − u w ) ,$

where

• u a and u w are pore air and water pressures, respectively
• χ is a factor that is often assumed to be directly related to the degree of saturation (S r)

At saturation (u w = 0 kPa), χ = 1, while at u w = −106 kPa, χ = 0.

For sandy, less compressible, and nonaggregated soils, χ can be estimated as

3.3 $χ = 0.22 + 0.78 S r .$

For silty and clayey soils, the values of the parameters in Equations 3.1 and 3.2 depend on soil aggregation, pore arrangement and strength, and hydraulic properties. Thus, the material function of the components in structured soils is only valid as long as the internal soil strength is not exceeded by the externally applied stresses. It changes if, for example, aggregates are destroyed during soil deformation and the structural properties reduce to those that depend merely on texture.

The extent of soil deformation can be described by stress–strain relationships and by their relative proportions. In the absence of gravitational and other applied forces, effective stresses in three-phase soil systems can be expressed as a tensor containing three normal stresses and six shearing components; assuming symmetry of the stress tensor, the shear stresses are reduced to three independent components. Therefore, three normal stress (σ x , σ y , σ z ) and three shear stress (τ xy , τ xz , τ yz ) components must be determined to fully define the stress state at every location in a solid body (Nichols et al., 1987; Horn et al., 1992). For an unsaturated soil, the stress state can be described completely by a symmetric matrix, which can be written as two independent stress tensors (Fredlund and Rahardjo, 1993):

3.4 $[ σ x − u a τ y x τ z x τ x y σ y − u a τ z y τ x z τ y z σ z − u a ]$

and

3.5 $[ ( u a − u w ) 0 0 0 ( u a − u w ) 0 0 0 ( u a − u w ) ] .$

A graphical representation of the stress state variables and the action of water menisci at intergranular contacts in an unsaturated soil is provided in Figure 3.2.

Figure 3.2   (a) Stress state variables for an unsaturated soil. (b) Schematic sketch of water menisci at interparticle/interaggregate contacts illustrating the effect of pore water pressure (u w) on intergranular tension stresses.

(Supplemented after Fredlund, D.G., and H. Rahardjo. 1993. Soil mechanics for unsaturated soils. John Wiley & Sons, New York.)

For symmetric tensors, it is always possible to find a coordinate system in which the shear components become 0 and the tensor becomes diagonal. In this principal axis system and assuming that the pore air pressure u a = atmospheric pressure, the stress tensor in Equation 3.4 simplifies to

3.6 $[ σ 1 0 0 0 σ 2 0 0 0 σ 3 ]$

with principal stresses σ1, σ2, and σ3 acting on the solid. For a single-valued characterization of the stress state, two invariants of the stress matrix are often used—the mean normal stress (MNS) and the octahedral shear stress (OCTSS; Koolen, 1994):

3.7 $MNS = 1 3 ( σ 1 + σ 2 + σ 3 ) ,$
3.8 $OCTSS = 1 3 ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 .$

#### 3.2.1.2.2  Stress Propagation

Each applied force per contact area is transmitted into the soil in three dimensions and can alter the physical, chemical, and biological properties of the soil (e.g., water infiltration, rootability) if the internal mechanical strength is exceeded. Stress propagation theories are rather old and have been often modified and adapted to in situ situations. The fundamental theory of Boussinesq (1885) is only valid for completely elastic material, while Fröhlich (1934) and Soehne (1958) included elastoplastic properties through the introduction of concentration factor values (υ k). More comprehensive descriptions of these models are given by Koolen and Kuipers (1983), Bailey et al. (1986), Johnson and Bailey (1990), and Bailey et al. (1992). Horn et al. (1989) introduced precompression stress (P v )-dependent values for the concentration factor, which are smaller in the recompression stress range, while they increase in the virgin compression stress range. The latter can be explained by the plastic deformation behavior, which causes a deeper stress transmission closer to the perpendicular line.

#### 3.2.1.3  Hydraulic Stresses

The effective stress (Equation 3.1) defines the forces per given area, which can stabilize the soil particles against any kind of soil deformation. The hydraulic stress component (u w) can either be negative (concave water menisci) or positive (convex water menisci). In case of convex menisci, it can result in weakening of the total soil system, especially when shear forces are applied. In case of pure compression under saturated conditions, however, effective and hence compressive stresses are reduced while the neutral stress (water pressure) bears part or even the entire externally applied load. In most cases, air pressure (u a) in Equation 3.2 is ignored, assuming that gas pressure in soil during loading is in equilibrium with atmospheric conditions.

If, on the other hand, hydraulic stresses become negative (=suction), the contractive forces even increase the effective stress (Figure 3.2). In addition to capillary forces associated with water, salt effects (i.e., water potential = sum of matric and osmotic potential) and hydrophobic substances can also increase soil strength by altering wetting angle, cohesion, internal friction, as well as elastic and viscous properties of the soil (Barré and Hallett, 2009; Markgraf and Horn, in press).

Hydraulic stresses result in shrinkage in almost all soils. In an initially homogenized state, soils undergo proportional and thereafter residual shrinkage during the first drying phase. Depending on the history of the formerly applied hydraulic and mechanical stresses, the shrinkage curve pattern, however, shifts and it must be now differentiated between structural shrinkage (=structural rigidity), virgin, and residual shrinkage behavior (Groenevelt and Grant, 2001; Braudeau et al., 2004; Peng and Horn, 2005). Structural shrinkage defines the internal soil strength caused by deformation related to capillary forces. Structural shrinkage is more pronounced, the more often and intensive soil dries out. Horn (1994a), for example, described the effect of the drying frequency and intensity on shrinkage behavior and pointed out that soil strength increases the more often and longer contracting water menisci occur while only a few but very intense drying processes are mostly not capable of rearranging particles to form a rigid structure with smaller entropy (Horn and Dexter, 1989).

The power of capillary forces and their strengthening potential for soil architectures can be demonstrated by a simple computation. Given the mass of the earth (~6 × 1024 kg) and the area of, for example, Germany (~360,000 km2) of which 50% is arable soil (=180,000 km2) and assuming a thickness of the plowed topsoil of 30 cm with a hypothetical specific surface area of 800 m2 g−1 (for simplicity we assume the soil to consist of smectitic clay and 2%–4% of organic carbon), we would end up with a total surface area of ca. 6.5 × 1019 m2. If we furthermore presume that water menisci at a potential of pF 4 would be effective over the total surface (χ → 1, assuming a very high negative air-entry pressure), then the sum of menisci forces acting within the topmost 30 cm of arable soils in Germany could carry the total mass of the earth.

#### 3.2.1.4  Stress Coupling

Based on the findings by Baumgartl (2003), who described the similarity of mechanical stress–strain curves and shrinkage curves resulting from hydraulic stresses (Figure 3.3), Peng and Horn (2008) proved that the link between mechanical and hydraulic “pre”stresses results in nearly identical changes in shrinkage curves. Additionally, Baumgartl (2003) explained the differences between mechanically induced collapse and matric potential-dependent shrinkage patterns by the dimensionless χ factor, which, however, is difficult to determine experimentally. Accounting for the interactions between mechanical and hydraulic processes is very important to understand the deformation behavior of unsaturated soils. Nevertheless, their description is inherently complex involving effective stress theory, loading rate dependency, and nonlinear hysteretic coupled transport and stress–strain functions.

Figure 3.3   Stress–strain relationships showing the similarity between compaction (right ordinate) and shrinkage curves (left ordinate).

(Modified from Baumgartl, T. 2003. Koppelung von mechanischen und hydraulischen Bodenzustandsfunktionen zur Bestimmung und Modellierung von Zugspannungen und Volumenänderungen in porösen Medien. Habilitation Thesis. Schriftenreihe Inst. Pflanzenernaehrung und Bodenkunde, Germany, No. 62, 133 pp. ISSN: 0933-680X.)

According to Richards and Peth (2009), mechanical and hydraulic processes can be coupled by treating the initially independent load-deformation and transport processes by an incremental approach as shown in Figure 3.4. This coupling is most important in modeling the interaction between water flow and deformation/failure as, for example, encountered in swelling soils, landslides on slopes during rainstorms, and consolidation of high water content soils. In a stress-coupling routine, changes in pore water pressure (h) are calculated as a function of changes in stress (σ) or strain (ε), and in a moisture-coupling routine, changes in stress or strain are calculated as a result of pore water pressure changes related to both water transport and load deformation. A detailed mathematical description of the finite element model (FEM) used for mechanical and hydraulic stress coupling and the underlying constitutive relations and material parameters is given in Richards and Peth (2009).

Figure 3.4   Conceptual framework for modeling coupled soil mechanical and hydraulic processes. The stress-coupling routine models pore water pressure changes (Δh) as a function of stress (σ) or strain (ε0). The moisture-coupling routine models strain changes (Δε0) as a function of pore water pressure (h) and effective stress (σ′).

(From Richards, B.G., and S. Peth. 2009. Modelling soil physical behaviour with particular reference to soil science. Soil Till. Res. 102:216–224.)

#### 3.2.1.5  Soil Strength Changes due to Stress-Dependent Effects on Hydraulic Properties

Figure 3.5   Changes in shear stress, sample height, and pore water pressure during shearing. The samples were predried to −6 kPa and pre-consolidated prior shearing. Upon shearing at the same applied stress, the sample further consolidated (height change) and pore water pressure increased reaching even positive values during shearing. After normal and shear stress were removed, pore water pressure decreased to a final value more negative than the initial pore water pressure.

(From Horn, R., T. Baumgartl, R. Kayser, and S. Baasch. 1995. Effect of aggregate strength on changes in strength and stress distribution in structured bulk soils, p. 31–52. In K.H. Hartge and R. Stewart (eds.) Soil structure: Its development and function. CRC Press, Boca Raton, FL.)

#### 3.2.2  Strain Theory

Mechanical processes in soil are generally stress dependent. All stresses exceeding the internal soil strength result in failure, which can be defined either by an irreversible plastic volume decrease (=soil compaction) or by a rearrangement of soil particles at a constant bulk soil volume (=shearing). Analogous to stress, strain can be described by normal (ε x , ε y , ε z ) and shear strain (ε xy , ε xz , ε yz ) components, which are also described completely by a symmetric strain tensor:

3.9 $[ ε x ε x y ε x z ε x y ε y ε y z ε x z ε y z ε z ] .$

In the corresponding principal axis system, the strain tensor reduces to

3.10 $[ ε 1 0 0 0 ε 2 0 0 0 ε 3 ] .$

The volumetric strain (εvol) can be calculated as

3.11 $ε vol = ε 1 + ε 2 + ε 3 = ε x + ε y + ε z .$

Note that the tensor trace is invariant under coordinate transformations. Furthermore, the strain components and their proportions depend on internal and external parameters and require the determination of all components in a 3D volume.

#### 3.2.2.1  Volumetric Deformation

Compression refers to a process that describes the increase in soil mass per unit volume (increase in bulk density) under an externally applied load or under changes in internal pore water pressure. Considering the soil as a continuum, soil movement can be described as a translation field (d) whose local properties are usually characterized by three components of its spatial derivative:

3.12 $∇ ⋅ d → = [ ∂ ∂ x d x ∂ ∂ x d y ∂ ∂ x d z ∂ ∂ y d x ∂ ∂ y d y ∂ ∂ y d z ∂ ∂ z d x ∂ ∂ z d y ∂ ∂ z d z ] .$

Every change in stress state therefore results in soil deformation. This divergence (volumetric strain) can be expressed as the divergence of the deformation field:

3.13 $ε vol = div ( d → ) = ∇ ⋅ d → = tr ( ∇ ⋅ d → ) = ∂ ∂ x d x + ∂ ∂ y d y + ∂ ∂ z d z ,$

and results in volumetric compression or extension. The terms of the right-hand side of Equation 3.13 show the corresponding strains in x, y, and z directions and how they are related to respective deformation components.

#### 3.2.2.2  Shear Deformation

Shearing a soil volume always leads to a pronounced volume constant rearrangement of particles often resulting in significant changes in pore functions by decreasing pore connectivity or continuity and changes in interparticle strength due to a loss of cohesion (Figure 3.6). The stress/strain at which the interparticle strength is lost and the soil begins to fail is referred to as yield stress/strain. The resistance of a soil toward shear deformation is described by shear moduli (G), which will be explained in more detail in Section 3.3 (rheometrical parameters).

Figure 3.6   Schematic sketch of the effect of shear displacement on pore continuity (b) in comparison with an “unstressed” pore system (a).

#### 3.2.3  Stress–Strain Processes

Generally, mechanical processes in soils are described by the stress/strain equation:

3.14 $[ σ x ( x → , t ) τ x y ( x → , t ) τ x z ( x → , t ) τ x y ( x → , t ) σ y ( x → , t ) τ y z ( x → , t ) τ x z ( x → , t ) τ y z ( x → , t ) σ z ( x → , t ) ] = f [ ε x ( x → , t ) ε x y ( x → , t ) ε x z ( x → , t ) ε x y ( x → , t ) ε y ( x → , t ) ε y z ( x → , t ) ε x z ( x → , t ) ε y z ( x → , t ) ε z ( x → , t ) ] ,$

where f defines the soil-specific material properties. In the case of a linear elastic isotropic material, the tensor notation reads as follows:

3.15 $[ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] = E 1 + v [ ε x − ε m ε x y ε x z ε x y ε y − ε m ε y z ε x z ε y z ε z − ε m ] + E 3 ( 1 − 2 v ) [ ε m 0 0 0 ε m 0 0 0 ε m ]$

where

• E = Δσ/Δε (Young’s modulus or modulus of elasticity)
• ν is the Poisson’s ratio

The first tensor on the right-hand side of Equation 3.15 describes the volume constant shear deformation and the second tensor the volumetric strain with the mean volume strain ε m defined as

3.16 $ε m = 1 3 ( ε x + ε y + ε z ) = 1 3 ε vol .$

Further details on the notions of stress–strain relationships can be found in Fredlund and Rahardjo (1993) and Warrick (2002).

To describe nonlinear elastic–plastic soil behavior as a function of stress, soil–water suction, and stress history, Richards (1978) introduced, based on the analysis of experimental data, a constitutive model, where the mechanical material properties are defined as hyperbolic functions allowing to treat soil as nonlinear and irreversible hysteretic material. The stress-dependent mechanical material properties are defined based on standard soil mechanical material parameters (bulk modulus, shear modulus, and Mohr–Coulomb failure criteria), which can be readily derived from soil tests (e.g., consolidation test, shear test, triaxial test; Section 3.3.4). Bulk modulus K and shear modulus G characterize the stiffness and shear strength of the soil, respectively. The basic relationships for the two simple stress-dependent moduli are (Richards and Peth, 2009)

3.17 $K = k 1 σ m n + k 2 h m + k 0$

and

3.18 $G = ( g 1 σ m p + g 2 h r ) ( 1 − ( τ τ f ) q ) + g 0 ,$

where

• σm is the previous maximum of the mean normal stress ((σ1 + σ2 + σ3)/3)
• h is the soil water suction
• τ is the shear stress
• τf is the yield stress according to the employed yield criteria

The shape of the material functions is further defined by empirical material constants k 0–2, g 0–2, n, m, p, r, and q, which are derived directly from fitting soil test data or by back-analyzing soil test results with an FEM approach (for further details see Section 3.7.3 and Richards and Peth (2009)).

#### 3.3  Soil Strength–Stability Indicators and Their Measurement

Soil strength and elasticity are mechanical key determinants for maintaining efficient functionality and ultimately quality of soils. Indicators describing and quantifying the mechanical resistance, deformation, and resilience of agricultural soils are important for several reasons. One is to assess if external mechanical loads applied to soils during field operations modify solid–void architectures and volume relationships through particle relocations and how far this may detrimentally affect pore functions. Another is to evaluate the penetrability of soil horizons and aggregates by roots and soil organisms while prospecting soils for water, nutrients, and food. A third reason is to estimate the risk for soil mass movements and soil erosion processes along slopes.

Soil scientists and civil engineers are likewise interested in measuring soil mechanical stability parameters, although with sometimes contrasting goals. In fact, many of the mechanical parameters that are commonly used in agricultural soil science originated in the field of civil engineering. On the other hand, civil engineers have adapted methods from agricultural soil science (e.g., Atterberg’s limits) and nowadays realize the importance of understanding soils as bioactive media where interacting physical and biological processes in turn have a strong influence on soil strength. However, recognizing soils as deformable (elastoplastic) multiphase systems with an internal structure, we need to define boundary-dependent strength/stability limits in order to estimate and ultimately predict deformation-induced changes in physical, chemical, and biological soil functions.

Most commonly used and well-established parameters characterizing the mechanical strength/stability of soils on multiple scales ranging from interparticle to pedon levels will be described in the following, including methods for their determination. Where appropriate, we supplemented this section with new techniques and methodologies in the field of soil mechanical research.

#### 3.3.1  Soil Consistency

The consistency of soils is a result of cohesive and adhesive forces acting between particles and between fluid-phase molecules and the solid surface, respectively. Together, they determine the interparticulate resistance to deformation, rupture, and flow. Soil wetness obviously has a strong influence on the kind of reaction against external forces (e.g., hard, friable, soft, plastic, sticky, and viscous). Atterberg (1911) has defined different classes of soil behavior based on soil wetness, which is a simple yet useful method to judge soil consistency and hence workability. The so-called Atterberg’s limits quantify the soil’s resistance to mechanical energy as a function of gravimetric water content (Θg):

#### 3.3.1.1  Liquid Limit

The liquid limit is determined with a Casagrande apparatus (Casagrande, 1932) where a homogenized moist sample is placed in a special bowl and a V-shaped trench is cut from the top to the bottom (Kézdi, 1974). The bowl with the sample is bounced up and down by a special crank onto a hard rubber block at a frequency of two strokes per second until the trench is closed at a length of <1 cm. The number of strokes when this condition is met is counted and the gravimetric water content determined on a subsample at the end of the test. The liquid limit is defined as the water content (Θg) where the test criteria are fulfilled at 25 strokes corresponding to a certain “standardized” amount of energy applied to the soil. The test is repeated for at least four water content values and the number of strokes plotted versus the water content on a semilogarithmic paper, finally producing a linear line. The liquid limit increases with clay and organic-matter contents, ionic strength, cation valency, and proportion of 2:1 type clay minerals in the soil.

#### 3.3.1.2  Plastic Limit

The plastic limit is defined as the water content (Θg) when homogenized soil samples begin to crack and crumble when rolled to a diameter <3 mm. It has practical implication in civil engineering and agriculture since it determines the lower critical water content at which soils become plastic and hence difficult to till or excavate.

#### 3.3.1.3  Plasticity Index

The difference between the water contents (Θg) at the liquid limit (LL) and plastic limit (PL) is the so-called plasticity index (PI = LL – PL), which is often used as an indicator for soil workability. With an increasing plasticity index, soils become more sensitive to plastic deformation. The higher the plasticity index the smaller is the water-content range at which soils can be tilled efficiently and trafficked without considerable soil deformation. The smaller is also the angle of internal friction over a wider range of water contents. Many attempts have been made to correlate PI to soil strength (Kézdi, 1974; Hartge and Horn, 1992; Kretschmer, 1997). In principle, this test only gives information on minimum strength values correlated with soil water content. It is furthermore limited to silty, loamy, and clayey soil samples while such correlation fails and is too insensitive for sandy material. Thus, especially coarse-textured soils cannot be classified by these data.

#### 3.3.1.4  Rheometrical Parameters

Advances in rheological research as an independent science have brought sophisticated techniques for quantifying flow behavior and deformation properties of viscoelastic substances (mainly fluids)—the most important parameters are shear modulus (G), plastic viscosity (ηp), and yield stress (τy).

Although first rotational viscometers (1888) and later rheometers (~1951) have been developed, early modern microprocessor-based systems, which are available since the 1980s, added more options for rheological testing including creep, relaxation, and oscillation tests. The methods are today well established and widely used in polymer, chemical, and material sciences as well as in the paper and food industries.

Despite its potential applicability, however, they are still not commonly employed in soil science research. One of the earliest contributions has been made though by Yasutomi and Sudo (1967) who used a low-frequency forced oscillation viscometer to study viscoelastic properties of soil. Later, Keren (1988) and Hesterberg and Page (1993) used a viscometer to investigate the influence of electrolyte composition and pH on rheological characteristics of clay mineral suspensions. Studies based on mineral suspensions certainly aid the understanding of the principle influence of surface chemistry on mechanical solid–interface interactions but have the drawback of ignoring the pertinent action of water as a lubricant during deformation processes. More recently, Ghezzehei and Or (2001) have used a rotational rheometer with a parallel-plate sensor system to determine these rheometrical parameters on natural soils and clay samples under more realistic field-moisture conditions. Higher water contents and high stress amplitudes resulted in lower viscosity of all investigated soils. Although this is not surprising, the authors could demonstrate that rheometry is a suitable and sensitive technique that enables us to quantify rheological properties of soils based on physically well-defined boundary conditions. Also Markgraf et al. (2006) investigated rheological characteristics of natural soil samples (Avdat loess, smectitic vertisol, and kaolinitic oxisol) using a parallel-plate rheometer. By conducting so-called amplitude sweep tests (oscillatory test with continuously increasing deformation), they were able to reveal differences in structural stabilities between the smectitic vertisol and the kaolinitic oxisol and the influence of electrolyte concentrations on rheological parameters of Avdat loess samples.

It would go beyond the scope of this chapter to describe the principles of rheometry in great detail. We therefore will only outline the basic test and analysis procedures in the following. Readers that are more interested in this technique will find useful staring points in Ghezzehei and Or (2001), Markgraf et al. (2006), and Mezger (2006).

The sample (usually a remolded paste at defined water content) is placed between two parallel plates (diameter 2.5 cm) with an adjustable gap size commonly between 2 and 4 mm (Figure 3.7). To set up the test, the upper plate is slightly pressed against the sample to ensure contact and time is given for stress dissipation. To avoid slip at the interface between the sample and the plates, both the upper and lower plates are furrowed. In an amplitude sweep test, the upper plate rotates (oscillatory) at a predefined frequency (e.g., 0.5 Hz) and variable amplitude beginning with very small shear deformations (corresponding to angular deflection) up to the maximum shear deformation where the angular deflection is equal to the gap spacing between the plates (e.g., 4 mm deflection at a gap spacing of 4 mm). Shear strain γ is expressed as percentage ranging from 0% (no deflection) to 100% (maximum deflection).

Figure 3.7   Rheometer with a parallel-plate measurement system. A remolded soil paste is placed between two furrowed plates by moving down the upper rotational shaft. (a) The upper plate rotates (arrows indicate the rotation in an oscillatory test) while the lower plate remains static. During oscillation, shear force (torque) and angular velocity are measured by sensors placed in the shaft of the upper plate. (b) Shows the sample after the test. A typical result of a rheometrical test is shown in Figure 3.8.

Figure 3.8   Schematic diagram showing a typical result of a rheo-metrical test. G′ = storage modulus (Pa) and G″ = loss modulus (Pa) correspond to the elastic and viscous deformation, respectively. Tan δ indicates the viscoelasticity of the sample: for tan δ = 0, deformation is ideal elastic; for tan δ = ∞, deformation is ideal viscous; for tan δ = 1 (yield point), deformation is equally viscous and elastic. Note that both moduli (G′ and G″) and strain are plotted on log scale.

A typical rheological parameter calculated from an amplitude sweep test is the shear modulus G = τ/γ, where τ is the shear stress and γ the shear strain. The shear modulus is distinguished into a storage modulus G′ = τAA cos δ and a loss modulus G″ = τAA sin δ representing the elastic and viscous part of deformation, respectively (Figure 3.8). The subscript A denotes the amplitude of the deflection that corresponds to the shear stress and shear deformation pair. The phase shift angle δ describes the viscous reaction of the sample to the prescribed shear deformation or shear stress. At δ = 0° (tan δ = 0), the sample deforms ideal elastic (system response is instantaneous, that is, stress is proportional to strain); at δ = 90° (tan δ = ∞), the sample shows ideal viscous behavior (system response is shifted by 1/4th of a full oscillation). In other words, the higher the phase shift angle δ the higher is the viscous and the lower is the elastic part of deformation, respectively. Hence, at small tan δ values, the storage modulus G′ exceeds the loss modulus G″ and vice versa for tan δ values >1. The magnitude of G′ or G″ in turn represents the resistance of the sample against deformation irrespective of whether it is elastic or viscous. However, at tan δ = 1, the yield point is reached indicating that the sample reacts predominantly viscous. For shear deformation exceeding the yield point, the sample becomes quickly more viscous (Figure 3.8). Tan δ is also referred to as loss factor since it indicates the fraction of mechanical energy introduced into the system that is lost in deformation. Note that possibilities for variations in boundary conditions during rheometer tests (stress or strain controlled, different stress–strain paths, stress relaxation, oscillation frequency) are numerous and choice of either methodology depends on the material under investigation. For soils, however, amplitude sweep tests have produced interpretable results and seem to be sensitive for a wide range of soil conditions (salt effects, water effects, and organic-matter effects) and allow expressed as yield stress a quantification of micromechanical soil strength (Markgraf et al., 2006; Markgraf and Horn, 2007).

#### 3.3.2  Bulk and Proctor Density

Bulk density is defined as mass per unit volume ignoring the internal arrangement of solids and voids, which limits its use to interpret strength properties and functions of structured soils. For example, two soils at the same bulk density, one aggregated and the other not, can have significantly different strengths. Employing bulk density for deriving site-specific soil properties should therefore in general be done with care (Horn and Kutilek, 2009).

Nevertheless, as far as remolded or freshly deposited soil material is concerned, bulk density may be a suitable indicator for the degree of compaction of such soils. Especially, in civil engineering, often the state of maximum compaction of a given material is of interest and the water content where compactibil-ity of a soil material is at optimum, that is, where least energy input results in maximum effect (=maximum density). To evaluate the soil moisture condition where compaction of homogenized soil samples is most efficient, Proctor tests are commonly employed (Proctor, 1933). Here, a remolded soil at various water contents (Θg) is filled into a container and compacted by dropping a weight (~2.5 kg) on the soil from a height of ~30 cm and for a defined number of strokes (usually 25). A series of tests are performed at different water contents and the resulting bulk density measured after each test. Bulk densities and corresponding water contents (Θg) are plotted on a graph, which represents the so-called Proctor curve. The water content where the compaction reaches a maximum is called the optimum water content (Θopt) and the maximum density is referred to as Proctor densityp). The coarser the soil sample, the higher ρp and the smaller Θopt. For example, sandy soils have higher ρp values than silty or clayey soils, while the latter require a higher water content (Θopt) to reach the optimum bulk density. Further, the more heterogeneous the grain-size distribution, the higher are ρp and Θopt. Strongly aggregated soils behave like coarser soil materials, that is, also here the proctor density is higher and the corresponding optimal water content lower than in less-aggregated soils. If aggregates themselves are destroyed during the test, ρp gets smaller while Θopt increases compared with completely homogenized material.

#### 3.3.3  Penetration Resistance

Soil resistance to any kind of deformation is determined by various types of penetrometers. The most simple is a thin metal rod (<0.5 cm diameter) with a defined tip shape. Frequently, the tip angle is 30° to simulate root properties or earthworm shapes. The penetrometer can be either pushed into the ground by the constant weight of a falling hammer or it can be driven by a motor at constant speed. The output readings can be either penetration depth per hammer stroke or depth stress depletions, which have to be overcome by the penetrating body in more sophisticated models. Penetration resistance is correlated with root growth, earthworm activity, and tillage effects. When penetration resistance exceeds 2 MPa, root growth is often reduced by half, while values >3 MPa often prevent root growth. Tillage may increase the critical stress value of a hardpan to >3.5 MPa depending on the nature of the pore system and the type of soil structure. Because the penetrometer needle is not as flexible as a root, which can choose planes of weakness for growth, penetrometer readings quantify resistances mainly in the vertical direction. In addition, the penetrometer readings only integrate impeding effects and cannot identify its causes. Despite a voluminous literature on the effect of increasing bulk density and water content on penetration resistance, extrapolation to land management situations is limited. Penetrometer readings can be used to create maps of derived properties (e.g., definition of sites with a given strength irrespective of its origin). Such data can be interpreted using statistical variograms, fractal analyses, or simply by stating that values are spatially different.

#### 3.3.4  Precompression Stress and Compressibility Indices

The compression of a soil means a change not only in total pore volume but is generally accompanied by a change in pore-size distribution including related soil functions (e.g., permeability, storage of water, aeration, gas exchange, heat diffusivity). Hence, the determination of the soil’s resistance to compaction and the magnitude of volume changes in soils exposed to external mechanical stresses are of paramount importance for preserving soil physical quality. A number of laboratory tests under precisely defined boundary conditions may be used to quantify soil strength and volume change behavior of soils (Table 3.1).

#### 3.3.4.1  Uniaxial Compression Test

Uniaxial compression tests are used to obtain approximate values of shear strength and settlement behavior of fine-grained soils at unconfined conditions. A vertical normal stress (σ1) is applied to the specimen (cylindrical or cubic sample), while the stresses on the planes mutually perpendicular to the σ1 direction (σ2 = σ3) are zero. The vertical stress is increased until the specimen fails, which determines the strength of the soil at the given water content. The results of the test can also be used to define the tensile strength of single soil aggregates (Section 3.3.6).

#### 3.3.4.2  Confined Compression Test

Soil stress–strain relationships of undisturbed structured and homogenized soils with respect to volume change behavior are quantified in confined compression tests (oedometer tests). In contrast to uniaxial compression tests, the stresses σ2 and σ3 are undefined (rigid wall of the soil cylinder) and the respective strains are defined and equal to zero. Both time- and load-dependent changes in soil deformation are measured (recorded as settlement). The slopes of the virgin compression and the recompression line in a void ratio e versus log σ plot are referred to as the compression index (C c = −δe/δ log σ) and swelling index (C s = −δe/δ log σ), respectively. The transition from the region of overconsolidation (recompression) to normal consolidation (virgin compression) is defined by the precompression stress, which separates the stress range where soil deformation is considered fully elastic (i.e., reversible) from the stress range where soil deformation is elastoplastic (i.e., partly irreversible). Precompression stress (σp) is most frequently determined based on the method by Casagrande (Casagrande, 1936); however, a number of alternative approaches are found in the literature. For a review of different σp estimates refer to Grozic et al. (2003).

#### 3.3.4.3  Triaxial Test

Triaxial tests mimic in situ stress distributions and are most realistic since horizontal and vertical stresses are defined and adjustable. However, they are also more difficult to conduct and more laborious than, for example, oedometer tests, where a high number of samples can be measured, and, hence, spatial heterogeneity of natural soils is better reflected. Triaxial tests on the other hand have the advantage that both volume and shape changes are determined simultaneously on the same sample and that failure occurs at the naturally weakest plane, which is important for deriving material parameters used in modeling approaches. In triaxial tests, undisturbed cylindrical soil samples are loaded with an increasing vertical principal stress (σ1), while the horizontal principal stresses (σ2 = σ3) are kept constant (equal to the cell pressure). Shear stresses occur in any plane other than those of the principal stresses. Shear parameters (cohesion and angle of internal friction) can be determined from the slope of Coulomb’s failure line embracing the Mohr circles at failure for various combinations of σ1 and σ2 = σ3 (e.g., Kézdi (1974), Mitchell (1993), Wulfsohn et al. (1998)). Due to aggregate deterioration and prevented drainage of excess soil water, the Mohr–Coulomb failure line is bent toward smaller slope values at higher mean normal stresses. However, the numbers of contact points, strength per contact point, and pore geometry also affect triaxial test results.

There are three types of triaxial tests: (1) In the consolidated drained (CD) test, the soil sample is equilibrated with the mean normal stresses prior to an increase in the vertical stress (σ1); the pore water drains off when the decrease in volume exceeds the air-filled pore space. Therefore, the applied stresses are assumed to be transmitted as effective stresses via the solid phase. However, Baumgartl (1991) found an additional change in the pore water pressure during extended CD triaxial tests depending on soil hydraulic properties. Thus, shear speed and low hydraulic conductivity, high tortuosity, and small hydraulic gradients further affect the drainage of excess soil water and the effective stresses (Horn et al., 1995). (2) In the consolidated undrained (CU) test, pore water is prevented from draining off the soil as vertical stress increases. Thus, high hydraulic gradients occur and the pore water reacts as a lubricant with a low surface tension. Thus, in the CU test (considering total stresses), shear parameters are much smaller and pore water pressure values are much greater than those in the CD test. (3) The highest neutral stresses and, therefore, the lowest shear stresses are measured in the unconsolidated undrained (UU) test, where neither the effective stresses nor the neutral stresses are equilibrated with the applied principal stresses at the beginning of the test. Thus, cohesion and angle of internal friction in terms of total stresses are strongly dependent on the compression and drainage conditions during deformation. In terms of strength of agricultural soils under traffic, texture, pore water pressure (in unsaturated conditions referred to as soil matric potential), nature of aggregates, and soil structure determine the preference for a particular test.

### Table 3.1   Methods of Determining Soil Strength

Method

Dimension

Derived

Soil Condition

Uniaxial compression

Pressure (Pa)

Homogenized soil, structured bulk soil, and single aggregates

Confined compression test

Pressure (Pa)

Precompression stress (Pa)

Homogenized soil, structured bulk soil

Triaxial test

Pressure (Pa)

Cohesion (Pa), angle of internal friction (°)

Homogenized soil, structured bulk soil, and single aggregates

Direct shear test

Pressure (Pa)

Cohesion (Pa), angle of internal friction (°)

Homogenized soil, structured bulk soil, and single aggregates

#### 3.3.4.4  Cyclic Compression Test

Figure 3.9   Compressibility as a function of repeated (“cyclic”) loading. The slope C n (cyclic compressibility index) denotes the sen-sitivity of the soil toward cyclic compression at the same magnitude of external load.

(After Peth, S., and R. Horn. 2006b. The mechanical behavior of structured and homogenized soil under repeated loading. J. Plant Nutr. Soil Sci./Z. Pflanzenernahr. Bodenkd. 169:401–410.)

3.19 $e ( N ) = − C n × log N + e u 1 .$

Here, e and N are the void ratio and number of loading cycles, respectively. The axis intercept $e u 1$ denotes the void ratio in the unloaded condition after the first loading event. Only the following loading events are considered to belong to the cyclic loading process. The slope C n in Equation 3.19 is referred to as cyclic compressibility and indicates the sensitivity of a soil toward cyclic compression. The higher C n the higher is the compression under repeated loading.

#### 3.3.5  Shear Strength Parameters

The shear strength parameters, cohesion (c), and angle of internal friction (ϕ), may be determined by triaxial tests as described above or in a direct shear test. During direct shear tests, the type and direction of the shear plane, which is assumed to be affected only by normal and shear stresses, are fixed. Normal stress is applied to the specimen in the vertical and shear stress in the horizontal direction. To determine the Mohr–Coulomb failure line, at least four to five samples need to be tested each with a different normal stress. The maximum shear resistance (τmax) is determined from a shear stress–displacement curve and plotted against the corresponding normal stress (σn). Plotting all pairs of τmax versus σn gives the Mohr–Coulomb failure line in which the slope and intercept are the angle of internal friction (ϕ) and cohesion (c), respectively. As in triaxial tests, cohesion and angle of internal friction are influenced by shear speed and drainage conditions for a given soil.

#### 3.3.6  Tensile Strength of Aggregates and Bulk Soil

Tensile strength of soils is a measure of the direct strength of interparticle bonds, which depend on chemical bonding, clay type and content, water potential (menisci forces), and organic matter. Also the internal geometry of particle arrangements and preexisting shrinkage cracks determine shear failure and crack propagation upon mechanical and hydraulic loads generating tensile forces. Tensile strength is an important indicator for the structural stability from the aggregate scale to the bulk soil level and a decisive factor for the preservation of physical quality of soils subject to mechanical stresses. The most widely used technique to measure tensile strength of soil aggregates is the crushing test. Soil aggregates are placed between two parallel plates in their most stable position (flat side downward) and the upper plate is loaded either with a weight (e.g., water dripping slowly into a bucket, Horn and Dexter, 1989, or by a load frame). The force when failure occurs is recorded and tensile strength is calculated according to Dexter and Kroesbergen (1985):

3.20 $Y = 0.576 × F d 2 ,$

where

• Y is the tensile strength
• F is the applied force
• d is the mean diameter of the aggregate given by
3.21 $d = ( d x + d y + d z ) 3$

with indices x, y, and z denoting the aggregate diameter of the longest, intermediate, and shortest axes, respectively.

Other indirect techniques have been proposed to investigate crack growth and propagation in soils subject to tensile forces. One method is based on approaches adapted from fracture mechanics where crack formation and propagation are measured by so-called deep notch three-point bending tests (Hallett and Newson, 2001, 2005). Also direct tensile tests have been suggested to study crack propagation and tensile strength of soils, for example, using bone-shaped bulk soil samples that are torn apart by tension forces perpendicular to a predefined crack (Junge et al., 2000).

#### 3.3.7  Other Methods to Determine Aggregate Stability

Aggregate stability may also be determined by wet sieving. Here, soil samples are prewetted to a given pore water pressure and then sieved through a set of sieves from 8 to 2 mm diameter. The difference between the aggregate-size distribution at the beginning and end of sieving under water for a given time is calculated as the mean weight diameter (MWD). This value is qualitatively related to aggregate strength and increases with increasing aggregate stability. The wettability of the solid interfaces has a pronounced effect on the test results in wet sieving and direct comparison with strength parameters derived from mechanical stress applications through polar loads is limited. Aggregate stability has also been measured by the application of ultrasonic dispersion techniques where aggregate stability can be expressed by the input of ultrasound energy resulting in the breakup of aggregates into smaller microaggregate units up to single particles (Raine and So, 1994). A comparison of different methods to determine aggregate stability is given by Baumgartl and Horn (1993).

#### 3.4  Effect of Soil Structure on Soil Strength and Deformation Processes

As mechanical stresses are applied, soil deformation occurs first at the weakest point(s) in the soil matrix and further increases in stress resulting in the formation of failure zones. The strength of the failure zone is equal to the energy required to create a new unit of surface area or to initiate a crack (Skidmore and Powers, 1982) and is called the apparent surface energy (Hadas, 1987). Consequently, soil stability against shear or tensile stresses is related to strength distribution in failure zones. In principle, soil structure will be stable if the applied stress is smaller than the strength of the failure zone, that is, if the bond strength at the points of contact exceeds the shear or tensile stresses generated by external loads. If the resisting forces are smaller than the active forces, stresses are in disequilibrium and hence soil deformation will occur to generate more contact points until stress equilibrium is reached again. Reorientation of particles is accompanied by a change in soil structure and consequently functions. In extreme stress situations (especially high shear stresses) or when soil is subject to mechanical loads in unfavorable moisture conditions (near the liquid limit), soil structure may be almost completely returned to an immature state (homogeneous soil). In contrast to shear stresses that lead to volume constant deformation, normal stresses result in volume change, that is, compression, which is plastic as soon as the soil stability is exceeded. In situ stress conditions during field operations (traffic, soil–tool interactions) are generally characterized by a combination of both shear and compressive stresses.

#### 3.4.1  Mechanical Properties of Homogenized Soils

If only mechanical properties of homogenized soils are compared, gravelly soils are less compressible than sandy or finer-textured soils (Horn, 1988), and any deviation of the particles from the spherical shape results in an increase in the shearing resistance and soil strength (Gudehus, 1981; Hartge and Horn, 1999). Ellies et al. (1995) found that the angle of internal friction for medium and coarse sands at comparable bulk density and water content but of different origins (quartz, basalt, volcanic ash, and shells) ranged from 25° to 50°. Soil strength also depends on the type and content of clay minerals and exchangeable cations. At the same bulk density and pore water pressure, compressibility of homogenized soil increases with clay content and decreases with soil organic-matter (SOM) content. Vice versa soils at the same clay content are more readily compressed at lower bulk density or higher moisture content. Cohesive forces between illite, smectite, and vermiculite are greater than for kaolinite, which in turn has a larger angle of internal friction because of its size and particle shape. Additionally, the higher the valency of the adsorbed cations and the higher the ionic strength of the soil solution, the greater is the shearing resistance under otherwise similar conditions (Yong and Warkentin, 1966; Mitchell, 1993).

Shearing resistance also increases depending on both the nature and content of SOM. In agricultural soils with the same physical and chemical properties, the mechanical parameters (precompression stress, angle of internal friction, and cohesion) are greater when the relative contents of carbohydrates, condensed lignin subunits, bound fatty acids, and aliphatic polymers are higher (Hempfling et al., 1990).

#### 3.4.2  Mechanical Properties of Aggregated Soils

Mechanical strength of structured soils depends on aggregation and history of hydraulic stresses (degree of maximum predrying), as well as the shape and internal arrangement of particles, microaggregates, and voids. At comparable grain-size distribution, bulk density, and pore water pressure, soil strength increases with aggregation (i.e., coherent < prismatic < blocky < subangular blocky < crumbly). In the case of a platy structure, the strength depends on the direction of shear forces relative to the preferred orientation of the particles. In the direction of the elongated axes of aligned particles, shear strength is lower than perpendicular to it. This means that not only mechanical stresses are anisotropic but also strength properties that in turn are a function of aggregation. However, both the magnitude of previous hydraulic stresses (predrying intensity) and their dynamic changes determine soil strength. Therefore, mechanical properties also depend on the frequency of swell/shrink and wet/dry events and on the actual pore water pressure at the time of loading. Soils become stronger when redried and rewetted and when pore water pressure gradients over longer distances promote particle movement and rearrangement until entropy is minimized (Semmel et al., 1990). The effect of pore water pressure on strength itself is governed by the theory of effective stresses in unsaturated soils (e.g., Bishop’s effective stress equation (Equation 3.2)). It has been extended from the early considerations of Terzaghi (Terzaghi, 1936) for saturated soils; however, its application remains difficult since the evaluation of the parameter × in Equation 3.2 is still complicated. It accounts for the tradeoff between pore water pressure (increases interparticle forces with curvature of menisci) and effective area of interparticle contact (when decreasing, the effect of pore water pressure is reduced). The interrelationship of pore water pressure and effective area of interparticle contact is structure dependent and hence also χ is related to structure and not as sometimes assumed purely a function of the degree of saturation. If the relative reduction in water-filled pores is smaller than the actual decrease in pore water pressure (more negative matric potential), soil strength increases while if the decrease in the effective surface of water menisci exceeds the decrease in matric potential (getting more negative), the effective stress of such drier soils gets absolutely smaller. In principle, soil strength is promoted by two different mechanisms: (1) the increase in strength results from an increase in the total number of contact points between single particles (i.e., an increase in effective stress) and (2) the increase in shear resistance per contact point (Hartge and Horn, 1984). Therefore, even if soil bulk densities are similar, strength properties may be quite different. Also freezing and thawing affect soil strength, because aggregates become either denser or they are destroyed by ice lens formation during freezing. The exfoliation process starts from the outer skin. Both effects are called soil curing, but result in completely different strength values and physical properties of the bulk soil (Horn, 1985b; Kay, 1990). Pedogenic effects on natural mechanical strength of three soils are illustrated in Figure 3.10. Luvisols derived from loess are characterized by clay illuviation, which results in strength decrease in the Al horizon and an increase in the Bt horizon due to aggregation. On the other hand, calcium precipitation in the Cca horizon of the mollisol also leads to a strength increase. In all three soils, the parent material (C horizon) was weakest although overburden pressure increases with soil depth suggesting the opposite should have been the case. This underlines the offset of soil strength parameters by pedogenic processes. On the other hand, anthropogenic processes such as annual plowing and tractor traffic may create very strong plow pan layers with precompression stress values similar to the contact pressure of a tractor tire or even higher (up to 300%) due to lug effects. The transition from the region of overconsolidation to the virgin compression line depends to a great extent on internal parameters and further on the hydraulic and mechanical stress history. In addition, strength decreases in the Ap horizon due to plowing and seedbed preparation can be observed until texture-dependent values are reached.

Figure 3.10   Precompression stress values for three soil profiles at soil matric potential of Ψm = −6kPa.

(Modified from Horn, R. 1994b. Stress transmission and recompaction in tilled and segmently disturbed soils under trafficking, p. 53–87. In N.S. Jayawardane and E. Stewart (eds.) Subsoil management techniques. Lewis Publishers, Boca Raton, FL.)

Pronounced strength decreases are also found in soil profiles that have been deep-loosened or deep-plowed (up to 60 cm) or loosened and/or mixed with other soil material. In the case of a partial deep loosening by a trenching slit plow (Reich et al., 1985; Blackwell et al., 1989), traffic must be restricted by the use of smaller machinery or controlled tracks perpendicular to the homogenized soil volume.

Soils with a well-developed vertical pore system are stronger than those with randomized or horizontal pore arrangements since elongated pores are more stable with their symmetry axis oriented in the direction of the major principle stress (σ1) than perpendicular to it. Therefore, vertical (bio) pores are less prone to compression than oblique or horizontally oriented pores and, thus, untilled or minimum tilled soils are stronger than conventionally tilled soils (Horn, 1986).

Each soil deformation requires air-filled pore space and sufficiently high hydraulic conductivity during compression to drain off the excess pore water. The smaller the hydraulic conductivity, the hydraulic potential gradient, and the pore continuity, the more stable is the soil especially during short-term loading. In sandy soils, this effect is small and initial settlement equals the total strain. With increasing clay content, however, the proportion of initial to primary and secondary consolidation is reduced and time-dependent soil settlement becomes important. This results in an increase in precompression stresses for short-term loading due to an increase in pore water pressure that dissipates only slowly depending on the local hydraulic conductivity. Such drainage effects on the consolidation process are smaller in more strongly aggregated soils and with coarser soil textures.

The development of structure generally results in increasing soil strength since the formation of secondary large pores is always accompanied by the development of denser aggregates. Although the aggregate assemblages carry the same stresses over fewer contact points, the aggregates themselves are more resistant to shear stresses. Strength differences may also result from changes in the angle of internal friction and cohesion as indicated in Figure 3.11 for a single aggregate, an undisturbed aggregate, and a homogenized bulk soil, respectively. More intense aggregation increases soil strength at comparable hydraulic stresses.

Figure 3.11   Changes in shear strength at constant soil matric potential Ψm = −6 kPa for various applied stress ranges for single aggregates, bulk soil, and homogenized material.

(Modified from Horn, R., T. Baumgartl, R. Kayser, and S. Baasch. 1995. Effect of aggregate strength on changes in strength and stress distribution in structured bulk soils, p. 31–52. In K.H. Hartge and R. Stewart (eds.) Soil structure: Its development and function. CRC Press, Boca Raton, FL.)

Angle of internal friction for a single aggregate (∼45°) is very high compared to bulk soil samples. However, aggregated bulk soil reveals higher friction angles than the homogenized soil at the same bulk density. This shear strength gain is lost when stresses exceed the overall stability of the aggregated sample leading to a reduction in the angle of internal friction to values similar for homogenized soil. When the maximum value of stress exceeds the maximum strength such that structure is homogenized, only texture-dependent properties remain.

#### 3.4.3  Soil Deformation under Dynamic Stresses

Standard mechanical stability indicators such as shear strength parameters (ϕ and c) and precompression stress (σp) are without dispute suitable for characterizing the mechanical strength/ stability of soils and to assess if significant irreversible volumetric and shear deformation under given stress conditions are likely to occur. Horn et al. (2005), for example, have derived stability maps on various scales down to farm level where sensitive and less sensitive areas are outlined, which could help farmers to avoid harmful subsoil compaction. We must, however, consider more in detail the repeated short-term loading because multiple loading events cumulatively add to the total compression, although concerning precompression stresses no further compression maybe expected. Zapf (1997), for example, argued that the increase in irreversible deformation of arable soil by agricultural machinery is not only related to the increasing mass of machines but also related to the increase in wheeling frequency. Peth and Horn (2006b) estimated that, based on the calculations of Olfe (1995), for an average-sized wheat-production farm in a time period of 5 years, the number of load repetitions may add up to 50 events for 85% of the field and up to 100 events for permanent wheeling tracks.

Figure 3.12   Changes in porosity for different soil textures (homogenized samples) due to cyclic loading with a load of 40 kPa at the same initial bulk density (1.62 g cm−3). Samples have been equilibrated to a standard matric potential of −6 kPa prior to the tests. Note that the change in porosity after the first loading-unloading event is omitted here (for sand = 1.28, for silty sand = 1.22, and for sandy loam = 2.12 vol%). Error bars indicate standard deviation (n = 14). Corresponding C n values are given in brackets (mean ± SD).

(From Peth, S., and R. Horn. 2006b. The mechanical behavior of structured and homogenized soil under repeated loading. J. Plant Nutr. Soil Sci./Z. Pflanzenernahr. Bodenkd. 169:401–410.)

Figure 3.13   Correlation between initial soil bulk density and cyclic compressibility coefficients (C n) of a haplic luvisol (Marienstein/ Germany) with silt loam texture for different tillage systems: CT1 and CT2 are two different conventionally tilled sites; CnT is conservation tillage.

(From Peth, S., and R. Horn. 2006b. The mechanical behavior of structured and homogenized soil under repeated loading. J. Plant Nutr. Soil Sci./Z. Pflanzenernahr. Bodenkd. 169:401–410.)

(After Peth, S., J. Rostek, A. Zink, A. Mordhorst, and R. Horn. 2010. Soil testing of dynamic deformation processes of arable soils. Soil Till. Res. 106:317–328.)

#### 3.5.1  Stress Application and Variations in the Contact Area

Any load applied at the surface is transmitted to the soil in three dimensions by the solid, liquid, and gas phases. If air permeability is high enough to allow immediate deformation of the air-filled pores, soil settlement is mainly affected by fluid flow. However, fluid flow may be delayed because changes in water content or pore water pressure depend on the hydraulic conductivity, gradient, and pore continuity. Thus, the intensity and form of pressure transmission are again affected by soil strength. In the following discussion, stress distribution in both homogenized soils and aggregated systems will be defined. Based on the theory of Boussinesq (1885) who solved the problem of a point load acting perpendicular on an infinite half-space consisting of homogeneous, isotropic, and linear elastic material, Fröhlich (1934) introduced a textural-dependent concentration factor (v k) to give more “flexibility” to the original Boussinesq solution for determining stress propagation in soils. Under saturated conditions, v k ranges between 3 for very strong material (which is defined as linear elastic and corresponds to the Boussinesq solution) and 9 for soft soil becoming higher with increasing clay content. In weak soils with high concentration factor, stresses are transmitted to greater depth but remain concentrated around the perpendicular line of the load center. On the other hand, in strong soils with low values of the concentration factor, stresses are transmitted more horizontally and shallower.

At the same contact pressure, stresses are transmitted deeper when the contact area is larger. Furthermore, stress distribution patterns in the soil are not only different for tire lugs and the intervening area but are also affected by the stiffness of the carcass (Horn et al., 1987). Thus, there are no well-defined equipotential stress lines in soils, but the concentration factor values must be related to precompression stresses in relation to applied stress and contact area for a given texture (DVWK, 1995). If additionally applied stresses do not exceed the internal soil strength defined by the precompression stress, the concentration factor is smaller as compared to the situation when the stresses result in a plastic deformation. The latter is defined by higher values (Berli et al., 2003).

The speed effect of wheeling, as well as the kind of stress application (tire, rubber belt tire-inflation pressure), affects the stress path at the soil surface and also causes enormous differences in the stress propagation inside the soil volume. It is well understood that at a given surface load, the stress propagation reaches the deeper the larger the contact area and that at a given contact area stresses are transmitted deeper for higher ground contact pressures (Peth and Horn, 2004; Figure 3.15a and b).

Figure 3.15   (a) Stress distribution under a tire with the same ground contact pressure of 100 kPa but increasing contact area (tire size increases from left to right) and (b) for the same tire size (100 cm diameter) but increasing ground contact pressure (σ0) due to higher wheel load, respectively. Vertical stresses σz (kPa) have been modeled with an FEM approach assuming rotational symmetry of the contact area (i.e., circular tire-soil contact shape). Note that only a part of the model domain is shown. The width of the model domain was 4.5 m to reduce boundary effects. The horizontal white-dashed line indicates the usual plowing depth of 30 cm. Mechanical properties for the topsoil and the subsoil were assumed to be the same. Following mechanical parameters were used for the simulation: cohesion c = 10 kPa, angle of internal friction ϕ = 20°, stress-dependent bulk modulus K ( σ ) kPa = 800 × σ m 0.5 + 50 , stress-dependent shear modulus G ( σ ) kPa = ( 270 × σ m 0.5 ) ( 1 − ( τ / τ f ) 0.37 ) + 50 , where the yield stress τf (σn) = c + σn tan ϕ. For further details on material functions and parameters, see Section 3.2.3 and Richards and Peth (2009).

When we consider the stress vector field during wheeling, the tangential redirection of stresses increases with stronger wheel slip and less rigid tire or rubber belt (Figure 3.16a and b). Keller (2004), Arvidsson and Keller (2007), Cui et al. (2007), and Keller et al. (2007a) proved the effect of various tire forms and tire-inflation pressure on soil stress distribution and not only showed the heterogeneity of the stress distribution but also quantified the effect of changes in the tire forms and inflation pressure on stress distribution. Burt et al. (1989) already described the enormous heterogeneity in the stress distribution within the contact area that differed up to 300% and was higher beneath the lugs and the lowest in between them close to the middle part of the area while due to carcass effects it reached extremely high values at the outer edge of the tire.

Figure 3.16   (a) Vertical and horizontal movement of a displacement sensor installed in situ at 10 cm depth during wheeling with a rubber belt tractor. (b) Modeled (FEM) distribution of shear/tensile stresses under a rubber belt tractor during wheeling. Note that material properties used for the FEM are not the same as for the site where soil displacement has been measured in situ (a). Simulated stress distribution has been conducted for a hypothetical case. Material properties and boundary conditions used in (b) were as follows: belt length = 240 cm, tractor weight = 12.6 Mg, ground contact pressure σ0 = 67 kPa, horizontal stress transmitted via the belt = 32.5 kPa, cohesion c = 10 kPa, angle of internal friction ϕ = 20°, stress-dependent bulk modulus K ( σ ) kPa = 800 × σ m 0.5 + 50 , stress-dependent shear modulus G ( σ ) kPa = ( 270 × σ m 0.5 ) ( 1 − ( τ / τ f ) 0.37 ) + 50 , where the yield stress τf (σn) = c + σn tan ϕ. For further details on material functions and parameters, see Section 3.2.3 and Richards and Peth (2009).

The question about the effect of traffic on stress distribution and its consequences on soil properties remains, even after more than three decades of intensive discussion, still controversial, which is most often caused by different scientific views or interests in the subject. It is to be understood that the machine industry is mostly interested in the development of more effective and more powerful machines resulting in heavier traffic with higher wheel load and wheel contact area. The benefits of new and smaller more intelligent self-steering GPS-controlled agricultural machines are still under debate, although it is a positive way to minimize the still increasing and deeper stress transmission into soils taking place when using conventional (heavier) machines. On the other hand, when soil scientists deal with traffic–soil interactions, they are mostly concerned about the effects of stress application on soil properties and the future effects for crop production, water infiltration, filtering and buffering, and groundwater recharge. Recently, enhanced greenhouse gas emissions have been more often associated with the interaction of soil stress and soil deformation, which results in changes in aeration and microbial activity. Legislation finally has to link both interests where stakeholders have to decide based on their administrative perspectives and given laws. The debate on soil use and soil protection has resulted in Germany in the German Soil Protection Law (BBodSchG, 1998), while at European level such a law is still under discussion. However, the liability varies to a great extent even in between different provinces within Germany and the definition of main soil parameters (indicators) is not finalized (Horn and Fleige, 2009).

#### 3.5.2.1  Internal Parameters

At a given bulk density and pore water pressure, applied stress at the soil surface will be transmitted deeper as silt and clay contents increase while stress attenuation will be greater in a smaller volume as the soil dries. For a given particle-size distribution, water content, and bulk density, stress attenuation will increase with increasing SOM content. Based on many stress distribution measurements in the field and in monoliths, a general correlation scheme involving texture, precompression stress, and tire contact area to derive the concentration factor v k has been used to approximately predict the stress distribution in soils (DVWK, 1995); for more information see also Horn and Fleige (2003).

#### 3.5.2.2  Structure Effects on Stress Attenuation

The effect of aggregation on stress distribution and its consequences for ecological parameters are well understood. Under in situ conditions for soils with the same internal parameters, stress attenuation is greater the more aggregated soils are. Concentration factor values expressed as precompression stress are smaller for better aggregated and drier soils (DVWK, 1995). Not only the pressure but also the size and shape of the contact area affect stress distribution in unsaturated structured soils.

Stress distribution calculations in a luvisol derived from loess under natural forest and its comparison with the identical computation under arable management show the effect of soil strength on stress attenuation (Figure 3.17a and b). Under natural forest, especially the clay-enriched Bt horizon with a prismatic–blocky structure (matric potential ∼ −30 kPa) attenuated the applied stresses of the tractor and both combine harvesters while the topsoil (Ah- and the clay-depleted Al horizons), respectively, with a crumbly or coherent structure was deformed.

Figure 3.17   Modeled stress distribution in a luvisol derived from loess under forest (b) and under conventional tillage (a). The concentration factors were defined according to the degree of aggregation.

Under arable conditions, approximately 1/4th of the pores within the top 100 cm were lost compared with the forest site, which also resulted in higher precompression stress at all depths except for the Ap horizon and a more complete stress attenuation apart from the homogenized seedbed. However, the first introduction of the heavier sugar beet harvesters caused a deeper and a more intense soil deformation while exceeding the corresponding actual soil strength (Peth et al., 2006). The effect of stress attenuation depending on soil structure and strength (as a function of depth) can be also derived from Figure 3.18 (Zink, 2009). The luvisol derived from loess under long-term conventional and conservation tillage management shows limited stress attenuation at given wheel loads with and without reduced inflation pressure, which causes irreversible soil deformation (stresses exceed the precompression stress).

Figure 3.18   Comparison of stress attenuation for different wheel loads and tire-inflation pressures in a luvisol derived from loess under conservation (a) and conventional tillage (b). Dashed and solid lines with symbols indicate stresses derived from SST (stress rate transducer) measurements; solid line without symbol indicates horizon-specific precompression stresses.

(From Zink, A. 2009. Bodenstabilität und Auswirkungen dynamischer Lasteinträge auf physikalische Eigenschaften von Ackerböden unter konservierender und konventioneller Bodenbearbeitung. Ph.D. Thesis. Schriftenreihe Institut für Pflanzenernährung und Bodenkunde, Germany, No. 84, 180 pp. ISSN: 0933-680X.)

Repeated short-term stress application results in an increase in the matric potential and a higher water saturation within the various horizons. Consequently, the repeated short-term loading not only weakens the soil structure by deformation but also pumps additional soil water from deeper depths (repeated elastic rebound creates a suction mechanism), which further enhances soil deformation. Semmel (1993) could prove that due to repeated wheeling-induced soil softening, horizontal minor stresses decreased while the major vertical stress increased during 50 repeated wheeling events with a 3.7 Mg tractor. The soil softening was also reflected by reduced soil strength and an increase in the concentration factor due to repeated wheeling. The same processes also occur in grassland ecosystems under arid conditions during sheep trampling. Krümmelbein et al. (2008) could show that in grassland soils, which are intensely trampled during winter, soil strength is reduced due to “kneading,” which negatively affected hydraulic properties. Also, tree harvesting in forest soils causes an intense increase in the pre-compression stresses, formation of deeper ruts, reduced aeration, and a restricted plant growth as a result of dynamic soil loading (Vossbrink and Horn, 2004; Horn et al., 2007b). The controlled traffic lane concept could prevent large-scale soil deformation by restricting mechanical stresses to defined lanes, hence protecting arable and forest land even during intensive future land use (Alakukku et al., 2003; Chamen et al., 2003; Watts et al., 2005).

An interesting aspect of soil deformation caused by extremely heavy loads is shown in Figure 3.19. During a first time wheeling event with a sugar beet harvester (35 Mg), a formerly formed rigid plow pan layer lying on top of a weaker subsoil may even be destroyed by breaking the platy structure due to shear stress concentrations in the plow pan (Fazekas, 2005; Peth et al., 2006). Repeated wheeling of such a site may then result in deeper stress penetration and consequently aggravates subsoil compaction.

Figure 3.19   Schematic sketch of stress-induced changes in the plow pan layer as a consequence of the initial wheeling with a heavy sugar beet harvester.

(Modified from Fazekas, O. 2005. Bedeutung von Bodenstruktur und Wasserspannung als stabilisierende Kenngrößen gegen intensive mechanische Belastungen in einer Parabraunerde aus Löss unter Pflug- und Mulchsaat. Ph.D. Thesis. Schriftenreihe des Institut für Pflanzenernährung und Bodenkunde, Germany, No. 67, 170 pp. ISSN: 0933-680X.)

#### 3.5.3  Effect of Stress Application on Soil Strain

If external stress is smaller than internal soil strength, no deformation results and vice versa. However, because the latter includes soil compaction and shear processes, both the rut depth and the vertical movement of a given soil volume below the rut must be known if we want to distinguish between both processes. The extent to which soil deformation occurs during traffic and the extent to which various tillage implements (conventional/conservation) deform a soil at a given pore water pressure are shown in Figure 3.20. In the conventional tillage treatment in a loessial luvisol, passage of a tractor (front/rear wheel) results in a pronounced vertical (up to 8 cm) and horizontal forward and backward (up to 2 cm) displacement. Under conservation tillage, the same tractor results in smaller soil deformations because of a higher internal soil strength leading to a maximum vertical displacement of <4 cm after three traffic events and a much less pronounced horizontal displacement. With increasing aggregate development, soil strength increases and aggregate deterioration is less pronounced during displacement and alteration of the pore system due to the infilling of interaggregate pores by smaller particles. Nevertheless, all stresses that are not attenuated to levels below soil strength result in volume alterations, even if the applied stresses vary for different soil types, land uses, tillage systems, and environmental conditions. The intense effect of wheel load on soil strain down to 40 cm depth in luvisols derived from loess under conventional and conservation management depicts the sensitivity of stress application on changes in the pore system (Figure 3.21). With increasing wheel load at a given soil stress, the vertical height change is more intense under conventional than under long-term conservation tillage practice and affects also deeper soil depths irreversibly, which was also stated by Richards and Peth (2009).

Figure 3.20   Strain distribution in structured soils due to traffic. Particle movement is more pronounced in the conventionally tilled luvisol (a) while soil deformation is less intensive in the conservation tillage plot (b).

(From Wiermann, C., D. Werner, R. Horn, J. Rostek, and B. Werner. 2000. Stress/strain processes in a structured unsaturated silty loam Luvisol under different tillage treatments in Germany. Soil Till. Res. 53:117–128.)

Figure 3.21   Vertical-elastic (reversible) and plastic (irreversible) soil displacement in a luvisol derived from loess for different wheel loads and management systems. Soil movement has been measured with a DTS/SST (displacement transmitter/stress state transducer system).

(From Zink, A. 2009. Bodenstabilität und Auswirkungen dynamischer Lasteinträge auf physikalische Eigenschaften von Ackerböden unter konservierender und konventioneller Bodenbearbeitung. Ph.D. Thesis. Schriftenreihe Institut für Pflanzenernährung und Bodenkunde, Germany, No. 84, 180 pp. ISSN: 0933-680X.)

#### 3.6  Soil Deformation and its Effects on Physical and Physicochemical Properties

At mechanical failure, physical, chemical, and biological as well as physicochemical properties are affected. Stepniewski et al. (1994), Pagliai and Jones (2002), Lipiec and Hatano (2003), Lipiec et al. (2004), and Horn et al. (2006) summed up the present knowledge and underline the detrimental effects of soil deformation on physical, chemical, and biological soil properties and functions. Larink et al. (2001) and Langmaack et al. (2002) emphasize the stress effect on biological activity and regeneration after stress application and observed an intense change in species abundances. Numerous papers defined in addition the effect of soil deformation of penetration resistance, which is often primarily linked with root growth, but these data can also be used to define in situ “stresses at rest” as a measure of the actual compression status of plots, fields, or even landscapes (Horn et al., 2007a).

Concerning changes in hydraulic properties under static loading, which are of main importance for numerous processes, it becomes obvious that both capacity properties like pore-size distribution (water retention characteristic) and intensity relations like the unsaturated hydraulic conductivity function show a severe change with applied stress (Figure 3.22; Horn et al., 1995).

Figure 3.22   Change in water retention characteristic (a) and unsaturated hydraulic conductivity (b) as a function of applied mechanical stresses.

Identical diagrams are also prepared for gas diffusion (Glinski and Stepniewski, 1985), thermal properties, and also redox reactions where a more intense stress dependency with remaining smaller values is shown when the internal soil strength is exceeded (Figure 3.23). A direct link between the mechanical stress application and increased denitrification rates was presented by Flessa et al. (1998) and Liu et al. (2007).

Figure 3.23   Changes in redox potential as a function of load in two soils depths.

(Data from Horn, R. 1985a. Auswirkung mechanischer Belastungen auf die Redoxpotentiale von 3 Bodenmonolithenein Laborversuch. J. Plant Nutr. Soil Sci./Z. Pflanzenernahr. Bodenkd. 148:47-53.)

However, not only these general changes, but also alterations in vector properties of, for example, the hydraulic conductivity (Doerner, 2005; Doerner and Horn 2009), reflect the strong effect of stress–strain processes on soil functions. Wheeling-induced formation of platy structures results in a pronounced horizontal anisotropy of the hydraulic conductivity and intensifies lateral water and solute movement, which enhances soil erosion and nutrient export. Particularly, shear-induced strain processes lead to changes in transport functions also with consequences for gas fluxes and compositions due to more pronounced flow path tortuosities and reduced pore connectivity, which can be observed down to the aggregate scale (Ball et al., 1999; Peth et al., 2008).

#### 3.7  Modeling Soil Deformation and Coupled Processes

In unsaturated structured soils, hydraulic, thermal, and/or gas transport processes must be mutually linked in order to fully account for deformation-induced changes in soil functions. Especially, hydraulic and mechanical processes are tied up closely. This is evident for example in the action of hydraulic stresses during shrinking and swelling and their consequences for soil strength or in the role of pore water pressures on consolidation and compression (effective stress theory). On the other hand, soil deformation and reorientation of particles and aggregates influence pore-size distribution and porenetwork geometries. Hence, hydraulic conductivity within the matrix and structural pores is changed, which finally affects, in turn, hydraulic stresses and subsequent deformation behavior. The mathematical equations for water, gas, and heat transport in soils and the effects of tillage on changes in structure are available (Jury et al., 1991). However, often physical soil behavior is still investigated by treating either process in isolation rather than accounting for their interdependency (Kirby, 1994; Kirby et al., 1997; Keller et al., 2007b). Richards (1992) proposed a finite element model that was able to interactively couple load deformation and flow processes in unsaturated and swelling soils. The model was later further improved by Gräsle (1999) and tested by simulating standard mechanical laboratory tests. The modeling concepts Richards developed over decades are summarized in Richards and Peth (2009), including some model applications for civil engineering and soil science problems. A detailed conceptual framework for modeling coupled soil mechanical and hydraulic processes in soils is represented in Figure 3.24 (Gräsle et al., 1995). O’Sullivan and Simota (1995) and Defossez and Richard (2002) have reviewed some soil compaction models and discussed problems when combining them with environmental impact and crop production models.

Figure 3.24   Schematic representation of a coupled hydraulic elastoplastic model to define and quantify coupled processes in soils.

(From Gräsle, W. T. Baumgartl, R. Horn, and B.G. Richards. 1995. Interaction between soil mechanical properties of structured soils and hydraulic process—Theoretical fundamentals of a model. Balkema 2, p. 719-724. In E.E. Alonso and P. Delage (eds.) Proc. First Int. Conf. Unsaturated Soils/ UNSAT’95, September 6–8, Paris, France. Balkema, Rotterdam, the Netherlands.)

#### 3.7.1  Deformation and Failure

This process describes the load/deformation/failure response of soil to a change in stress, strain, and/or displacement with all other factors held constant. It includes (1) nonlinear elastic behavior of the material as a function of stress and its history and matric potential (Richards, 1978); (2) changes in initial stress or strain as a result of swell/shrink behavior caused by changes in soil water suction, solute content, or temperature (Richards, 1986); (3) stress/strain path dependency or hysteretic behavior in load-deformation response (Richards, 1979); and (4) shear and tensile failure with dilatance or compression and with strain softening.

#### 3.7.2  Coupling Soil Mechanical and Hydraulic Processes

During any kind of soil deformation, changes in hydraulic properties will occur, which is an important factor influencing the further deformation process as pointed out in the effective stress equation of Terzaghi Equation 3.1. In addition, changes in soil stress due to swelling or shrinkage will also cause changes in soil water potential. This effect of stress on water potential is sometimes referred to as the stress potential or the stress component of field measurements of soil water suction (Richards, 1986). Load-deformation analysis must therefore be coupled with changes in soil water potential, including subsequent water flow (Figure 3.24). The displacements can also be used to calculate the new geometry and material velocities for the analysis of water flow in soils undergoing strain. Such analysis can be also extended to 3D flow problems, which are of interest in the plow pan and in well-structured horizons below.

#### 3.7.3  Example Application of the Finite Element Model

Modeling soil stiffness for various horizons with differing mechanical properties and the overall stress–strain relationships of a complete soil profile requires the derivation of stress-dependent moduli (bulk K and shear G modulus, see Section 3.2.3). The mechanical material properties defined by K and G may be determined by back-analyzing consolidation and shear test data (Richards et al., 1997). In this approach, the soil test is simulated with an FEM using an initial estimate of the material parameters, which are subsequently modified until a good agreement between modeled and measured test data is achieved. An example where a consolidation test has been simulated is shown in Figure 3.25.

Figure 3.25   Measured and simulated settlement of a consolidation test (oedometer). The ordinate has been scaled to the original height of the tested undisturbed soil sample (30 mm).

(From Richards, B.G., and S. Peth. 2009. Modelling soil physical behaviour with particular reference to soil science. Soil Till. Res. 102:216–224.)

Such stress-dependent mechanical soil properties may be used to calculate stress distribution under loads within a soil profile more accurately accounting for differences in soil stiffness and strength with depth including the mechanical history of the soil (precompression stress). A comparison of measured stress (stress state transducer [SST]) data from a traffic experiment on a conservation tillage plot (shallow chiseling up to 8 cm, luvisol derived from glacial till) with stress distributions simulated by FEM using the mechanical parameters derived for three soil horizons of the corresponding soil profile is shown in Figure 3.26. Simulations including and excluding the plow pan layer, which was encountered in 20–40 cm soil depth, indicate that no good agreement between the measured and simulated stresses was obtained when the plow pan was not included in the FEM simulation. In contrast, if the corresponding mechanical data of the plow pan were included in the simulation, then a reasonably good agreement between the stress measurements and model prediction would be obtained (Figure 3.26). Similar examples of the simulation of stress distributions under tires using this FEM approach in conjunction with parameter estimations from mechanical laboratory testing are given in Peth et al. (2006) and Richards and Peth (2009).

Figure 3.26   Determined and modeled vertical stress versus depth in a luvisol derived from glacial till under conservation tillage (type Horsch). The properties of a plow pan layer, which had been created until 1989 and which had a thickness of 20 cm

(from 20 to 40 cm were not (a) and were (b) included in the model). (Data from Kühner, S. 1997. Simultane Messung von Spannungen und Bodenbewegungen bei statischen und dynamischen Belastungen zur Abschätzung der dadurch induzierten Bodenbeanspruchung. Ph.D. Thesis. Schriftenreihe des Institut für Pflanzenernährung und Bodenkunde, Germany, 39, 190 pp. ISSN: 0933-680X.)

#### 3.8  Concluding Remarks

1. Mechanical soil properties are useful indicators for the development and physical quality of soil structure.
2. The determination of soil strength requires the measurement of volumetric stress and strain.
3. Structure development in arable and forest soils always results in increased soil strength. With increased aggregation, strength increases and the characteristics of the water retention curve become more pronounced.
4. In structured soils, hydraulic properties of inter- and intraaggregate pores are different. Differences in hydraulic conductivity, plant available water, and air capacity become more pronounced as structure development increases. In addition, root penetrability is reduced into aggregate interiors but may be enhanced in cracks of the interaggregate pore space.
5. The increase in strength with decreasing pore water pressure depends on the pattern of the water retention curve for single aggregates and bulk soil.
6. With increasing degree of aggregation, pore water pressure at the same water content in the bulk soil becomes more negative. Single aggregates are always stronger than the bulk soil or the homogenized material. This is related to smaller increases in pore water pressure during loading and higher aggregate bulk densities. Shear strength parameters (angle of internal friction and cohesion) are always higher for single aggregates than for bulk soil. However, as soon as applied stress exceeds the internal soil strength, single aggregate and aggregated bulk soils will become homogenized accompanied by a decline in strength.
7. Soil strength is strongly affected by spatial differences in pore water pressure. Hydraulic properties of single aggregates and their arrangement in the bulk soil are therefore most important for predicting soil strength and stress distributions in soils. Stress attenuation, stress distribution, and soil deformation due to loading under running wheels or belts can vary to a great extent depending on in situ soil conditions, machinery parameters, and frequency of loading. Soil deformation should therefore be considered as a dynamic rather than a static process.
8. Structure development always involves reduced accessibility to exchange sites for soil solutions, retarded exchange processes, and stronger chemical disequilibrium. Only at the lowest entropy (at an increased accessibility), structured soils become macroscopically homogeneous. With reduced pore water potential (reduced flux rate), the exchange process becomes more complete. At higher saturations, anaerobic soil volumes in aggregates become more extensive for longer periods. Thus, soil physico-chemical properties vary intensely throughout the year.
9. The coupled analysis of multifunctional and interactive processes in unsaturated structured soils is an important topic for future soil research but very complex. Long-term soil stability involves materials, which are unsaturated, nonlinear hysteretic, and the soil composition changes with time. During tillage operations, many physical and mechanical properties are altered requiring a more precise definition of the actual conditions and their expected temporal changes.
10. Measurements and modeling of coupled ion and water fluxes through unsaturated structured soils are most important for future work because aggregation affects ion diffusion, dispersion, and mass flow. Changes in pore systems and functions are also affected by mechanical stresses. Thus, model coupling is required and could enhance the estimate of time-dependent effects on nutrient, heavy metal, or pesticide transport through structured soils also including the effect of drying or wetting or deformation (compaction) processes.

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