Cavity contraction solution in a non-linear elastic drained material

$amax$

maximum cavity radius at end of loading (m)

$a0$

initial cavity radius (m)

$a1$ and $b1$

hyperbolic parameters for pre-stress reversal (1/kPa)

$a2$ and $b2$

hyperbolic parameters for post-stress reversal (1/kPa)

$c′$

drained cohesion (kPa)

h

power law coefficient of non-linear stiffness decay (Bolton & Whittle, 1999)

M

dilation ratio

N

constant in Nova & Wood (1982) flow rule

$NEOL$

stress ratio mobilised at end of loading

$Ny$

stress ratio at the point of yield during unloading

$PEOL′$

plane-strain mean effective stress at end of loading (kPa)

$Pmax′$

maximum applied effective cavity pressure (kPa)

$P0$

lift-off pressure resembling in situ lateral pressure (kPa)

$p′$

mean effective stress (kPa)

q

deviatoric stress (kPa)

R

ratio of shear strength during loading and unloading

$Rc$

radius of plastic zone during cavity contraction (m)

$Re$

radius of plastic zone during cavity expansion (m)

s

(σ′_r + σ′_θ)

t

(σ′_r − σ′_θ)

$umax$

maximum displacement at cavity wall experienced at the end of loading (m)

$u0$

ambient pore water pressure (kPa)

$εc$

cavity strain

$εcref$

$εmax−εc$

$εe$

elastic strains

$εmax$

maximum induced cavity strain during cavity expansion

$εp$

plastic strains

$εr$

radial strain

$εUR$

cavity strain required to induce plasticity at cavity wall

$εURref$

$εmax−εUR$

$εθ$

circumferential strain

$ν$

Poisson’s ratio

$ξ$

cavity displacement (m)

$σh0′$

in situ horizontal effective stress (kPa)

$σn′$

applied normal stress (kPa)

$σr′$

effective radial stress (kPa)

$σUR′$

effective radial stress required to induce plasticity at cavity wall (kPa)

$σv0′$

in situ vertical effective stress (kPa)

$σz′$

effective vertical stress (kPa)

$σθ′$

effective circumferential stress (kPa)

$τ$

shear stress (kPa)

$τEOL$

shear stress defined at end of loading (kPa)

$τref$

corrected shear stress with $τEOL$ as reference (kPa)

$ϕcv′$

angle of critical state friction (degrees)

$ϕEOL$

angle of friction at end of loading (degrees)

$ϕp′$

angle of peak friction (degrees)

$ϕy$

angle of friction at yield during unloading (degrees)

$χ$

constant used in estimating elastic perfectly plastic stiffness matrix

$ψ$

angle of dilation (degrees)

Cavity expansion is a widely used method for assessing in situ geo-material properties because of its clear boundary conditions and stress path. For low-disturbance, self-bored pressuremeter tests, reliable parameters may be obtained using closed-form solutions like the Hughes & Whittle (2022) adaptation of Carter et al. (1986). However, for either pre-bored or full displacement pressuremeter tests there is substantial disturbance to the surrounding soil during instrument installation. This ground disturbance then requires assumptions about the in situ horizontal stress and strain origin that affect interpreted results. The well-defined boundary conditions at the end of expansion reduce uncertainties from installation, making accurate unloading models in a cohesive–frictional material a valuable tool for site characterisation.

Studies conducted by Wroth (1984), Robertson & Hughes (1985), Bellotti et al. (1989) and Whittle & Liu (2013) found that an increase in cavity stress results in a non-linear variation in stiffness, shear strength and volume change throughout the plastic zone. When subjected to stress reversal and unloading, sands are found to exhibit additional compressive behaviour (Tatsuoka & Ishihara, 1974; Sadrekarimi & Olson, 2010). Currently, there are no known analytical solutions for evaluating the non-linear elastic behaviour of drained materials for the unloading phase of the pressuremeter test. This research presents an approach based on a non-linear elastic, perfectly plastic, cohesive–frictional material model for the unloading phase of a pressuremeter test. The yield locus of the material is defined by the Mohr–Coulomb surface and incorporates the Nova & Wood (1982) flow rule as suggested by Jefferies (1997). Yielding is governed by power law, while strain is defined using a hyperbolic rule. Once yielding initiates, the model transitions into a perfectly plastic state, whose response is governed by the derived unloading solution. The applicability of this method is currently constrained to pressuremeter tests conducted in dense soils due to the assumption of constant dilation. The calculated values are validated against high-quality loading analyses utilising the Hughes & Whittle (2022) adapted Carter et al. (1986) model.

It is important to note that the pre-yield response in a pressuremeter test is not, strictly speaking, one of non-linear elasticity. The term is used here solely to maintain consistency with prior publications developed for the loading branch of a pressuremeter test (Briaud et al., 1983; Wood, 1990, 1991; Jardine, 1992; Pye, 1995; Bolton & Whittle, 1999; Lopes dos Santos et al., 2021; 2022a; 2022b; Hughes & Whittle, 2022; Byrne & Whittle, 2023; Habert & Lopes dos Santos, 2024; Lopes dos Santos & Habert, 2025). It also serves as a convenient descriptor when referring to advanced ‘non-linear elastic’ constitutive models such as the ‘Hardening soil’ and ‘Hardening soil with small-strain stiffness’, which are increasingly adopted in engineering practice.

Carter et al. (1986) developed a linear elastic, closed-form solution for cavity expansion in cohesive–frictional materials. Their solution considered both cylindrical and spherical geometries while assuming the Rowe (1962) stress–dilatancy theory to define the flow rule. Unlike the model by Hughes et al. (1977), this approach accounted for elastic strains within the plastic region, where total shear strains had previously been considered responsible for volume change. The Carter et al. (1986) solution assumes cavity expansion starts from an undisturbed in situ stress state with a known strain origin. As a result, it is only applicable for self-boring pressuremeter tests. Hughes & Whittle (2022) refined the Carter et al. (1986) solution to better capture sand’s non-linear behaviour during loading by applying power-law elastic moduli trends originally developed for clays (Bolton & Whittle, 1999). Whittle & Liu (2013) demonstrated that sands exhibit increasing stiffness with cavity stress, driven by volumetric changes within the plastic zone during increased cavity expansion. Convergence of non-linear stiffness behaviour with increased expansion reflects diminishing volumetric change, indicating that near-field soils are approaching the critical state. At this point, stiffness curves begin to overlap, resembling an undrained response (Bolton & Whittle, 1999). These methods result in highly accurate estimations of the in situ conditions and material parameters when applied to self-bore pressuremeter tests. However, pressuremeter tests are rarely carried out using self-bored methods, given the necessity for relatively uniform, aggregate-free ground conditions. Instrument installation methods such as pre-boring and full displacement erase the previous stress history and strain origin and therefore cannot be modelled using closed-form solutions.

Houlsby et al. (1986) developed a linear elastic, perfectly plastic solution for cavity contraction in purely frictional material. Their model neglected elastic strains within the plastic zone, while assuming that dilation increases with shear strain during unloading. Disregard of small strains tends to an underestimation of friction angle (⁠$ϕp$⁠) as demonstrated by Whittle & Byrne (2020) and Hughes & Whittle (2022). Test data showed the best fit when the soil was assumed to be at the critical state, yielding at $ϕ′cv$ with the peak dilation, $ψp=0$⁠, therefore providing a reliable estimate of $ϕ′cv$⁠.

The assumption of linear elastic behaviour adopted by Houlsby et al. (1986) and Carter et al. (1986) yields lower-bound estimates of shear stiffness compared to the non-linear response observed in sands. Houlsby et al. (1986) reported that the linear modulus determined from the unloading arm was lower than values estimated from unload–reload cycles during expansion, despite being recorded at a higher stress state. This indicates that stiffness depends not only on stress state but also on the measured stress range.

The proposed method extends the closed-form solution of Houlsby et al. (1986) by incorporating non-linear stiffness and enabling identification of the onset of yielding during unloading. Post-yield behaviour is described using the framework of Carter et al. (1986), which includes the contribution of the elastic deformations post-yielding.

With respect to the volumetric changes within the plastic region, Silvestri et al. (2009) compared flow rules given by Rowe (1962), Nova & Wood (1982) and Bolton (1986), finding similar dilation behaviour during cavity expansion. This suggests that dilation angle is relatively insensitive to the choice of flow rule under pressuremeter expansion. However, the models Rowe (1962) and Bolton (1986) fail to capture sand behaviour during stress reversal and unloading. Jefferies (1997) considered the Nova & Wood (1982) rule for unloading, and demonstrated that sands store inelastic volumetric work during triaxial compression, which dissipates upon unloading, resulting in further volumetric contraction in the elastic zone. This additional compression during unloading was also observed by Windle & Wroth (1975) and by Liu et al. (2025) through strain field measurements during cone penetration testing.

In order to develop the proposed closed-form solution, the behaviour of the ground must first be examined both during pressuremeter expansion and contraction. Fig. 1 illustrates the stress states of a soil element at the cavity wall throughout the loading phase of a pressuremeter test. Point A represents the undisturbed in situ condition, defined by principal stresses $σv0′$ and $σh0′$⁠. Point B marks the onset of yielding during initial expansion, governed by the Mohr–Coulomb failure surface at $ϕ′p$⁠. Point C represents the end of expansion and the start of unloading, which may, but not necessarily, correspond to the critical state friction angle, $ϕ′cv$⁠. At point D, unloading begins, leading to stress reduction and strain reversal. Point E approximates the onset of reverse plasticity, where the soil fails in extension; E' represents the actual yield point. Finally, point F marks the end of unloading, completing the test.

To derive a solution for cavity contraction, the surrounding material is initially assumed to be linear elastic, simplifying the formulation before incorporating non-linear elasticity. The domain is treated as transversely isotropic and infinite in the plane of contraction, minimising boundary effects. Axisymmetric plane-strain conditions are assumed, and the overburden stress is neglected, making the analysis valid only for test pockets where the vertical stress acts as an intermediate stress. These assumptions allow the problem to be expressed in radially axisymmetric form using polar coordinates $(r,θ,z)$⁠. The material is considered a single-phase system with effective stresses only, assuming no water table. If water is present, a constant hydraulic head is assumed, and total stresses are converted to effective stresses by subtracting ambient pore water pressure $(u0)$⁠.

The actual stress path during unloading of a pressuremeter test is shown as the dashed line C–D–E′ in Fig. 1 (Hughes & Robertson, 1985) and is based on the behaviour of sand to show yield at the point of stress reversal mentioned in Tatsuoka & Ishihara (1974). For simplicity, the mean effective stress is assumed to remain constant throughout the elastic branch of the unloading curve until the stress ratio matches the yield stress ratio (C–D–E). Upon yielding, the cavity wall follows a constant stress ratio, defined by line E–F.

Figure 2(a) illustrates a cross-sectional view of the cavity at the end of expansion, with initial radius $(a0)$⁠, maximum radial displacement $umax$ and plastic zone radius $Re$⁠. Fig. 2(b) shows the stress path of soil elements during loading in s′ and t space, while Fig. 2(c) presents the same in p′ and q space. Soil behaviour is depicted for both contractile (i) and dilative (ii) responses of a cohesive–frictional material within a critical state framework. Fig. 2(d) displays the radial variation of volumetric strain in the plastic zone at the end of expansion for both behaviours. Finally, Fig. 2(e) illustrates the variation of non-linear shear modulus in the shear strain range captured in a high-resolution pressuremeter test (⁠$10−4$ to $10−2$⁠) during cavity expansion, corresponding to the shear strain distribution shown in Fig. 2(d).

At the end of the loading phase, cavity expansion reaches its maximum and is defined as t = 0. The cavity expands from its initial radius $a0$ by a maximum displacement $umax$ under an applied cavity pressure $P′max$⁠. This stress induces a corresponding cavity strain $εmax$ and the plastic zone extends to a radius, $Re$ as shown in Fig. 2(a). Six locations within the plastic and elastic zones are considered to illustrate how material properties vary with distance from the borehole wall. The stress state at each location is indicated, as illustrated in Fig. 1. The greatest stress change occurs at the cavity wall, decreasing with radial distance, as illustrated in s′ and t space in Fig. 2(b). The yield surface follows a curved shear strength envelope, with its slope decreasing from $sin−1(ϕ′p)$ to its final value of $sin−1(ϕ′cv)$⁠. Fig. 2(c) shows the stress paths in p′–q space for each soil element at the end of expansion, capturing both dilative (strain weakening) and contractive (strain hardening) behaviours of a cohesive–frictional soil within a critical state framework. For dilative materials, the yield surface initially contracts, transitioning to a looser state during yielding. At the critical state, the borehole expands under constant shear stress, and no further changes in mean effective stress or material stiffness occur near the cavity wall (point 1 in Fig. 2(a)). In contrast, contractile soils exhibit an expanding yield surface with increasing shear stress (q), gaining strength and densifying until reaching the critical state, where plastic flow continues at constant volume and shear stress.

At the end of loading, each element within the plastic zone has a distinct yield surface due to varying degrees of volumetric change, resulting in spatial variability in stiffness and shear strength. It is important to note that the assumption of constant void ratio at the critical state is not entirely accurate, as grain crushing and asperity abrasion may alter particle angularity and grain size distribution, thus affecting pore structure. Torsional ring shear tests have shown continued compression and increase in $ϕcv$ value beyond the critical state in sands, consistent with findings by Sadrekarimi & Olson (2010). Evidence of particle crushing is also apparent in the present pressuremeter tests, as indicated by the reduction in dilation angle observed during unloading analysis, as presented in subsequent sections.

Figure 2(d)(i) illustrates the variation in radial stress and shear strain experienced by soil elements across the plastic zone during cavity expansion. Hypothetical volumetric strains at each location are shown, highlighting the progressive change in soil behaviour with distance from the cavity wall. Point 1, closest to the wall, undergoes the highest shear strain and reaches critical state, shearing at constant volume. Point 2 marks the onset of critical state, while point 3 corresponds to the peak rate of dilation. Point 5 lies at the elastic–plastic interface $(Re)$⁠, indicating the onset of dilation, and point 6 represents far-field conditions, where the soil remains undisturbed under in situ stress.

Figure 2(e) illustrates the increase in shear modulus across four unload–reload cycles during cavity expansion. The initial in situ stiffness, $G0$⁠, with its corresponding non-linear decaying curves are presented in grey. These decay curves extend over shear strains ranging from $10−4$ to $10−2$⁠, consistent with the strain resolution generally achievable in high-resolution pressuremeter testing. Since the sand behaves as a drained material, soil elements experience increasing confining stress and volumetric change throughout yielding. Stiffness, being both strain- and stress-dependent, increases with cavity pressure (Whittle & Liu, 2013). As volumetric change slows with increasing shear strain, the rate of stiffness increase also diminishes, resulting in flatter non-linear stiffness curves. Once the critical state is reached, volume change ceases, halting further stiffness growth and causing subsequent curves to converge, similar to undrained behaviour (Bolton & Whittle, 1999).

Following the completion of the loading phase, it is assumed that the unloading branch of the test begins. For solution derivation, time is considered as t$>0$⁠, during which the cavity pressure decreases from $P′max$ to a value $P′i$⁠. This reduction causes the entire domain to revert to an elastic state until it reaches peak shear strength during unloading. Concurrently, the cavity wall radius contracts from $amax$ = $a0$ + $umax$ to a value $ai$⁠, expanding the plastic zone outward to point 3 at radius $Rc$⁠, referred to as the plastic radius in contraction (see Fig. 3(a)).

In this context, $P′iandai$ represent incremental changes in cavity pressure and radius, respectively, during either loading or unloading. Figs 3(a)–3(e) illustrate six locations within the plastic and elastic zones surrounding the cavity wall, analogous to those in Fig. 2. Each location is arranged radially outward from the cavity wall and corresponds to distinct stress states, as shown in Fig. 1. As in the loading scenario, point 1 remains fixed at the cavity wall during unloading.

Figure 3(b) presents the stress states of each point in s′–t space demonstrating a curved yield surface in the positive s′ and negative t plane. This surface reflects frictional behaviour approaching constant shear stress at the critical state. The unloading stress path, shown as dashed lines, follows the framework of Robertson & Hughes (1985) and the stress reversal behaviour in sand described by Tatsuoka & Ishihara (1974). The current stress state at each load step is depicted similarly to Fig. 2, but now for cavity contraction. Unloading paths are marked with hollow symbols, contrasting with the solid markers used for expansion.

Point 3’s stress path is magnified to highlight the yield stress variation due to the assumption of a linear unloading path. Shear strength varies with radial distance from the cavity wall, as each element experiences a different reduction in confining stress based on the maximum radial stress reached during expansion. This indicates that the yield surface size varies with radial position due to non-uniform volumetric changes. This behaviour is illustrated in Fig. 3(c)(i) for dilative and Fig. 3(c)(ii) for contractive responses.

Soil elements that experienced plastic deformation during the expansion phase are observed initially to contract upon shear reversal, regardless of their behaviour during expansion. This observation aligns with the findings of Jefferies (1988, 1997). Upon reaching the yield stress in the reverse direction, the volume changes induced during cavity expansion influence the sand’s response, causing further compression at varying rates and magnitudes across the plastic region. These volume changes during strain reversal, coupled with unloading, are illustrated in Fig. 3(d)(ii).

This initial contraction within the elastic zone reduces the void ratio and increases grain-to-grain contact, thereby restoring the energy required for dilation. As a result, both the yield surface and peak shear strength increase compared to the end of the expansion phase. For contractile materials, however, despite the enlarged yield surface, the void ratio continues to decrease post-yield due to regression and steepening of the critical state line in e–ln(p′) space, attributed to grain crushing (Sadrekarimi & Olson, 2010). This behaviour is demonstrated in Fig. 3(c)(iii), and supported by the ring shear test results on 98% silica sand shown in Fig. 4.

The silica sand exhibited contraction upon strain reversal and incremental reduction in applied vertical stress. A strong linear relationship was observed between the reduction in confining stress and sample compression. This response was validated through multiple unloading tests conducted using a conventional Bromhead ring shear apparatus, both in dry conditions and submerged in water. The slope of the regression line is expected to depend on material properties that influence grain crushing, such as grain size, mineralogy and particle angularity.

Figure 3(e) illustrates the evolution of the non-linear shear modulus at the cavity wall during the contraction phase of the test. Upon unloading, the shear modulus resets to its small-strain value $(G0)$⁠, which is governed by the maximum mean effective stress attained during cavity expansion. Within the plastic zone, $G0$ decreases radially and approaches the in situ modulus, $G0$ at the elastic–plastic interface at radius $Re$⁠.

During elastic unloading, the mean effective stress remains unchanged, and the stiffness degrades with increasing shear strain. This degradation rate is a function of the stress state and volumetric conditions established at the end of the loading phase. Once a soil element reaches the failure surface, confinement begins to reduce, leading to a decline in the stress-dependent non-linear behaviour.

The influence of grain crushing diminishes beyond a certain stress threshold, after which the sand begins to dilate as confinement continues to decrease. This transition marks a shift from compressive to dilative behaviour, reflecting the complex interplay between stress path, grain-scale mechanics and evolving stiffness characteristics.

At any given radius r, the effective hoop stress $(σθ′)$ must remain in equilibrium with the effective radial stress $(σr′)$⁠. During the elastic phase of unloading, the overburden pressure $(σz′)$ acts as the intermediate principal stress and is neglected in the elastic calculations. The intermediate principal stress does, however, play a role for establishing the onset of stress reversal and yielding, as will be described later. Assuming that stress continuity must be maintained, the equation of equilibrium around the expanding cavity is described by equation (1).

The assumed boundary conditions for calculation of the stresses are that the radial stress: (i) is equal to cavity stress at the cavity wall, as shown in equation (1a); and (ii) is equal to the far-field, in situ stress at an infinite distance from the cavity wall, as given in equation (1b).

Assuming plane-strain conditions, during yield, and provided that the vertical stress $(σz′)$ remains as the intermediate stress and the strains in the vertical (z) direction are minor relative to the in-plane strains, the change in $(σz′)$ can be determined using equation (2).

where $ν$ is the Poisson’s ratio.

During unloading, as cavity stress decreases, the vertical effective stress, $σz′$ reduces to $σθ′$⁠. The intermediate principal stress then reduces at the same rate as $σθ′$ throughout the unloading phase. This stress reduction results in dilation of the soil elements above the cavity wall and results in increased confinement when compared to not accounting for the intermediate principal stress.

The elasto-plastic matrix is formulated following Carter et al. (1986), with the constitutive equations derived after neglecting the convective term of the stress rate, expressed in equation (3)

where

$ξ$ represents the outward radial displacement of an element at radius r, measured from its initial radius, $r0$⁠. The dot notation on variables indicates rate of change with respect to time.

During unloading, stress reversal occurs first within the elastic region, rendering the constitutive relation invalid under pre-stress reversal conditions. Therefore, the derived solution provided below is only applicable post-yield, where plastic deformation governs the material response.

Given the derived analytical solution is only valid for post-yield stresses, in order to simplify the derivation, the material is assumed to exhibit linear elastic–perfectly plastic behaviour and will be later adapted for non-linear elasticity without affecting the analytical solution. Initially, for a purely frictional material, the yield ratio is governed by the Mohr–Coulomb failure criterion, expressed as equation (4).

where

The subscript ‘y’ denotes yield, and represents the angle of friction that governs the onset of yielding. During unloading-induced yielding, the cavity wall must undergo a stress reversal, resulting in $σθ′$ and $σr′$ becoming the major and minor principal stresses, respectively. Under the post-stress reversal yield conditions, the yield ratio can be reformulated as in equation (6).

The total strain at yield is assumed to consist of elastic and plastic components, expressed in equation (7).

The flow rule of negative dilatancy (dilation) considered in Houlsby et al. (1986) in plane-strain cavity contraction can be written for the case where plastic volumetric strains are governed by plastic shear strains given in equation (8).

Rearranging equation (8) and solving for dilation, M, gives

where dilation is defined by M and is calculated based on the dilation angle, $ψ$⁠, using

The Nova & Wood (1982) flow rule is employed to characterise the volume change behaviour of dense sand during elastic unloading in triaxial stress space. Jefferies (1997) introduced a method to quantify the reduction in shear modulus during unloading, denoted as $hu$⁠, which serves as an additional material property. The Nova & Wood (1982) flow rule is then applied to describe the volumetric changes occurring throughout the elastic phase of the unloading branch in a pressuremeter test.

The analysis assumes that the volume changes during unloading are negligible. This assumption would appear reasonable when evaluating the non-linear elastic behaviour of an unload–reload cycle, but may not strictly be true during unloading given the relatively high shear strains. The lack of explicit quantification of volume change during unloading, however, does not appear to have a significant impact on the overall analysis or the final outcome of the solution to fit field data. Accordingly, the Nova & Wood (1982) flow rule is used here solely to define the increase in peak shear strength $(ϕp′)$ resulting from dilation $(ψ)$ and the critical state shear strength $(ϕcv′)$⁠. The angle of dilation can be determined using the Nova & Wood (1982) stress–dilatancy relationship, adapted for plane strain cavity contraction, as presented in equation (11).

Here, N will be assumed as 0·2, as recommended by Shuttle & Jefferies (1998) for triaxial compression tests. According to Silvestri et al. (2009), the value of N obtained from triaxial compression testing has been shown to effectively simulate plane strain conditions with high accuracy.

The stiffness matrix (D) for cavity contraction under purely elastic deformations can be expressed as follows, following the formulation adapted from Carter et al. (1986) in equation (12).

where

During cavity expansion, the cavity radius increases from $a0$ to $amax$ and results in a plastic radius around the cavity at radial distance, $Re$⁠. As unloading begins, the soil surrounding the cavity transitions back to an elastic state. The domain remains elastic until the stress ratio reaches the yield surface in extension. Further unloading results in outward radial growth of the plastic zone. At the elastic–plastic interface, the radial stress is defined as $σUR′$⁠, while the cavity strain is given as $εUR=εmax−εURref$⁠.

The cavity strain required to invoke plasticity during unloading, denoted as $εURref$⁠, is measured from the strain origin $εmax$⁠. It is defined assuming linear elasticity, as expressed in equation (13).

At the current instance the applied cavity stress $(P')$ satisfies $P′≤σUR′$⁠, and the cavity wall radius (r) lies within $a0≤r≤Rc$⁠. $Rc$ is the radius of the elastic–plastic interface in contraction. For $r>Rc$⁠, the material remains elastic.

Assuming that there is no volume change within the elastic phase, the cavity displacement, adapted from the Hughes et al. (1977) analysis for unloading can be expressed as equation (14).

and the rate of displacement can be expressed as

Assuming the stress path (C–D–E) illustrated in Fig. 1, the mean effective stress remains constant throughout the elastic unloading. At the elastic–plastic interface, the stress ratio defined by equation (6), combined with the assumption of constant mean effective stress, leads to determination of $σUR′$ and $εURref$⁠. The value of $σUR′$ can then be calculated from the applied stress as shown in equation (15),

and the corresponding cavity strain, $εURref$ is determined from equation (16)

Substituting the yield criterion for unloading defined by equation (6) into the equation of equilibrium equation (1) results in

Integration of equation (17) between the bounds of the cavity wall, $ai$⁠, and the plastic radius, $Rc$⁠, results in

To maintain consistency with the solution provided by Carter et al. (1986), $β=Ny$ is assumed, and equation (18) becomes

The rate of stress change is then obtained by differentiating equation (19) with respect to time, and is given as

The constitutive relationship within the plastic zone, derived by Carter et al. (1986), may be reduced to a single differential equation and for unloading it is expressed as

Because $Pmax′$ remains constant over time (it is the maximum pressure realised during cavity expansion), equation (21) can be reformulated as

Here

and substituting values from equations (20), (15) and (16) into equation (22) yields

Equation (23) is a first-order non-homogeneous, plastic differential equation. By integrating it while applying equation (14a) as the boundary condition for the plastic interface

where

Equation (24) is observed to hold the exact same structure as presented in Carter et al. (1986), but with different values of coefficients for T and Z, as shown in equations (24a) and (24b).

Integrating equation (24) with respect to time, and provided that the cavity strain (⁠$εmax−εc$⁠) remains within 10% (small-strain), use of equation (14) as a boundary condition results in

where

Substituting the value of $Rc/r$ from equation (19) for any radius, r, within the plastic zone yields

At the cavity wall, the applied radial stress $(σr′)$ is equal to $P′$⁠, while the cavity strain $(εθ)$ corresponds to $εcref$⁠. Equation (26) can then be expressed as

Resetting the strain origin to the initial conditions where $P′=Pmax′$ and $εc=0$ results in,

where

This equation is valid only for defining strains within the plastic arm of the unloading branch of a pressuremeter test in a purely cohesionless material. The solution is not applicable for strains within the elastic zone owing to a stress reversal, which affects model accuracy.

If cohesion is present, the above equation (28) can be modified by raising the stresses by $c′cotϕ′$ within the stress ratios, yielding

To characterise the non-linear behaviour of soil within the elastic zone, a hyperbolic non-linear elastic model will be considered and evaluated using self-bored pressuremeter test data from a clean sand on Canvey Island. Hughes et al. (1975) reported that the sand was medium dense with a typical standard penetration test (SPT) N value of 30 blows/0·3 m of penetration. The test was conducted at a depth of 14·2 m and the test data are presented in Fig. 5. The analysis of the non-linear elastic parameters considers two stages, as follows:

• end of loading (EOL): this section describes the state of the near-field material at the conclusion of cavity expansion

• power law method: this section provides the method to estimate yield stress during unloading using power law equations

• hyperbolic method: this section details the method for evaluating hyperbolic parameters both before and after the point of stress reversal.

Finally, the results obtained from the non-linear elastic hyperbolic model are compared to the field data to assess their relative performance and applicability.

At the end of cavity expansion, herein referred to as EOL, represented as point C in Figs 1 and 5, the effective cavity pressure is $P′max$ and the corresponding cavity strain is defined as $εmax$⁠.

The subscript ‘EOL’ also stands for ‘end of loading’. The mobilised angle of friction, $ϕEOL′$ at the end of loading is determined using the non-linear adaptation of the Carter et al. (1986) method described by Hughes & Whittle (2022), for an assumed $ϕcv$⁠, as shown in Fig. 6. The initial total horizontal stress, $P0$⁠, was obtained from the mean lift-off stress calculated across all six strain arms, while the ambient pore water pressure has been inferred from the total stress measured upon cavity closure. Assuming a constant-volume friction angle $ϕcv$ = 32°, the $ϕEOL′$ is estimated to be 42°, as shown in Fig. 6. The corresponding dilation angle, $ψEOL$⁠, is determined using the Rowe (1962) stress–dilatancy relationship and is estimated to be 12°, as also illustrated in Fig. 6.

The mean effective stress $PEOL'$ can be established through the stress ratio defined by the Mohr–Coulomb envelope, given by: $σr′=NEOL σθ′$⁠. Assuming that the mean effective stress remains unchanged within the elastic zone, and taking $Pmax′$ as the effective radial stress at the end of expansion, the equation for $PEOL′$ is expressed as

where $NEOL$ is the stress ratio achieved at the end of cavity expansion and is stated as

Whittle & Liu (2013) demonstrated that the non-linear elastic behaviour of sands can be effectively characterised using power law (Bolton & Whittle (1999)) relationships normalised to a reference stress level.

It is important to note that the power law assumption does not fully capture the complete stress–strain behaviour of sands. As a result, the modelled non-linear response tends to deviate from field data following stress reversal. Recognising that the model exhibits an abrupt transition to plasticity after reversal, the power law framework remains a useful tool for estimating the yield stress during unloading, denoted as $σUR′$⁠.

The magnitude of $σUR′$ is determined as per Bolton & Whittle (1999) and is expressed as

where

R = 2, when $ϕEOL$ determined from Hughes & Whittle (2022) is equal to $ϕy$⁠.

The non-linear coefficient ‘h’ used in this study is consistent with the formulation of Hughes & Whittle (2022). It is important to note that in the original power law formulation by Bolton & Whittle (1999), the symbol $β$ was used to represent the non-linear exponent. However, the use of ‘h’ here is intentional to avoid confusion with the variables employed in the Carter et al. (1986) solution for cohesive–frictional materials.

The value of h is estimated using the methodology described by Bolton & Whittle (1999). However, analysis of the unloading arm from multiple pressuremeter datasets indicates a mismatch with the value of h obtained from the reloading branch of an unload–reload cycle, as proposed by Bolton & Whittle (1999). Specifically, the values of h estimated from the unloading arm are consistently low, typically close to the lower bound of 0·5. It was found that adopting h = 0·5 provides a reliable estimate of $σUR′$ for the analyses presented in this study.

Defining the hoop stress as per stress ratio during yielding described in equation (6) results in:

For R = 2, the yield stress becomes

If cohesion is present

For R = 2 the above equation (39) takes the form:

Equations (33) and (34) are in general form and can incorporate conditions when shear strength during unloading is different compared to loading $(ϕy≠ϕEOL)$⁠.

Kondner (1963) identified a strong hyperbolic stress–strain relationship in sands and clays during triaxial compression testing, formulated between deviatoric stress and the major principal strain (i.e. axial strain).

During loading and the initial phase of unloading, radial stress acts as the major principal stress, while cavity strain, recorded at the cavity wall, corresponds to the minor principal strain. However, in the latter half of unloading, this relationship reverses: radial stress becomes the minor principal stress, and cavity strain becomes the major principal strain. As a result, the original formulation by Kondner (1963) is not directly applicable in this context.

To address this incompatibility, a hyperbolic relationship between shear stress and shear strain is developed for both the pre- and post-stress reversal portions of the curve. This two-stage approach captures the material softening that occurs after stress reversal and provides insight into the variation of shear modulus throughout the unloading phase. These calculated stiffness variations can be used to define input parameters for advanced non-linear elasto-plastic constitutive models.

Assuming no volume change in the elastic branch of the unloading arm, shear strain is taken as twice the magnitude of cavity strain. Due to stress reversal (point D in Figs 5 and 7), shear stress must be defined separately for the pre- and post-reversal stages. Prior to stress reversal, radial stress is the major principal stress, and shear stress is defined as

The mean effective stress in the elastic phase of the unloading arm is defined using equation (30). Assuming a constant mean effective stress, equation (35) can be reformulated as

Following the stress reversal, the circumferential stress becomes the major principal stress, resulting in the shear stress being defined as

Substitution of the mean effective stress (⁠$PEOL′$⁠) from equation (30) into equation (37) yields

Analysis of the equations for shear stress shows that prior to the point of stress reversal, the shear stresses in the ground decrease with decreasing cavity stress. This reduction in shear stress is due to a lower magnitude of hoop stress around the cavity. Once the stress reversal occurs, the shear stresses begin to increase despite an overall reduction in applied radial cavity stress as the hoop stresses within the plastic region exceed the applied stress at the cavity wall. Fig. 7 presents the analysis results of the unloading arm of the self-boring pressuremeter test in the Canvey Sand shown in Fig. 5, using equation (36) for pre-stress reversal and equation (38) for post-stress reversal. For both Figs 5 and 7, the following points correspond with different stages of unloading as follows:

• C represents the maximum shear stress at the end of loading

• D is the point of stress reversal

• E denotes the radial stress required to induce plasticity at the cavity wall.

A strong hyperbolic stress–strain response is observed in both the pre-stress reversal phase (C to D) and the post-stress reversal phase (D to E). Consequently, two separate hyperbolic analyses are required to accurately define the material response and offer insight into post-stress reversal behaviour. Furthermore, the shear stress and hyperbolic parameters determined are sensitive to $ϕEOL$⁠, necessitating reiteration for any incremental change (⁠$δϕEOL$⁠).

The hyperbolic relationship between shear stress and shear strain is expressed as

The parameters a and b represent hyperbolic constants that define the material properties, as established by Kondner (1963). These parameters emerge through the transformation of the hyperbolic stress–strain relationship, which is expressed as

For this analysis, a is the intercept and b is the slope of the trend line between $γ$ and $τ$⁠.

Pre-stress reversal refers to the unloading phase, beginning at the end of loading to the onset of stress reversal. The maximum shear stress at the end of loading (⁠$τEOL$⁠) serves as the reference point (origin) for transforming the C–D portion of the unloading curve into a standard hyperbola. The transformed shear stress, $τref$⁠, representing the change in shear stress by definition, is expressed as follows:

where $τc$ is the current shear stress following $τEOL$ and prior to stress reversal. Substituting $τ$ from equation (36) in the above equation (41) leads to

For non-linear elastic strains, equation (39) is reformulated in terms of cavity strains, using $γ=2εc$⁠. Applying this relationship, equation (39) is expressed as

Correcting the point of reference back to the origin results in

Here $a1$ and $b1$ represent hyperbolic parameters that characterise the material behaviour in the pre-stress reversal phase. The shear stress (⁠$τ$⁠) is defined using equation (42). Fig. 8 illustrates the hyperbolic parameters obtained for pre-stress reversal using the method outlined earlier in this section. The calculated hyperbolic fit for the Canvey data is presented in Fig. 9, and the corresponding strains calculated using equation (43a) are shown in Fig. 10 referenced as C–D.

The post-stress reversal condition starts at point (D) and remains valid until the yield stress, marked as point E in Figs 1, 5 and 7. Because the stress and strain inherently follow a hyperbolic trend, no strain or stress origin correction is required. The shear stress in this phase is determined using equation (38) and the strains post-stress reversal are computed by summing the total strains accumulated up to the stress reversal point (D) using equation (43a). The final equation is expressed as

Correcting the point of reference back to the origin results in

Here $a2$ and $b2$ represent hyperbolic parameters that characterise the material behaviour in the post-stress reversal phase. Fig. 8 illustrates the hyperbolic parameters obtained for post-stress reversal using the method outlined earlier in this section. The hyperbolic fit achieved is presented in Fig. 9, whereas strains obtained using equation (44a) are presented in Fig. 10, referenced as D–E.

To adapt the derived analytical solution in the earlier section ‘Cavity contraction assumptions’ for non-linear elastic behaviour, the approach outlined in Hughes & Whittle (2022) is followed. The strains in the elastic zone will be governed by the non-linear elastic formulations derived in the earlier section ‘Hyperbolic method’ for the hyperbolic law. From the instance of yield, the strains will be governed using the derived analytical solution. For convenience the unloading arm is divided into three sections: (a) pre-stress reversal; (b) post-stress reversal; and (c) post yield.

In the pre-stress reversal phase (⁠$P′max≥σr′≥P′EOL$⁠), $σr′$ is reduced from $P′max$ but is still greater than the stress reversal point that is mean effective stress (⁠$P′EOL$ as defined by equation (31)). The strain in this zone is governed by equation (43a):

In the post-stress reversal phase (⁠$PEOL′≥σr′≥σUR′$⁠), $σr′$ is reduced below the mean effective stress $PEOL′$ but still greater than yield stress in unloading (⁠$σ′UR$ as defined by equation (34) using power law behaviour). The strains are estimated by equation (44a) and are expressed as

In post-yield behaviour, $σr'$ is reduced below the yield stress (⁠$σUR'$⁠), inducing plasticity in the material. The strains will be defined using the derived analytical solution in the earlier section ‘Cavity contraction assumptions’ and are expressed using equation (29) as

where $εURref$ is cavity strain at the onset of plasticity without origin correction and is determined using $σUR′$ (defined as per power law). For hyperbolic law, it is defined as

Figure 11 presents the result of complete analysis and it is observed that a strong fit can be achieved utilising same values of $ϕ′p$ and $ϕ′cv$⁠. However, the dilation angles obtained from the two analyses differ. This discrepancy is attributed to differences in flow rules and is interpreted as loss in dilative potential resulting from the accumulation of shear strains during loading. Further evaluation of $ϕ′p$ and $ψ$ using the flow rule proposed by Bolton (1986) for plane-strain conditions indicates that the $ϕ′cv$ increases from 32° during loading to 36° during unloading, corresponding to a 12% increment in $ϕ′cv$⁠. This increase is consistent with experimental observations reported by Sadrekarimi & Olson (2010). Additional worked examples using different pressuremeter test methods including pre-bored and full-displacement field data are shown in the Appendix.

This research presents a non-linear elastic closed-form solution for evaluating stiffness and strength properties from the unloading branch of a pressuremeter test in cohesive–frictional soils. The proposed model incorporates the Nova & Wood (1982) shear–dilatancy relationship as a flow rule, alongside a Mohr–Coulomb yield surface, to account for volume changes during stress reversal and unloading. While the magnitude of volumetric strain is not explicitly quantified, the Nova & Wood (1982) framework is used to estimate the gain in peak shear strength $σ′p$ due to dilation $ψ$⁠, relative to the critical state shear strength $σ′cv$⁠.

Strength parameters obtained from the unloading phase show strong agreement with those derived from the Hughes & Whittle (2022) solution applied to the expansion phase.

Overall, the proposed framework offers a robust method for interpreting pressuremeter data during unloading, capturing key aspects of stress reversal, dilatancy and stiffness degradation, and enabling improved calibration of advanced soil models.

• The proposed model is currently valid only for data obtained from pressuremeter tests conducted in heavily overconsolidated materials. This limitation arises from the assumption of a constant angle of dilation, which is only true for dense sands.

• By assuming a linear stress path and neglecting volumetric strains during the elastic portion of the unloading phase, the model tends to overestimate shear strength for a given $ϕ′$ as evident by the discrepancy in yield stress illustrated in Fig. 3(b).

• The hyperbolic non-linear analysis is applied solely to an element located at the cavity wall. As a result, it does not account for spatial non-linear variations in material properties that occur throughout the plastic zone.

The authors declare that there is no conflict of interest regarding the publication of this work.

The first worked example is a pre-bored high-pressure pressuremeter test conducted in the Thanet Sand (TS) in London, UK. The test was conducted as part of the Canary Wharf redevelopment and was published in Whittle & Liu (2013). Observation of the field data suggests that the test pocket was over-drilled and the strain origin was shifted approximately 3·6% and the best-fit in situ horizontal stress was calculated as 655 kPa, corresponding with a $K0$ value of 2, which is in good agreement with Menkiti et al. (2015) for the shallow TS. Based on the unloading, the estimated peak friction angle, $ϕp′$⁠, was found to be 43° while the peak dilation angle, $ψp$⁠, was found to be 7°. The measured field data and corresponding analyses are included in Fig. 12.

The data presented in Fig. 13 are from a pre-bored pressuremeter test conducted at a depth of 52 m in silty sands (Unit 2) (Shahin, 2022) near the Padma Bridge, Bangladesh. The test cavity was oversized and exhibited localised relaxation at the cavity wall, which disturbed the in situ stress state of the soil. This disturbance shifted the strain origin by 7·1%, making the loading (expansion) arm unreliable for analysis. The silty sands are of Holocene age and were deposited through alluvial processes (De Silva et al., 2010). A series of self-bored and pre-bored pressuremeter tests indicated a friction angle, $ϕ$⁠, of 34° (Shahin, 2022). The groundwater table was observed at a depth of 1–2 m during the dry season (De Silva et al. (2010)), and this condition was assumed in the analysis.

Analysis of the unloading arm resulted in a friction angle of $ϕ$ = 35° assuming a critical state friction angle of $ϕcv$ 32°, which is consistent with values reported in the literature. Back-analysis of the loading arm using these friction angles yielded an in situ horizontal stress, $P0$⁠, of 710 kPa, corresponding to a pore water pressure of 490 kPa and a unit weight of 18 kN/m³. The resulting coefficient of earth pressure at rest, $K0$⁠, was 0·5, which is consistent with published values for Unit 2.

Figure 14 presents a self-bored pressuremeter test conducted at a depth of 47 m at an anonymous tailings facility. Analysis of the closing pressure indicates an ambient pore water pressure of 50 kPa. Interpretation of the loading arm using Hughes & Whittle (2022) solution yielded a $P0$ of 230 kPa, corresponding to a $K0$ = 0·25 and a $ϕp′$ of 40°, assuming a $ϕcv′$ = 31°. Analysis of the unloading arm using the proposed unloading solution resulted in the same $ϕp′$= 40°, for the assumed $ϕcv′$= 31°. These estimated values of $ϕp′$ and $ϕcv′$ are consistent with the values estimated during triaxial testing, thereby validating the applicability of the proposed method.

Finally, Fig. 15 presents a cone pressuremeter test carried out at a depth of 38·5 m in the Halton Till Formation (HTF) in Vaughan, Ontario, Canada. During cone penetration, very large strains were generated, causing the cavity wall to reach the limit pressure. As a result, the loading arm data could not be used, and the analysis relied on the unloading arm. The ambient pore water pressure was estimated to be 130 kPa from the closing pressure of the unloading arm. Previous studies report an internal friction angle, $ϕ$⁠, for HTF in the range of 34–36° (YRT, 2009). Analysis of the unloading arm gave a peak effective friction angle, $ϕp′$⁠, of 34°, assuming a critical state friction angle of $ϕcv′$ = 30°. This result is consistent with the values reported in the literature.