Estimation of Effective Internal Friction Angle by Ball Penetration Test: Large-Deformation Analyses

Abstract

The ball penetrometer is a full-flow penetrometer used as an alternative to the traditional cone penetrometer for characterizing the strength of soft sediments, particularly for offshore engineering, due to its large projection area. However, if the ball is penetrated under partially drained conditions, the resistance of the ball changes with the penetration velocity. The performance of ball penetration is examined employing a coupled large-deformation finite-element method. The reliability of numerical simulations under undrained and partially drained penetrations is verified by comparing with previous tests in the chamber and centrifuge. The backbone curve determining the penetration resistance on the spherical probe is proposed to quantify the influence of partially drained conditions, which captures the effect of the ratio of the diameters of the shaft and ball. Base on the backbone curve derived, an interpretation approach is proposed to predict the effective internal friction angle using the net resistance measured in the ball penetration tests with different penetration velocities.

1. Introduction

The reliability of a few soil laboratory tests is limited in offshore investigations since the soil samples are prone to be seriously disturbed during their extraction and transportation. Therefore, several types of penetrometers, such as those with conical or spherical probes, and the free-falling penetrometer, have been developed to obtain the in-situ stiffness, strength or permeability of soil [1,2,3]. Compared to the cone penetrometers used widely in practice, the ball penetrometer is more favorable for soft cohesive soils because of its much larger projection area. Another advantage of the ball penetrometer is that the effect of the overburden pressure on the penetration resistance can be minimized due to the spherical shape of the probe [4,5].

The cone resistance measured in the cone penetration test (CPT) is employed to estimate the effective internal friction angle of soil, φ′ [6,7]. The penetration velocity, v, in standard CPTs is 20 mm/s, with a conical probe of diameter 35.7 mm or 43.7 mm, indicating that the clayey soil around the moving probe is under undrained conditions. Most of the previous correlations between φ′ and the corrected cone resistance qt were against the penetrations under undrained conditions. For example, Lim et al. (2020) [8] proposed an empirical equation to estimate φ′ based on the normalized cone resistance Q.

$$Q=\frac{q_{\mathrm{t}}-{\sigma }_{\mathrm{v}0}}{{{\sigma }^{\prime }}_{\mathrm{v}0}}=1.8\frac{e_0{}^{0.15}}{{(\lambda \kappa )}^{0.1}}{M}^{0.7}{\mathrm{OCR}}^{\mathsf{\Lambda }}$$

(1)

where σ_v0 and σ′_v0 were total and effective vertical stress, respectively; e₀ was the initial void ratio; M = 6sinφ′/(3 − sinφ′); OCR represented the over-consolidation ratio; and the power Λ = 1 − κ/λ, and λ and κ were the slopes of the normal consolidation line and the swelling line, respectively. Accounting for the practical ranges of e₀, λ and κ, Zhang et al. (20234) [9] simplified Equation (1) with OCR = 1

$$Q=2.8M^{0.7}$$

(2)

When the CPTs are carried out in silty clays, partially drained conditions may be formed around the probe, as reported in Lehane et al. (2009) [10], Chow et al. (2020) [11] and Mo et al. (2020) [12]. The drainage conditions were distinguished through the normalized velocity of penetration, V, as suggested in DeJong and Randolph (2012) [13]

$$V=\frac{v{D}_{\mathrm{c}}}{c_{\mathrm{v}}}$$

(3)

where D_c was the cone diameter and c_v was the coefficient of consolidation. Drained and undrained conditions were approximately within V < 0.01 and V > 30, respectively. V = 0.01–30 indicated partial drainages, where the cone resistance was increased with the decreasing penetration velocity until the drained conditions were achieved [10,14]. Ouyang and Mayne (2021) [7] suggested an equation to predict φ′ (in degree) through CPTs under partially drained or undrained conditions, expressed as

$${\phi }^{\prime }=29.5B_q{}^{0.121}(0.256+0.336B_q+\log Q), B_q=\frac{\Delta u_2}{q_{\mathrm{t}}-{\sigma }_{\mathrm{v}0}}$$

(4)

where B_q was the normalized pore pressure and Δu₂ the excess pore pressure measured at the shoulder of the cone. The experimental evidence showed that the above equation is applicable for cohesive soils with φ′ ranging from 20° to 45°.

Compared to the studies of estimating φ′ through CPTs, few experimental or numerical results about the ball penetrometer have been reported publicly. The net ball resistance, q_bn, was defined as [5]

$$q_{\mathrm{bn}}=q_{\mathrm{b}}–{\sigma }_{\mathrm{v}0}\frac{d_{\mathrm{s}}}{D_{\mathrm{b}}}$$

(5)

where q_b was the corrected resistance of the spherical probe against the excess pore pressure measured at the equator of the ball; d_s was the shaft diameter; and D_b was the ball diameter. Similar to the cone penetrometer, the normalized resistance factors and penetration velocities of balls were expressed as

$$Q=q_{\mathrm{bn}}/{{\sigma }^{\prime }}_{\mathrm{v}0}$$

(6)

$$V=\frac{v{D}_{\mathrm{b}}}{c_{\mathrm{v}}}$$

(7)

There have been several experimental and theoretical studies on the penetration resistance of the spherical probe for soils under partially drained and undrained conditions [10,15]. Most of these studies focused on estimating the undrained shear strength of cohesive soil through the penetration resistance, such as Low et al. (2011) [16], Zhou et al. (2016) [17] and Chow et al. (2023) [18]. Additional efforts encompass numerical analyses of the tests in strain-softening and rate-dependent clays [19]. However, there is no publicly reported approach to predicting φ′ for the ball penetrometers, as far as we know. As more ball penetrometers have been used in offshore geotechnical investigations, the correlation between q_bn and φ′ under various drainage conditions is required.

In this paper, a large-deformation finite-element (LDFE) approach incorporating the modified Cam-Clay (MCC) model is employed to simulate the penetration of the ball in cohesive soil, with an aim to predict the effective internal friction angle through the resistance of the spherical probe. The existing model tests in the chamber and the centrifuge are reproduced to validate the approach developed, where the penetrations were under nearly undrained or partially drained conditions. The change in ball resistance with the drainage condition can be quantified by a unique curve as a function of the normalized penetration velocity. Finally, an equation is proposed to derive the effective internal friction angle by accounting for the influences of the ball geometry and soil type.

2. Materials and Methods

2.1. Strategies of Numerical Analysis

To prevent significant distortions of soil elements around the ball, an LDFE approach within the framework of effective stress-porous water coupled analysis, termed the remeshing and interpolation technique with small strain [20,21,22], was used. The effective stress–strain behaviors of cohesive soil were described by the MCC model. The penetration of the ball was divided into a series of incremental steps. During each of these steps, an updated Lagrangian calculation was conducted in the commercial package Abaqus, with an implicit integration scheme. At the start of every step, a new mesh was generated on the deformed soil, and the field variables, including the effective stresses and void ratio at each integration point and the excess pore pressures at each corner node, were mapped to this new mesh.

The penetration of the shafted ball was an axisymmetric problem, and the region of the soil was discretized using quadrilateral elements, which was referred to as CAX8RP within Abaqus. The sizes of the soil area were taken as 50 D_b radially and 70 D_b vertically, to avoid potential boundary effects, as presented in Figure 1. The base and vertical boundaries of the soil were constrained vertically and horizontally, respectively, and the soil surface was permeable. The ball–soil interaction was simplified as frictionless. The diameter of the ball D_b was 15 mm, and the diameter of the shaft d_s was 5 mm, unless otherwise indicated. To save computational efforts, the balls were pre-buried at a soil depth of no less than 8 D_b and were then moved downwards with a depth of 2 D_b. The strategy has been demonstrated to be sufficiently accurate for obtaining stable penetration resistance. The ball displacement of 0.05 D_b per incremental step was selected through trial calculations to guarantee the computational convergence and accuracy. More details of the LDFE analyses for ball or cone penetrometers can be found in Zhang et al. (2023) [3].

2.2. Soil Properties

Three types of clay or clayey silt were taken as examples here: (a) UWA kaolin. This was used extensively in the centrifuge tests—for example, in the tests of penetrometers by Mahmoodzadeh and Randolph (2014) [23]. (b) Malaysian kaolin. Its permeability was one order higher than that of UWA kaolin [22], suggesting that partially drained conditions may appear easily during ball penetration. (c) Burswood clay. This soil had a high effective internal friction angle of 31.5°, which facilitated an insightful verification of the large-deformation analyses. UWA kaolin is typical clay, while the latter two soil types are classified as clayey silts [24,25,26]. All three soil types were regarded as MCC materials, and their properties were reported by Purwana et al. (2006) [27], Mahmoodzadeh and Randolph (2014) [23] and Low and Randolph (2010) [26], as shown in

Table 1. Properties of soils used in large-deformation simulations.

Property	UWA Kaolin	Malaysian Kaolin	Burswood Clay
Angle of internal friction, ${\phi }^{\prime }$: degree	23	23	31.5
Void ratio at p′ = 1 kPa on virgin consolidated line, $e_{\mathrm{N}}$	2.25	2.35	2.25
Slope of normal consolidation line, $\lambda$	0.205	0.244	0.330
Slope of swelling line, $\kappa$	0.044	0.053	0.036
Poisson’s ratio, $\nu$	0.3	0.3	0.3
Submerged unit weight, ${\gamma }^{\prime }$: kN/m3	6	6	5.2

. The experimental findings for these commercial kaolins, such as those for CPT [10,11], spudcan footings [28,29], pipelines [30] and anchors [31,32], have been employed in recent offshore geotechnical practices. Therefore, the LDFE studies are to be for the Malaysian kaolin, UWA kaolin, and Burswood clay. According to Equation (8) in Wroth (1984) [33], for UWA and Malaysian kaolins at normally consolidated (NC) conditions, the profiles of undrained shear strength s_u deduced from the MCC model were both s_u = 1.46 z kPa, where z was soil depth. s_u = 1.55 z kPa for NC Burswood clay [23,26,27].

$$s_{\mathrm{u}}=\frac{\left(1+2K_0^{}\right)}{3}{\sigma }_{\mathrm{v}}{}_0\frac{M}{2}{\left(\frac{{\eta }_{}{}^2+M^2}{2M^2}\right)}^{\mathsf{\Lambda }}$$

(8)

where K₀ was the coefficient of earth pressure at rest and η = 3(1 − K₀)/(1 + 2 K₀).

A void ratio-dependent permeability function was suggested by Mahmoodzadeh and Randolph (2014) [23] to describe the permeabilities of soils. The permeability of soil is a function of the void ratio [28,34,35], and the void ratio-dependent permeability function has been used extensively in the coupled effective stress-pore pressure simulations of penetrometers [23] and spudcan footings [29]. The permeability, k, was assumed to be isotropic, calculated as

$$k={\gamma }_{\mathrm{w}}{m}_{\mathrm{v}}{c}_{\mathrm{v}}=\frac{{\gamma }_{\mathrm{w}}\lambda c_{\mathrm{v}}}{(1+e){\sigma }^{\prime }{}_{\mathrm{v}0}}$$

(9)

where γ_w was the unit weight of porous water, m_v was the compressibility during one-dimensional consolidation, and e was the void ratio. The coefficient of consolidation c_v is a traditional and widely used property in soil mechanics. It is a phenomenon covering the seepage and deformation of the porous material. c_v is obtained from one-dimensional consolidation tests and could be fitted, as described by Low and Randolph (2010) [26] and Wang and Bienen (2016) [22].

$$c_{\mathrm{v}}=39.7{\left({{\sigma }^{\prime }}_{\mathrm{v}0}/p_{\mathrm{a}}\right)}^{0.45} {\mathrm{m}}^2/\mathrm{y} \mathrm{for} \mathrm{Malaysian} \mathrm{kaolin}$$

(10a)

$$c_{\mathrm{v}}=\sqrt{1+14{{\sigma }^{\prime }}_{\mathrm{v}0}/p_{\mathrm{a}}} {\mathrm{m}}^2/\mathrm{y} \mathrm{for} \mathrm{UWA} \mathrm{kaolin}$$

(10b)

$$c_{\mathrm{v}}=1.56({{\sigma }^{\prime }}_{\mathrm{v}0}/p_{\mathrm{a}}{)}^{0.824} {\mathrm{m}}^2/\mathrm{y} \mathrm{for} \mathrm{Burswood} \mathrm{clay}$$

(10c)

where p_a was the atmospheric pressure. A flowchart about the processes of the methods used is shown in Figure 2.

3. Results

3.1. Validation

The model tests in the chamber and centrifuge conducted by Low and Randolph (2010) [26] and Mahmoodzadeh and Randolph (2014) [23], respectively, were reproduced to validate the reliability of the LDFE approach employed. The model balls were penetrated in NC Burswood clay or UWA kaolin, and the ball resistances measured were varied with the penetration velocities. In the following discussions, the normalized velocity of penetration V was obtained from Equation (7), and the values of c_v were calculated by Equation (10). According to extensive trial calculations, it was found that the backbone curve discussed later can be fitted by including V, although c_v is a function of soil depth. Therefore, the c_v at the final depth of the ball penetrometer was adopted for normalizations.

3.1.1. Chamber Test by Low and Randolph (2010)

The effective internal friction angle was measured as φ′ = 31.5° through the triaxial compression tests against the Burswood clay [26]. The effective internal friction angle was measured as φ′ = 31.5° through the undrained triaxial tests against the Burswood clay, where the soil sample was consolidated anisotropically and was sheared in triaxial compression mode. More details of triaxial tests and the corresponding results can be found in [16]. A ball with D_b = 11.9 mm and d_s = 5 mm was penetrated at a velocity of 1 mm/s up to a depth of 13.3 D_b. To prepare the sample, a surcharge of 35 kPa was applied at the soil surface. The surcharge represented a soil depth of 6.7 m with γ′ = 5.2 kN/m³ (see Table 1); therefore, the penetration depth of 13.3 D_b indicated an actual depth of 6.9 m. The corresponding c_v value was thus calculated as 0.6 m²/y by Equation (10c). V was found to be 619.8 by using Equation (7), suggesting that the penetration was essentially carried out under undrained conditions.

The experimental and numerical curves, q_bn/s_u versus penetration depth normalized by D_b, are plotted in Figure 3. The solid curve in the figure represents the results of the chamber test reported by Low and Randolph (2010) [26], while the dashed curve is the penetration profile found using the LDFE approach. A general agreement between the two curves is achieved. The experimental value of q_bn/s_u increased quickly at the early stage of the penetration and then reached a stable value of 10.7 rapidly at around 1 D_b. This rapid increase was attributed to the overburden stress of 35 kPa applied on the soil surface, as indicated by the experimental setup [26]. There existed an insignificant fluctuation of q_bn/s_u within the depth of 2.5 D_b–10 D_b, potentially due to the fact that the soil in the chamber was not ideally uniform. To save computational efforts, the LDFE simulation was performed with the spherical probe pre-embedded at 8.67 D_b. The penetration displacement was 5 D_b in the LDFE simulation, to guarantee a sufficiently steady value of q_bn/s_u. The undrained shear strength measured by the vane shear test was 14.2 kPa in Low and Randolph (2010) [26], and the resistance factor of the ball penetrometer was 10.7 in terms of s_u = 14.2 kPa. Zhou and Randolph (2009) [36] suggested a resistance factor of 10.5. In Figure 3, the experimental and numerical stable values of q_bn/s_u are both around 10.7 at the penetration distance larger than 1 D_b. The centrifuge data, LDFE results and existing empirical solutions agreed well with each other, validating the reliability of the LDFE approach.

3.1.2. Centrifuge Tests by Mahmoodzadeh and Randolph (2014) [23]

The centrifuge tests by Mahmoodzadeh and Randolph (2014) [23] were in terms of UWA kaolin, with a spinning acceleration of 110 g (g represents the gravitational acceleration). The effective internal friction angle of UWA kaolin was measured as 23° in the triaxial compression test. The model ball with D_b = 15 mm and d_s = 5 mm was moved deliberately at velocities of 0.01, 0.03, 0.05, 0.1 and 1 mm/s, respectively, to generate different drainage conditions in the soil around the probe. The final depth of the probe was around 10.67 D_b, i.e., equivalent depth z = 17.6 m. Based on the properties of UWA kaolin in Table 1, c_v was 4 m²/y at this depth. The normalized penetration velocities were thus calculated as 1.2, 3.6, 5.9, 11.8 and 118, indicating that the soil is under partially drained or undrained conditions. The spherical probe was pre-embedded at 5 D_b in the LDFE simulations, followed by a penetration distance of 5.67 D_b.

The variations of q_bn/s_u with z/D_b under different drainage conditions are compared in Figure 4, where the penetration velocities are selected deliberately to cover a wide range from 0.01 mm/s to 1 mm/s. The solid and dash curves represent the experimental and LDFE results, respectively. Fluctuations of the experimental curves were observed again, as in the chamber test discussed above [26], due to the non-uniform soil sample. Slight or moderate divergences between the numerical and experimental results were observed, but it was argued that the numerical curves agreed reasonably well with the experimental ones. Either experimental or numerical results highlighted a tendency that the normalized ball resistance was reduced with increasing V. The soil depth where the steady value of q_bn/s_u was reached depended on the penetration velocity: this critical depth was shallower under undrained conditions, and the depth became deeper with decreasing penetration velocity. For example, in the LDFE simulations, the steady q_bn/s_u value was reached at a penetration distance of 1 D_b for V = 118 (v = 1 mm/s), while a penetration distance of 4 D_b was required for V = 1.2 (v = 0.01 mm/s).

3.2. Interpretation of Ball Resistance under Partial Drainage Condition Penetrations

To explore the influence of the drainage condition on the normalized penetration resistance and then on the estimation of φ′, a parametric study based on the LDFE analyses was conducted. The velocity of penetration v range between 0.1 mm/s and 10 mm/s in Malaysian kaolin and v = 0.001–1 mm/s in Burswood clay was used to achieve the partially drained or undrained conditions. To investigate the penetration of the ball penetrometer in deep soils, a pressure of 100 kPa was applied to the soil surface in all groups. The numerical simulations were in terms of the ball, with D_b = 15 mm at the acceleration of 1 g. The soil types and the diameter ratio of the shaft and probe of the ball, d_s/D_b, were varied deliberately. As shown in

Table 2. Numerical analyses of balls.

Group	Soil	z/Db	ds/Db	v: mm/s	V = vDb/cv
1	Malaysian kaolin	10.67	1/3	10	115.3
				5	57.6
				3	34.6
				2	23.1
				1	11.5
				0.5	5.8
				0.2	2.4
				0.1	1.2
2	Malaysian kaolin	10.67	1/4	10	115.3
				5	57.6
				3	34.6
				2	23.1
				1	11.5
				0.5	5.8
				0.2	2.4
				0.1	1.2
3	Malaysian kaolin	10.67	2/5	10	115.3
				5	57.6
				3	34.6
				2	23.1
				1	11.5
				0.5	5.8
				0.2	2.4
				0.1	1.2
4	Burswood clay	10.67	1/3	1	300
				0.5	150
				0.2	60
				0.1	30
				0.05	15
				0.02	6
				0.01	3
				0.001	0.3

Note: the diameter of the probe is 15 mm in all groups.

, the LDFE analyses are classified into four groups.

3.2.1. Backbone Curve

The results of Malaysian kaolin, group 1 of Table 2, are plotted in Figure 5. Similar to that observed in the centrifuge tests of UWA kaolin, the normalized resistance is a function of V. The steady q_bn value decreases slightly as V is increased from 34.6 to 115.3, which suggests that the undrained conditions are nearly approached at V = 34.6. The q_bn value at V = 34.6 is thus defined as q_bn^un, the resistance under undrained conditions.

For groups 1 and 4, which are in terms of two soil types, the variations of the normalized ball resistances with the normalized penetration rate are demonstrated in Figure 6a. The experimental data of UWA kaolin from Mahmoodzadeh and Randolph (2014) [23] and Bassendean clay from Suzuki (2015) [37] are plotted in Figure 6a as well. Obviously, the numerical and experimental data are scattered within a wide range. This indicates a significant divergence in ball resistances due to the soil types, effective internal friction angle of soil, and depth of the probe: (a) The effective internal friction angles of both UWA kaolin and Malaysian kaolin reported in Mahmoodzadeh and Randolph (2014) [23] are 23°, and hence, the q_bn values of the two are close to each other for a given penetration velocity and a given depth. The angle of the Burswood clay (group 4 in Table 2) is 31.5°, which is much higher than that of UWA or Malaysian kaolin, resulting in larger q_bn values. (b) Different q_bn values of the spherical probe in Bassendean clay were observed at soil depths of 3–4 m and 4–5 m in Suzuki (2015) [37]. As demonstrated in Figure 6a, for a penetrometer in a given soil, a larger penetration resistance is reached due to higher overburden stress in a deeper soil. This is proven as well by the higher resistance in Malaysian kaolin in group 1, where the probe is penetrated to an equivalent depth of about 17 m (pressure 100 kPa applied on the soil surface). The higher overburden stress, similar to the higher confining stress in triaxial tests or vertical stress in direct simple shear tests, results in an enhanced shear strength under undrained or drained conditions.

If q_bn is normalized by q_bn^un, all results are located near a unique curve, as shown in Figure 6b. This backbone curve may be expressed as

$$\frac{q_{\mathrm{bn}}}{q_{\mathrm{bn}}{}^{\mathrm{un}}}=1+\frac{a}{1+{\left(V/V_{50}\right)}^{\mathrm{b}}}$$

(11)

where the coefficients are fitted as V₅₀ = 3, a = 1.1 and b = 1.2 for ball penetrometers. According to Equation (11), the net ball resistance with extremely low penetration velocity (i.e., drained conditions are formed around the ball) is 2.1 times that under undrained conditions, which is similar to the observation in Figure 4. Based on the experimental and numerical observations, DeJong et al. (2012) [12] and Martinez et al. (2018) [38] adopted expressions similar to Equation (11) for cone penetrometers, with V₅₀ ranged between 3 and 4, a between 1.5 and 2.25, and b between 1.0 and 1.6. In Figure 6b, the effective internal friction angles of the soils discussed are in the range of between 23° and 32°. Clearly, the numerical and existing experimental data can be fitted well by Equation (10).

The backbone curve in Figure 6b is composed of three parts: (a) q_bn/q_bn^un = 1 occurs approximately at V > 50, indicating the undrained condition. This phenomenon is similar to that reported in Mahmoodzadeh and Randolph (2014) [23] and Chung et al. (2006) [39]. (b) As reported in Colreavy et al. (2016) [40], V < 0.7 represents nearly drained conditions. Here, q_bn/q_bn^un varies slightly with V as V < 0.7, which is in accordance with the previous finding. and (c) The partially drained conditions occur at 0.7 < V < 50, therefore, V = 50 and 0.7 denote transitions of the drainage conditions.

3.2.2. Effect of the Penetrometer Geometry

To investigate the influence of penetrometer geometry, the ratio between the diameters of the shaft and ball, d_s/D_b, is varied by following Zhou and Randolph (2009) [36]. The ball diameter is maintained as 15 mm, while the values of d_s/D_b are taken as 1/3, 1/4 and 2/5, respectively, in groups 1–3 in Table 2. As shown in Figure 7, the ratio of q_bn and q_bn^un is changed slightly with varying d_s/D_b for a given normalized velocity of penetration V. All numerical data are scattered within a narrow range, suggesting that the backbone curve expressed by Equation (11) is near, regardless of the penetrometer geometry over the range of d_s/D_b ratios studied.

3.3. A New Method to Predict φ′ by Q and V

Based on the backbone curve, the expression between φ′ and Q under different drainage conditions is further developed. The net ball resistance factor of the ball is 10.5 [36], which is close to the factor of 10 for the cone. Therefore, the normalized ball resistances, Q, for the penetrometers of the ball and cone are similar. Equation (1) provides the empirical relationship between Q and φ′ for CPTs under undrained conditions. Here, the equation is expanded to the ball penetrometers under undrained conditions. By combining Equation (2), Equation (10) can be reorganized as q_bn/q_bn^un = Q/(2.8 M^0.7). Further, a relationship between Q and φ′ can be established, accounting for the effect of drainage conditions for NC soils

$$Q=2.8M^{0.7}\left(1+\frac{1}{1+(V/3{)}^{1.2}}\right)$$

(12)

Once Q and V of the ball penetrometer are known, φ′ can be derived directly from Equation (12). Equation (12) fits for both undrained and partially drained conditions of soil around the probe.

The experimental and numerical data under different drainage conditions covering the φ′ ranged from 23° to 32°. These cases include the ball penetration tests in Malaysian kaolin, Burswood clay and Bassendean clay under different drainage conditions. The LDFE approach is used in the first two soil types and the data for Bassendean clay are reported by Suzuki (2015) [37]. The values of φ′ obtained by Equation (12) and measured from the triaxial compression tests for these soil types are plotted in Figure 8. The horizontal axis is the normalized penetration velocity on a logarithmic scale, and the vertical axis represents the effective friction angle (in degree) estimated by Equation (12). The red dashed lines in Figure 8 represent the φ′ values measured from the triaxial compression tests. The effective internal friction angles estimated are ranged between 21° and 24°, 26° and 31°, and 30° and 33°, respectively. It is found that the predicted values of φ′ at low velocities are underestimated for Malaysian kaolin (see Figure 8a), where various d_s/D_b ratios are distinguished by markers of different colors: φ′ is predicted as 21.1° at V = 1.2 and 21.7°–22.2° at V = 2.4. As V > 3, the estimated values of φ′ are in the range of 22.2°–23.7°, close to 23° measured in the triaxial compression tests. The predicted values of φ′ are located around the red dash line for Bassendean clay and Burswood clay, which ranged between 26.6°–30° and 30.3°–32.8°, respectively, i.e., the estimation errors are no more than ±2° (see Figure 8b,c). Totally, all estimated φ′ values of three soil types are scattered in narrow ranges, while the divergences between the measured and predicted become relatively large under nearly drained conditions.

4. Discussions

Almost all existing experimental, numerical and analytical studies for ball penetrometers were aimed at obtaining the undrained shear strength or coefficient of consolidation of cohesive soil [16,17,18,22]. The new method developed here is based on insights from Lim et al. (2020) [8] and Zhang et al. (2024) [9] on predicting φ′ through the in-situ tests. The advantages of the new method are: (a) Penetrations of the ball under undrained or partially drained conditions can be tackled within a general framework, and (b), The measurement of pore pressure is not mandatory. However, this method is proved against normally consolidated soils only, and it remains unclear if the method can be expanded to overconsolidated soils.

The procedure below is presented to estimate the effective internal friction angle through in-situ tests using ball penetrometers:

(a)

Given that the ball diameter, the coefficient of consolidation of soil, and the penetration velocity are known, the normalized penetration rate is calculated through Equation (7).

(b)

As the net resistance (Equation (5)) and the effective vertical stress are recorded, the normalized ball resistance Q is determined through Equation (6).

(c)

Equation (12) is employed to estimate the slope of the critical state line M and then the value of φ′.

5. Conclusions

A large-deformation finite-element approach is developed to investigate the ball penetration resistance under varying drainage conditions, followed by estimating the effective internal friction angle of cohesive soil through the penetration resistance. The numerical approach is validated by comparing with the model tests in the chamber and in the centrifuge under both undrained and partially drained conditions. A comprehensive parametric study is conducted to explore the key factors, including the soil type, the ratio between the shaft and probe diameters, the penetration velocity, and the soil depth. From the large-deformation analyses and previous data from tests, the following conclusions can be drawn.

The net penetration resistance under partially drained conditions is normalized against that under undrained conditions, i.e., q_bn/q_bn^un. It is found that the value of q_bn/q_bn^un is a function of the normalized penetration velocity V. A unique backbone curve between q_bn/q_bn^un and V is proposed, suitable for a wide range of drainage conditions of cohesive soil around the advancing spherical probe. According to a large quantity of parametric studies, the backbone curve is nearly independent of the ratio between the shaft and probe diameters and the soil types. By expanding the backbone curve, a novel empirical equation denoted as Equation (12) is suggested to estimate the effective internal friction angle.