### SUMMARY

- Top of page
- SUMMARY
- INTRODUCTION
- DEFINITION OF PROBLEM
- CAVITY EXPANSION IN TWO CONCENTRIC REGIONS OF SOIL
- VALIDATION WITH FINITE ELEMENT SIMULATION
- RESULTS
- COMMENTS ON GEOTECHNICAL APPLICATIONS
- CONCLUSIONS
- LIST OF NOTATION
- APPENDIX
- REFERENCES

An analytical solution of cavity expansion in two different concentric regions of soil is developed and investigated in this paper. The cavity is embedded within a soil with finite radial dimension and surrounded by a second soil, which extends to infinity. Large-strain quasi-static expansion of both spherical and cylindrical cavities in elastic-plastic soils is considered. A non-associated Mohr–Coulomb yield criterion is used for both soils. Closed-form solutions are derived, which provide the stress and strain fields during the expansion of the cavity from an initial to a final radius. The analytical solution is validated against finite element simulations, and the effect of varying geometric and material parameters is studied. The influence of the two different soils during cavity expansion is discussed by using pressure–expansion curves and by studying the development of plastic regions within the soils. The analytical method may be applied to various geotechnical problems, which involve aspects of soil layering, such as cone penetration test interpretation, ground-freezing around shafts, tunnelling, and mining. © 2014 The Authors. *International Journal for Numerical and Analytical Methods in Geomechanics* published by John Wiley & Sons Ltd.

### INTRODUCTION

- Top of page
- SUMMARY
- INTRODUCTION
- DEFINITION OF PROBLEM
- CAVITY EXPANSION IN TWO CONCENTRIC REGIONS OF SOIL
- VALIDATION WITH FINITE ELEMENT SIMULATION
- RESULTS
- COMMENTS ON GEOTECHNICAL APPLICATIONS
- CONCLUSIONS
- LIST OF NOTATION
- APPENDIX
- REFERENCES

Cavity expansion theory has been extensively developed and widely used for the study of many engineering problems. Since its early application to geotechnical problems in the 1960s [1], many analytical solutions have been proposed using increasingly sophisticated constitutive soil models. Vesic [2] presented an approximate solution for spherical cavity expansion in an infinite soil mass using a compressible Mohr–Coulomb material. The analysis was applied to evaluate the bearing capacity factors of deep foundations. Carter *et al.* [3] derived closed-form solutions for cavity expansion from zero initial radius in an ideal cohesive-frictional material with a small-strain restriction. The deformations in the elastic region were assumed to be infinitesimal, and the convected term of the stress rate was neglected in the governing equation, which provided an approximate limit pressure solution. Yu and Houlsby [4] provided a unified analytical solution of cavity expansion in dilatant elastic-plastic soils, using the Mohr–Coulomb yield criterion with a non-associated flow rule. The complete large-strain analysis, with the aid of a series expansion, was introduced to derive a rigorous closed-form solution without any additional restrictions or assumptions. The limitation of their analysis was that the material properties were assumed to be constant and independent of stress–strain history. Salgado *et al.* [5] reported a cylindrical cavity expansion solution and produced a stress rotation analysis for the interpretation of the cone penetration test (CPT). A numerical formulation was used in the plastic region to achieve a variable stiffness, friction angle, and dilation angle.

As reviewed by Yu [6], the cavity expansion theory has mainly been applied in the geotechnical engineering areas of in-situ soil testing [7-9, 5, 10, 11], deep foundations [12-15], tunnels and underground excavations [16-18], and recently, for an interaction analysis between tunnels and piles [19, 20]. Despite the wide application of the theory to geotechnical problems, very little work has been carried out to consider the effect of distinct soil regions within the framework of cavity expansion analyses. The work of Xu and Lehane [21] is notable for its use of a numerical analysis of spherical cavity expansion for investigating pile or probe resistance in two-layered soil profiles.

Analytical cavity expansion solutions for two concentrically layered media were developed by Bernard [22, 23] for the study of projectile penetration. The analysis considered an incompressible material as well as the assumption of a finite locking strain and was used to solve for dynamic solutions of penetration depth and impact velocity. Sayed and Hamed [24] were the first to apply analytical cavity expansion analyses of concentrically layered media to the field of geomechanics. However, in their analysis, the medium was assumed to be a frictionless linear-elastic solid and did not account for the plastic behaviour of soils. In this paper, the analytical solution described by Yu and Houlsby [4] is extended in order to consider a cavity embedded within a profile of two different concentric regions of soil. The soil is treated as an isotropic dilatant elastic-perfectly plastic material with a Mohr–Coulomb yield criterion and a non-associated flow rule. Large-strain quasi-static expansion of both spherical and cylindrical cavities is considered. The complete large-strain expansion for non-associated Mohr–Coulomb materials in two concentric media has not previously been presented in the literature. The development of an analytical cavity expansion method for application to geotechnical problems involving aspects of soil layering is the main motivation for the work described in this paper. The focus here is on the development of the analytical method; its application to practical geotechnical problems will be explored in future publications.

The paper begins with a general definition of the problem and the necessary geometric parameters. The following section considers the most general expansion problem for a cavity embedded in two different concentric regions of soil and derives expressions for stresses, strains, and displacements within elastic and plastic zones. The cavity expansion solution is then validated against results obtained using the Finite Element (FE) method. Further results and parametric analyses are then presented with focus placed on the resulting pressure–expansion curves and the development of plastic regions within the two soil zones. A discussion of the application of the proposed method and its limitations is provided, followed by concluding remarks.

### DEFINITION OF PROBLEM

- Top of page
- SUMMARY
- INTRODUCTION
- DEFINITION OF PROBLEM
- CAVITY EXPANSION IN TWO CONCENTRIC REGIONS OF SOIL
- VALIDATION WITH FINITE ELEMENT SIMULATION
- RESULTS
- COMMENTS ON GEOTECHNICAL APPLICATIONS
- CONCLUSIONS
- LIST OF NOTATION
- APPENDIX
- REFERENCES

The problem involves three concentric zones; (i) an inner zone representing the expanding cavity, (ii) a second zone representing Soil A, and (iii) a bounding region, which extends to infinity and represents Soil B, as shown in Figure 1(a). Initially, the cavity has a radius *a*_{0}, and the interface between Soils A and B is located at a radial distance *b*_{0} from the centre of the cavity. The soils are assumed to be isotropic homogeneous media; therefore, an initial hydrostatic stress *P*_{0} acts throughout both Soils A and B as well as within the cavity. Note that a tension positive notation is used in this paper.

When the cavity pressure *P*_{a} increases slowly from its initial value *P*_{0}, the radius of cavity and Soil A/B interface are expanded to *a* and *b*, respectively (Figure 1(b)). The pressure at the Soil A/B interface is given by *P*_{b}. Depending on material properties (and adopting the Mohr–Coulomb yield criterion), a plastic region may form within either of the Soils A or B and extend to some radial distance *c*_{A} or *c*_{B}, respectively. For a given increment of cavity expansion, the initial plastic-elastic interfaces in Soils A and B are given by *c*_{0A} and *c*_{0B}, respectively. The radial stresses at the plastic-elastic interfaces for Soils A and B are defined as and , respectively.

As in the work of Yu and Houlsby [4], the soils are modelled as an isotropic dilatant elastic-perfectly plastic material, obeying Hooke's law for elastic analysis and the Mohr–Coulomb yield criterion with a non-associated flow rule for plastic analysis. The properties of Soils A and B are denoted by subscripts 1 and 2, respectively: Young's modulus (*E*_{1}, *E*_{2}), Poisson's ratio (*ν*_{1}, *ν*_{2}), cohesion (*C*_{1}, *C*_{2}), friction angle (*φ*_{1}, *φ*_{2}), and dilation angle (*ψ*_{1}, *ψ*_{2}).

To combine both spherical and cylindrical analyses, the parameter *k* is used to indicate spherical analysis (*k* = 2) or cylindrical analysis (*k* = 1). It should be noted that for the cylindrical case, plane strain conditions in the *z* direction are assumed and the axial stress is assumed as the intermediate principal stress, which is satisfied for most realistic values of soil parameters, as discussed by Yu and Houlsby [4]. In accordance with Yu [6], the following parameters are used for mathematical convenience (definitions provided in the Appendix): *G*, *M*, *Y*, *α*, *β*, *γ*, *δ*.

During cavity expansion, plastic regions in the two concentric soils are generated and develop depending on the relevant properties and profiles of Soils A and B. Considering all possible situations, the expansion process would follow one of the routes in the flow chart illustrated in Figure 2, which also provides a definition of some notation. Generally, during expansion of the cavity from *a*_{0} to *a*, an elastic stage (*A*_{E}*B*_{E}) appears initially, followed by plastic regions developing in both Soils A and B as *a* increases (*A*_{PE}*B*_{PE}). Ultimately, as *a* is increased further, Soil A becomes fully plastic (*A*_{P}*B*_{PE}) (Soil B extends to infinity and therefore never becomes fully plastic). The events at the circular nodes in the flow chart describe the situation of expansion and determine the appropriate state of soil to be considered. The solutions provided here are for the most general case of expansion (*A*_{PE}*B*_{PE}); all the scenarios described in Figure 2 can be deduced from this general solution.

### VALIDATION WITH FINITE ELEMENT SIMULATION

- Top of page
- SUMMARY
- INTRODUCTION
- DEFINITION OF PROBLEM
- CAVITY EXPANSION IN TWO CONCENTRIC REGIONS OF SOIL
- VALIDATION WITH FINITE ELEMENT SIMULATION
- RESULTS
- COMMENTS ON GEOTECHNICAL APPLICATIONS
- CONCLUSIONS
- LIST OF NOTATION
- APPENDIX
- REFERENCES

The accuracy of the analytical model was initially confirmed against results obtained with the fundamental solutions from Yu and Houlsby [4] for the case where the properties of Soils A and B were identical. To further validate the analytical model, two FE numerical models were developed in Abaqus/Standard and used to simulate the expansion of both spherical and cylindrical cavities, as shown in Figure 3. The axis-symmetric option was used in Abaqus in order to achieve spherical and cylindrical analyses using the 2D models. The cavity was expanded from an initial radius of 6 mm under an initial pressure of 1 kPa. The initial radius of the Soil A/B interface (given by *b*_{0}) was 30 mm, while Soil B had a radius *D*, which was large enough to make boundary effects negligible. In the numerical simulations, the properties of both Soils A and B were set as follows: *ν* = 0.2, *φ* = 10 °, *ψ* = 10 °, *C* = 10 kPa.

A total of four expansion tests were carried out using the numerical model in which the Young's modulus (*E*) of Soils A and B was either 1 or 10 MPa (results presented in Figure 4). The labels on the figure indicate the model (analytical = CEM; numerical = FEM), followed by the value of Young's modulus of Soil A and B, respectively. Hence, the label CEM-10-1 relates to the analytical cavity expansion analysis results in which Soil A has *E*_{1} = 10 MPa and Soil B has *E*_{2} = 1 MPa. Figure 4 shows that very good agreement between analytical and numerical results was obtained.

As indicated in Figure 4(a) for spherical expansion, for the uniform soil tests (‘-10-10’ and ‘-1-1’), the cavity pressure (*P*_{a}) increases gradually with cavity displacement and asymptotically approaches a limit pressure. The limit pressure of the soil with *E* = 10 MPa is shown to be nearly twice as large as that with *E* = 1 MPa. For the tests with two different soils (two-region tests), the pressure–expansion curves initially follow the trend in which the *E* of the uniform soil tests matches the value of *E* in Soil A of the two-region tests (i.e. ‘-10-1’ matches ‘10-10’ and ‘-1-10’ matches ‘-1-1’). At a certain stage, the existence of Soil B begins to have an effect, and the pressure–expansion curve of the two-region analysis tends towards the limit pressure obtained from the uniform soil test in which *E* matches that of Soil B of the two-region test (i.e. ultimately ‘-10-1’ approaches ‘1-1’ and ‘-1-10’ approaches ‘-10-10’). Figure 4(b) shows equivalent results for cylindrical cavity expansion and illustrates that cylindrical pressures are about 60 % of those from the spherical analysis.

The effect of a distinct change in soil stiffness on the pressure expansion curves is shown to be significant in Figure 4. The limit pressure is often applied to investigate pile capacity or probe resistance in conventional cavity expansion solutions (e.g. [13]). This approach is appropriate for uniform soils because the limiting pressure is only affected by the parameters of a single soil. For concentric regions of two different soils, Figure 4 shows that the limiting pressure depends only on the properties of Soil B. For penetration problems such as CPT or pile capacity analysis, the resistance of a probe located in Soil A will certainly depend on the properties of Soil A, so the limit pressure approach may not be adequate for cases where two different soils affect results (e.g. layered soils). A more suitable approach for layered soils, as suggested by Xu and Lehane [21], is to consider a realistic increase in cavity size (given by *a* ∕ *a*0) and to evaluate the cavity pressure required to achieve this expansion. The results of such an approach for the case of cavity expansion in two concentric regions of different soils (rather than horizontally layered soils) are explored further in the next section.

### RESULTS

- Top of page
- SUMMARY
- INTRODUCTION
- DEFINITION OF PROBLEM
- CAVITY EXPANSION IN TWO CONCENTRIC REGIONS OF SOIL
- VALIDATION WITH FINITE ELEMENT SIMULATION
- RESULTS
- COMMENTS ON GEOTECHNICAL APPLICATIONS
- CONCLUSIONS
- LIST OF NOTATION
- APPENDIX
- REFERENCES

This section considers the cavity expansion method in two concentric regions of different soils and investigates the effect of various parameters on model results. Results are based on the expansion of a cavity from *a*_{0} = 0.1 mm to *a* = 6 mm (*a* ∕ *a*_{0} = 60). As illustrated in Figure 4, the two-region tests are highly sensitive to the ratio *a* ∕ *a*_{0} (the value of *a*_{0} has no effect on the normalised pressure expansion curves as long as the ratio of *b*_{0} ∕ *a*_{0} is maintained). The selection of these cavity parameters was based on geotechnical centrifuge experiments being carried out as part of this research (to be published in a future paper) in which a 6 mm radius penetrometer is pushed into sand with an average grain size of approximately 0.1 mm. The cavity expansion analysis was conducted with a Soil A/B interface at *b*_{0} = 30 mm and initial hydrostatic stress *P*_{0} = 1 kPa. The following material parameters are taken for baseline comparison (note that subscripts 1 and 2 refer to Soils A and B, respectively): *ν*_{1} = *ν*_{2} = 0.2; *φ*_{1} = *φ*_{2} = 40 °; *ψ*_{1} = *ψ*_{2} = 10 °; *C*_{1} = *C*_{2} = 0 kPa. As in the previous section, results here focus mainly on the effect of varying the value of Young's modulus *E* of the two soils (*E*_{1} = 10 or 1 MPa; *E*_{2} = 10 or 1 MPa).

Figure 5 shows the distribution of radial (a, b) and tangential (c, d) stresses respectively, for both spherical and cylindrical cavity expansion, as radial distance from the cavity (*r*) is increased. The results from the two-region tests are bounded by the results from the uniform soil tests (‘-10-10’ and ‘-1-1’). A sharper decrease in stresses is noted for the spherical cases compared to the cylindrical cases. There is an interesting difference between the spherical and cylindrical analysis results. For the cylindrical tests, the results for the two-region analysis appear to be mainly controlled by the value of *E* of Soil B (‘-10-1’ effectively matches ‘-1-1’ and ‘-1-10’ is close to ‘-10-10’). For the spherical tests, however, the data from both the two-region tests are close to the uniform test ‘-1-1’. The reason for this behaviour is the different degree of interaction between Soils A and B within the spherical and cavity expansion analyses, which will be discussed later in the text using pressure–expansion curves (Figure 8).

Normalised displacement distributions are presented in Figure 6 and show that results for all tests closely agree. This is due to the kinematic nature of the expansion problem; the differences between the lines shown in Figure 6 (for constant values of friction and dilation angles in Soils A and B) are due only to the effect of yielding. For purely elastic behaviour, the displacements are insensitive to the elastic parameters (as in the elastic half-plane analysis of Verruijt and Booker [26] for displacements around tunnels).

The spherical test ‘CEM-1-10’ is selected to investigate the variation of displacement with strength and plastic-flow parameters (i.e. friction and dilation angles), as shown in Figure 7. For tests with uniform parameters in Soils A and B, the displacements increase with an increase in dilation angle (Figure 7(b)), whereas displacements decrease only marginally with an increase in friction angle (Figure 7(a)). The effect of varying friction angle of the two soils in the two-region tests is difficult to observe because the overall effect on displacements is small. The magnified zone in Figure 7(a) shows that the effect of friction angle is bounded by the uniform tests. The magnitudes of the differences are of little practical concern. For dilation angle, the behaviour for the two-region tests is dominated by the value of dilation angle in Soil A, where the lines with equal values of *ψ*_{1} are shown to overlap in Figure 7(b).

The pressure–expansion curves in Figure 8 show the effects of the two different concentric regions of soil, as discussed previously where analytical results were validated against FE simulations. As the cavity size (*a* ∕ *a*_{0}) is increased, the curves from the uniform soil tests reach a limit pressure. The limit pressure is reached quite quickly (in terms of *a* ∕ *a*_{0}) for the uniform soil tests (*a* ∕ *a*_{0} < 20 for spherical and cylindrical tests), while the two-region tests reach the limit pressure after a much greater expansion (*a* ∕ *a*_{0} ranging from 250 to > 500 for the spherical tests and from about 100 to 500 for the cylindrical tests).

The distinction between the two-region effects in the spherical and cylindrical analyses mentioned in discussion of Figure 5 can be explained using Figure 8. For the analysis, in which *a* ∕ *a*_{0} = 60, Figure 8 shows that the cavity pressure is generally dominated by the stiffness of Soil B, except for the spherical test ‘CEM-1-10’. The two concentric zones have a significant effect in this spherical expansion test at the considered expansion state, whereas in the cylindrical analysis, the effect is minimal. This explains the difference in stress distributions between the spherical and cylindrical tests in Figure 5.

In Figure 9, the development of normalised plastic radius (*c*_{A} ∕ *a*, *c*_{B} ∕ *a*) in Soils A and B as the normalised cavity radius increases is presented for the case of spherical cavity expansion, as well as the Soil A/B interface *b* ∕ *a*, plotted with dotted lines. The uniform soil test results in Figure 9(a and b) show that plastic radius increases linearly with expansion after a small initial stage of nonlinear development (*a* ∕ *a*_{0} < 5). The growth of the plastic region is noted to be much faster in the test with higher stiffness, resulting in Soil A becoming fully plastic (*A*_{P}) at a much lower expansion ratio in test ‘10-10’ (*a* ∕ *a*_{0} = 12) compared to test ‘-1-1’ (*a* ∕ *a*_{0} = 32). For the two-region tests ‘-10-1’ and ‘-1-10’, the results in Figure 9(c and d) show the development of plastic radius within the different expansion stages (refer to Figure 2 for definition of labels). In test ‘-10-1’, for *a* ∕ *a*_{0} between 11 and 22, Soil A is fully plastic while Soil B remains fully elastic (*A*_{P}*B*_{E}). In test ‘-1-10’, there is a stage during which Soil B becomes partially plastic prior to Soil A becoming fully plastic (*A*_{PE}*B*_{PE}). The nonlinear behaviour of the plastic radius in the two-region tests is much more obvious compared to the uniform soil tests. All tests eventually tend towards an ultimate state in which further expansion generates a linear increase of the plastic radius (i.e. *c*_{B} ∕ *a* levels off, which is discernible in the figures).

Figure 10 shows the equivalent results for the cylindrical cavity expansion. The cylindrical results show a significantly faster development (in terms of *a* ∕ *a*_{0}) and higher value of plastic radius (*c*_{A}, *c*_{B}) compared to the spherical analysis results.

The spherical test ‘CEM-1-10’ is selected to study the effect of the variation of friction angle on the pressure–expansion curves and the development of cavity radius in Figure 11. The effect of the two different concentric regions of soil on cavity pressure (Figure 11(a)) is clearly shown where cavity pressure is initially controlled by Soil A but is then controlled by Soil B at larger expansion ratios. Plastic radius of Soil A (*c*_{A}) is dominated by Soil A (‘ *φ*_{1} = 40 °; *φ*_{2} = 40 °’ is close to ‘ *φ*_{1} = 40 °; *φ*_{2} = 20 °’, and ‘ *φ*_{1} = 20 °; *φ*_{2} = 20 °’ overlaps ‘ *φ*_{1} = 20 °; *φ*_{2} = 40 °’), as shown in Figure 11(b). The tests with a lower friction angle in Soil A have larger values of *c*_{A}, earlier appearance of *c*_{B}, and larger values of *c*_{B}. Figure 12 shows similar results for the effect of variation of dilation angle from spherical test ‘CEM-1-10’. The development of plastic radius *c*_{A} and *c*_{B} are mainly controlled by Soil A, while a lower dilation angle in Soil A leads to a smaller value of *c*_{A} before Soil A becomes fully plastic (*A*_{P}).

The results of the two-region analyses also depend to a large degree on the size of Soil A. Indeed, for some critical size of Soil A, Soil B should have no effect on the results of the analysis. Figure 13 shows the variation of cavity pressure with the size of Soil A (given by *b*_{0}) for cavities expanded from *a*_{0} = 0.1 mm to *a* = 6 mm. The results for the uniform soil tests are, as expected, unaffected by the variation of *b*_{0}. For the two-region tests, when *b*_{0} is small, the cavity pressure is close to the uniform soil test where *E* matches the value of *E* in Soil B of the two-region test. As *b*_{0} increases, the two-region effects diminish, and the cavity pressure approaches the uniform soil test pressure in which *E* matches the value of *E* in Soil A of the two-region test. The value of *b*_{0} at this stage can be considered as defining the critical size of Soil A, referred to as *b*_{0,crit}; for Soil A larger than *b*_{0,crit}, there will be no effect of the outer region of soil. For example, for the spherical test ‘-1-10’ in Figure 13(a), the cavity pressure decreases from about 290 kPa (equivalent to the ‘-10-10’ test) and approaches the pressure of the ‘-1-1’ test when *b*_{0} ∕ *a* is about 25. This value of *b*_{0} ∕ *a* defines the critical size of Soil A in order for the two regions to have an effect in the spherical cavity expansion analysis. In contrast, the critical size for test ‘-10-1’ is about three times larger than that of test ‘-1-10’ (*b*_{0} ∕ *a* ≅ 90 where ‘-1-10’ line approaches ‘-10-10’ line), illustrating the effect of soil stiffness on the critical size. The cylindrical analysis results in Figure 13(b) show a much larger critical size compared to the spherical results.

Figure 14 shows the variation of plastic radius (*c*_{A}, *c*_{B}) with *b*_{0} for both spherical and cylindrical analyses for cavity expansion from *a*_{0} = 0.1 mm to *a* = 6 mm. The grey areas indicate values of the plastic radius in Soil B (*c*_{B}). The right-side boundary of the shaded area defines a line describing the linear increase of *c*_{A} with *b*_{0} for all tests. The value of *c*_{A} eventually deviates from this line for all tests. Outside of the shaded area, *c*_{B} does not exist; the size of Soil A (defined by *b*_{0}) is great enough that plasticity does not commence within Soil B.

As expected, for the uniform soil tests, the plastic radius is unaffected by the variation of *b*_{0}. Considering the spherical test ‘-10-1’ in Figure 14(a), *c*_{B} increases initially with *b*_{0}, although at a lower rate than *c*_{A}. The plastic region in Soil B disappears when *b*_{0} ∕ *a* ≅ 15 (where the ‘10-1’ line for *c*_{B} meets the right-side boundary of the shaded area). Soil A is fully plastic until *b*_{0} ∕ *a* ≅ 30, after which the value of *c*_{A} decreases towards and finally reaches the value obtained from the ‘-10-10’ test at *b*_{0} ∕ *a* ≅ 90 (as the effects of Soil B gradually dissipate). In test ‘-1-10’, *c*_{B} decreases initially with *b*_{0} and *c*_{A} gradually increases and reaches the value from test ‘-1-1’ at *b*_{0} ∕ *a* ≅ 90. This again defines the critical size of Soil A (*b*_{0,crit}) for the spherical analysis with the assumed material parameters. The cylindrical results in Figure 14(b) show similar trends to the spherical test.