Skip to main content
SpringerLink
Log in

Continuum Modeling of Partially Saturated Soils

  • Chapter
  • First Online:
Shock Phenomena in Granular and Porous Materials

Part of the book series: Shock Wave and High Pressure Phenomena ((SHOCKWAVE))

  • 1087 Accesses

Abstract

Essential features in continuum-scale models for sand and/or clay under high-rate loading conditions are summarized. The scope of this chapter is limited to adiabatic load/unload conditions in order to focus on model features that are most crucial for simulations of buried explosives and similar phenomena that involve shock compression followed by free expansion (possibly with recompression when ejecta impacts an object). Evidence is provided that such conditions fall in a realm for which there is expected to be no substantial difference between additive and multiplicative inelasticity approaches. A new constitutive model, Arena, for fully and partially saturated soils is presented and validated with split-Hopkinson pressure bar experiments. Similarities of the presented cap-plasticity model with quasistatic critical-state theories are briefly mentioned. Simplifying assumptions for a tractable and robust computational model, as well as avenues for further research, are identified. Though the model is calibrated with data that is largely from quasistatic experiments on dry soils it is found to perform remarkably well in predicting the behavior of partially saturated Colorado Mason sand at high strain rates, with mild discrepancies typical of what can be handled via conventional viscoplasticity enhancement.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 149.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Other examples can be found in the following publications and the references cited in them. In [12] the validity of the effective stress model for quasistatic loading of soils with high pore pressure (6 MPa) is reconfirmed. A complete thermodynamic description of multiphase soils and connections between microscopic and macroscopic quantities can be found in [13,14,15,16,17]. An alternative to the “pressure equilibrium” assumption in mixtures and a rigorous definition of fluid pressure are explored in [18]. Alternative approaches based on effective medium theory can be found in [19,20,21]. In [22, 23], the “zero air-pressure” assumption is used to extend the small deformation theory of partially saturated soils to quasistatic large deformations. In more recent papers, the importance of suction is acknowledged but the water and air phases are typically assumed to be incompressible [24,25,26]. The mixture theory developed by Hassanizadeh was extended to finite strains and dynamics in [27,28,29,30]. Phenomenological models have also been developed at a regular pace [31,32,33]. For instance, models for the interaction between damage and fluid flow in porous rocks can be found in [34,35,36]. A cap evolution model for partially saturated soils is described in [37]. Configurational mechanics-based theories and models have also been developed, e.g., [38]. Scant attention has been paid to the deformability of the fluid phases but a few works do address that issue, e.g., [39, 40].

  2. 2.

    Arena (available under open source [48]) has been verified against several analytical tests [53] and validated against data for a variety of sands at different moisture and initial states.

  3. 3.

    Not part of Kayenta, these governing equations are based on work of Uzuoka and Borja [56] (see also [27,28,29,30, 56,57,58]) which is, in turn, a simplification of the averaging theory proposed by Hassanizadeh and Gray [13, 14, 16, 17, 59].

  4. 4.

    This assumption can significantly increase computational overhead if a highly expanded domain impacts an obstacle after a blast event. The problem arises because an RVE does not exist on this distended scale, and might be alleviated via enriched basis functions [61].

  5. 5.

    This model is applied in the “unrotated” configuration to satisfy frame indifference. The applications of interest are assumed to have negligible rotation of reference stretch directions, thus making the symmetric part of the velocity gradient, d, a very good approximation to the rate of Hencky strain and hence (in this approximation) conjugate to Cauchy stress σ.

  6. 6.

    As this theory is described in the context of an additive decomposition of strain rates, it carries with it an implicit potential limitation that reference stretch directions remain approximately stationary or that, by the time such rotations become large, the stress in the material is negligible due to disaggregation. These assumptions are quite reasonable in high-rate buried-explosive applications for which the model was designed. In this context, the unrotated symmetric part of the velocity gradient d equals the rate of reference Hencky strain \(\dot {{\boldsymbol {\varepsilon }}}\) and is conjugate to the unrotated Cauchy stress σ. Because multiplicative decompositions of the deformation gradient should (for initially isotropic media) become additive in these conditions, any claim of their superiority must be backed with (1) demonstrated equivalence to simpler additive models if stretch directions are stationary and (2) compellingly better agreement with validation data when stretch directions rotate.

  7. 7.

    For high-rate applications, a crush curve that is measured in quasistatic conditions must be converted to a form that removes low-rate creep (and other effects from heat transfer, fluid seepage, etc.) that would not occur in dynamic loading. The constitutive model must be furthermore supplemented with viscoplasticity parameters needed to predict apparent strengthening (beyond hardening) that does pertain to slow loading.

  8. 8.

    This particular model has been developed by the authors to fit experimental data on Colorado Mason sand. Existing bulk modulus models, e.g., the Kayenta model, were found to be inadequate for sand. Even if the numerical model is revised to use tabular data for this type of function, this approximate form can potentially serve as an interpolation function that would be more accurate (filling unknown gaps in data better) than a piecewise-linear fit—especially for low-data situations and for extrapolation beyond available data.

  9. 9.

    Experiments at quasistatic strain rates indicate that the saturation can affect the shear modulus [66]. However, these effects are important only at low confining pressures. We are not aware of any analogous experimental studies for high-rate loading.

  10. 10.

    To obtain a numerically tractable set of equations, several simplifying assumptions (such as neglecting elastic contributions to pore pressure) are adopted in [62], suggesting avenues appropriate for future research to justify or replace these choices.

  11. 11.

    Details of the derivation of internal variable evolution equations can be found in [62].

  12. 12.

    Such efforts should be seen as no more objectionable than long-accepted similar procedures in gas dynamics, where the low-rate (isothermal) pressure–volume curve along with the constant-pressure specific heat and thermal expansion properties is sufficient to infer the isentropic properties required for a purely mechanical model to accurately model acoustic waves. The key in such simplifications is to recognize that the resulting mechanical model is, by definition, limited in its application scope.

  13. 13.

    Note that the Arena model does not include damage at this stage and therefore we are only concerned with the rise part of the stress–time curves.

  14. 14.

    These invariants, denoted r and z, do not refer to radial and axial directions of an SHPB rod. Instead, they are radial and axial directions in six-dimensional stress space, defined such that z is the projection of stress tensor onto the hydrostat, while r is the hyper-distance of the stress from the hydrostat, making these, respectively, the magnitudes of the isotropic and deviatoric parts of stress. Unlike more traditional invariants (such as equivalent stress \(q=\sqrt {3J_2}\) and pressure p = I 1∕3), the (r, z) stress invariants are isomorphic to stress space, making lengths and angles in an (r, z) plot identical to those in stress space. For axisymmetric problems, r is a multiple of the difference between axial and lateral SHPB stresses, and therefore discrepancies in r will be magnified whenever the stress difference is small in comparison to the mean stress. Other applications involving larger shearing stresses, on the other hand, would likely exhibit errors that would highlight a long-standing need for induced anisotropy (requiring greater resources to develop and calibrate than what is available in a typical engineering budget).

References

  1. Omidvar M, Iskander M, Bless S (2014) Response of granular media to rapid penetration. Int J Impact Eng 66:60–82

    Article  Google Scholar 

  2. Iskander M, Omidvar M, Bless S (2015) Behavior of granular media under high strain-rate loading. In: Rapid Penetration into Granular Media: Visualizing the Fundamental Physics of Rapid Earth Penetration. Elsevier, Amsterdam

    Chapter  Google Scholar 

  3. Tong X, Tuan CY (2007) Viscoplastic cap model for soils under high strain rate loading. J Geotech Geoenviron Eng 133(2):206–214

    Article  Google Scholar 

  4. An J, Tuan CY, Cheeseman BA, Gazonas GA (2011) Simulation of soil behavior under blast loading. Int J Geomech 11(4):323–334

    Article  Google Scholar 

  5. Lu G, Fall M (2017) Modelling blast wave propagation in a subsurface geotechnical structure made of an evolutive porous material. Mech Mater 108:21–39

    Article  Google Scholar 

  6. Reguerio R, Pak R, McCartney J, Sture S, Yan B, Duan Z, Svoboda J, Mun W, Vasilyev O, Kasimov N, Brown-Dymkoski E, Hansen C, Li S, Ren B, Alshibli K, Druckrey A, Lu H, Luo H, Brannon R, Bonifasi-Lista C, Yarahmadi A, Ghodrati E, Colovos J (2013) ONR MURI project on soil blast modeling and simulation. In: Song B (ed) Dynamic Behavior of Materials, vol 1, Chapter 42. Springer, Vienna, pp 341–353

    Google Scholar 

  7. Fan H, Li S (2017) A peridynamics-SPH modeling and simulation of blast fragmentation of soil under buried explosive loads. Comput Methods Appl Mech Eng 318:349–381

    Article  ADS  MathSciNet  Google Scholar 

  8. Grujicic M, He T, Pandurangan B, Bell WC, Cheeseman BA, Roy WN, Skaggs RR (2009) Development, parameterization, and validation of a visco-plastic material model for sand with different levels of water saturation. Proc Inst Mech Eng Pt L J Mater Des Appl 223(2):63–81

    Google Scholar 

  9. Suescun-Florez E, Kashuk S, Iskander M, Bless S (2015) Predicting the uniaxial compressive response of granular media over a wide range of strain rates using the strain energy density concept. J Dynam Behav Mat 1(3):330–346

    Article  Google Scholar 

  10. Coussy O (2007) Revisiting the constitutive equations of unsaturated porous solids using a Lagrangian saturation concept. Int J Numer Anal Methods Geomech 31(15):1675–1694

    Article  Google Scholar 

  11. Pesavento F, Schrefler BA, Sciumè G (2017) Multiphase flow in deforming porous media: a review. Arch Comput Meth Eng 24(2):423–448

    Article  MathSciNet  Google Scholar 

  12. Bishop AW, Skinner A (1977) The influence of high pore-water pressure on the strength of cohesionless soils. Philos Trans Royal Soc Lond A Math Phys Eng Sci 284(1318):91–130

    Article  ADS  Google Scholar 

  13. Hassanizadeh SM, Gray WG (1979) General conservation equations for multi-phase systems: 1. Averaging procedure. Adv Water Resour 2:131–144

    Article  ADS  Google Scholar 

  14. Gray WG, Hassanizadeh SM (1989) Averaging theorems and averaged equations for transport of interface properties in multiphase systems. Int J Multiphase Flow 15(1):81–95

    Article  Google Scholar 

  15. Alonso EE, Gens A, Josa A, et al (1990) Constitutive model for partially saturated soils. Géotechnique 40(3):405–430

    Article  Google Scholar 

  16. Hassanizadeh SM, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resour 13(4):169–186

    Article  ADS  Google Scholar 

  17. Gray WG, Hassanizadeh SM (1991) Unsaturated flow theory including interfacial phenomena. Water Resour Res 27(8):1855–1863

    Article  ADS  Google Scholar 

  18. Svendsen B, Hutter K (1995) On the thermodynamics of a mixture of isotropic materials with constraints. Int J Eng Sci 33(14):2021–2054

    Article  MathSciNet  Google Scholar 

  19. Berg CR (1995) A simple, effective-medium model for water saturation in porous rocks. Geophysics 60(4):1070–1080

    Article  ADS  Google Scholar 

  20. Pietruszczak S, Pande G (1996) Constitutive relations for partially saturated soils containing gas inclusions. J Geotech Eng 122(1):50–59

    Article  Google Scholar 

  21. Vlahinić I, Jennings HM, Andrade JE, Thomas JJ (2011) A novel and general form of effective stress in a partially saturated porous material: the influence of microstructure. Mech Mater 43(1):25–35

    Article  Google Scholar 

  22. Meroi EA, Schrefler BA, Zienkiewicz OC (1995) Large strain static and dynamic semisaturated soil behaviour. Int J Numer Anal Methods Geomech 19(2):81–106

    Article  Google Scholar 

  23. Gawin D, Schrefler BA, Galindo M (1996) Thermo-hydro-mechanical analysis of partially saturated porous materials. Eng Comput 13(7):113–143

    Article  Google Scholar 

  24. Houlsby G (1997) The work input to an unsaturated granular material. Géotechnique 47(1):193–196

    Article  Google Scholar 

  25. Gray WG, Schrefler BA (2001) Thermodynamic approach to effective stress in partially saturated porous media. Eur J Mech A Solids 20(4):521–538

    Article  Google Scholar 

  26. Grasley ZC, Rajagopal KR (2012) Revisiting total, matric, and osmotic suction in partially saturated geomaterials. Z Angew Math Phys 63(2):373–394

    Article  MathSciNet  Google Scholar 

  27. Borja RI (2004) Cam-Clay plasticity. Part V: A mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media. Comput Methods Appl Mech Eng 193(48):5301–5338

    Article  ADS  MathSciNet  Google Scholar 

  28. Li C, Borja RI, Regueiro RA (2004) Dynamics of porous media at finite strain. Comput Methods Appl Mech Eng 193(36):3837–3870

    Article  ADS  MathSciNet  Google Scholar 

  29. Borja RI (2006) On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int J Solids Struct 43(6):1764–1786

    Article  MathSciNet  Google Scholar 

  30. Song X, Borja RI (2014) Mathematical framework for unsaturated flow in the finite deformation range. Int J Numer Methods Eng 97(9):658–682

    Article  MathSciNet  Google Scholar 

  31. Fuentes W, Triantafyllidis T (2013) Hydro-mechanical hypoplastic models for unsaturated soils under isotropic stress conditions. Comput Geotech 51:72–82

    Article  Google Scholar 

  32. Nedjar B (2013) Formulation of a nonlinear porosity law for fully saturated porous media at finite strains. J Mech Phys Solids 61(2):537–556

    Article  ADS  MathSciNet  Google Scholar 

  33. Lakeland DL, Rechenmacher A, Ghanem R (2014) Towards a complete model of soil liquefaction: the importance of fluid flow and grain motion. Proc Royal Soc A Math Phys Eng Sci 470(2165):20130453

    Article  MathSciNet  Google Scholar 

  34. Hamiel Y, Lyakhovsky V, Agnon A (2004) Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks. Geophys J Int 156(3):701–713

    Article  ADS  Google Scholar 

  35. Arson C, Gatmiri B (2009) A mixed damage model for unsaturated porous media. C R Mec 337(2):68–74

    Article  ADS  Google Scholar 

  36. Le Pense S, Arson C, Pouya A (2016) A fully coupled damage-plasticity model for unsaturated geomaterials accounting for the ductile–brittle transition in drying clayey soils. Int J Solids Struct 91:102–114

    Article  Google Scholar 

  37. Kohler R, Hofstetter G (2008) A cap model for partially saturated soils. Int J Numer Anal Methods Geomech 32(8):981–1004

    Article  Google Scholar 

  38. Papastavrou A, Steinmann P (2010) On deformational and configurational poro-mechanics: dissipative versus non-dissipative modelling of two-phase solid/fluid mixtures. Arch Appl Mech 80(9):969–984

    Article  Google Scholar 

  39. Gajo A (2011) Finite strain hyperelastoplastic modelling of saturated porous media with compressible constituents. Int J Solids Struct 48(11):1738–1753

    Article  Google Scholar 

  40. Madeo A, Dell’Isola F, Darve F (2013) A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J Mech Phys Solids 61(11):2196–2211

    Article  ADS  MathSciNet  Google Scholar 

  41. Gray WG, Schrefler BA, Pesavento F (2009) The solid phase stress tensor in porous media mechanics and the Hill–Mandel condition. J Mech Phys Solids 57(3):539–554

    Article  ADS  MathSciNet  Google Scholar 

  42. Gray WG, Schrefler BA (2007) Analysis of the solid phase stress tensor in multiphase porous media. Int J Numer Anal Methods Geomech 31(4):541–581

    Article  Google Scholar 

  43. Nikooee E, Habibagahi G, Hassanizadeh SM, Ghahramani A (2012) The effective stress in unsaturated soils: insights from thermodynamics. In: Unsaturated Soils: Research and Applications. Springer, Berlin, pp 5–11

    Chapter  Google Scholar 

  44. Homel MA, Guilkey JE, Brannon RM (2017) Mesoscale validation of the effective stress approach for modeling the plastic deformation of fluid-saturated porous materials. J Dyn Behav Mat 3(1):23–44

    Article  Google Scholar 

  45. Itskov M (2004) On the applicability of generalized strain measures in large strain plasticity. Mech Res Commun 31(5):507–517

    Article  Google Scholar 

  46. Rubin MB, Ichihara M (2010) Rheological models for large deformations of elastic-viscoplastic materials. Int J Eng Sci 48:1534–1543

    Article  Google Scholar 

  47. Dienes J (1979) On the analysis of rotation and stress rate in deforming bodies. Acta Mechanica 32:217

    Article  MathSciNet  Google Scholar 

  48. Davison de St Germain J, McCorquodale J, Parker SG, Johnson CR (2000) Uintah: a massively parallel problem solving environment. In: Proceedings the Ninth International Symposium on High-Performance Distributed Computing, 2000. IEEE, Piscataway, pp 33–41. This has a generic bibtex key (just Uintah) so that it can be edited to be whatever is currently considered the main reference for Uintah, not any particular publication.

    Google Scholar 

  49. Sulsky D, Chen A, Schreyer H (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196

    Article  ADS  MathSciNet  Google Scholar 

  50. Kamrin K, Bazant MZ (2007) Stochastic flow rule for granular materials. Phys Rev E 75:1–28 (preprint), file is Kamrin07.pdf

    Google Scholar 

  51. Strack O, Leavy R, Brannon R (2014) Aleatory uncertainty and scale effects in computational damage models for failure and fragmentation. Int J Numer Methods Eng 102(3–4):468–495

    Google Scholar 

  52. Huq F, Brannon R, Brady L (2015) An efficient binning scheme with application to statistical crack mechanics. Int J Numer Meth Eng 105:33–62

    Article  MathSciNet  Google Scholar 

  53. Kamojjala K, Brannon R, Sadeghirad A, Guilkey J (2015) Verification tests in solid mechanics. Eng Comput 2:193–213

    Article  Google Scholar 

  54. Homel MA, Guilkey JE, Brannon RM (2015) Continuum effective-stress approach for high-rate plastic deformation of fluid-saturated geomaterials with application to shaped-charge jet penetration. Acta Mech 227(2):279–310

    Article  Google Scholar 

  55. Brannon RM, Fuller T, Strack O, Fossum A, Sanchez J (2015) Kayenta: Theory and user’s guide. Sandia National Laboratories report SAND2015-0803

    Google Scholar 

  56. Uzuoka R, Borja RI (2012) Dynamics of unsaturated poroelastic solids at finite strain. Int J Numer Anal Methods Geomech 36(13):1535–1573

    Article  Google Scholar 

  57. Borja RI, White JA (2010) Conservation laws for coupled hydromechanical processes in unsaturated porous media: theory and implementation. In: Mechanics of unsaturated geomaterials, pp 185–208

    Google Scholar 

  58. Song X, Borja RI (2014) Finite deformation and fluid flow in unsaturated soils with random heterogeneity. Vadose Zone J 13(5)

    Google Scholar 

  59. Gray WG, Hassanizadeh SM (1991) Paradoxes and realities in unsaturated flow theory. Water Resour Res 27(8):1847–1854

    Article  ADS  Google Scholar 

  60. Brannon RM (2007) Elements of phenomenological plasticity: geometrical insight, computational algorithms, and topics in shock physics. In: Shock Wave Science and Technology Reference Library. Springer, Berlin, pp 225–274

    Chapter  Google Scholar 

  61. Mo es N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    Google Scholar 

  62. Banerjee B, Brannon R (2017) Theory, verification, and validation of the ARENA constitutive model for applications to high-rate loading of fully or partially saturated granular media. Tech. Rep. PAR-10021867-1516.v1, Parresia Research Limited and University of Utah, https://doi.org/10.13140/RG.2.2.10671.53922

  63. Norris AN (2008) Eulerian conjugate stress and strain. J Mech Mater Struct 3(2):243–260

    Article  Google Scholar 

  64. Brannon R, Leelavanichkul S (2010) A multi-stage return algorithm for solving the classical damage component of constitutive models for rock, ceramics, and other rock-like media. Int J Fracture 163(1–2):133–149

    Article  Google Scholar 

  65. Fuller T, Brannon R (2012) On the effects of deformation-induced anisotropy on classical elastic-plastic materials. Int J Numer Anal Methods Geomech 37(9):1079–1094

    Article  Google Scholar 

  66. Qian X, Gray DH, Woods RD (1993) Voids and granulometry: effects on shear modulus of unsaturated sands. J Geotech Eng 119(2):295–314

    Article  Google Scholar 

  67. Berryman JG, Milton GW (1991) Exact results for generalized Gassmann’s equations in composite porous media with two constituents. Geophysics 56(12):1950–1960

    Article  ADS  Google Scholar 

  68. Berryman JG (2006) Effective medium theories for multicomponent poroelastic composites. J Eng Mech 32(5):519–531

    Article  Google Scholar 

  69. Dvorkin J, Moos D, Packwood JL, Nur AM (1999) Identifying patchy saturation from well logs. Geophysics 64(6):1756–1759

    Article  ADS  Google Scholar 

  70. Pučik T, Brannon R, Burghardt J (2015) Nonuniqueness and instability of classical formulations of nonassociated plasticity, I: case study. J Mech Mater Struct 10(2):123–148

    Article  MathSciNet  Google Scholar 

  71. Pabst W, Gregorová E (2015) Critical assessment 18: elastic and thermal properties of porous materials–rigorous bounds and cross-property relations. Mater Sci Technol 31(15):1801–1808

    Article  Google Scholar 

  72. Mun W, Teixeira T, Balci M, Svoboda J, McCartney J (2016) Rate effects on the undrained shear strength of compacted clay. Soils Found 56(4):719–731

    Article  Google Scholar 

  73. Mun W, McCartney JS (2017) Roles of particle breakage and drainage in the isotropic compression of sand to high pressures. J Geotech Geoenviron Eng 143(10):04017071

    Article  Google Scholar 

  74. Mun W, McCartney JS (2017) Constitutive model for drained compression of unsaturated clay to high stresses. J Geotech Geoenviron Eng 143(6):04017014

    Article  Google Scholar 

  75. Mun W, McCartney JS (2017) Compression of unsaturated clay under high stresses. J Geotech Geoenviron Eng 143(7):02817003

    Article  Google Scholar 

  76. Lu H, Luo H, Cooper WL, Komanduri R (2013) Effect of particle size on the compressive behavior of dry sand under confinement at high strain rates. In: Dynamic behavior of materials, vol 1, Springer, Berlin, pp 523–530

    Google Scholar 

  77. Luo H, Du Y, Hu Z, Lu H (2015) High-strain rate compressive behavior of dry mason sand under confinement. In: Dynamic behavior of materials, vol 1, Springer, Berlin, pp 325–333

    Google Scholar 

  78. Luo H, Hu Z, Xu T, Lu H (2017) High-strain rate compressive behavior of a clay under uniaxial strain state. In: Dynamic behavior of materials, vol 1, Springer, Berlin, pp 117–122

    Google Scholar 

  79. Jensen E (2017) Hierarchical multiscale modeling to inform continuum constitutive models of soils. PhD thesis, University of Colorado at Boulder

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Parresia Research Limited, Auckland, New Zealand

    Biswajit Banerjee

  2. University of Utah, Salt Lake City, UT, USA

    Rebecca Brannon

Authors
  1. Biswajit Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rebecca Brannon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rebecca Brannon .

Editor information

Editors and Affiliations

  1. Sandia National Laboratories, Livermore, CA, USA

    Tracy J. Vogler

  2. Los Alamos National Laboratory, Los Alamos, NM, USA

    D. Anthony Fredenburg

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Banerjee, B., Brannon, R. (2019). Continuum Modeling of Partially Saturated Soils. In: Vogler, T., Fredenburg, D. (eds) Shock Phenomena in Granular and Porous Materials. Shock Wave and High Pressure Phenomena. Springer, Cham. https://doi.org/10.1007/978-3-030-23002-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation