Acta Geotechnica

, Volume 13, Issue 2, pp 227–242 | Cite as

Explicit meshfree solution for large deformation dynamic problems in saturated porous media

  • Pedro Navas
  • Lorenzo Sanavia
  • Susana López-Querol
  • Rena C. Yu
Research Paper


In this paper a new methodology to simulate saturated soils subjected to dynamic loadings under large deformation regime (locally up to 40% in equivalent plastic strain) is presented. The coupling between solid and fluid phases is solved through the complete formulation of the Biot’s equations. The additional novelty lies in the employment of an explicit time integration scheme of the \(u-w\) (solid displacement–relative fluid displacement) formulation which enables us to take advantage of such explicit schemes. Shape functions based on the principle of maximum entropy implemented in the framework of Optimal Transportation Meshfree schemes are utilized to solve both elastic and plastic problems.


Biot’s equation Complete formulation Meshfree Explicit approach Large strains 

List of symbols

\(\varvec{a}^s \equiv \ddot{\varvec{u}}\)

Acceleration vector of the solid = material time derivative of \(\varvec{v}^s\)


Relative water acceleration vector with respect to the solid = material time derivative of \(\varvec{v}^{ws}\) with respect to the solid


Left Cauchy–Green tensor


Body forces vector


Cohesion (equivalent to the yield stress, \(\sigma _Y\))


Right Cauchy–Green tensor

\(\varvec{C}\) (time integration scheme)

Damping matrix

\(\frac{D^s\Box }{Dt}\equiv \dot{\Box }\)

Material time derivative of \(\square \) with respect to the solid

\(\varvec{F}=\frac{\partial \varvec{x}}{\partial \varvec{X}}\)

Deformation gradient


Gravity acceleration vector


Shear modulus


Nodal spacing


Hardening modulus, derivative of the cohesion against time


Second-order unit tensor

\(J=\hbox {det}\varvec{F}\)

Jacobian determinant


Intrinsic permeability


Permeability tensor


Bulk modulus


Bulk modulus of the solid grains


Bulk modulus of the fluid


Mass matrix



\(N(\varvec{x})\), \(\nabla N(\varvec{x})\)

Shape function and its derivatives


Solid pressure


Pore pressure

\(\varvec{P}\) (time integration scheme)

External forces vector


Volumetric compressibility of the mixture


Internal forces vector

\(\varvec{s}=\varvec{\sigma }^{dev}\)

Deviatoric stress tensor




Displacement vector of the solid


Displacement vector of the water


Velocity vector of the solid


Relative velocity vector of the water with respect to the solid


Relative displacement vector of the water with respect to the solid

\(Z(\varvec{x},\varvec{\lambda })\)

Denominator of the exponential shape function

\(\alpha _{_F}\), \(\alpha _{_Q}\) and \(\beta \)

Drucker–Prager parameters

\(\beta \), \(\gamma \)

Time integration schemes parameters

\(\beta \), \(\gamma \)

LME parameters related with the shape of the neighborhood

\(\Delta \gamma \)

Increment of equivalent plastic strain

\(\overline{\varepsilon }^p\)

Equivalent plastic strain

\(\varvec{\varepsilon }\)

Small strain tensor

\(\varepsilon _0\)

Reference plastic strain

\(\kappa \)

Hydraulic conductivity

\(\lambda \)

Lamé constant

\(\varvec{\lambda }\)

Minimizer of log\(Z(\varvec{x},\varvec{\lambda })\)

\(\mu _w \)

Viscosity of the water

\(\nu \)

Poisson’s ratio

\(\rho \)

Current mixture density

\(\rho _w\)

Water density

\(\rho _s\)

Density of the solid particles

\(\varvec{\sigma }\)

Cauchy stress tensor

\(\varvec{\sigma }'\)

Effective Cauchy stress tensor

\(\varvec{\tau }\)

Kirchhoff stress tensor

\(\varvec{\tau '}\)

Effective Kirchhoff stress tensor

\(\Phi \)

Plastic yield surface

\(\phi \)

Friction angle

\(\psi \)

Dilatancy angle

Superscripts and subscripts


Superscript for deviatoric part


Superscript for elastic part


Subscript for the previous step


Subscript for the current step


Superscript for plastic part


Superscript for the solid part


Superscript for trial state in the plastic calculation


Superscript for volumetric part


Superscript for the fluid part


Superscript for the fluid part relative to the solid one



The financial support from the Ministerio de Ciencia e Innovación, under Grant Numbers, BIA2012-31678 and BIA2015-68678-C2-1-R, and the Consejería de Educación, Cultura y Deportes de la Junta de Comunidades de Castilla-La Mancha, Fondo Europeo de Desarrollo Regional, under Grant No. PEII-2014-016-P, Spain, is greatly appreciated. The first author also acknowledges the fellowship BES2013-0639 as well as the fellowship EEBB-I-17-12624 which supported him on his stay in DICEA, University of Padova, Italy. The second author also would like to thank the University of Padova, (Research Grant DOR1725272/17).


  1. 1.
    Armero F (1999) Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions. Comput Methods Appl Mech Eng 171:205–241MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arroyo M, Ortiz M (2006) Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int J Numer Methods Eng 65(13):2167–2202MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bandara S, Soga K (2015) Coupling of soil deformation and pore fluid flow using material point method. Comput Geotech 63:199–214CrossRefGoogle Scholar
  4. 4.
    Biot MA (1956) General solutions of the equations of elasticity and consolidation for a porous material. J Appl Mech 23(1):91–96MathSciNetzbMATHGoogle Scholar
  5. 5.
    Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J Acoust Soc Am 28(2):168–178MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bonet J, Wood R (1997) Nonlinear continuum mechanics fot finite element analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  7. 7.
    Borja RI, Alarcón E (1995) A mathematical framework for finite strain elastoplastic consolidation. Part 1: Balance laws, variational formulation, and linearization. Comput Methods Appl Mech Eng 122:145–171CrossRefzbMATHGoogle Scholar
  8. 8.
    Borja RI, Tamagnini C, Alarcón E (1998) Elastoplastic consolidation at finite strain. Part 2: Finite element implementation and numerical examples. Comput Methods Appl Mech Eng 159:103–122CrossRefzbMATHGoogle Scholar
  9. 9.
    Camacho G, Ortiz M (1997) Adaptive Lagrangian modelling of ballistic penetration of metallic targets. Comput Methods Appl Mech Eng 142:269–301CrossRefzbMATHGoogle Scholar
  10. 10.
    Cao T, Sanavia L, Schrefler B (2016) A thermo-hydro-mechanical model for multiphase geomaterials in dynamics with application to strain localization simulation. Int J Numer Methods Eng 107:312–337MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ceccato F, Simonini P (2016) Numerical study of partially drained penetration and pore pressure dissipation in piezocone test. Acta Geotech 12:195–209CrossRefGoogle Scholar
  12. 12.
    Cuitiño A, Ortiz M (1992) A material-independent method for extending stress update algotithms from small-strain plasticity to finite plasticity with multiplicative kinematics. Eng Comput 9:437–451CrossRefGoogle Scholar
  13. 13.
    Diebels S, Ehlers W (1996) Dynamic analysis of fully saturated porous medium accounting for geometrical and material non-linearities. Int J Numer Methods Eng 39:81–97CrossRefzbMATHGoogle Scholar
  14. 14.
    Ehlers W, Eipper G (1999) Finite elastic deformations in liquid-saturated and empty porous solids. Transp Porous Media 34:179–191MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jeremić B, Cheng Z, Taiebat M, Dafalias Y (2008) Numerical simulation of fully saturated porous materials. Int J Numer Anal Methods Geomech 32:1635–1660CrossRefzbMATHGoogle Scholar
  16. 16.
    Lewis R, Schrefler B (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, New YorkzbMATHGoogle Scholar
  17. 17.
    Li B, Habbal F, Ortiz M (2010) Optimal transportation meshfree approximation schemes for fluid and plastic flows. Int J Numer Methods Eng 83:1541–1579MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li C, Borja RI, Regueiro RA (2004) Dynamics of porous media at finite strain. Comput Methods Appl Mech Eng 193:3837–3870MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    López-Querol S, Blazquez R (2006) Liquefaction and cyclic mobility model in saturated granular media. Int J Numer Anal Methods Geomech 30:413–439CrossRefzbMATHGoogle Scholar
  20. 20.
    López-Querol S, Fernández-Merodo J, Mira P, Pastor M (2008) Numerical modelling of dynamic consolidation on granular soils. Int J Numer Anal Methods Geomech 32:1431–1457CrossRefzbMATHGoogle Scholar
  21. 21.
    Navas P, López-Querol S, Yu R, Li B (2016) B-bar based algorithm applied to meshfree numerical schemes to solve unconfined seepage problems through porous media. Int J Numer Anal Methods Geomech 40:962–984CrossRefGoogle Scholar
  22. 22.
    Navas P, Yu RC, López-Querol S, Li B (2016) Dynamic consolidation problems in saturated soils solved through u-w formulation in a LME meshfree framework. Comput Geotech 79:55–72CrossRefGoogle Scholar
  23. 23.
    Navas P (2017) Meshfree methods applied to dynamic problems in materials in construction and soils. University of Castilla-La ManchaGoogle Scholar
  24. 24.
    Ortiz A, Puso M, Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Methods Eng 61(12):2159–2181MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ravichandran N, Muraleetharan K (2009) Dynamics of unsaturated soils using various finite element formulations. Int J Numer Anal Methods Geomech 33:611–631CrossRefzbMATHGoogle Scholar
  26. 26.
    Sanavia L, Pesavento F, Schrefler B (2006) Finite element analysis of non-isothermal multiphase geomaterials with application to strain localization simulation. Comput Mech 37(4):331–348CrossRefzbMATHGoogle Scholar
  27. 27.
    Sanavia L, Schrefler B, Stein E, Steinmann P (2001) Modelling of localisation at finite inelastic strain in fluid saturated porous media. In: Ehlers W (ed), Proceedings of IUTAM symposium on theoretical and numerical methods in continuum me- chanics of porous materials, Kluwer Academic Publishers, pp 239–244Google Scholar
  28. 28.
    Sanavia L, Schrefler BA, Steinmann P (2002) A mathematical and numerical model for finite elastoplastic deformations in fluid saturated porous media. In: Capriz G, Ghionna VN, Giovine P (eds) Modeling and mechanics of granular and porous materials. Modeling and simulation in science, engineering and technology. Birkhäuser, Boston, MA, pp 293–340. CrossRefGoogle Scholar
  29. 29.
    Sanavia L, Schrefler B, Steinmann P (2002) A formulation for an unsaturated porous medium undergoing large inelastic strains. Comput Mech 28:137–151CrossRefzbMATHGoogle Scholar
  30. 30.
    Terzaghi KV (1925) Principles of soil mechanics. Eng News Rec 95:19–27Google Scholar
  31. 31.
    Ye F, Goh S, Lee F (2010) A method to solve Biot’s u-U formulation for soil dynamic applications using the ABAQUS/explicit platform. In: Benz T, Nordal S (eds) Numerical methods in geotechnical engineering. CRC Press, London, pp 417–422Google Scholar
  32. 32.
    Zienkiewicz O, Chan A, Pastor M, Paul D, Shiomi T (1990) Static and dynamic behaviour of geomaterials: a rational approach to quantitative solutions. Part I: fully saturated problems. Proc R Soc Lond A429:285–309CrossRefzbMATHGoogle Scholar
  33. 33.
    Zienkiewicz O, Chan A, Pastor M, Schrefler B, Shiomi T (1999) Comput Geomech. John Wiley, ChichesterGoogle Scholar
  34. 34.
    Zienkiewicz O, Chang C, Bettes P (1980) Drained, undrained, consolidating and dynamic behaviour assumptions in soils. Géotechnique 30(4):385–395CrossRefGoogle Scholar
  35. 35.
    Zienkiewicz O, Shiomi T (1984) Dynamic behaviour of saturated porous media: the generalized biot formulation and its numerical solution. Int J Numer Anal Geomech 8:71–96CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Civil EngineeringUniversity of Castilla La-ManchaCiudad RealSpain
  2. 2.Dipartimento di Ingegneria Civile, Edile e AmbientaleUniversità degli Studi di PadovaPadovaItaly
  3. 3.Department of Civil, Environmental and Geomatic EngineeringUniversity College LondonLondonUK

Personalised recommendations