Acta Geotechnica

, Volume 13, Issue 2, pp 227–242

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

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

## Abstract

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.

## Keywords

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 $$\varvec{v}^s$$

$$\varvec{a}^{ws}$$

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

$$\varvec{b}=\varvec{F}\varvec{F}^{T}$$

Left Cauchy–Green tensor

$$\varvec{\overline{b}}$$

Body forces vector

c

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

$$\varvec{C}=\varvec{F}^{T}\varvec{F}$$

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

$$\varvec{F}=\frac{\partial \varvec{x}}{\partial \varvec{X}}$$

$$\varvec{g}$$

Gravity acceleration vector

G

Shear modulus

h

Nodal spacing

H

Hardening modulus, derivative of the cohesion against time

$$\varvec{I}$$

Second-order unit tensor

$$J=\hbox {det}\varvec{F}$$

Jacobian determinant

k

Intrinsic permeability

$$\varvec{k}$$

Permeability tensor

K

Bulk modulus

$$K_s$$

Bulk modulus of the solid grains

$$K_w$$

Bulk modulus of the fluid

$$\varvec{M}$$

Mass matrix

n

Porosity

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

Shape function and its derivatives

p

Solid pressure

$$p_w$$

Pore pressure

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

External forces vector

Q

Volumetric compressibility of the mixture

$$\varvec{R}$$

Internal forces vector

$$\varvec{s}=\varvec{\sigma }^{dev}$$

Deviatoric stress tensor

t

Time

$$\varvec{u}$$

Displacement vector of the solid

$$\varvec{U}$$

Displacement vector of the water

$$\varvec{v}^s=\dot{\varvec{u}}$$

Velocity vector of the solid

$$\varvec{v}^{ws}$$

Relative velocity vector of the water with respect to the solid

$$\varvec{w}$$

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

dev

Superscript for deviatoric part

e

Superscript for elastic part

k

Subscript for the previous step

k+1

Subscript for the current step

p

Superscript for plastic part

s

Superscript for the solid part

trial

Superscript for trial state in the plastic calculation

vol

Superscript for volumetric part

w

Superscript for the fluid part

ws

Superscript for the fluid part relative to the solid one

## Notes

### Acknowledgements

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).

## References

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 171:205–241
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):2167–2202
3. 3.
Bandara S, Soga K (2015) Coupling of soil deformation and pore fluid flow using material point method. Comput Geotech 63:199–214
4. 4.
Biot MA (1956) General solutions of the equations of elasticity and consolidation for a porous material. J Appl Mech 23(1):91–96
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):168–178
6. 6.
Bonet J, Wood R (1997) Nonlinear continuum mechanics fot finite element analysis. Cambridge University Press, Cambridge
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 122:145–171
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 159:103–122
9. 9.
Camacho G, Ortiz M (1997) Adaptive Lagrangian modelling of ballistic penetration of metallic targets. Comput Methods Appl Mech Eng 142:269–301
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 107:312–337
11. 11.
Ceccato F, Simonini P (2016) Numerical study of partially drained penetration and pore pressure dissipation in piezocone test. Acta Geotech 12:195–209
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 9:437–451
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 39:81–97
14. 14.
Ehlers W, Eipper G (1999) Finite elastic deformations in liquid-saturated and empty porous solids. Transp Porous Media 34:179–191
15. 15.
Jeremić B, Cheng Z, Taiebat M, Dafalias Y (2008) Numerical simulation of fully saturated porous materials. Int J Numer Anal Methods Geomech 32:1635–1660
16. 16.
Lewis R, Schrefler B (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, New York
17. 17.
Li B, Habbal F, Ortiz M (2010) Optimal transportation meshfree approximation schemes for fluid and plastic flows. Int J Numer Methods Eng 83:1541–1579
18. 18.
Li C, Borja RI, Regueiro RA (2004) Dynamics of porous media at finite strain. Comput Methods Appl Mech Eng 193:3837–3870
19. 19.
López-Querol S, Blazquez R (2006) Liquefaction and cyclic mobility model in saturated granular media. Int J Numer Anal Methods Geomech 30:413–439
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 32:1431–1457
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 40:962–984
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 79:55–72
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):2159–2181
25. 25.
Ravichandran N, Muraleetharan K (2009) Dynamics of unsaturated soils using various finite element formulations. Int J Numer Anal Methods Geomech 33:611–631
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):331–348
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.
29. 29.
Sanavia L, Schrefler B, Steinmann P (2002) A formulation for an unsaturated porous medium undergoing large inelastic strains. Comput Mech 28:137–151
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 A429:285–309
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):385–395
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 8:71–96

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

## Authors and Affiliations

• Pedro Navas
• 1
• Lorenzo Sanavia
• 2
• Susana López-Querol
• 3
• Rena C. Yu
• 1
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