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.
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Abbreviations
- \(\varvec{a}^s \equiv \ddot{\varvec{u}}\) :
-
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
- \(\varvec{F}=\frac{\partial \varvec{x}}{\partial \varvec{X}}\) :
-
Deformation gradient
- \(\varvec{g}\) :
-
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
- 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
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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).
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Dedicated to the $$75\mathrm{th}$$ 75 th birthday of Professor Bernhard A. Schrefler.
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Navas, P., Sanavia, L., López-Querol, S. et al. Explicit meshfree solution for large deformation dynamic problems in saturated porous media. Acta Geotech. 13, 227–242 (2018). https://doi.org/10.1007/s11440-017-0612-7
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DOI: https://doi.org/10.1007/s11440-017-0612-7