National TechnicalUniversity of Athens
Nonlinear Modeling of Geotechnical Problems: From Theory to Practice,Johns Hopkins University, Maryland, 3-4 November 2005
Gradient elasticity and plasticity:Numerical modelling
I. Vardoulakis1 and A. Zervos2
1National Technical University of Athens, Greece2University of Southampton, United Kingdom
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 2
IntroductionFailure due to localization of deformation (F. Oka, Prediction methods in large strain and failure ingeomechanics, ISSMGE TC34, SoA report, 2005)
The scale of the problem changes: if the microstructure is ignored, the problem becomes “ill-posed”.The microstructure dominates the post-localisation behaviour (shear band thickness).
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 3
Continua with microstructure
Cosserat theoriesIndependent degrees of freedom of rotation introduce a material length scale.Appeal to practitioners: rotation and it’s conjugate “couple stress”have well-established equivalents in beam and plate bending theories.
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 4
Continua with microstructure
Gradient theoriesHigher-order gradients of strains in the constitutive relations introduce a material length scale.Generalised “stresses” and boundary conditions are introduced, which again have clear links to concepts from beam and plate bending.
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 5
Continua with microstructure
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 6
General framework
Maintain the ellipticity of the problemPreclude the existence of strong discontinuities.Shear bands are considered discontinuities of the incremental displacement gradient.
Use a numerical approach that is robust and mainstream
Finite Element Analysis is favoured by the majority of practitioners.The displacement formulation is easy to grasp intuitively and is the standard formulation used by almost all existing codes.
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 7
Gradient Elasticity & PlasticityGradient elasticity with surface energy
Based on work by Mindlin and Eshel (1964)Introduces a scalar material length scale and a director field that allows for surface tension effects.
Gradient elasticity with gradient plasticityIntroduces two scalar material length scales, associated with elastic and plastic deformations respectively.
( ) ( )dSnQPdVS
kikiiiV
kijkijijij ∫∫ +=+σ
υυδεδµεδσ ,,ˆˆ &&&&
( ) ( )dSntdVmS
kikiiiV
kijkijijij ∫∫ +=+σ
υδµυδεδεδσ ,,)0( &&&&
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 8
Finite element formulation
Displacement formulation: u = N · ûDisplacement is the only interpolated field
Strain gradients are present in the weak formu must be C1 continuous
C1 triangle(Argyris et al. 1968, Zervos 2001)
3 nodes36 DOFsComplete quintic polynomialCubic normal derivative
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 9
Elasticity: oedometer test
“Rigid” piston: zero strain at the interface.Cannot be accommodated by linear elasticity.
Boundary layer in which the vertical strain varies.Layer thickness dependent on internal length scale.
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 10
Elasticity: “bolted” layer
Uniform distribution of pre-tensioned bolts: constant double force at the boundary.
Cannot be accommodated by linear elasticity.
Boundary layer in which the vertical strain varies.Layer thickness dependent on internal length scale and the length of the bolts.
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 11
Elastoplasticity: biaxial test
Robust simulations capture details of the failure mechanism:
Shear-band inclination in the core:
Reorientation near a stress boundary:
Contours of plastic strain for various dilatancy angles
56.5°
54.0°
52.5°
44.5°
020=ψ 00=ψ
( ) 4/450 ψφθ ++≅2/450 ψθ +≅
60.0°
60.5°
032=ψ
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 12
Elastoplasticity: thick-cylinders
Borehole shear failure and scale effect
Spontaneous loss of symmetry and final localized pattern
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 13
Elastoplasticity: cavity expansion
Spontaneous loss of symmetry and final localized pattern
Borehole shear failure
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 14
Conclusions
Gradient Elasticity and ElastoplasticityAccount for material microstructure.Accommodate details of external load transfer mechanisms.Model localized shear failure mechanisms in a robust manner.Capture experimentally observed scale effects.
Open issuesThe physical meaning of extra boundary conditions.The measurement of additional b.c. constitutive parameters.Contact mechanics testing.
Workshop on Non-linear Modeling of Geotechnical Problems, Maryland 2005I. Vardoulakis and A. Zervos 15
Personal remarks
Mechanics and mathematics are necessary language skills which enable the engineer to communicate better his/her ideas and to follow published progress.
Research is a creative act that has a serious side-effect (=education), that of improving an engineer’s reflexes in making rational decisions in-situ.
Theoretical bias is necessary pre-requisite for recognising what the engineering problem is.
