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1
Finite Deformations in
Geomechanics
Luís André Berenguer Todo-BomPhD student
Supervisor – Prof. Arezou Modaressi
Laboratoire MSSMAT, ECP
2
INDEX
Description
Core Issues
Mechanical Formulations
Constitutive Models
Finite Element Method
Application
EDF 2012
3EDF 2012
Description
Finite deformations in geomechanics are common
Modelling failure mechanisms
4EDF 2012
Finite Deformations – Core Issues
Generalization of behaviour laws to the finite deformation range
Choice of formulation ( Eulerian or Lagrangian )
Choice of work-conjugate stress-strain pair
Frame indifference of tensors
Core Issues – Mechanical Formulation
5EDF 2012
Deformation Gradient tensor
Using the polar decomposition theorem
R – Rotation tensor
U , V - Stretch tensors (right and left, respectively)
Core Issues – Mechanical Formulation
6EDF 2012
Velocity Gradient tensor
Decomposing in a symmetric and skew-symmetric part
Rate of deformation tensor
Spin / Vorticity tensor
Core Issues – Mechanical Formulation
7EDF 2012
Strain Tensors ( Hill strains ) – most commonly used
Green-Lagrange
Logarithmic
Biot
Hencky
Lagrangian
Eulerian
Core Issues – Mechanical Formulation
8EDF 2012
Stress Tensors
• Cauchy (true) stress tensor
• 2nd Piola – Kirchhoff stress tensor
Lagrangian
Eulerian
Core Issues – Mechanical Formulation
Commonly known pairs of work (or energy) conjugates:
LagrangianEulerian
9EDF 2012
Infinitesimal Theory
Undeformed state ≈ Deformed state
Eulerian formulation ≈ Lagrangian formulation
No distinctions among different stress and strain measures
No distinctions in work-conjugate stress-strain pairs
Important limitations on geometrical changes are imposed
Core Issues – Mechanical Formulation
10EDF 2012
Infinitesimal Theory – Historical choices
“Engineering” Stress - Cauchy (true) Stress
“Engineering” Strain - “Reduced” Green-Lagrange
Most behaviour laws are written in these terms (or their rates)
Generally, not a work-conjugate stress-strain pair
Core Issues – Mechanical Formulation
11EDF 2012
Infinitesimal Theory – Geometrical changes
Infinitesimal Theory
Infinitesimal Strains
Infinitesimal Rotations
Rigid-body rotation is negligible
( Objectivity is always verified )
Core Issues – Mechanical Formulation
12EDF 2012
Objective Stress rates – Cauchy Stress tensor
Objectivity requirements (2nd order tensor)
WARNING :
Cauchy Stress tensor
Cauchy stress rate tensor
Core Issues – Mechanical Formulation
13EDF 2012
Finite Deformations
Historical evolution of mechanical formulations
Core Issues – Mechanical Formulation
14EDF 2012
• Lagrangian formulation
Elastoplasticity
Work-conjugate stress-strain pair(frame indifferent)
2nd Piola–Kirchhoff stress tensor
Green-Lagrange strain tensor
Due to non-linear terms, the strain tensor cannot be decomposed as
Core Issues – Mechanical Formulation
15EDF 2012
• Lagrangian formulation
The stress tensor does not have a clear physical meaning
Consideration of a “preferred state” for the calculation of strain has no direct physical pertinence in the flow-like behaviour of elastoplasticity
Extremely complex to introduce the concept of elastic/plastic strain( e.g. use of the plastic strain as primitive variable )
Fully justified mathematically but no direct pertinence to physical reality
Core Issues – Mechanical Formulation
16EDF 2012
- Objective Stress rate Cauchy tensor
- Rate of deformation tensor
• Eulerian formulation
Hypoelastic equation of grade zero
Core Issues – Mechanical Formulation
17EDF 2012
Elastoplasticity ( de Souza et al. [2008] )
The rate of deformation tensor has a linear expression with the velocity gradient
Direct physical pertinence, conceptual clarity and structural simplicity
The expression for the elastoplastic stiffness matrix is the same as for small deformation for corotational stress rates since
Core Issues – Mechanical Formulation
18EDF 2012
Objective Stress rates – Cauchy Stress tensor
Infinite possible objective stress rates – Uniqueness issue
Core Issues – Mechanical Formulation
19EDF 2012
Simple Shear test
Hypoelasticity
Core Issues – Mechanical Formulation
20EDF 2012
Hypoelasticity – Integrability issue
Hypoelastic rate equation is not exactly integrable
Path-dependent and dissipative
Always negligible in the infinitesimal range as long as:
No yielding occurs
Small number of cycles in cyclic loading
Core Issues – Mechanical Formulation
21EDF 2012
Hypoelasticity – Integrability issue
Core Issues – Mechanical Formulation
22EDF 2012
• Formulations with unstressed configurations ( Hyperelastic )
Multiplicative decomposition
Local imaginary “unstressed” intermediate configuration
Based on the slip theory of crystals
Non-uniqueness of the separation of the gradient deformation tensor
Core Issues – Mechanical Formulation
23EDF 2012
Non-uniqueness of the decomposition
Does not fulfil the objectivity requirement
Therefore,
Isotropy of the elastic domain must be assumed
and
An extra “ad-hoc” assumption must be made, e.g.
Core Issues – Mechanical Formulation
24EDF 2012
Separation of the rate of deformation tensor
or
or
“Ad-hoc” assumption such as
( ... )
Core Issues – Mechanical Formulation
25EDF 2012
• Eulerian formulation “ re-visited ” – Logarithmic rate
H. Xiao, O.T. Bruhns, and A. Meyers. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica, 124:89–105, 1997.
Core Issues – Mechanical Formulation
26EDF 2012
• Eulerian formulation – Logarithmic rate
Direct physical pertinence, conceptual clarity and structural simplicity
Hypoelastic rate equation is NOW exactly integrable
Elastic integrability ( Prager’s criterion )
Uniqueness of the logarithmic rate
Core Issues – Mechanical Formulation
27EDF 2012
Core Issues – Mechanical Formulation
28EDF 2012
Core Issues – Mechanical Formulation
29EDF 2012
Constitutive Models
Granular materials
Core Issues – Constitutive Model
30EDF 2012
Core Issues – Constitutive Model
Reference behaviour
Volumetric and Deviatoric Hardening
Phase transformation (contractive -> dilative)
State dependent material behaviour
Critical State soil mechanics
31EDF 2012
Core Issues – Constitutive Model
Reference behaviour
32EDF 2012
Finite Deformations
Behaviour laws for a range of deformations ≤ 30 %
Are there new physical phenomena to consider ?
shear banding grain breakage new phase transformation etc.
Does the “ Critical State ” exist as/when defined ?
Core Issues – Constitutive Model
33EDF 2012
Core Issues – Constitutive Model
Critical State Line
34EDF 2012
Core Issues – Constitutive Model
Proposed variations to existing models
Critical state friction angle
Slope (or slopes) of the compression and/or the critical state line
“Size” of the elastic domain
Dilatancy law
Variables associated with the breakage index
Multiple yield surfaces
35EDF 2012
ECP constitutive model
Shear band formation
Grain crushing phenomena
Altering the constitutive model
Core Issues – Constitutive Model
Okada, Y., Sassa, K., Fukuoka, H.. Undrained shear behaviour of sands subjected to large shear displacement and estimation of
excess pore-pressure generation from drained ring shear tests. Canadian Geotechnical Journal 2005;42:787–803.
36EDF 2012
Core Issues – Applications
Revised ECP constitutive model – Undrained Ring shear test
37EDF 2012
Core Issues – Applications
Revised ECP constitutive model – Undrained Ring shear test
38EDF 2012
Core Issues – Applications
Revised ECP constitutive model – Undrained Ring shear test
39EDF 2012
Finite Deformations
Finite Element Method
Core Issues – Numerical Methods
40EDF 2012
Finite Element Method
Eulerian formulation
Lagrangian formulation
Updated-Lagrangian Method
Arbitrary Lagrangian-Eulerian formulation
Core Issues – Numerical Methods
41EDF 2012
Core Issues – Numerical Methods
Formulations Advantages Disadvantages
Eulerian No mesh distortions
Fluid mechanics
Numerical diffusion
Interface definition
Lagrangian Solid mechanics
Interface definition Mesh distortions
Updated Lagrangian
Easily to implement
Reduces mesh distortion
Frequent re-meshing required in cases of
localized deformation
ALE Advantages of both methods No formal definition
Mesh refinement (motion)
42EDF 2012
M. Nazem. Numerical algorithms for large deformation problems in geomechanics. PhD thesis, Un. of Newcastle, 2006.
Core Issues – Applications
43EDF 2012
M. Nazem. Numerical algorithms for large deformation problems in geomechanics. PhD thesis, Un. of Newcastle, 2006.
Core Issues – Applications
44EDF 2012
M. Nazem. Numerical algorithms for large deformation problems in geomechanics. PhD thesis, Un. of Newcastle, 2006.
Core Issues – Applications
45EDF 2012
M. Nazem. Numerical algorithms for large deformation problems in geomechanics. PhD thesis, Un. of Newcastle, 2006.
Core Issues – Applications
46EDF 2012
M. Nazem. Numerical algorithms for large deformation problems in geomechanics. PhD thesis, Un. of Newcastle, 2006.
Core Issues – Applications
47EDF 2012
Application
Core Issues – Applications
48EDF 2012
Core Issues – Applications
Diameter = 0.5 m
Length = 5 m
Tip angle = 60 deg
Jacked installation
( monotonic loading )
Pile Installation simulation
49EDF 2012
Core Issues – Applications
Horizontal displacements
50EDF 2012
Core Issues – Applications
Vertical displacements
51EDF 2012
Core Issues – Applications
Vector of displacements
52EDF 2012
Core Issues – Applications
53EDF 2012
Core Issues – Applications
54EDF 2012
Core Issues – Applications
Pile Design - Idealized Initial conditions
55EDF 2012
Core Issues – Applications
56EDF 2012
Core Issues – Applications
57INDEX
Conclusions
Installation procedure defines the initial conditions of the problem
Valid mechanical formulation must be considered
Generalize the constitutive structure to model high strain behaviour
Future Work
Thermodynamic considerations of the constitutive model
Finish GEFDYN coding and result analysis
Conclusions and Future Work
58
Thank you
Laboratoire MSSMAT, ECP
59EDF 2012
H.-C. Wu. Continuum mechanics and plasticity. Chapman and Hall, 2005.
H.-S. Yu. Plasticity and Geotechnics. Springer, 2006.
H. Xiao, O.T. Bruhns, and A. Meyers. Elastoplasticity beyond small deformations. Acta Mechanica,
182:31–111, 2006.
H. Xiao, O.T. Bruhns, and A. Meyers. Logarithmic strain, logarithmic spin and logarithmic rate. Acta
Mechanica, 124:89–105, 1997.
J. Wang. Arbitrary Lagrangian-Eulerian method and its application in solid mechanics. PhD dissertation,
University of British Columbia, 1998.
J.C. Simo and K. S. Pister. Remarks on rate constitutive equations for finite deformation problems:
Computational implications. Comput. Meth. Appl. Mech. Engng, 46:201–215, 1984.
M. Nazem. Numerical algorithms for large deformation problems in geomechanics. PhD dissertation,
University of Newcastle, 2006.
R. Hill. A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids, 6:236–
249, 1958.
E.A. de Souza, D. Peric, and D.R.J. Owen. Computational methods for plasticity. Theory and Applications.
Wiley, 2008.
Main References
60EDF 2012Luís André Berenguer Todo-Bom
December 2012
Displacement Gradient
Cauchy-Green Extension tensors (right and left, respectively)
Need to quantify relative length changes – Strain measures
Core Issues – Mechanical Formulation
61EDF 2012Luís André Berenguer Todo-Bom
December 2012
Conservation Laws
The Conservation of Mass
The Conservation of Momentum ( linear and angular )
The Conservation of Energy
Core Issues – Mechanical Formulation
62EDF 2012Luís André Berenguer Todo-Bom
December 2012
Work / Energy conjugates – Specific rate of work per unit mass
Commonly known pairs of work (or energy) conjugates:
This term represents part of work rate that affects the strain energy of a material element.
The conjugate stress and strain should be used in any formulation of continuum mechanics problems.
LagrangianEulerian
Core Issues – Mechanical Formulation
63EDF 2012Luís André Berenguer Todo-Bom
December 2012
Objective Stress rates – Prager’s criterion
“ The simultaneous vanishing of the stress rate, back stress and hardening parameters should render the yield function stationary "
“ The definitions of the stress rate and the back stress should be the same and corotational “
From the classical rates only the Jaumann stress rate is admissible
Core Issues – Mechanical Formulation
64EDF 2012Luís André Berenguer Todo-Bom
December 2012
Non-uniqueness of the decomposition
Considering the decomposition true for
Decomposition also true for
- Arbitrary rotation
are rendered indetermined failing to totally separate these two rotations achieving only a partial separation
Mahrenholtz, O., Wilmanski, K.: Note on simple shear of plastic monocrystals. Mech. Res. Commun. 17, 393–402 (1990)
Core Issues – Mechanical Formulation
65EDF 2012Luís André Berenguer Todo-Bom
December 2012
The movement of the continuum is specified as a function of the spatial coordinate and time
Eulerian reference mesh which remains undistorted is needed to trace the motion of the material in the Eulerian domain
Materials can move freely through an Eulerian mesh
Eulerian formulation
Core Issues – Numerical Methods
66EDF 2012Luís André Berenguer Todo-Bom
December 2012
No element distortions occur
Field description that is often applied in fluid mechanics
Numerical diffusion possible in the case of two or more materials in the Eulerian domain
Eulerian formulation
Core Issues – Numerical Methods
67EDF 2012Luís André Berenguer Todo-Bom
December 2012
The movement of the continuum is specified as a function of the material coordinates and time
Particle description that is often applied in solid mechanics.
The nodes of the Lagrangian mesh move together with the material
Lagrangian formulation
Core Issues – Numerical Methods
68EDF 2012Luís André Berenguer Todo-Bom
December 2012
The interface between two parts is precisely tracked and defined
Lagrangian formulation
Large deformations may lead to an unpromising mesh and large element distortions
Core Issues – Numerical Methods
69EDF 2012Luís André Berenguer Todo-Bom
December 2012
Same characteristics as the Lagrangian approach
The mesh is modified after each incremental step calculation
Ability to re-mesh or re-zone - stresses and strains are taken from the old mesh and introduced in the new one
Large element distortions are still possible
In cases of localized deformation, very frequent re-meshing is required
Updated-Lagrangian formulation
Core Issues – Numerical Methods
70EDF 2012Luís André Berenguer Todo-Bom
December 2012
Arbitrary Lagrangian-Eulerian formulation
Attempt to join the advantages of both formulations
No formal definition as of yet ( only “reduction” verification )
Consists on uncoupling nodal point displacements and velocities and
material displacements and velocities
No mesh distortions
Material can “flow” trough the elements
Requires a “mesh refinement” procedure
Core Issues – Numerical Methods
71EDF 2012Luís André Berenguer Todo-Bom
December 2012
Due to the uncoupling convection must be taken into account to update the state at the nodal points
(between material and mesh displacements and velocities)
The reference system (computational mesh) is not a priori fixed in space or attached to the body, but an arbitrary computational reference system
The finite element mesh need not adhere to the material or be fixed space but may be moved arbitrarily relative to the material.
The number of unknowns surpasses the number of equations
Mesh motion must be specified !
Core Issues – Numerical Methods
72EDF 2012Luís André Berenguer Todo-Bom
December 2012
Coupled ALE
The two sets of unknown displacements (mesh and material) are solved simultaneously
New set of unknowns: equations due to unknown mesh displacements in addition to the already existent material displacements
Decoupled ALE - Operator Split technique
Solve the material displacements via the equilibrium equations
Compute the mesh displacements through a mesh refinement technique
Eulerean step
( Updated )Lagrangian step
Core Issues – Numerical Methods
73EDF 2012Luís André Berenguer Todo-Bom
December 2012
Core Issues – Numerical Methods
74EDF 2012Luís André Berenguer Todo-Bom
December 2012
UL step
Solving incremental displacements
Integrating constitutive equations for the stresses
Verify equilibrium
State variables satisfy both global equilibrium and local consistency requirements
Mesh may be distorted since it moves along with the material
Core Issues – Numerical Methods
75EDF 2012Luís André Berenguer Todo-Bom
December 2012
Eulerean step
Mesh is optimized based on initial topology but without element distortion
All kinematic and state variables are transferred to the new mesh using the relation between material time derivative and mesh derivation
The Eulerean step does not always verify objectivity
Additional corrections may be required
Core Issues – Numerical Methods
76EDF 2012Luís André Berenguer Todo-Bom
December 2012
Advantages of the de-coupled
Cost of implementation:
Only the Eulerian step algorithm needs to be added.
Simpler equations to be solved
From the theoretical point of view, the fully coupled ALE approach represents a true kinematical description in which material
deformation is described relative to a moving reference configuration.
Core Issues – Numerical Methods
77EDF 2012
Volumetric and Deviatoric Hardening
Phase transformation (contractive -> dilative)
State dependent material behaviour
Critical State soil mechanics
Luís André Berenguer Todo-Bom
December 2012
Core Issues – Constitutive Model
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