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1. Objectives and
Applications >
1.1 Defining a
Problem >
1.1.1 Deciding what to
calculate
1.1.2 Defining geometry
1.1.3 Defining loading
1.1.4 Choosing physics
1.1.5 Defining material
behavior
1.1.6 A representative
problem
1.1.7 Choosing a method
of analysis
2. Governing Equations >
2.1 Deformation
measures >
2.1.1 Displacement and
Velocity
2.1.2 Deformation gradient
2.1.3 Deformation gradient
from two deformations
2.1.4 Jacobian of deformation
gradient
2.1.5 Lagrange strain
2.1.6 Eulerian strain
2.1.7 Infinitesimal Strain
2.1.8 Engineering Shear
Strain
2.1.9 Volumetric and
Deviatoric strain
2.1.10 Infinitesimal rotation
2.1.11 Principal strains
2.1.12 CauchyGreen
deformation tensors
2.1.13 Rotation tensor,
Stretch tensors
2.1.14 Principal stretches
2.1.15 Generalized strain
measures
2.1.16 Velocity gradient
2.1.17 Stretch rate and spin
2.1.18 Infinitesimal
strain/rotation rate
2.1.19 Other deformation
rates
2.1.20 Strain equations of
compatibility
2.2 Internal forces >
2.2.1 Surface traction/body
force
2.2.2 Internal tractions
2.2.3 Cauchy stress
2.2.4 Kirchhoff, Nominal,
Material stress
2.2.5 Stress for
infinitesimal motions
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2.2.6 Principal stresses
2.2.7 Hydrostatic,
Deviatoric, Von Mises stress
2.2.8 Stresses at a boundary
2.3 Equations of
motion >
2.3.1 Linear momentum
balance
2.3.2 Angular momentum
balance
2.3.3 Equations using
other stresses
2.4 Work and Virtual
Work >
2.4.1 Work done by Cauchy
stress
2.4.2 Work done by other
stresses
2.4.3 Work for
infinitesimal motions
2.4.4 Principle of
virtual work
2.4.5 Virtual work with
other stresses
2.4.6 Virtual work for
small strains
3. Constitutive Equations
>
3.1 General requirements
3.2 Linear elasticity >
3.2.1 Isotropic elastic
behavior
3.2.2 Isotropic stressstrain
laws
3.2.3 Plane stress & strain
3.2.4 Isotropic material data
3.2.5 Lame, Shear, & Bulk
modulus
3.2.6 Interpreting elastic
constants
3.2.7 Strain energy density
(isotropic)
3.2.8 Anisotropic stress
strain laws
3.2.9 Interpreting
anisotropic constants
3.2.10 Anisotropic strain
energy density
3.2.11 Basis change formulas
3.2.12 Effect of material
symmetry
3.2.13 Orthotropic materials
3.2.14 Transversely isotropic
materials
3.2.15 Transversely isotropic
data
3.2.16 Cubic materials
3.2.17 Cubic material data
3.3 Hypoelasticity
3.4 Elasticity w/ large
rotations
3.5 Hyperelasticity >
3.5.1 Deformation
measures
3.5.2 Stress measures
3.5.3 Strain energy
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density
3.5.4 Incompressible
materials
3.5.5 Energy density
functions
3.5.6 Calibrating
material models
3.5.7 Representative
properties
3.6 Viscoelasticity >
3.6.1 Polymer behavior
3.6.2 General
constitutive equations
3.6.3 Springdamper
approximations
3.6.4 Prony series
3.6.5 Calibrating
constitutive laws
3.6.6 Calibrating
material models
3.6.7 Representative
properties
3.7 Rate independent
plasticity >
3.7.1 Plastic metal behavior
3.7.2 Elastic/plastic strain
decomposition
3.7.3 Yield criteria
3.7.4 Graphical yield
surfaces
3.7.5 Hardening laws
3.7.6 Plastic flow law
3.7.7 Unloading condition
3.7.8 Summary of stress
strain relations
3.7.9 Representative
properties
3.7.10 Principle of max.
plastic resistance
3.7.11 Drucker's postulate
3.7.12 Microscopic
perspectives
3.8 Viscoplasticity >
3.8.1 Creep behavior
3.8.2 High strain rate
behavior
3.8.3 Constitutive equations
3.8.4 Representative creep
properties
3.8.5 Representative high
rate properties
3.9 Large strain
plasticity >
3.9.1 Deformation
measures
3.9.2 Stress measures
3.9.3 Elastic stress
strain relations
3.5.4 Plastic stress
strain relations
3.10 Large strain
viscoelasticity >
3.10.1 Deformation
measures
3.10.2 Stress measures
3.10.3 Stressstrain
energy relations
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3.10.4 Strain relaxation
3.10.5 Representative
properties
3.11 Critical state
soils >
3.11.1 Soil behavior
3.11.2 Constitutive laws
(Camclay)
3.11.3 Response to 2D
loading
3.11.4 Representative
properties
3.12 Crystal plasticity
>
3.12.1 Basic
crystallography
3.12.2 Features of
crystal plasticity
3.12.3 Deformation
measures
3.12.4 Stress measures
3.12.5 Elastic stress
strain relations
3.12.6 Plastic stress
strain relations
3.12.7 Representative
properties
3.13 Surfaces and
interfaces >
3.13.1 Cohesive interface
models
3.13.2 Contact and friction
4. Solutions to simple
problems >
4.1 Axial/Spherical linear
elasticity >
4.1.1 Elastic governing
equations
4.1.2 Spherically symmetric
equations
4.1.3 General spherical
solution
4.1.4 Pressurized sphere
4.1.5 Gravitating sphere
4.1.6 Heated spherical
shell
4.1.7 Axially symmetric
equations
4.1.8 General axisymmetric
solution
4.1.9 Pressurized cylinder
4.1.10 Spinning circular
disk
4.1.11 Interference fit
4.2 Axial/Spherical
elastoplasticity >
4.2.1 Plastic governing
equations
4.2.2 Spherically symmetric
equations
4.2.3 Pressurized sphere
4.2.4 Cyclically
pressurized sphere
4.2.5 Axisymmetric
equations
4.2.6 Pressurized cylinder
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4.3 Spherical
hyperelasticity >
4.3.1 Governing equations
4.3.2 Spherically symmetric
equations
4.3.3 Pressurized sphere
4.4 1D elastodynamics >
4.4.1 Surface subjected to
pressure
4.4.2 Surface under
tangential loading
4.4.3 1D bar
4.4.4 Plane waves
4.4.5 Wave speeds in
isotropic solid
4.4.6 Reflection at a
surface
4.4.7 Reflection at an
interface
4.4.8 Plate impact
experiment
5. Solutions for elastic
solids >
5.1 General Principles >
5.1.1 Governing
equations
5.1.2 Navier equation
5.1.3 Superposition &
linearity
5.1.4 Uniqueness &
existence
5.1.5 SaintVenants
principle
5.2 2D Airy function
solutions >
5.2.1 Airy solution in
rectangular coords
5.2.2 Demonstration of Airy
solution
5.2.3 Airy solution in polar
coords
5.2.4 End loaded cantilever
5.2.5 Line load
perpendicular to surface
5.2.6 Line load parallel to
surface
4.4.7 Pressure on a surface
4.4.8 Uniform pressure on a
strip
4.4.8 Stress near a crack
tip
5.3 2D Complex variable
solutions >
5.3.1 Complex variable
solution
5.3.2 Demonstration of CV
solution
5.3.3 Line force
5.3.4 Edge dislocation
5.3.5 Circular hole in
infinite solid
5.3.6 Slit crack
5.3.7 Bimaterial
interface crack
5.3.8 Rigid flat punch on
a surface
5.3.9 Parabolic punch on
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a surface
5.3.10 General line
contact
4.3.11 Frictional sliding
contact
4.3.12 Dislocation near a
surface
5.4 3D static problems >
5.4.1 PapkovichNeuber
potentials
5.4.2 Demonstration of PN
potentials
5.4.3 Point force in
infinite solid
5.4.4 Point force normal to
surface
5.4.5 Point force tangent
to surface
5.4.6 Eshelby inclusion
problem
5.4.7 Inclusion in an
elastic solid
5.4.8 Spherical cavity in
infinite solid
5.4.9 Flat cylindrical
punch on surface
5.4.10 Contact between
spheres
4.4.11 Relations for
general contacts
4.4.12 Pd relations for
axisymmetric contact
5.5 2D Anisotropic
elasticity >
5.5.1 Governing equations
5.5.2 Stroh solution
5.5.3 Demonstration of Stroh
solution
5.5.4 Stroh matrices for
cubic materials
5.5.5 Degenerate materials
5.5.6 Fundamental elasticity
matrix
5.5.7 Orthogonality of Stroh
matrices
5.5.8 Barnett/Lothe &
Impedance tensors
5.5.9 Properties of matrices
5.5.10 Basis change formulas
5.5.11 BarnettLothe
integrals
5.5.12 Uniform stress state
5.5.13 Line load/dislocation
in infinite solid
5.5.14 Line load/dislocation
near a surface
5.6 Dynamic problems >
5.6.1 Love potentials
5.6.2 Pressurized
spherical cavity
5.6.3 Rayleigh waves
5.6.4 Love waves
5.6.5 Elastic waves in
waveguides
5.7 Energy methods >
5.7.1 Definition of potential
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energy
5.7.2 Minimum energy theorem
5.7.3 Simple example of
energy minimization
5.7.4 Variational approach to
beam theory
5.7.5 Estimating stiffness
5.8 Reciprocal theorem >
5.8.1 Statement and proof
of theorem
5.8.2 Simple example
5.8.3 Boundaryinternal
value relations
5.8.4 3D dislocation loops
5.9 Energetics of
dislocations >
5.9.1 Potential energy of
isolated loop
5.9.2 Nonsingular
dislocation theory
5.9.3 Dislocation in
bounded solid
5.9.4 Energy of interacting
loops
5.9.5 PeachKoehler formula
5.10 Rayleigh Ritz method
>
5.10.1 Mode shapes, nat.
frequencies, Rayleigh's
principle
5.10.2 Natural frequency of a
beam
6. Solutions for plastic
solids >
6.1 Slipline
fields >
6.1.1 Interpreting slip
line fields
6.1.2 Derivation of slip
line fields
6.1.3 Examples of
solutions
6.2 Bounding
theorems >
6.2.1 Definition of plastic
dissipation
6.2.2 Principle of min
plastic dissipation
6.2.3 Upper bound collapse
theorem
6.2.4 Lower bound collapse
theorem
6.2.5 Examples of bounding
theorems
6.2.6 Lower bound shakedown
theorem
6.2.7 Examples of lower bound
shakedown theorem
6.2.8 Upper bound shakedown
theorem
6.2.9 Examples of upper bound
shakedown theorem
7. Introduction to FEA >
7.1 Guide to FEA >
7.1.1 FE mesh
7.1.2 Nodes and elements
7.1.3 Special elements
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7.1.4 Material behavior
7.1.5 Boundary conditions
7.1.6 Constraints
7.1.7 Contacting
surface/interfaces
7.1.8 Initial
conditions/external fields
7.1.9 Soln procedures / time
increments
7.1.10 Output
7.1.11 Units in FEA
calculations
7.1.12 Using dimensional
analysis
7.1.13 Scaling governing
equations
7.1.14 Remarks on dimensional
analysis
7.2 Simple FEA
program >
7.2.1 FE mesh and
connectivity
7.2.2 Global displacement
vector
7.2.3 Interpolation functions
7.2.4 Element strains &
energy density
7.2.5 Element stiffness
matrix
7.2.6 Global stiffness matrix
7.2.7 Boundary loading
7.2.8 Global force vector
7.2.9 Minimizing potential
energy
7.2.10 Eliminating prescribed
displacements
7.2.11 Solution
7.2.12 Post processing
7.2.13 Example code
8. Theory & Implementation
of FEA >
8.1 Static linear
elasticity >
8.1.1 Review of virtual
work
8.1.2 Weak form of
governing equns
8.1.3 Interpolating
displacements
8.1.4 Finite element
equations
8.1.5 Simple 1D
implementation
8.1.6 Summary of 1D
procedure
8.1.7 Example 1D code
8.1.8 Extension to 2D/3D
8.1.9 2D interpolation
functions
8.1.10 3D interpolation
functions
8.1.11 Volume integrals
8.1.12 2D/3D integration
schemes
8.1.13 Summary of element
matrices
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8.1.14 Sample 2D/3D code
8.2 Dynamic
elasticity >
8.2.1 Governing equations
8.2.2 Weak form of governing
eqns
8.2.3 Finite element
equations
8.2.4 Newmark time
integration
8.2.5 Simple 1D
implementation
8.2.6 Example 1D code
8.2.7 Lumped mass matrices
8.2.8 Example 2D/3D code
8.2.9 Modal time integration
8.2.10 Natural
frequencies/mode shapes
8.2.11 Example 1D modal
dynamic code
8.2.12 Example 2D/3D modal
dynamic code
8.3 Hypoelasticity >
8.3.1 Governing equations
8.3.2 Weak form of governing
eqns
8.3.3 Finite element
equations
8.3.4 NewtonRaphson
iteration
8.3.5 Tangent moduli for
hypoelastic solid
8.3.6 Summary of Newton
Raphson method
8.3.7 Convergence problems
8.3.8 Variations on Newton
Raphson
8.3.9 Example code
8.4 Hyperelasticity >
8.4.1 Governing equations
8.4.2 Weak form of
governing eqns
8.4.3 Finite element
equations
8.4.4 NewtonRaphson
iteration
8.4.5 NeoHookean tangent
moduli
8.4.6 Evaluating boundary
integrals
8.4.7 Convergence
problems
8.4.8 Example code
8.5 Viscoplasticity >
8.5.1 Governing equations
8.5.2 Weak form of
governing eqns
8.5.3 Finite element
equations
8.5.4 Integrating the
stressstrain law
8.5.5 Material tangent
8.5.6 NewtonRaphson
solution
8.5.7 Example code
8.6 Advanced elements
>
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8.6.1 Shear
locking/incompatible modes
8.6.2 Volumetric
locking/Reduced integration
8.6.3 Incompressible
materials/Hybrid elements
9. Modeling Material
Failure >
9.1 Mechanisms of failure
>
9.1.1 Monotonic
loading
9.1.2 Cyclic
loading
9.2 Stress/strain based
criteria >
9.2.1 Stress based
criteria
9.2.2 Probabilistic
methods
9.2.3 Static fatigue
criterion
9.2.4 Models of
crushing failure
9.2.5 Ductile failure
criteria
9.2.6 Strain
localization
9.2.7 High cycle
fatigue
9.2.8 Low cycle fatigue
9.2.9 Variable
amplitude loading
9.3 Elastic fracture
mechanics >
9.3.1 Crack tip fields
9.3.2 Linear elastic
fracture mechanics
9.3.3 Calculating stress
intensities
9.3.4 Using FEA
9.3.5 Measuring toughness
9.3.6 Values of fracture
toughness
9.3.7 Stable tearing
9.3.8 Mixed mode fracture
9.3.9 Static fatigue
9.3.10 Cyclic fatigue
9.3.11 Finding cracks
9.4 Energy methods in
fracture >
9.4.1 Definition of energy
release rate
9.4.2 Energy based fracture
criterion
9.4.3 GK relations
9.4.4 Gcompliance relation
9.4.5 Calculating K with
compliance
9.4.6 Integral expression
for G
9.4.7 The J integral
9.4.8 Calculating K using J
9.5 Plastic fracture
mechanics >
9.5.1 DugdaleBarenblatt
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model
9.5.2 HRR crack tip
fields
9.5.3 J based fracture
mechanics
9.6 Interface fracture
mechanics >
9.6.1 Interface crack tip
fields
9.6.2 Interface fracture
mechanics
9.6.3 Stress intensity
factors
9.6.4 Crack path
selection
10. Rods, Beams, Plates &
Shells >
10.1 Dyadic notation
10.2 Deformable rods 
general >
10.2.1 Characterizing the
xsection
10.2.2 Coordinate systems
10.2.3 Kinematic relations
10.2.4 Displacement,
velocity and acceleration
10.2.5 Deformation gradient
10.2.6 Strain measures
10.2.7 Kinematics of bent
rods
10.2.8 Internal forces and
moments
10.2.9 Equations of motion
10.2.10 Constitutive
equations
10.2.11 Strain energy
density
10.3 String / beam
theory >
10.3.1 Stretched string
10.3.2 Straight beam (small
deflections)
10.3.3 Axially loaded beam
10.4 Solutions for rods
>
10.4.1 Vibration of a
straight beam
10.4.2 Buckling under
gravitational loading
10.4.3 Post buckled shape of
a rod
10.4.4 Rod bent into a helix
10.4.5 Helical spring
10.5 Shells  general >
10.5.1 Coordinate systems
10.5.2 Using non
orthogonal bases
10.5.3 Deformation
measures
10.5.4 Displacement and
velocity
10.5.5 Deformation
gradient
10.5.6 Other strain
measures
10.5.7 Internal forces
and moments
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10.5.8 Equations of
motion
10.5.9 Constitutive
relations
10.5.10 Strain energy
10.6 Plates and
membranes >
10.6.1 Flat plates (small
strain)
10.6.2 Flat plates with in
plane loading
10.6.3 Plates with large
displacements
10.6.4 Membranes
10.6.5 Membranes in polar
coordinates
10.7 Solutions for
shells >
10.7.1 Circular plate bent by
pressure
10.7.2 Vibrating circular
membrane
10.7.3 Natural frequency of
rectangular plate
10.7.4 Thin film on a
substrate (Stoney eqs)
10.7.5 Buckling of heated
plate
10.7.6 Cylindrical shell
under axial load
10.7.7 Twisted open walled
cylinder
10.7.8 Gravity loaded
spherical shell
A: Vectors & Matrices
B: Intro to tensors
C: Index Notation
D: Using polar coordinates
E: Misc derivations
Problems
1. Objectives and
Applications >
1.1 Defining a
Problem
2. Governing Equations >
2.1 Deformation
measures
2.2 Internal forces
2.3 Equations of
motion
2.4 Work and Virtual
Work
3. Constitutive Equations
>
3.1 General requirements
3.2 Linear elasticity
3.3 Hypoelasticity
3.4 Elasticity w/ large
rotations
3.5 Hyperelasticity
3.6 Viscoelasticity
3.7 Rate independent
plasticity
3.8 Viscoplasticity
3.9 Large strain
plasticity
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3.10 Large strain
viscoelasticity
3.11 Critical state
soils
3.12 Crystal plasticity
3.13 Surfaces and
interfaces
4. Solutions to simple
problems >
4.1 Axial/Spherical linear
elasticity
4.2 Axial/Spherical
elastoplasticity
4.3 Spherical
hyperelasticity
4.4 1D elastodynamics
5. Solutions for elastic
solids >
5.1 General Principles
5.2 2D Airy function
solutions
5.3 2D Complex variable
solutions
5.4 3D static problems
5.5 2D Anisotropic
elasticity
5.6 Dynamic problems
5.7 Energy methods
5.8 Reciprocal theorem
5.9 Energetics of
dislocations
5.10 Rayleigh Ritz method
6. Solutions for plastic
solids >
6.1 Slipline
fields
6.2 Bounding
theorems
7. Introduction to FEA >
7.1 Guide to FEA
7.2 Simple FEA
program
8. Theory & Implementation
of FEA >
8.1 Static linear
elasticity
8.2 Dynamic
elasticity
8.3 Hypoelasticity
8.4 Hyperelasticity
8.5 Viscoplasticity
8.6 Advanced elements
9. Modeling Material
Failure >
9.1 Mechanisms of failure
9.2 Stress/strain based
criteria
9.3 Elastic fracture
mechanics
9.4 Energy methods in
fracture
9.5 Plastic fracture
mechanics
9.6 Interface fracture
mechanics
10. Rods, Beams, Plates &
Shells >
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10.1 Dyadic notation
10.2 Deformable rods 
general
10.3 String / beam
theory
10.4 Solutions for rods
10.5 Shells  general
10.6 Plates and
membranes
10.7 Solutions for
shells
A: Vectors & Matrices
B: Intro to tensors
C: Index Notation
D: Using polar coordinates
E: Misc derivations
FEA codes
Maple
Matlab
Report an error
Chapter 8
Theory and Implementation of the Finite Element Method
8.6 Advanced element formulations Incompatible modes; reduced
integration; and hybrid elements
Techniques for interpolating the displacement field within 2D and 3D finite elements were discussed in Section
8.1.9 and 8.1.10. In addition, methods for evaluating the volume or area integrals in the principle of virtual work
were discussed in Section 8.1.11. These procedures work well for most applications, but there are situations
where the simple element formulations can give very inaccurate results. In this section
1.
We illustrate some of the unexpected difficulties that can arise in apparently perfectly well designed finite
element solutions to boundary value problems;
2.
We describe a few more sophisticated elements that have been developed to solve these problems.
We focus in particular on `Locking’ phenomena. Finite elements are said to `lock’ if they exhibit an unphysically
stiff response to deformation. Locking can occur for many different reasons. The most common causes are (i)
the governing equations you are trying to solve are poorly conditioned, which leads to an ill conditioned system of
finite element equations; (ii) the element interpolation functions are unable to approximate accurately the strain
distribution in the solid, so the solution converges very slowly as the mesh size is reduced; (iii) in certain element
formulations (especially beam, plate and shell elements) displacements and their derivatives are interpolated
separately. Locking can occur in these elements if the interpolation functions for displacements and their
derivatives are not consistent.
8.6.1 Shear locking and incompatible mode elements
Shear locking can be illustrated by attempting to find a finite
element solution to the simple boundary value problem illustrated
in the picture. Consider a cantilever beam, with length L, height
2a and outofplane thickness b, as shown in the figure. The top
and bottom of the beam are traction free, the left hand
end is subjected to a resultant force P, and the right hand end is
clamped. Assume that b<<a, so that a state of plane stress is
developed in the beam. The analytical solution to this problem is given in Section 5.2.4.
The figures below compare this result to a finite element solution, obtained with standard 4 noded linear
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quadrilateral plane stress elements. Results are shown for two different ratios of , with
for both cases.
For the thick beam, the finite element and exact solutions agree nearly perfectly. For the thin beam, however,
the finite element solution is very poor even though the mesh resolution is unchanged. The error in the finite
element solution occurs because the standard 4 noded quadrilateral elements cannot accurately approximate the
strain distribution associated with bending. The phenomenon is known as `shear locking’ because the element
interpolation functions give rise to large, unphysical shear strains in bent elements. The solution would eventually
converge if a large number of elements were added along the length of the beam. The elements would have to
be roughly square, which would require about 133 elements along the length of the beam, giving a total mesh size
of about 500 elements.
Shear locking is therefore relatively benign, since it can be detected by refining the mesh, and can be avoided by
using a sufficiently fine mesh. However, finite element analysts sometimes cannot resist the temptation to
reduce computational cost by using elongated elements, which can introduce errors.
Shear locking can also be avoided by using more sophisticated element interpolation functions that can accurately
approximate bending. `Incompatible Mode’ elements do this by adding an additional strain distribution to the
element. The elements are called `incompatible’ because the strain is not required to be compatible with the
displacement interpolation functions. The approach is conceptually straightforward:
1.
The displacement fields in the element are interpolated using the standard scheme, by setting
where are the shape functions listed in Sections 8.1.9 or 8.1.10, are a
set of local coordinates in the element, denote the displacement values and
coordinates of the nodes on the element, and is the number of nodes on the
element.
2.
The Jacobian matrix for the interpolation functions, its determinant, and its inverse are defined in the
usual way
3.
The usual expression for displacement gradient in the element is replaced by
where p=2 for a 2D problem and p=3 for a 3D problem, are a set of unknown displacement
gradients in the element, which must be determined as part of the solution.
4.
Similarly, the virtual displacement gradient is written as
where is a variation in the internal displacement gradient field for the element.
5.
These expressions are then substituted into the virtual work equation, which must now be satisfied for
all possible values of virtual nodal displacements and virtual displacement gradients . At first
sight this procedure appears to greatly increase the size of the global stiffness matrix, because a set of
unknown displacement gradient components must be calculated for each element. However, the
unknown are local to each element, and can be eliminated while computing the element stiffness
matrix. The procedure to do this can be shown most clearly in a sample code.
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A sample smallstrain, linear elastic code with
incompatible mode elements is provided in
Femlinelast_incompatible_modes.mws. When run with
the input file shear_locking_demo.txt it produces the
results shown in the figure. The incompatible modes
clearly give a spectacular improvement in the
performance of the element.
HEALTH WARNINGS:
(i) The procedure outlined here only works for smallstrain problems. Finite strain
versions exist but are somewhat more complicated. (ii) Adding strain variables to elements can dramatically
improve their performance, but this procedure must be used with great care to ensure that the strain and
displacement degrees of freedom are independent variables. For further details see Simo, J. C., and M. S. Rifai,
Int J. Num. Meth in Eng.
29
, pp. 1595 1638, 1990. Simo, J. C., and F. Armero, Int J. Num. Meth in Eng.,
33
,
pp. 1413 1449, 1992.
8.6.2 Volumetric locking and reduced integration elements
Volumetric locking can be illustrated using a simple boundary value problem.
Consider a long hollow cylinder with internal radius a and external radius b as
shown in the figure. The solid is made from a linear elastic material with Young’s
modulus and Poisson’s ratio . The cylinder is loaded by an internal pressure
and deforms in plane strain.
The analytical solution to this problem is given in Section 4.1.9.
The figures below compare the analytical solution to a finite element solution with
standard 4 noded plane strain quadrilateral elements. Results are shown for two
values of Poisson’s ratio . The dashed lines show the analytical solution, while
the solid line shows the FEA solution.
The two solutions agree well for , but the finite element solution grossly underestimates the displacements
as Poisson’s ratio is increased towards 0.5 (recall that the material is incompressible in the limit ). In this
limit, the finite element displacements tend to zero this is known as `volumetric locking’
The error in the finite element solution occurs because the finite element interpolation functions are unable to
properly approximate a volume preserving strain field. In the incompressible limit, a nonzero volumetric strain at
any of the integration points gives rise to a very large contribution to the virtual power. The interpolation
functions can make the volumetric strain vanish at some, but not all, the integration points in the element.
Volumetric locking is a much more serious problem than shear locking, because it cannot be avoided by refining
the mesh. In addition, all the standard fully integrated finite elements will lock in the incompressible limit; and
some elements show very poor performance even for Poisson’s ratios as small as . Fortunately, most
materials have Poisson’s ratios around 0.3 or less, so the standard elements can be used for most linear elasticity
and smallstrain plasticity problems. To model rubbers, or to solve problems involving large plastic strains, the
elements must be redesigned to avoid locking.
Reduced Integration
is the simplest way to avoid locking. The basic idea is simple: since the fully integrated
elements cannot make the strain field volume preserving at all the integration points, it is tempting to reduce the
number of integration points so that the constraint can be met. `Reduced integration’ usually means that the
element stiffness is integrated using an integration scheme that is one order less accurate than the standard
scheme. The number of reduced integration points for various element types is listed in the table below. The
coordinates of the integration points are listed in the tables in Section 8.1.12.
Number of integration points for reduced integration schemes
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Linear triangle (3 nodes) 1 point
Quadratic triangle (6 nodes): 3 points
Linear quadrilateral (4 nodes): 1 point
Quadratic quadrilateral (8 nodes): 4 points
Linear tetrahedron (4 nodes): 1 point
Quadratic tetrahedron (10 nodes): 4 points
Linear brick (8 nodes): 1 point
Quadratic brick (20 nodes): 8 points
HEALTH WARNING:
Notice that the integration order cannot be reduced for the linear triangular and
tetrahedral elements these elements should not be used to model near incompressible materials, although in
desperation you can a few such elements in regions where the solid cannot be meshed using quadrilaterals.
Remarkably, reduced integration completely resolves locking in some elements (the quadratic quadrilateral and
brick), and even improves the accuracy of the element. As an example, the figure below shows the solution to
the pressurized cylinder problem, using both full and reduced integration for 8 noded quadrilaterals. With
reduced integration, the analytical and finite element results are indistinguishable.
Reduced integration does not work in 4 noded quadrilateral elements or 8
noded brick elements. For example, the figure on the right shows the
solution to the pressure vessel problem with linear (4 noded) quadrilateral
elements with reduced integration. The solution is clearly a disaster. The
error occurs because the stiffness matrix is nearly singular the system of
equations includes a weakly constrained deformation mode. This
phenomenon is known as `hourglassing’ because of the characteristic shape
of the spurious deformation mode.
Selectively Reduced Integration
can be used to cure hourglassing. The
procedure is illustrated most clearly by modifying the formulation for static linear elasticity. To implement the
method:
1.
The volume integral in the virtual work principle is separated into a deviatoric and volumetric part by
writing
Here, the first integral on the right hand side vanishes for a hydrostatic stress.
2.
Substituting the linear elastic constitutive equation and the finite element interpolation functions into
the virtual work principle, we find that the element stiffness matrix can be reduced to
3.
When selectively reduced integration is used, the first volume integral is evaluated using the full
integration scheme; the second integral is evaluated using reduced integration points.
Selective reduced integration has been implemented in the sample program
fem_selective_reduced_integration.mws. When this code is run with the
input file volumetric_locking_demo.txt it produces the results shown in the
figure. The analytical and finite element solutions agree, and there are no
signs of hourglassing.
In many commercial codes, the `fully integrated’ elements actually use
selective reduced integration.
The ‘Bbar’ method
: Like selective reduced integration, the Bbar method works by treating the volumetric
and deviatoric parts of the stiffness matrix separately. Instead of separating the volume integral into two parts,
however, the Bbar method modifies the definition of the strain in the element. Its main advantage is that the
concept can easily be generalized to finite strain problems. Here, we will illustrate the method by applying it to
smallstrain linear elasticity. The procedure starts with the usual virtual work principle
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Hourglass base vectors
Linear quadrilateral
Linear brick
In the Bbar method
1.
We introduce a new variable to characterize the volumetric strain in the elements. Define
where the integral is taken over the volume of the element.
2.
The strain variation in each element is replaced by the approximation
3.
Similarly, the virtual strain in each element is replaced by
This means that the volumetric strain in the element is everywhere equal to its mean value. The virtual work
principle is then written in terms of and as
Finally, introducing the finite element interpolation and using the constitutive equation yields the usual system of
linear finite element equations, but with a modified element stiffness matrix given by
This expression can be integrated using a standard, full integration scheme.
The Bbar method has been implemented in the sample code FEM_Bbar.mws. The code can be run with the
input file volumetric_locking_demo.txt. Run the code for yourself to verify that the analytical and finite element
solutions agree, and there are no signs of hourglassing.
Reduced integration with hourglass control:
Hourglassing in 4 noded quadrilateral and 8 noded
brick elements can also be cured by adding an
artificial stiffness to the element that acts to
constrain the hourglass mode. The stiffness must
be carefully chosen so as to influence only
the
hourglass mode
. Only the final result will be given
here for details see D.P. Flanagan and T.
Belytschko, International J. Num Meth in Engineering,
17
, pp. 679 706, (1981). To compute the corrective term:
1.
Define a series of `hourglass base vectors’ which specify the displacements of the ath node in the ith
hourglass mode. The 4 noded quadrilateral element has only one hourglass mode; the 8 noded brick has 4
modes, listed in the table.
2.
Calculate the `hourglass shape vectors’ for each mode as follows
where denotes the number of nodes on the element.
3.
The modified stiffness matrix for the element is written as
where denotes the volume of the element, and is a numerical parameter that controls the stiffness of
the hourglass resistance. Taking where is the elastic shear modulus
works well for most applications. If is too large, it will seriously overstiffen the solid.
Sample code:
Reduced integration with hourglass control has been
implemented in the sample code Fem_hourglasscontrol.mws. When run
with the input file volumetric_locking_demo.txt it produces the results
shown in the picture. Hourglassing has clearly been satisfactorily
eliminated.
HEALTH WARNING:
Hourglass control is not completely effective: it
can fail for finite strain problems and can also cause problems in a
dynamic analysis, where the low stiffness of the hourglass modes can
introduce spurious low frequency vibration modes and low wave speeds.
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8.6.3 Hybrid elements for modeling nearincompressible materials
The bulk elastic modulus is infinite for a fully incompressible material, which leads to an infinite stiffness matrix
in the standard finite element formulation (even if reduced integration is used to avoid locking). This behavior
can cause the stiffness matrix for a nearly incompressible material to become illconditioned, so that small
rounding errors during the computation result in large errors in the solution.
Hybrid elements are designed to avoid this problem. They work by including the hydrostatic stress distribution as
an additional unknown variable, which must be computed at the same time as the displacement field. This allows
the stiff terms to be removed from the system of finite element equations. The procedure is illustrated most
easily using isotropic linear elasticity, but in practice hybrid elements are usually used to simulate rubbers or
metals subjected to large plastic strains.
Hybrid elements are based on a modified version of the principle of virtual work, as follows. The virtual work
equation (for small strains) is rewritten as
Here
1.
is the deviatoric stress, determined from the displacement field;
2.
is the hydrostatic stress, again determined from the displacement field;
3.
p is an additional degree of freedom that represents the (unknown) hydrostatic stress in the solid;
4.
is an arbitrary variation in the hydrostatic stress;
5.
is the bulk modulus of the solid.
The modified virtual work principle states that, if the virtual work equation is satisfied for all kinematically
admissible variations in displacement and strain and all possible variations in
pressure , the stress field will satisfy the equilibrium equations and traction boundary conditions, and the
pressure variable p will be equal to the hydrostatic stress in the solid.
The finite element equations are derived in the usual way
1.
The displacement field, virtual displacement field and position in the each element are interpolated
using the standard interpolation functions defined in Sections 8.1.9 and 8.1.10 as
2.
Since the pressure is now an independent variable, it must also be interpolated. We write
where are a discrete set of pressure variables, is an arbitrary change in these pressure variables,
are a set of interpolation functions for the pressure, and is the number of pressure variables
associated with the element. The pressure need not be continuous across neighboring elements, so that
independent pressure variables can be added to each element. The following schemes are usually used
a.
In linear elements (the 3 noded triangle, 4 noded quadrilateral, 5 noded tetrahedron or 8 noded
brick) the pressure is constant. The pressure is defined by its value at the centroid of each
element, and the interpolation functions are constant.
b.
In quadratic elements (6 noded triangle, 8 noded quadrilateral, 10 noded tetrahedron or 20
noded brick) the pressure varies linearly in the element. Its value can be defined by the
pressure at the corners of each element, and interpolated using the standard linear interpolation
functions.
3.
For an isotropic, linear elastic solid with shear modulus and Poisson ratio the deviatoric stress is
related to the displacement field by , while the hydrostatic
stress is , where is the bulk modulus.
4.
Substituting the linear elastic equations and the finite element interpolation functions into the virtual
work principle leads to a system of equations for the unknown displacements and pressures of
the form
where the global stiffness matrices are obtained by summing the following element stiffness
matrices
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and the force
F
is defined in the usual way. The integrals defining may be evaluated using the full
integration scheme or reduced integration (hourglass control may be required in this case). The
remaining integrals must be evaluated using reduced integration to avoid element locking.
5.
Note that, although the pressure variables are local to the elements, they cannot be eliminated from the
element stiffness matrix, since doing so would reduce the element stiffness matrix to the usual, non
hybrid form. Consequently, hybrid elements increase the cost of storing and solving the system of
equations.
(c) A.F. Bower, 2008
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