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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 Cauchy-Green

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 stress-strain

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 Spring-damper

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 Stress-strain

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

(Cam-clay)

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 1-D 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 Saint-Venants

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 Papkovich-Neuber

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 P-d 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 Barnett-Lothe

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 Boundary-internal

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 Peach-Koehler 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 Slip-line

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 Newton-Raphson

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 Newton-Raphson

iteration

8.4.5 Neo-Hookean 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

stress-strain law

8.5.5 Material tangent

8.5.6 Newton-Raphson

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 G-K relations

9.4.4 G-compliance 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 Dugdale-Barenblatt

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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

x-section

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 Slip-line

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 out-of-plane 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 small-strain, 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 small-strain 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 small-strain 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 ‘B-bar’ method

: Like selective reduced integration, the B-bar 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 B-bar 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

small-strain linear elasticity. The procedure starts with the usual virtual work principle

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Hourglass base vectors

Linear quadrilateral

Linear brick

In the B-bar 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 B-bar 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 over-stiffen 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 near-incompressible 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 ill-conditioned, 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 re-written 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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