G FE FORMULATIONS FOR PLASTICITYtaminmn/MMJ 1153/MMJ1153_E_FE... · FE FORMULATIONS FOR PLASTICITY M.N. Tamin, ... Finite Elements in Plasticity –Theory and Practice, ... To learn

FE FORMULATIONS FOR PLASTICITY M.N. Tamin, CSMLab, UTM

MMJ1153 – COMPUTATIONAL METHOD IN SOLID MECHANICS

G – FE FORMULATIONS FOR PLASTICITY

1

These slides are designed based on the book:

Finite Elements in Plasticity – Theory and Practice, D.R.J. Owen and E. Hinton,

1970, Pineridge Press Ltd., Swansea, UK.



2

Course Content:

A – INTRODUCTION AND OVERVIEW

Numerical method and Computer-Aided Engineering; Physical

problems; Mathematical models; Finite element method.

B – REVIEW OF 1-D FORMULATIONS

Elements and nodes, natural coordinates, interpolation function, bar

elements, constitutive equations, stiffness matrix, boundary

conditions, applied loads, theory of minimum potential energy; Plane

truss elements; Examples.

C – PLANE ELASTICITY PROBLEM FORMULATIONS

Constant-strain triangular (CST) elements; Plane stress, plane

strain; Axisymmetric elements; Stress calculations; Programming

structure; Numerical examples.



3

D – ELASTIC-PLASTIC PROBLEM FORMULATIONS

Iterative solution methods,1-D problems, Mathematical theory of

plasticity, matrix formulation, yield criteria, Equations for plane stress

and plane strain case; Numerical examples

E – PHYSICAL INTERPRETATION OF FE RESULTS

Case studies in solid mechanics; FE simulations in 3-D; Physical

interpretation; FE model validation.

.



4

OBJECTIVE

To learn the use of finite element method for

the solution of problems involving elastic-

plastic materials.



Terikan

0.0 0.1 0.2 0.3 0.4 0.5

Tegasan (

MP

a)

0

200

400

600

Ujikaji A

Ujikaji B

5

Stress-strain diagram

P

P

lo+lloAo

• Effects of high straining rates

• Effects of test temperature

• Foil versus bulk specimens



6

Elastic-Plastic Behavior

Characteristics:

(1) An initial elastic material

response onto which a plastic

deformation is superimposed after

a certain level of stress has been

reached

(2) Onset of

yield governed

by a yield

criterion

(3) Irreversible on

unloading

(4) Post-yield deformation occurs

at reduced material stiffness



7

Monotonic stress-strain behavior Engineering stress

oA

PS

A

P

oo

o

l

l

l

lle

l

dld

ol

l

l

dlln

eS 1

eA

Ao 1lnln

Typical low carbon steelTrue stress

Engineering strain

True or natural strain

For the assumed constant

volume condition,

Then;

oo lAlA



8

Plastic strains

Elastic region – Hooke’s law:

E

Terikan

0.0 0.1 0.2 0.3 0.4 0.5

Tegasan (

MP

a)

0

200

400

600

Ujikaji A

Ujikaji B

Engineering stress-strain diagram

log)1(loglog E

Plastic region:

npK

logloglog nK

True stress-strain diagram

SS316 steel



9

PLASTIC STRAIN, p

0.0 0.1 0.2 0.3 0.4 0.5 0.6

ST

RE

SS

,

(M

Pa

)

0

100

200

300

400

500

600

700

PLASTIC STRAIN, p

0.01 0.1 1

ST

RE

SS

,

(M

Pa)

100

1000

= Kn

log K = 2.8735n = 0.1992

r2 = 0.9772

199.03.747 p

Plastic strains - Example

Non-linear /Power-law

SS316 steel npK



10

Ideal Materials behavior

Elstic-perfectly plastic

perfectly plastic

Elstic-linear hardening



11

Simulation of elastic-plastic problem

Requirements for formulating the theory to model elastic-plastic

material deformation:

• explicit stress-strain

relationship under elastic

condition.

• a yield criterion

indicating the onset

of plastic flow.

• stress-strain relationship

for post yield behavior



12

Elastic-plastic problem in 2-D

In elastic region:

klijklij C

jkiljlikklijijklC

, are Lamé constants

jiif

jiifij

0

1

Kronecker’s delta:



Elastic case for plane stress and plane strain (Review)

13

x

y

xy

u

x

v

y

u v

y x

Strain

Stress

x

y

xy

D

Hooke’s Law

2

12

1 0

1 01

0 0 v

vE

D vv

12

1 0

1 01 1 2

0 0

v vE

D v vv v

v

Elasticity matrix

Plane stress

Plane strain



14

Yield criteria:

Stress level at which plastic deformation begins

kf ij

k() is an experimentally determined function of hardening parameter

The material of a component subjected to complex loading will start to

yield when the (parametric stress) reaches the (characteristic stress) in

an identical material during a tensile test.

Other parameters:

Strain

Energy

Specific stress component (shear stress, maximum principal stress)



15

Yield criteria:

Stress level at which plastic deformation begins

kf ij

Yield criterion should be independent of coordinate axes orientation,

thus can be expressed in terms of stress invariants:

kijkij

ijij

ii

J

J

J

3

1

2

1

3

2

1



16

Plastic deformation is independent of hydrostatic pressure

kJJf 32 ,

kkijijij 3

1

are the second and third invariant of deviatoric stress: 32 , JJ

+

deviatoric hydrostatic



17

Yield Criteria

For ductile materials:

•Maximum-distortion (shear) strain energy criterion (von-Mises)

• Maximum-shear-stress criterion (Tresca)

For brittle materials:

• Maximum-principal-stress criterion (Rankine)

• Mohr fracture criterion



18

Maximum-distortion-energy theory

(von Mises)

22

13

2

32

2

21 2 Y

22

221

2

1 Y

+

deviatoric hydrostatic

For plane problems:



19

Maximum-normal-stress theory

(Rankine)

ult

ult

2

1



20

Tresca and von Mises yield criteria



21

Tresca and von Mises yield criteria



22

Strain Hardening

behavior



23

Normality rule of associated plasticity



Notations:

24

0

FQK

0 FQ K

Nonlinear problem:

Thus, iterative solution is required

Linear elastic problem:

0

fH

0f H

For 1-D (1 variable) problem:

0 fH



25

• Direct iteration / successive approximation

• Newton-Raphson method

• Tangential stiffness method

• Initial stiffness method

The problem:

0

fH

0f H

0 fH

Iterative Solution Methods

For single variable



Iterative Solution Methods

(a) Direct iteration or successive approximation

26

fH-1

H

In each solution step, the previous solution for the

unknowns is used to predict the current values of the

coefficient matrix

For (r+1)th approximation:

fH-11 rr

Convergence is not guaranteed and cannot be predicted at initial

solution stage. Converged iteration when r-1 and r are close.

Initial guess 0 is based on solution for an

average material properties.

0f H



Direct iteration method

for a single variable

Convex H- relation

27

Task:

to illustrate the process.



28

Steps for Direct Iteration Method

Start with {0}

Determine [H(0)]

Calculate

Repeat until {r+1} {r}

fH101

0

H(0)

1 = -H(0) f

r+1 r



Direct iteration method


29

Concave H- relation



30

Assignment # 5

A one degree of freedom problem can be represented by

H + f = 0

where f = 10 and H() = 10 (1+e3)

Design and run a computer algorithm to determine the solution

of the problem using direct iteration method and Newton-

Raphson method.

(a) Show the flow chart of the algorithm and description of the

steps.

(b) Tabulate the iterative values of H and and compare the

converged solution from the two methods. Specify a

convergence tolerance of 1%.

Try with 0 = 0.2Should converge at 7 = -9.444



31

Review of Newton-Raphson Method(for finding root of an equation f(x)= 0)

Taylor Series expansion:

...!2

11

2

1111 xfxxxfxxxfxf

Consider only the first 2 terms:

111 xfxxxfxf

Set f(x)=0 to find roots:

1

11

111 0

xf

xfxx

xfxxxf



32

Review of Newton-Raphson Method(for finding root of an equation f(x)= 0)

Iterative procedures:

1

11

xf

xfxx ii



Newton-Raphson Method

33

0

0

fH

fHΨ

rr

The problem

If true solution exists at

Converged iteration when n-1 and n are close

Initial guess is based on 0 is based on solution

for an average material properties.

r

j

iN

1j

r

j

r

i

ψΔψ

rrr J

is the residual force



Jacobian Matrix

34

HHJ

ψhJ

m

1k

r

k

r

j

ir

ij

j

iij

rrr

rrr

HH

J

1

1

rrr 1



Newton-Raphson

method for a single

variable

35



Newton-Raphson

method for a single

variable

36



37

Tangential Stiffness Method

(Generalized Newton-Raphson Method)

Assume a trial value {0}

Calculate [H(0)]

Calculate {(0)}

Calculate

Iterate until {r} {0}

0100

H

fH

001

Linearize {} in [H()] such that the term [H’()] can be omitted

Steps:



Tangential stiffness

method for a single

variable

38



39

Initial Stiffness Method

fH rr 11 Recall that in Direct Stiffness Method:

In Tangential Stiffness Method: rrr H 1

This requires complete reduction and solution of the set of

simultaneous equations for each iteration.

Use the initial stiffness [H(0)] for subsequent approximation.

rr H 10



Initial stiffness method


40



Uniaxial stress-strain curve for elastic-linear hardening

41

Assume

Define strain-hardening

parameter:

pe ddd

pd

dH



42

11

11)(

L

EAK

L

EAFK

e

e

e

AdHAddFA

F

LddLddL

pe

;

;

HE

E

L

EA

LdE

d

AdH

d

dFK

p

p

ep 1

11

111

)(

HE

E

L

EAK

e

ep

Element stiffness for elastic-plastic material behavior

1 2

L

F

Elastic behavior:

Elastic-plastic behavior:

Element stiffness matrix for elastic-

plastic material behavior



43

Numerical singularity issue

Slope, H’ = 0SY

11

111

)(

HE

E

L

EAK

e

ep

After initial yielding, H’=0,

Thus, [Kep](e) = [0]

Use initial stiffness method

to ensure positive definite

[K]



44

Program structure

for 1-D problemInput geometry, loading, b.c.

INITIALIZATION Zero all arrays for data storing

INC. LOAD

SET INDICATOR

STIFFNESS

Choose type of solution algorithm – direct iteration,

tangential stiffness, etc

ASSEMBLY

[K](e)

[K] {} = {F}

REDUCTION Solve for {}

RESIDUALCalculate residual force {} for Newton-Raphson,

Initial and Tangential stiffness method

Check for convergence

YN

Output results

ITE

RA

TIO

N L

OO

P

LO

AD

IN

CR

EM

EN

T L

OO

P



Incremental stress and strain changes at initial yielding

45

Task:

to illustrate the process.

r

e

rr

e

rr

e

r

rrr

rr

E

H

fH

1

1

11

11 r

pY

r

Y H

Check if the element has

previously yielded:



46

Uniaxial elastic-plastic strain hardening behavior



47

Engineering stress-strain curve for Al2219

G FE FORMULATIONS FOR PLASTICITYtaminmn/MMJ 1153/MMJ1153_E_FE... · FE FORMULATIONS FOR PLASTICITY M.N. Tamin, ... Finite Elements in Plasticity –Theory and Practice, ... To learn

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