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• 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.

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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.

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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.

.

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

4

OBJECTIVE

To learn the use of finite element method for

the solution of problems involving elastic-

plastic materials.

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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+llo

Ao

Effects of high straining rates

Effects of test temperature

Foil versus bulk specimens

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

(4) Post-yield deformation occurs

at reduced material stiffness

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

7

Monotonic stress-strain behavior Engineering stressoA

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

8

Plastic strains

Elastic region Hookes 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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

10

Ideal Materials behavior

Elstic-perfectly plastic

perfectly plastic

Elstic-linear hardening

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

12

Elastic-plastic problem in 2-D

In elastic region:

klijklij C

jkiljlikklijijklC

, are Lam constants

jiif

jiifij

0

1

Kroneckers delta:

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

Hookes Law

21

2

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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)

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

18

Maximum-distortion-energy theory

(von Mises)

22132

32

2

21 2 Y

22

221

2

1 Y

+

deviatoric hydrostatic

For plane problems:

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

19

Maximum-normal-stress theory

(Rankine)

ult

ult

2

1

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

20

Tresca and von Mises yield criteria

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

21

Tresca and von Mises yield criteria

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

22

Strain Hardening

behavior

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

23

Normality rule of associated plasticity

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

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

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS