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### Transcript of G FE FORMULATIONS FOR taminmn/MMJ 1153/MMJ1153_E_FE formulations fo · FE FORMULATIONS FOR...

• 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

The dependence of stress level for further plastic deformation,

after initial yielding on the current degree of plastic straining

Perfect plasticity

- Yield stress level does not depend on current

degree of plastic deformation

Strain Hardening models

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

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS

23

Isotropic strain hardening

-Current yield surfaces are a uniform expansion of the original yield curve,

without translation

-For strain soften material, the yield stress level at a point decreases with

increasing plastic strain, thus th