MECHANICS OF POLYMERS - Technische Universität Darmstadt · contact: [email protected]...

contact: [email protected]

MECHANICS OF POLYMERS

by

Stefan Kolling

Technische Universität Darmstadt, WS 2013/14

Course Contents and Goals

Topics

Basics on continuum mechanics

(no chemistry!)

Classification of polymers

Material modeling

Damage and failure

Composites

(glass fiber reinforces plastics)

Goals

General aspects of structural analysis

with focus on numerical simulation

Background and fundamentals

Experimental requirements and input

data for numerical simulation

Limits of the chosen formulations

2 Mechanics of Polymers

1. Introduction

Polymer Structures

Plastics is a group name comprising many different materials

Mechanical response at room temperature may be glassy or

rubbery

str

ess

1 2

3

4

5

strain

ε

1 glasslike behaviour

2 plastic or viscous flow

3 low ductility

4 high ductility

5 rubbery

amorphous

thermoplastic

crystalline

thermoplastic

thermoset plastic

(duroplast)

elastomer thermoplastic


Glass Transition and Temperature – Strain Rate Relation

Example: polyethylene

TTc Tg

glass

rubber

G

0

AKTe

Ref

Ref

1 1log C

T T

Roughly: for certain thermoplastics, 10°C decrease of temperature

corresponds to an increase of one order of magnitude in strain rate,

so the rate effects have a higher relative importance than in the case of

metals

Arrhenius’ law


Glass transition of PVB (Saflex):

Temperature


„Room temperature“

Source: Hooper, Blackman, Dear – The mechanical behaviour of poly(vinyl butyral) at different strain magnitudes and strain rates, J Mater Sci (2012)

Temperature


Consistency

Glass transistion and melting temperature

8

amorphous

= glassy

Tg [°C]

PS 105

PMMA 120

PVC 80

PC 150

PET 85

semicrystalline

Tm [°C]

PET 265

PBT 225

PA6 220

PA66 265

PE 110-135

PP 165

elastomers

Tg [°C]

BR -90

SBR -50

PEA -25

PBA -60

thermoplastics

Mechanics of Polymers

Usage of Polymers

Overview PE

PVC

PS PS PP


Usage of Polymers

In Automotive Industry

Percentage of polymeric

materials in a

middle class car (2003)

63,8%17,0%

4,7%

0,3%2,2%

0,5%

4,4%

7,1%

Stahl und Eisen

Polymere

Leichtmetalle

Betriebsstoffe

Sonstige Werkstoffe

Buntmetalle

Pozeßpolymere

Elektrik/Elektronik

steel

polymers

Al / Mg

other materials

63,8%17,0%

4,7%

0,3%2,2%

0,5%

4,4%

7,1%

Stahl und Eisen

Polymere

Leichtmetalle

Betriebsstoffe

Sonstige Werkstoffe

Buntmetalle

Pozeßpolymere

Elektrik/Elektronik

steel

polymers

Al / Mg

other materials


Usage of Polymers

Construction

PMMA

PE-PTFE


Summary

Polymers make life easier, cheaper and more comfortable

They have a wide range of application in engineering

Basic knowhow of polymers is just a must for an analysis engineer

(at least for a good one)

For structural parts made from polymers, computational methods

are still topic of ongoing research and development projects

In what follows we care for analysis methods in the context of

modern mechanics, i,e. with focus on numerical simulations


2. Basics on Continuum Mechanics

Basics on Continuum Mechanics - Contents

Introduction

Measure of Stress

Beam

Definition – Plane Stress – Principal Stress

True and Engineering Stress

Measure of Strain

Definition – Plane Strain – Principal Strain

True and Engineering Strain – Strain Rate


Introduction

Newton‘s 2nd Law

„The alteration of motion is ever proportional to the

motive force impressed, and is made in the direction

of the right line in which that force is impressed.“

Consequences hereof:

the force depends on direction and is thus a vector quantity

there is no motive force in the case of statics, i.e. we have neither

translation nor rotation of the system!

This is the fundamental law of mechanics and contains the

conservation of momentum (1st law) and action-reaction (3rd law) as

special cases

Sir Isaac Newton

(1643 –1727)


Introduction

Newton‘s 2nd Law (modern interpretation)

“The change of momentum of a body is proportional

to the impulse impressed on the body, and happens

along the straight line on which that impulse is

impressed ”

force = rate of momentum:

Case of statics: which leads to

In the case of statics vanishes the resulting force acting on a

body (equilibrium of forces).

Sir Isaac Newton

(1643 –1727)

( )dp d

F mvdt dt

0v 0F


Introduction

And for rotation of a body:

moment = rate of angular momentum

Case of statics: which leads to

In the case of statics vanishes the resulting moment acting on a

body (equilibrium of moments).

Sir Isaac Newton

(1643 –1727) ( mv)

dL dM r F r

dt dt

0v 0M


Introduction

Cartesian coordinates

right-hand system, i.e.

here:

x

z

yy

x

z

x

0x y ze e e

1 0 0 0 0

0 1 0 0 0 1 0

0 0 1 1 1


Introduction

Definition of stress

beam under uniaxial tension

Stress is defined as the local force DN over local area DA

Usually, the unit of stress is 1N/mm² = 1MPa

Under uniaxial stress s we may thus compute the normal force N by

N

DN

DA

0limA

dN

A dA

Ns

D

D

D

A

dN dA N dAs s

A


True and Engineering Stress

Consider a bar with constant cross section along the entire length

Engineering stress

True stress (Cauchy stress)

Relationship

for incompressible materials, i.e. A0l0=Al, else

00 1 ss

0l

FF

l

initial cross section A0

actual cross section A

0

0

F

As

F

As

0,

0,

necking, i.e.

0 0( )

( )


2

0 01

s s

Engineering strain

Another definition of strain: true strain (Hencky’s strain)

An infinitesimal change of elongation dl related to the actual length l

defines the true strain increment d

True and Engineering Strain

0

0

0

0

ln ln ln ln

ll

ll

dld

l

dl ll l l

l l

FFx

0l lD

l

0

lnl

l

Heinrich Hencky

(1885-1951)

Mechanics of Polymers 21 * Hencky studied at the TH Darmstadt where he also received his PhD

0

0 0

l l l

l l

D

True Strain vs. Engineering Strain

Relationship between true and engineering strain

True and Engineering Values

0

0

l

l

D

small strain region

0, 0

0ll

0

00

0 0 0

1 1l ll l

l l l

D D

0ln 1

0

lnl

l


Increase of Volume during Tensile Tests

Relative Volume for =const. :

Example: Terblend® N NM19 (ABS/PA blend by BASF)

Mechanics of Polymers 23

rela

tive v

olu

me [

-]

1st principle strain

volu

me [m

m³]

total volume V(t)

relative volume V(t)/V0

0

exp 1 2V

V

Courtesy of FAT AK27, experiments performed at EMI, Freiburg

True and Engineering Values

Engineering values are always related to the initial geometry

True values are always related to the actual geometry

The difference vanishes for small strain problems (linear

calculations)

Nonlinear FE analysis (e.g. crash) is based on true values and

gives thus true values in the output data!

All material data have to be converted to true data for input in such

FE-computations (see example in section “mechanics of

materials”)


Stress as a Tensor Quantity

Consider now a beam under angular cross section and compute

Cauchy’s stress

Stress components depend on (the normal of the) cross section!

Later on we will therefore define stress by a second rank tensor

F

cosnF F

F

F

ns

F

F

sintF F


3D Stress State: Transformation

Transformation in y-z-plane

Transformation matrix 𝚽 is valid for all vectors in ℝ3

For transformation of matrices, e.g. 𝐀 ∈ ℝ3𝑥3: 𝚽T𝐀𝚽

It seems that stresses are transformed rather than

matrices than vectors!

, , , ,P x y z P x

cos siny z

sin cosy z

rotary matrix in y-z-plane Φ

1 0 0

0 cos sin

0 sin cos

x x

y

z

y

z

P


Stress Tensor

Definition of stress

Stress vector

The stress vector is defined as the local force over local area

Stress components depend on the normal of the cross section!

This mapping is given by the 2nd rank stress tensor s:

that is sometimes called Cauchy’s relation

y

x

z

FD

nt

AD

0limA

dFt

F

A dAD

D

D

t n σ


Stress Tensor

Three-dimensional stress state (Cauchy stress)

In general, we have the following stress components

Balance of angular momentum results in the symmetry of the stress

tensor, i.e. 6 components are independently:

y

x

z

dx

dy

dz

zs

ys

xs

zy

zxyz

yxxy

xz

yx zx

xy

x

zy

xz yz

y

z

s

s

s

σ

xy xz

xy yz

xz y

y

T

x

z z

s

s

s

σ σ

Augustin Louis Cauchy

1789-1857


12 13

12 23

13 2

11

22

33 3

T

s

s

s

σ σAlternatively:

Stress Tensor

Balance of angular momentum

zy

zy

yz

yz

sz

sz

sy

sy

y

z

x’

zx

zx

xz

xz

sz

sz

sx

sx

x

z

y’

sx

xy

xy

yx

yx

sx

sy

sy

z’

x

y

' 0 2 22 2

0

yz zy

yz zy

yz zy

dy dzM x dx dz dx dy

dx dy dz dx dy dz

' 0 2 22 2

0

zx xz

zx xz

zx xz

dz dxM y dx dy dz dy

dx dy dz dx dy dz

' 0 2 22 2

0

xy yx

xy yx

xy yx

dx dyM z dy dz dx dz

dx dy dz dx dy dz


Stress Tensor

Computation of the stress vector 𝑡

Stress vector

Example: stress in x-z-plane

y

x

z

FD

nt

AD

0 0

1 1

0 0

xy xz xy

xy yz

xz yz

x

yz

y y

z

n t n

s

s

s

s

σ

0limA

dFt

F

A dAD

D

D


Stress Tensor

Transformation (z-Rotation)

y

x

sy

yx

xy

sx

s

s

s s s s s

s s s s s

s s

1

2

1

22 2

1

2

1

22 2

1

22 2

x y x y xy

x y x y xy

x y xy

cos( ) sin( )

cos( ) sin( )

sin( ) cos( )

TΦ σΦ


Objectivity of the Stress Tensor

The components of the stress tensor depend on the section under

consideration (tensor property)

Transformation of the coordinate system, however, may not affect

the state of stress (principle of objectivity), i.e. uniaxial stress

remains uniaxial stress :

y

x*

x

y*

0 0 0 0 0

0 0 0 * 0 0

0 0 0 0 0 0


Stress Tensor

Principal stress

A unique transformation of the stress tensor s results in a stress

state for that no shear stress exist

We call this state principal stress and the corresponding axis principal

axis

xy xz

xy yz

xz yz

x

y

z

s

s

s

1

2

3

0 0

0 0

0 0

s

s

s

zs

ys

xs

zy

zxyz

yxxy

xz

1s

2s

3s

y

x

z

21

3

Φ

TΦ σΦ


Mohr‘s Circles for a three-dimensional state of stresses

Stress tensor in principal axes

Principal shear stresses

Centers of the three Mohr’s circles


2 3

12

s s

3 1

22

s s

1 2

32

s s

1 2 3diag , ,s s sσ

2 31

2m

s ss

1 32

2m

s ss

s

3s2s1s

1ms

1 23

2m

s ss

3ms2ms

13

2

Christian Otto Mohr

1835 – 1918

Mohr‘s Circles for a three-dimensional state of stresses

Uniaxial stress

Biaxial stress

Hydrostatic stress


1 2 30,2

s

diag ,0,0sσ

1 2 30,2

m m m m

ss s s s

s

sms

diag , ,0s sσ

1 2 3, 02

s

1 2 3,2

m m m m

ss s s s s

diag , ,s s sσ

1 2 3 0 1 2 3m m ms s s s

s

sms

Christian Otto Mohr

1835 – 1918

Equilibrium Conditions

Method of section: Consider a closed Volume V (boundary V) of a

deformed body B loaded by external forces and body forces

Equilibrium is fulfilled if

with Cauchy’s relation

Applying Gauß’ divergence theorem leads finally to


V

B

B

q

n

t

dAiF

if

0, 1,2,3i i

V V

t dA f dV i

i it n σ

div

div 0

i i i i i

V V V V V V

i

V

t dA f dV n dA f dV dV f dV

f dV

σ σ

σJohann Carl Friedrich Gauß

1777 - 1855

Equilibrium Conditions

The global equation

is fulfilled for arbitrary volumes if

and as fully written out symbol equation:

Note the time consuming derivation in engineering mechanics!


div 0i

V

f dV σ

div 0f σ , 0ji i ifs

xx dx

x

ss

xy

xy dxx

y

x

xy

xy

xs

ys

y

y dyy

ss

xy

xy dyy

xf

yfdx

dy

Stress Tensor

Principal stress

Eigenvalue problem

Characteristic equation / Cayley-Hamilton-Theorem

where I1, I2, I3 are the so-called invariants of the stress tensor s

Roots of the characteristic equation leads to 3 (real) eigenvalues, the

principal stresses s1 > s2 > s3

0

t n n

n

s

s

σ

σ 1

3 2

1 2 3

det 0

0 0

x xy xz

xy y yz

xz yz z

I I I

s

s s

s s s s s

s s

σ 1

dA


Arthur Cayley

1821 - 1895

William Rowan Hamilton

1805 - 1865

n

t

http://de.wikipedia.org/w/index.php?title=Datei:Arthur_Cayley.jpg&filetimestamp=20091118224125

http://de.wikipedia.org/w/index.php?title=Datei:WilliamRowanHamilton.jpeg&filetimestamp=20050301142105

Stress Tensor

Invariants of the stress tensor s

An invariant is a property of a system which remains unchanged

under some transformation

1 1 2 3tr( ) x y zI s s s s s s σ

222

2 2 2

1 2 1 3 2 3

1tr( ) tr( )

2

x y x z y z xy yz xz

I

s s s s s s s s s s s s

σ σ

3 1 2 3det( )

x xy xz

xy y yz

xz yz z

I

s

s s s s

s

σ


Stress Tensor

Volumetric-deviatoric split

The average normal stress gives the pressure as

Thus, the hydrostatic stress state is given by

The difference between stress tensor s and hydrostatic stress state

defines the deviatoric stress tensor s

1

3m x y zp s s s s

13

13

13

0 0

0 0

0 0

x y z

m x y z

x y z

s s s

s s s s

s s s

1

13

13

13

2

2

2

x y z xy xz

m xy x y z yz

xy yz x y z

s s s

s s s s

s s s

s σ 1


Stress Tensor

Invariants of the deviator s

An invariant is a property of a system which remains unchanged

under some transformation

I1, J2 (and J3) play an important role in the theory of plasticity

1 tr( ) 0x m y m z mJ s s s s s s s

2 2 2 2 2 22

2 2 21 2 2 3 3 1

1 1:

2 6

1

6

x y y z z x xy yz xzJ s s s s s s

s s s s s s

s s

3 det( )J s


Stress Tensor

Equivalent stress se

The equivalent stress is a single stress value that can be compared to

the admissible stress of the material

The equivalent stress represents yield condition for ductile materials

and failure criterion for brittle materials

In common use is the equivalent stress that is related to

• maximum principal stress

• maximum shear stress

• octahedral shear stress oct

• VonMises stress svm

oct

e 1 2 3 = max , , s s s s

e max 1 2 2 3 3 1 = 2 = max , , s s s s s s s


Stress Tensor

VonMises stress svm

Most popular criterion for equivalent stress

The vonMises stress is written in stress components as

The vonMises stress is directly related to the second invariant of the

deviatoric stress tensor and to the octahedral shear stress

2 2 21 2 2 3 3 1

2 2 2 2 2 212

2 2 2 2 2 2

2

2

6

3

vm

x y y z x z xy yz zx

x y z x y y z z x xy yz zx

s s s s s s s

s s s s s s

s s s s s s s s s

2

2

33 :

2vm Js s s

Richard von Mises

(1883-1953)


3 2

2vm octs

Stress Tensor

Illustration of vonMises stress

for two components s and (e.g. beam) the vonMises stress over

shear stress represents a circle

For arbitrary 3D-stress state vonMises

stress represents a cylinder in the

principal stress space. The direction

of the cylinder is given by the

hydrostatic axis, i.e.

2 23vms s

vonMises stress

experiments (steel)

1s

2s

3s

1 2 3s s s


Stress Tensor


A cut through the s1-s2-plane of the cylinder gives an ellipse that

represents the state of plane stress

1s

2s

3s2s

1s

plane stress

2 2

1 2 1 2vms s s s s


Stress Tensor


Invariant plane (BURZYŃSKI-plane)

State of pure shear at p=0

Uniaxial tension

vmsuniaxial

compression

uniaxial

tension

1

3

p

2

1 2

1,

3vmI p J s

0 0 1 1

0 0 0 33 3

0 0 0

x

x

vm

vm x

p trp

ss

s

s s

σσ

shear

31


Stress Tensor

Triaxiality

The relation of pressure over vonMises stress can be used as a

measure of triaxiality:

vm

p

s

0.333

0.577

vm

p

s00.20.40.6

0

plane strain

biaxial tension uniaxial tension shear


2

31

3

Stress Tensor

State of plane stress

Most of structural parts made from plastic are modeled by shell

elements that assume a so-called plane stress state,

i.e. in principal axis

What does this mean for the von Mises stress in the invariant plane?

1 1

2 2

3

( , ) 0 0

( , ) 0 0

0 0 00

s s

s s

s

x

y

sx

xy

xy

xy

xy sx

sy

sy


Plane Stress

1

2

1 1

0 0

0 0 1 ( 1)

0 0 0

vmk k k

s

s s s

σ

triaxiality

1

2

1

( 1) ( 1)

3 1 ( 1)3 1 ( 1)vm

kp k

k kk k

s

s s

bounds:

0

( 1) 1lim lim lim

33 1 ( 1)k k k

k

k k

1 1

( 1) 2lim lim

33 1 ( 1)k k

k

k k

k

vm

p

s

uniaxial tension

biaxial tension

1s

12 ss k

uniaxial tension

biaxial tension


Plane Stress

),(

),(

2

1

s

s

1s

lower bound: pvm2

3s (biaxial tension)

12 ss k


Plane Stress - Summary

For shell-like structures, i.e. plane stress state, the vonMises stress

is bounded by

pvm2

3s

Triaxiality thus becomes

3

2

vm

p

s

This defines the requirements/restrictions for experimental work

vmsuniaxial

compression

uniaxial

tension

1

3

p

shear

31

322

3

biaxial

compression

biaxial

tension


52

Strain Tensor

Reminder: 1D-definition of strain

Lets consider a bar with initial length l0 under uniaxial loading F

the local uniaxial strain is then defined as

If (and only if) the cross section of the bar and the material properties

(Young’s modulus) are constant along the length l0 , we may write

du

dxε

FF

dx

dx+du

FF

,x u

0 0

0

00 0

x l x l

x x

ldu dx l du dx dx l

l

D D ε ε ε ε ε

0l


Strain Tensor

In a fully 3-dimesional structural part, the local strain is defined as

And the strain tensor is given analogue to the stress tensor as

T1

2 ε u u

12 13 23

1 1 1 1 1 1, ,

2 2 2 2 2 2xy xz yz

u v u w v w

y x z x z y

11 22 33, ,x y z

u v w

x y z

,y v,x u

,z w

1 1

2 2

1 1

2 2

1 1

2 2

x xy xz

xy y yz

xz yz z

ε

• in total 6 independent strain components

• symmetric by definition

• rotation of a body does not cause any strain


Strain Tensor

Principal strains

For plane stress, we may expect the following deformation modes

This gives the state of plane strain, i.e. all z-components vanish:

sxsx

x

y

sy

sx

sx

sy

dx

dy

sy

sy

0.5yxdy

0.5xxdx

0.5yydy

0.5xydx0.5

1

2

1

2

x yx

yx y

ε

1

2

3

0 0

0 0

0 0

ε

1

2

0

0

ε respectively and


Comparison: Plane Stress / Plane Strain

Plane strain vs. plane stress

Dog bone model (fracture mechanics)

In practice: plane stress = shell structures; plane strain = thick shells

3D stress state can only be computed by solids!

plain

stress

plane

strain

x crack

y

plane

strain plain

stress

1

2

0 0

0 0

0 0 0

s

s

σ

1

2

0 0

0 0

0 0 0

ε


Generalizations

Initial (reference) and current (actual) configuration

We consider an undeformed body B0. A material point X within this

body is defined by its spatial coordinates. This state is called

reference configuration.

The deformed body B is characterized by its coordinates x. This state

is called current or actual configuration.


y

x

z

Generalizations

Displacement

Deformation gradient

Transformation of a volume element dV by the Jacobian J

Relation between the deformation gradient and the displacement


Generalizations

Measure of strain

the deformation gradient can be uniquely decomposed into a proper

orthogonal rotation tensor R = R−T with det R = 1 and a symmetric and

positive definite right stretch tensor U = UT and left stretch tensor V =

VT respectively

The squares of the right and left stretch tensor define the second rank

left and right Cauchy-Green tensor

With these geometrical measures we may define measures of strain


Green-Lagrange strain

Euler-Almansi strain

Logarithmic strain

If the displacement gradient is small enough,

the difference between current configuration

and reference configuration vanishes.

Then, the symmetric part of the displacement

gradient describes the infinitesimal strain

Generalizations

Leonhard Euler

(1707-1783)


Generalizations

Measure of stress

Stress vector in current and reference configuration

The symmetric Cauchy stress tensor σ is also referred

to as true stress tensor. The Piola-Kirchhoff stress

tensor P is the nominal or engineering stress:

Since P is unsymmetric, it is instructive to introduce the so-called

second Piola-Kirchhoff stress S as

This quantity has no physical meaning but is just defined as a

symmetric measure in the reference configuration

Gustav Robert Kirchhoff

(1824-1887)


Material Modeling

Rheological Models

Classification of Materials

Linear Elasticity

Hyperelasticity

Visco-Elasticity

Plasticity and Visco-Plasticity


Rheological Basic Models

Spring

Damper (dashpot)

Slider


s s

E

s

s

s ssy

s

E

linear (elasticity)

non-linear (hyperelasticity)

s

linear

non-linear

(not available in most FE packages)

non-linear hardening

linear hardening

perfect plasticity

sy

Rheological Models

These basic models in combination allow for the formulation of all

kind of stress-strain relations

Examples:


s s

visco-elasticity

s

elasto-visco-plasticity

s

s

s


From an engineering point of view (focus on simulation), it is

instructive to subdivide materials according to their mechanical

behavior into

elastic materials

viscous materials

plastic materials

Examples

Nearly all materials are elastic for small deformations. At large

deformations, rubber-like materials and recoverable foams are (non-

linear) elastic

Polymers are strain rate dependent to some degree. Under cyclic

loading they convert mechanical energy to heat which causes material

damping

Plastics show permanent deformation above yield stress

usually we have a combination of them



To characterize a material phenomenologically, we consider a

uniaxial tensile/compression test with unloading

Hereby we use engineering stresses and engineering strains for a

rough subdivision where A0 is the initial cross section and l0 the initial

length.

For the dynamic response, strain rate dependent tests are performed

subsequently (pendulum test, drop tower test, ...)

0

FA 0

1ll

t

FF

F

0l

cross section A0


Linear Elastic Materials

The loading path follows a straight stress-strain path

The deformations remain small and the unloading

path corresponds to the loading path

Slope corresponds to Young’s modulus1 E

This behaviour is called linear elastic

s

F

F

1E

Thomas Young

(1773-1829)

1the idea can be traced back to a paper by Leonhard Euler published in 1727, some 80 years before Young's publication


The linear relation between stress sij and

strain kl is mapped by the 4th rank

elasticity tensor Cijkl :

Symmetry:

Cijkl = Cjikl because of the stress tensor symmetry sij = sji

Cijkl = Cijlk because of the strain tensor symmetry kl = lk

Cijkl = Cklij holds if a strain energy density W exists, such that

(Schwarz’ theorem)

Linear Elasticity: Elasticity Tensor

σ Cε ij ijkl klCs


2 2

,ij

ij ijkl

ij kl ij kl kl ij

W W WC

ss

Karl Hermann Amandus Schwarz

(1843 –1921)

//upload.wikimedia.org/wikipedia/commons/d/d3/HermannSchwarz.jpeg

Because of the elasticity tensor symmetry

we may rearrange all stress tensor components in a

6-dimensional vector (and also for the strain

tensor components)

The relation between these two vectors is mapped

by a 6x6 matrix which contains all elastic constants:

Linear Elasticity: Voigt’s Notation

Woldemar Voigt

(1850-1919)


Linear Elasticity

In anisotropic materials, the number of constants depends on the

internal structure

triclinic

rhombic

tetragonal


Linear Elasticity

In anisotropic materials, the number of constants depends on the

internal structure

hexagonal (transverse isotropic)

cubic

isotropic


Linear Elasticity

Isotropic linear elastic materials can be expressed

by 2 parameters

where l and m are Lamé constants; alternatively

E = Young’s modulus and = Poisson’s ratio

m

m

m

mlll

lmll

llml

00000

00000

00000

0002

0002

0002

C

Gabriel Lamé

(1795-1870)

3 2E

m l m

l m

2

l

l m

Siméon Denis Poisson

(1781-1840)


Relation between Elastic Constants

whereby Mechanics of Polymers 72

Plane Stress vs. Plane Strain: Note the Difference!

Plane stress (ESZ)

Plane strain (EVZ)


Non-Linear Elastic Materials

The loading path follows a non-linear stress-strain curve

The unloading path ideally corresponds to the loading path.

This behaviour is called hypoelastic or hyperelastic depending on

the theoretical formulation

s

F

F


Non-Linear Elastic Materials

Examples: Rubber-like materials where internal damping can be

neglected

engine mount

Hardy-disc tire

head impactor

s


Hyperelasticity

Definition

In a hyperelastic material, both stress and strain energy are path-

independent functions of the current deformation

Consequently, the strain energy function W per unit undeformed

volume can be used as a potential function to compute the stress

by derivation:

In hyperelasticity it exists a strain energy function from which the

stresses can be computed by derivation with respect to the strain


2W W

E C

2

1

2 1: ( ) ( )

t

t

W dt W W E E E

Hyperelasticity

The given definition of hyperelasticity is equivalently fulfilled by the

first law of thermodynamics (energy balance, no dissipation):

rate of internal work

rate of free energy

Inserting yields:


2W W

C E

Hyperelasticity

W can be expressed in terms of total strain E or principal stretch

ratios λ𝑖 =𝑙𝑖

𝑙0𝑖 respectively

True stress in global reference frame:

True stress in principal (true stress) reference frame:

78

WE

xGrad x

X

F

ikj

iλ

W

λλσ

1


T1 1

2 2 E C 1 F F 1

T0

T

12

VW W

J V

σ F F

E F F 1

1

Jσ

1 2 3

2 3 1 1 3 2 2 1 3

1 1 1i.e. , and

W W Wσ σ σ

λ λ λ λ λ λ λ λ λ

Hyperelasticity

For an isotropic material the energy function should depend only

on the strain invariants , e.g. in terms of C = FTF:

or on the principal stretches:

Invariants of the right Cauchy-Green tensor:

79

1 2 3, ,W W I I I

2 2 2

1 1 2 3

2 2 2 2 2 2 2

2 1 1 2 2 3 3 1

22 2 2 2

3 1 2 3 2

0

: tr

1:

2

det( )

I

I I

VI J

V

l l l

l l l l l l

l l l

1 C C

C C

C


12 2 2 2C C

C C C

W W W WI III

I II III

1 1 C C

1

Jσ

1 2 3, ,W W l l likj

iλ

W

λλσ

1

Hyperelastic Material Laws

Conditions for incompressible behavior

Full incompressibility cannot be enforced in explicit codes

To treat the rubber as an unconstrained material, a hydrostatic

work term is included in the strain energy function, e.g.

This work term is also known as

“Equation of State” (EOS)

A confined compression test allows

validation of the hydrostatic penalty K

80

0.5 2 1 3

E EG

0.5 0.5lim lim

3 1 2

EK

1 2 3 1J l l l

1 lnvolW K J J


lateral

confinement

One-Parameter Laws: Blatz-Ko

General form for polyurethane foam rubbers (1962):

Simplification (e.g. implemented in LS-DYNA):

81

21

311

12

311

23

3

231

I

I

IGII

GW

11

32

463.0

1

31

IIG

WT 2 11

12 1

1 2

G JJ

σ FF 1


Two-Parameter Laws: Mooney-Rivlin

General form of the energy function:

Clearly a stress-free initial state requires:

D is a penalty coefficient related to hydrostatic response:

A Poisson ratio > 0.497 will usually work fine

82

22

1 2 3 3( 3) ( 3) 1 1W A I B I C I D I

BA

C 2

l

l

l

212

51125

00

00

00

BA

DF

T 2 2

1 3 33

2 2 14 1 4

BA BI DI I CI

J J J

σ FF C 1



engineering stress s in uniaxial tension or compression:

This allows to determine A and B by fitting a test result

Linearization:

gives the small strain modulus E=6(A+B)

83

1/2

1/22

0 0

F 0 01

20 0

BA

ls

ll

lll

6 1A Bs l



Uniaxial tensile test

84

2

12 1

1

s

B

A

1 1

1 l

s

l1

eng. stress vs. eng. strain


Multiple Parameter Models: Ogden’s Law

In Ogden’s energy function deviatoric and volumetric stresses are

uncoupled:

Principal true stresses follow from derivation

The summed part is clearly deviatoric

If we assume full incompressibility, the deviatoric stresses are a

function of the principal stretch ratios

85

3

* * 1/3

1 2 3 1/31 1

1 1 ln , where and j

nj i

i i i

i j j

W K J J J JJ

m ll l l l l l


J

JK

J

n

j k

ki

j

i

j

j1

31

3

1

**

l

lm

s

* * *1

, , , , ,i id i j k i id i j k

JJ K p

Js s l l l s s l l l

Multiple Parameter Models: Ogden’s Law

for n=2, 1=2 and 2=-2, we get Mooney-Rivlin’s material for the

incompressible case:


3 2 3 31 2

1 1 1 11 2

2 2 2 2 2 21 21 2 3 1 2 3

2 2 2 2 2 21 2 1 21 1 2 2 3 3 1 1 2

1 2

1 1 1

3 32 2

3 3 3 32 2 2 2

3 3

j j jj

i i i

i j i ij

A B

W

I I I

A I B I

m m ml l l

m ml l l l l l

m m m ml l l l l l

*

1 2 3 1, i iJ l l l l l

Example of a windshield interlayer (PVB)


Further Strain Energy Functions in Terms of Invariants

88

)1(21

3I2

21

2

v

v

Jv

vμW

b

2

1 2 v

vJ Jm

τ σ 1 b

Blatz-Ko Material

Mooney-Rivlin Material

25.01 )1D()1C()3B(II)3A(I),II,I( JJJWbbbb

2 2 42 (A BI ) 2B 4 (D ( 1) C ) J J J J 2

bτ σ b b 1

Yeoh Material / Neo-Hooke Material

3

3

2

21 )3(IC)3(IC)3(IC bbb

W

b)()(τ~bb

3IC63IC4C 2

321



89

Van der Waals Material (User Defined)

2

3

2

2

3~

3

21ln 3

IaW m lm

bb

II 1I ~

I

)3/()3~

( 2

mI l

2

bbbτ ' 1 2 )1(I 2' 2~ WW

2

1

2

3~

21

1

2

1~'

Ia

I

WW

m



90

27I

1050

119I

20

13I

2

1 3

2

2

bbbNN

W m

...243I

673750

51981I

7000

19 5

4

4

3 bbNN

bτbbbb

...134750

I 519

1750

I 19

350

I 11

10

I

2

12~

4

4

3

3

2

2

NNNNm

Arruda Boyce Material (MT 127)


Further Strain Energy Functions in Terms of Principal Stretches

91

11

3321

1

jjjj nbbbbm

j j

jJ

nb

CW lll

Simplified Rubber Model (material no. 181)

i 1,2,3

1211

1

1 n

nbb

m

j

j

jjCJ llτ

Uniaxial Tension:

nnearly incompressibility:


Further Strain Energy Functions in Terms of Principal Stretches

92

3

1

β

i2

2

2

2

1λ2

))3(I1ln()3(I1

)3(I )1(

2 i

ec GGW

b

b

b

Extended Tube Material (User Defined)

llll

ll 2

221-22

421 2

3))-2(1(

)1(1

2 ec GGJτ

Uniaxial Tension:


Applications: Non-Homogenous Shear

93

Simulation of complex deformation

Parameter identification of material models

based on additional uniaxial tensile tests

satisfactory approximation up to 50 %

strain

Mat_181 result identical to experiment



94

good agreement for displacements in x-direction

satisfactory agreement in z-direction



95

well reproduced displacements in x-direction

misleading results in z-direction

quite reasonable agreement with uniaxial experiments, but

not capable to simulate combined non-homogeneous deformation


Applications - Hardy Disc

96

Based on this uniaxial tests, parameter identification

compression test bending test System:

Iterative validation of compression test with Ogden material

Based on Ogden material parameters creation of uniaxial tests


Applications - Hardy Disc under Compression

97

Good approximation in x-direction

Neo-Hooke material shows slight stiffening


Applications - Hardy Disc under Bending Load

98

Well approximations in both loading cases with Extended Tube-, Yeoh- and

Van der Waals-material

Mooney-Rivlin / Mat181 based on uniaxial tests may lead to differences


Example of a transmission-disk

99

Hardy-disc:

Du Bois, Faßnacht & Kolling, LS-DYNA Forum, Bad Mergentheim,

2002

Validation

compression

test

Timmel, Kaliske & Kolling, LS-DYNA Forum, Bamberg 2004

deformation EuroNCAP

Courtesey of Daimler AG, Sindelfingen


Non-Linear Viscous Materials

The loading path follows again a non-linear stress-strain curve

Now, the unloading path does not correspond to the loading path

and a hysteresis loop is formed

We will call this behaviour visco-hyperelastic or “strain rate

dependent hyperelastic” respectively

F

F

s


Example: PU foam, strain rate dependency (RG50)

Dynamic test, loading only (no unloading)

nonlinear viscosity!

[EMI 1999, Mills 2003]


• Extremely high compression

up to 98%

• Stability problems

• Time step size!

• Contact problems

• Sharp impactors cause high

deformation gradients in foam

parts

• Lagrangean finite elements

cannot follow the

corresponding deformed

shapes unlimitedly

Example: PU-Foam


Linear Viscous Materials

The loading path follows a slightly non-linear stress-strain curve

Now, the unloading path does not correspond to the loading path

and a hysteresis loop is formed

We will call this behaviour visco-elastic

Example: thermoplastics at small strains

F

F

s


Linear Viscoelasticity

Maxwell1 element (1867)

1 James Clerk Maxwell (1831 – 1879), Scottish theoretical physicist and mathematician

s s

E

1 1

S D

S D

S S S S

D D

E E E

s s s

s s

s s

s

Equilibrium

Total strain

Material law


0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dimensionless time t E/ [-]

dim

en

sio

nle

ss s

tre

ss s

/ E

0

Maxwell element – sudden loading: 𝜀 𝑡 = 𝜀0 = 𝑐𝑜𝑛𝑠𝑡

Exponential decay equation


0 0

E 0E

Es s

0

0 0

( ) expE

t t

E

s s

s

In this dimensionless

diagram, the response is

independent on the chosen

material parameters


Maxwell element – sudden loading: 𝜀 𝑡 = 𝜀0 = 𝑐𝑜𝑛𝑠𝑡

Notation:

wherw 𝜏 =𝜂

𝐸=:

1

𝛽 is the exponential time constant and is the

decay rate. Thus the relaxation function yields


0 0

E


0 0 0( ) exp exp expE t

t t ts s s s

0

0

( )( ) exp exp

( )

tE t t E t

t

ss

Maxwell Element + Spring in parallel

logarithmic plotting

of the relaxation

function gives a

better overview

Alternatively:



0 0

E0

E

E0 / 0

0 0( ) expE t E E t

log ( )E t

log t

0

E0+E

E

0 0( ) expG t G G t

Generalized Maxwell Element

In 3D expressed by convolution integrals:

The relaxation functions are represented in

terms of Prony series consisting of a set of

material parameters that have to be identified (measured)



0G

1G

2G

3G

11 / G

22 /G

33 / G

00

0

1

( ) expN

i i

i

G t G G t

visc d

10

visc

10

s εi

ki

t Nt β

i

i

t Nt β

i v

i

t G e d

p t K e ε d

Gaspard de Prony

(1755 - 1839)

visc visc visc( ) ( ) ( )t t p t σ s 1

//upload.wikimedia.org/wikipedia/commons/8/8c/Gaspard_de_Prony.jpg

How to Measure the Relaxation Curves?

Time-temperature shift function (WLF shift)

Analytical approximation by Williams, Landel and Ferry1


1M.L. Williams, R.F. Landel, J.D. Ferry: The Temperature Dependence of Relaxation Mechanisms in Amorphous

Polymers and Other Glass-forming Liquids. Journal of the American Chemical Society 77, 1955, pp. 3701

( )T

tG t f

a

( )g

T

g

A T Ta

B T T

with

A = 17.44K and B = 51.6K

for amorphous polymers

t

G

t

G

WLF

T4

T3

T2

T1

Relaxation Tests

How to Measure the Relaxation Curves?

Dynamic mechanical analysis (DMA)


TTg

glass

rubber

G

ˆ( ) sinx t x t

( ), ( ),G T G

e.g. DMA/SDTA861e by Mettler Toledo

• fmax = 1000 Hz

• T= -150 °C - 500 °C

Plasticity: Thermoplastics

Visco-plastic Materials (Thermoplastics, Thermosets, …)

Permanent (plastic) deformation

in contrast to elastic materials

The unloading path ideally

follows a straight line (Hooke)

All polymers are strain rate

dependent to some degree, i.e.

they are visco-plastic

Von Mises yield criterion is usually used in practise

(OK for most of the metals not for plastics!)

F

F

s


Localization and Increase of Volume during plastic flow


Strain at break depends on the

chosen mean area

The decrease of the stress after

yield strength is a artefact of

the mean area, not a material

property.

Crazing

Crazing is the notion of formation of surface cracks

As a consequence thereof:

change of colour to white detectable

crazing leads to plastic (permanent) deformation with increase of

volume

crazing leads to low yield stress values in uniaxial/biaxial tension

seems to occur under high values of hydrostatic tension

113

Crazing


Mechanical Behavior of Thermoplastics (Uniaxial Loading)

114

0

5

1 0

1 5

2 0

2 5

0 0 ,0 5 0 ,1 0 ,1 5 0 ,2 0 ,2 5 0 ,3 0 ,3 5 0 ,4 0 ,4 5 0 ,5

wahre Dehnung

wahre

Spannung

Zug

Druck

Tru

e S

tress

True strain

Tru

e S

tress

Different yielding under

tension/compression (and

shear)

Plastic incompressibility for

compression only (p 0.5)

Under tension foam-like (p

0)

Yield curve AND Young‘s Modulus

are strain rate dependent

Non-linear elasticity

Uniform necking due to

stabilisation

No Von Mises type of plasticity!

Visco-elasto-visco-plasticity

True strain

Tensile Test

Compression Test

0

1 0

2 0

3 0

0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6

wahre Dehnung

wahre

Spannung

0.1mm/s

7mm/s

500mm/s


Material Models

Yield Criteria

Define a yield function f that is zero on the yield surface and less than

zero for elastic states:

• In terms of the stress tensor

• In terms of invariants

Pressure independent yielding: von Mises

where


ˆ ( )ijf f s 0

ˆ ( , , )f f I I I 1 2 3 0

ˆ ( )f f J 2 0

J 2

2 0ys 3

1s

2s

3s

116

Material Models

Pressure dependent yielding (Thermoplastics, …)

For materials that show dependency on triaxiality (most of the

thermoplastics) special yield criteria have to be used

tension

shear

p0

vms

ts

biaxial

tension

biaxial tension

t

shear

s

ss

3

1

compression

compression

c

cs

tension

t

ts

vonMises

SAMP


vmf A A p A ps 2 2

0 1 2 0

Experimental Data vs. SAMP - Polyvinyl Chloride (PVC)

117

exp. results taken from Bardenheier 1982


Experimental Data vs. SAMP - Polystyrene (PS)

118



Experimental Data vs. SAMP - Polycarbonate (PC)

119



Material Models

Isotropic Hardening Curve

f(s) = 0 corresponds the initial yield surface

(p) is a monotonically increasing function of the hardening

parameter p (usually the effective plastic strain)


( ) ( )pf s 0

1s

2s 3s

s

p

ys

121

displacement

forc

e

Example: Validation of a Component Test (PP-T10)


Example: Validation of a Component Test (PP-T10)

Typical behaviour for

thermoplastics:

material cards that are

fitted for uniaxial tension

yield a too soft responds

under bending

and compression

Different yield curves

under compression

and tension

are necessary!

displacement

forc

e


Associated and Non-Associated Flow Rules

Practical stability condition (Drucker’s first postulate)

Convex yield surface

Associated flow (Drucker’s second postulate)

Plastic strain increments and plastic stress increments are in same

direction:

i.e. plastic flow occurs perpendicular to the yield surface

Non-associated flow

Plastic potential is defined for the direction of yielding

Thus, plastic Poisson’s ratio can be influenced directly!


p

ij ijd ds 0

p fd d l

s

p gd d l

s

Material Models for Polymers – Summary

Many robust and reliable material models for polymer materials are

available (most of them in commercial FE-packages, too)

The most popular models are implemented in a tabular way due to their

user-friendliness (no parameter identification)

Using the presented models, the deformation behavior of the structure can

be reproduced pretty good

Consideration of the manufacturing process (molding for plastic

components) gives a further improvement of the simulation

Anisotropy is a topic of recent development (integrative simulation)


MECHANICS OF POLYMERS - Technische Universität Darmstadt · contact: [email protected]...

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