Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Material testingconstitutive equations

Experimental methods E181101 EXM6

Constitutive equations EXM6

Constitutive equations represent description of material properties

Kinematics (deformation) – stress (dynamic response to deformation)

kinematics is described by deformation of a body In case of solids

kinematics describes relative motion (rate af deformation) In case of fluids

Deformations and internal stresses are expressed as tensors in 3D case. Example: stress tensor describes distribution of internal stresses at an arbitrary cross section

x

y

z

x

y

z

x

y

z

zz

zy

zxyz

yy

yxxz

xy

xx

ijIndex of plane index of force component (cross section) (force acting upon the cross section i)

Stress in solids/fluids EXM6

Tensor of stresses is formally the same in solids and fluids (in both cases this tensor expresses forces acting to an arbitrary oriented plane at a point x,y,z) , however physical nature of these forces is different.

Solids – intermolecular forces (of electrical nature)

Fluids – stresses are caused by diffusional transfer of molecules (momentum flux) between layers of fluid with different velocities

ij

Total stress = pressure + dynamic stress

p

Viscous stresses affected by fluid flow. Stress is in fact momentum flux due to molecular diffusion

Unit tensor

Viscous Fluids (kinematics) EXM6

Delvaux

Viscous Fluids (kinematics) EXM6

))((2

1 Tuu

2

1

2

1)(

2

1

y

uuu x

xyyxxy

x

U=ux(H)y

Rate of deformation

(in words: rate of deformation is symmetric part of gradient of velocity)

Special kinematics: Simple shear flow ux(y) (only one nonzero component of velocity, dependent

on only one variable). Example: parallel plates, one is fixed, the second moving with velocity U

The only nonzero component of deformation rate tensor in case of simple shear flow

is called shear rate

Gradient of velocity is tensor with components

ji j

i

uu

x

Constitutive equations expressed for special case of simple shear flows

Newtonian fluid

Viscous Fluids (rheology) EXM6

nK

Model with one parameter – dynamic viscosity [Pa.s]

Model has two parameters K-consistency, n-power law index.

Power law fluid

Viscous Fluids (rheology) EXM6

)(2 II

3

1

3

1

:i j

jiijII

222

2

1)(

2

12

y

uII x

xy

1)2( nIIK

General formulation for fully three-dimensional velocity field

Viscosity is constant in Newtonian fluids and depends upon second invariant of deformation[1] rate (more specifically upon three scalar invariants I, II, III of this deformation rate tensor, but usually only the second invariant II is considered because the first invariant I is zero for incompressible liquids). General definition of second invariant II

(double dot product, give scalar value as a result)

The second invariant of rate of deformation tensor can be expressed easily in simple shear flows

Power law fluid

[1] Invariant is a scalar value evaluated from 9 components of a tensor, and this value is independent of the change (e.g. rotation) of coordinate system (mention the fact that the rotation changes all 9 components of tensor! but invariant remains). Therefore invariant is an objective characteristics of tensor, describing for example measure of deformation rate. It can be proved that the tensor of second order has 3 independent invariants.

Viscous Fluids (rheology) EXM6

More complicated constitutive equations exist for fluids exhibiting

yield stress (fluid flows only if stress exceeds a threshold),

thixotropic fluids (viscosity depends upon the whole deformation history)

viscoelastic fluids (exhibiting recovery of strains and relaxation of stresses).

Examples of Newtonian fluids are water, air, oils. Power law, and viscoelastic fluids are polymer melts, foods. Thixotropic fluids are paints and plasters. Yield stress exhibit for example tooth paste, ketchup, youghurt.

Oscillating rheometer: sinusoidaly applied stress and measured strain (not rate of strain!)

Hookean solid-stress is in phase with strain (phase shift =0)

Viscous liquid- zero stress corresponds to zero strain rate (maximum ) =900

Viscoelastic material – phase shift 0<<90

Rheograms (shear rate-shear stress)EXM6

0.01

0.1

1

0.01 0.1 1

0.01

0.11

0.21

0.31

0.41

0.51

0.61

0.71

0.81

0.01 0.11 0.21 0.31 0.41 0.51 0.61

n=1.5n=1

n=0.8

n=0.5

n=1.5n=1

n=0.8

n=0.5

[1/ ]s [1/ ]s

[ ]Pa

n=1 Newtonian fluid, n<1 pseudoplastic fluids (n is power law index)

Shear stress

Shear rate

DMA Dynamic Material Analysis and OscilogramsEXM6

)cos()(

)cos()(

0

0

t

t

sin''

cos'

0

0

0

0

E

E storage modulus

loss modulus

Elastic properties E’

Viscous properties E’’

polyoxymethylene

Viscoelastic effectsEXM6

Weissenberg effect (material climbing up on the rotating rod)

Barus effect (die swell)

Kaye effect

Deformation rate

Viscoelastic modelsEXM6

Oldroyd B model

Extra stress S

Upper convective derivative

Rheometry (identification of constitutive models).

-Rotational rheometers use different configurations of cylinders, plates, and cones. Rheograms are evaluated from measured torque (stress) and frequency of rotation (shear rate).

-Capillary rheometers evaluate rheological equations from experimentally determined relationship between flowrate and pressure drop. Theory of capillary viscometers, Rabinowitch equation, Bagley correction.

RheometersEXM6

Rotating cylinder

Plate-plate, or cone-plate

Capillary rheometerEXM6

1–glass cylinder, 2-metallic piston, 3-pressure transducer Kulite, 4-tested liquid, 5-plastic holder of needle, 6-needle, 7-calibrated resistor (electric current needle-tank), 8-calibrated resistor (current flowing in tank), 9-AC source (3-30V), 10-SS source for pressure transducer (10V), 11-A/D converter, 12-procesor, 13-metallic head,

14-push bar, 15-scale of volume

Capillary rheometerEXM6

Example: Relationship between flowrate and pressure drop for power law fluid

13)13

(2

n

nn

R

V

n

nK

dx

dp

L

p

Consistency variables

L

pRw 2

3

4

R

V

nnw n

nK

)

4

13(

Model parameters K,n are evaluated from diagram of consistency variables

Capillary rheometerEXM6

Ptotal Pres Pe Pcap

Elastic solidsEXM6

Lempická

Elastic solidsEXM6

Constitutive equations are usually designed in a different way for different materials: one class is represented by metals, crystals,… where arrays of atoms held together by interatomic forces (elastic stretches can be of only few percents). The second class are polymeric materials (biomaterials) characterized by complicated 3D networks of long-chain macromolecules with freely rotating links – interlocking is only at few places (cross-links). In this case the stretches can be much greater (of the order of tens or hundreds percents) and their behavior is highly nonlinear.

"Dogbone" sample

Elastic solids Deformation tensor EXM6

transforms a vector of a material segment from reference configuration to loaded configuration. Special case - thick wall cylinder

),,(),,( rzxRZX

t r and z are principal stretches (stretches in the principal directions). There are always three principal direction characterized by the fact that a material segment is not rotated, but only extended (by the stretch ratio). In this specific case and when the pipe is loaded only by inner pressure and by axial force, there is no twist and the principal directions are identical with directions of axis of cylindrical coordinate system. In this case the deformation gradient has simple diagonal form

Hh

Ll

Rr

F

r

z

t

/00

0/0

00/

00

00

00

R

rr

t

z

reference configuration

loaded configuration

),,( RZX

),,( rzx

Elastic solids Cauchy Green tensor EXM6

Disadvantage of deformation gradient F - it includes a rigid body rotation. And this rotation cannot effect the stress state (rotation is not a deformation). The rotation is excluded in the right Cauchy Green tensor C defined as

Deformed state can be expressed in terms of Cauchy Green tensor. Each tensor of the second order can be characterized by three scalars independent of coordinate system (mention the fact that if you change coordinate system all matrices F,C will be changed). The first two invariants (they characterize “magnitudes” of tensor) are defined as

FFC T

2

2

2

00

00

00

r

z

t

C

222rztc CtrI

22222222 )(

2

1zrzrztcc CtrIII

1 zrtreference

loaded

V

V

Material of blood vessel walls can be considered incompressible, therefore the volume of a loaded part is the same as the volume in the unloaded reference state. Ratio of volumes can be expressed in terms of stretches

Therefore only two stretches are independent and invariants of C-tensor can be expressed only in terms of these two independent (and easily measurable) stretches

2222

2222 11

1

ztztc

ztztc III

4

4

4

2

00

00

00

r

z

t

C

(unit cube is transformed to the brick having sides t r z)

Elastic solids Mooney Rivlin model EXM6

Using invariants it is possible to suggest several different models defining energy of deformation W (energy related to unit volume – this energy has unit of stress, Pa)

Pam

N

m

mN

m

J

233

Example: Mooney Rivlin model of hyperelastic material ).3()3( 21 cc IIcIcW

)311

()31

(),(22

22222

221

zt

zt

zt

ztzt ccW

Remark: for an unloaded sample are all stretches 1 (t =r = z=1) and Ic=3, IIc=3, therefore deformation energy is zero (as it should be).

))1

()1

((2),(

222

2222

1t

ztzt

tt

zttt cc

W

))1

()1

((2),(

222

2222

1z

ztzt

zz

ztzz cc

W

Stresses are partial derivatives of deformation energy W with respect stretches (please believe it wihout proof)

These equations represent constitutive equation, model calculating stresses for arbitrary stretches and for given coefficients c1, c2. At unloaded state with unit stretches, the stresses are zero (they represent only elastic stresses and an arbitrary isotropic hydrostatic pressure can be added giving total stresses).

t

z

Evaluation of stretches and stresses EXM6

Only two stretches is to be evaluated from measured outer radius after and before loading ro, Ro, from

initial wall thickness H, and lengths of sample l after and L before loading.

Corresponding stresses can be derived from balance of forces acting upon annular and transversal cross section of pipe

2

)/(2

2

2

HR

Hr

HR

hr

R

r

L

l

o

tzo

o

otz

t

)2

1()

2

1(

2

)2(

2

)(exp

H

rp

h

rp

h

hrp

h

rrp

h

pr ztoooiot

Therefore it is sufficient to determine Ro,H,L before measurement and only outer radius ro and length l after loading, so that the kinematics of deformation will be fully described.

2

)2(/

2

HR

HRHrr

o

ozoot

This is quadratic equation

)2(2

expexp

HRH

G

o

ztz

z

hp

r0

G-force

EXM6 Elastic solids instruments

Uniaxial testersSample in form af a rod, stripe, clamped at ends and stretched

Static test

Creep test

Relaxation test

EXM6 Elastic solids instruments

Biaxial testersSample in form of a plate, clamped at 4 sides to actuators and stretched

Anisotropy

Homogeneous inflation

EXM6 Elastic solids instruments

Inflation testsTubular samples inflated by inner overpressure.

Internal pressure load

Axial load

Torsion

Confocal probe

Laser scanner

CCD cameras of correlation system

Q-450

Pressure transducer

Pressurized sample (latex

tube)

Axial loading (weight)