Momentum Heat Mass TransferMHMT3

Kinematics of deformation, stresses, invariants and rheological constitutive equations. Fluids, solids and viscoelastic materials.

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Kinematics and dynamics. Constitutive equations

sourceDtD

Constitutive equationsMHMT3

Material (fluid, solid) reacts by inner forces only if the body is deformed or in nonhomogeneous flows.

l0 l0

deformation Stresses in elastic solids depend only upon the stretching of a short material „fiber“. Stretching of fibers depends upon the initial fiber orientation. Stresses are independent of the rate of stretching and of the rigid body motion (translation and rotation). Deformation is reversible, after removal of external forces, original configuration is restored.

Stresses in viscous fluids depend only upon the rate of changes of distance between neighboring molecules. The distance changes are caused by nonuniform velocity field (by nonzero velocity gradient). Stresses are again independent of the rigid body motion (translation and rotation). Process is irreversible and molecules have no memory of an initial configuration (stresses are caused by relative motion of instantaneous neighbors).

Material with memory

Material without memory

Many materials are something between – viscoelastic fluids have partial memory (fading memory). Example: polymers, food products…

Constitutive equationsMHMT3

l0 l0

Reference and deformed configuration is distinguished. Motion is described by displacement of material particles.

Only the current configuration is considered (changes of configuration during infinitely short time interval dt). Motion is described by velocities of material particles.

Solids

Fluids

d

u

Velocity is the time derivative of displacementdtddu

Kinematic variables in solids and fluids

displacement

stretch

velocity

time ttime t+dt

reference configuration

Constitutive equations FLUID

Macke

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Motion of viscous fluid is a fully irreversible process, mechanical energy is converted to heat.

Fluid has no memory to previous spatial configuration of fluid particle, there is no recoil after unloading external stress.

Newton’s law: dtd

Fluids: Kinematics of flowMHMT3

In case of fluids without a long time memory the role of displacements (differences between the current and the reference position of material particles) is taken over by the fluid velocities at near points. Viscous stresses are response to changing distances, for example between the near points A,B during the time dt.

dtxtu A ),(

xd

A

B

A’

B’

xd ( , )B Au t x x dx dt i

jij x

uu

|Gradient of velocity

Arbitrary tensor can be decomposed to the sum of symmetric and antisymmetric part

))((21))((

21 TT uuuuu

Spin (antisymmetric) Rate of deformation

( , ) ( , ) ( , )B A Au t x u t x du u t x dx u

( ) ( )du dt dx u dt

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Result can be expressed as Helmholtz kinematic theorem, stating that any motion of fluid can be decomposed to

translation + rotation + deformation

B Au u dx dx

)(21

)(21

j

i

i

jij

j

i

i

jij

xu

xu

xu

xu

Viscous stresses are not affected by translation nor by rotation (tensor of spin, vorticity), because these modes of motion preserve distance between the nearest fluid particles.

Vorticity tensor

Rate of deformation tensor

Fluids: Kinematics of flow

dtxtu A ),(

dx dt

xd

A

B

A’

B’

dtxd

xd ( , )B Au t x x dx dt

( ) ( )du dt dx u dt

Tensor of total stresses can be decomposed to pressure and viscous stresses

Hydrostatic pressure (isotropic, independent of relative motion of fluid particles). For pure fluids the pressure can be derived from thermodynamics relationships,

e.g. where u is internal energy and s – entropy.

Viscous stresses are fully described by a symmetric tensor independent of the rigid body motion. Viscous stress is in fact the momentum flux due to molecular diffusion.

p

MHMT3 Fluids: stresses

svup )(

Constitutive equations represent a relationship between

Kinematics (characterised by the rate of deformation for fluids)

Viscous stress (dynamic response to deformation)

( ) 2tr

Dynamic viscosity [Pa.s]

Second (volumetric)

viscosity [Pa.s]

MHMT3 Fluids: Constitutive equation

The simplest constitutive equation for purely viscous (Newtonian) fluids is linear relationship between the tensor of viscous stresses and the tensor of rate of deformation (both tensors are symmetric)

)(

2

j

i

i

j

m

mijij

ijmmijij

xu

xu

xu

Index notation

In terms of velocities

The coefficient of second viscosity represents resistance of fluid to volumetric expansion or compression. According to Lamb’s hypothesis the second (volumetric) viscosity can be expressed in terms of dynamic viscosity

23

( ) 3 2 0xx yy zztr u u

This follows from the requirement that the mean normal stresses are zero (this mean value is absorbed in the pressure term)

MHMT3 Fluids: Constitutive equation

2 ( )3u

Constitutive equation for Newtonian fluids (water, air, oils) is therefore characterized by only one parameter, dynamic viscosity

Viscosity is a scalar dependent on temperature.

Viscosity of gases can be calculated by kinetic theory as =(lmv)/3 (as a function of density, mean free path and the mean velocity of random molecular motion, see the previous lecture) and these parameters depend on temperature. This analysis was performed 150 years ago by Maxwell, giving temperature dependence of viscosity of gases

Viscosity of liquids is much more difficult. According to Eyring the liquid molecules vibrate in a “cage” of closely packed neighbors and move out only if an energy barrier is overcame. This energy level depends upon temperature by the Arrhenius term

Viscosity of gases therefore increases and viscosity of liquids decreases with temperature. Typical viscosities at room temperature

water=0.001 Pa.s

air=0.00005 Pa.s

MHMT3 Fluids: Constitutive equation

T

exp( / )E T

MHMT3 Fluids: Constitutive equationViscosity dependent on the rate of deformation and stress.

There exist many liquids with viscosity dependent upon the intensity of deformation rate (“apparent” viscosity usually decreases with the increasing shear rate), these liquids are calledgeneralized newtonian fluids (viscosity depends only upon the actual deformation rate, examples are food liquids, polymers…)

There are also materials which flow like liquids only as soon as the intensity of stress exceeds some threshold (and below this threshold the material behaves like solid, or an elastic solid)yield stress (viscoplastic) fluids (example is toothpaste, paints, foods like ketchup)

And there exist also liquids with viscosity dependent upon the whole history of previous deformation, changing an inner structure of liquid in timethixotropic fluids (examples are thixotropic paints, plasters, yoghurt).

Viscosity (T,t, rate of deformation, stress) is a scalar, so the intensity of deformation and the characteristic stress should be also scalars. However the rate of deformation and the stress are tensors

,

Fluids: InvariantsMHMT3

How large is a tensor? Magnitude of a stress tensor or intensity of the deformation rate are important characteristics of stress and kinematic state at a point x,y,z, information necessary for constitutive equations but also for decision whether a strength of material was exhausted (do you remember HMH criterion used in the structural analysis?) and many others.

Easy answer to this question is for vectors, it is simply the length of an arrow.

Magnitude of a tensor should be independent of the coordinate system, it should be INVARIANT. We will show, that there are just 3 invariants (3 characteristic numbers) in the case of second order tensors, telling us whether the material is compressed/expanded, what is the average value of the rate of deformation, density of deformation energy and so on (it depends upon the nature of tensor).

Fluids: InvariantsMHMT3

Any tensor of the second order is defined by 9 numbers arranged in a matrix. However these numbers depend upon rotation of the coordinate system. For the symmetric tensors (like stress, or deformation tensors) the rotation of coordinate system can be selected in such a way that the matrix representation will be a diagonal matrix ([[]], see the first lecture)

[[R]][[ ]][[R]] [[ ]]T

In view of orthogonality of [[R]] we obtain by multiplying the equation by [[R]]T

[[ ]] [[R]] [[R]] [[ ]]T T

This is so called eigenvalue problem: given the matrix [[]]3x3 calculate three eigenvectors (columns of the matrix [[R]]T=[[n1],[n2],[n3]]) and corresponding eigenvalues 1, 2, 3, that satisfy the previous equation.

3

2

1

000000

]][[

Fluids: InvariantsMHMT3

Physical interpretation (stresses):

The product represents the vector of forces acting at the cross section perpendicular to . In the case, that the normal is an eigenvector, the vector of normal will have the same direction as the vector of forces, and the corresponding eigenvalue is the value of the normal force. Eigenvectors are principal directions of the cross sections, where only normal stresses (and not shear stresses) act.

fn

n n

Physical interpretation (rate of deformations):

The product represents the vector of velocity differences at near points . (only velocities after the rigid body rotation removal are considered). As soon as the vector of distance is an eigenvector, the velocity difference vector will have the same direction.

x u

x

x

yDrag flow between parallel plates. Principal direction is at angle 450

x

y

nf Shear deformation of an elastic block.

Principal direction is at angle 450

x

ux(y)

x u

Fluids: InvariantsMHMT3

The eigenvalue problem can be reformulated to a system of linear algebraic equations for components of the eigenvector

[[ ]] [ ] [ ]n n This system is homogeneous (trivial solution n1=n2=n3=0) and non-trivial solution exists only if the matrix of system is singular, therefore if

11 12 13 1

21 22 23 2

31 32 33 3

000

nnn

11 12 13 1 1

21 22 23 2 2

31 32 33 3 3

n nn nn n

11 12 13

21 22 23

31 32 33

det 0

Expanding this determinant gives a cubic algebraic equation for eigenvalues

023 IIIIII

321321

323121

321

det

)(21

kjiijk

jiijjjii

ii

III

II

I

n

Fluids: InvariantsMHMT3

The values I , II , III are three principal invariants of tensor . Eigenvalues are also invariants and generally speaking any combination of principal invariants forms an invariant too. For example the second invariant of the deformation rate is frequently expressed in the following form and called intensity of the strain rate (it has a right unit 1/s).

3 3

1 1

2 : 2 2ij ji ij jii j

iiI u

22

2

1

2

12112 2

1)(21

21

xu

xu

ijij

shear rate

This is also the explanation why the constant 2 was introduced in the previous definition

Example: at simple shear flow u1(x2)0, u2=u3=0 holds

The first and third invariants of rate of deformation are no so important. For example the first invariant of incompressible liquid is identically zero and brings no information neither on shear nor elongational flows.

where( ), :II II

MHMT3 Fluids: GNF (power law)Generalised Newtonian Fluids are characterised by viscosity function dependent upon the second invariant of the deformation rate tensor

The most frequently used constitutive equation of GNF is the power law model (Ostwald de Waele fluid) 1

2 :n

K

Example: for the simple shear flow and incompressible liquid the power law reduces to

x

ux(y)

nxxyx y

uKyu )(

K-coefficient of consistency

n-flow behavior index

yx

yux

n=1 Newtonian

n<1 pseudoplastic fluid

Graph representing relationship between the shear rate and the shear stress is called rheogram

nK

MHMT3 Fluids: Yield stress (Bingham)Even for fluids exhibiting a yield stress it is possible to preserve the linear relationship between the stress and the rate of deformation tensor (Bingham fluid).

Constitutive equation for Bingham liquid (incompressible) can be expressed as

22 :

yp

Example: for the simple shear flow and incompressible liquid the Bingham model reduces to

xyx y p

uy

p-plastic viscosity y=yield stress

yx

yux

y=0 Newtonian

Bingham fluid

for

while for lower stresses the rate of deformation is zero.

2):(21

y (this is von Mises criterion of

plasticity)

This is orginal 1D model suggested by Bingham. The 3D

tensorial extension was suggested by Oldroyd

MHMT3 Fluids: Thixotropic (HZS)Thixotropic fluids have inner structure, characterised by a scalar structural parameter (=1 fully restored gel-like structure, while =0 completely destroyed fluid-like structure). Evolution of the structural parameter (identified with the flowing particle) can be described by a kinetic equation, depending upon the history of rate of deformation.

(1 ) 2 :mD a b

Dt

1

2(( ) 2 : )2 :

ny yK K

a-restoration parameter, b-decay parameter, m-decay index

Value of the structural parameter determines the actual viscosity or the consistency coefficient of the power law model and the yield stress (Herschel Bulkley)

The structural parameter increases at rest exponentially with the time constant 1/a. Rate of structure decay increases with the rate of deformation.

see Sestak J., Zitny R., Houska M.: Dynamika tixoropnich kapalin. Rozpravy CSAV Praha 1990

Rotating cylinder

Plate-plate, or cone-plate

MHMT3 Fluids: RheometryExperimental 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 the experimentally determined relationship between flowrate and pressure drop. Theory of capillary viscometers, Rabinowitch equation, Bagley correction.

MHMT3 Fluids: SummaryConstitutive equations for fluids assume that the stress tensor depends only upon the state of fluid (characterized by the rate of deformation and by temperature) at a given place x,y,z. It is also assumed that the same coefficient of proportionality holds for all components of the viscous stress tensor and the deformation rate tensor, therefore

It does not mean that the constitutive equations are always linear because the viscosity can depend upon the deformation rate itself (and there exist plenty of models for viscosity as a function of the second invariants of deformation rate and stresses: power law, Bingham, Herschel Bulkley, Carreau model, naming just a few). Nevertheless, some features are common, for example the absence of normal stresses (rr, zz,…) as soon as the corresponding component of the deformation rate tensor is zero. The exception are rheological models of the second order, for example (Rivlin), exhibiting features typical for viscoelastic fluids, like normal stresses, secondary flows in channels, nevertheless these models are not so important for engineering practice.

1 2

Solids:MHMT3

Macke

Reversible accumulation of external loads to internal deformation energy

Time is of no importance, unloaded material recoils immediately to initial configuration.

Hook’s law E

Solids: Kinematics-deformationMHMT3

In case of solids the deformation means that an infinitely short material fiber is stretched.

Kinematics of motion can be decomposed to stretching followed by a rotation:

A

B XdU dX

dx F dX R U dX

a

b

Reference configuration (unloaded body)

Deformed (current) configuration (loaded body)

Right stretch tensor

Rotation tensor

Deformation gradient

U

R

F

ntdisplaceme )( AXd

( )Ad X dX

Reference ( ) and deformed ( ) configurations are distinguished.X

x

(the same decomposition was done with fluids, but in solids the decomposition cannot be expressed in terms of velocities because time is not considered)

MHMT3

F-tensor (deformation gradient) transfers a short material line (vector dX) into new position (vector dx). Written in the index notation

jiji dXFdx Deformation gradient can be derived from relationship between coordinates of material points xi(X1,X2,X3) as partial derivatives

j

iij X

xF

Deformation gradient can be decomposed to stretch tensor U (symmetric and positive definite) and the rotation tensor R (orthogonal)

URF

Stretching firstFollowed by rotation

Solids: Kinematics-deformation

MHMT3

The stretch tensor U transforms a material fiber dX to the vector UijdXj which is extended or compressed (stretched). Length of the stretched fiber can be derived as

kjjkkjikTjikikjijii dXdXCdXdXFFdXFdXFdxdxdx 2

Cjk is the Green tensor of deformation (more precisely right Cauchy-Green deformation tensor), reflecting only the true deformation when the rigid body motion is eliminated

( )( )

Tjk ji ik ij ik im mj in nk

Tmj mi in nk mn mj nk nj nk

C F F F F R U R U

U R R U U U U U

The Cauchy-Green tensor C gives us the square of local change in distances due to deformation and equals the square of the stretch tensor C=U2. The C tensor can be calculated directly from the deformation gradient by matrix multiplication C=FTF. This is much easier than the calculation of stretch tensors U by decomposition.

Solids: Kinematics-deformation

MHMT3

Right Cauchy Green deformation tensor (called simply Green deformation, but there are many other names, e.g. Cauchy Green strain tensor by Tanner: Engineering Rheology)

Finger deformation tensor (but Holzapfel use the name Piola deformation tensor)

Left Cauchy Green deformation tensor also called Finger deformation tensor

Cauchy deformation tensor (but for example Fredrickson A.: Principles and application of rheology (1964) calls it Cauchy Green deformation tensor G)

k kij

i j

x xC

X X

1 jiij

k k

XXC

x x

TC F F

1 1 TC F F

jiij

k k

xxB

X X

TB F F

1 k kij

i j

X XB

x x

1 1TB F F

There exists a confusion in definitions of deformation tensors (there is no generally accepted unique nomenclature and naming the tensors), differences are caused by the fact that the decomposition of F tensor is not unique (F=RU stretching followed by rotation is not the same decomposition as the F=VR rotation followed by stretch). Tensors can be related to the unloaded reference configuration (denoted by capitals X) or to the deformed configuration (small letters x are usually used). In solid mechanics where the Lagrangian approach (tracking material particles) is preferred it is assumed that the reference configuration (X) is the starting one. This is not true In fluid mechanics, when the Eulerian approach is usually used, tracking history of motion starting from the current configuration. Therefore in the solid mechanics the Green’s deformation is preferred (given X(t=0) the x(t) position of particle is calculated) while its role is substituted by the Finger tensor or Cauchy deformation tensor in fluid mechanics (given the current position of material particle x(t) the values at previous times X(t’<t) are calculated).

Solids: Kinematics-deformation

Remark: I am confused not only by the nomenclature of kinematic tensors, but also by the conjugated stress tensors. Graphical representation of deformations and nine different stress tensors is available as a seminal work of Abbasi (student in UCI)

MHMT3

Given a material fiber vector dX in the reference configuration its length (square of length) after deformation is calculated as dX.C.dX. Principal components of tensor are therefore squares of stretches =ldeformed/lunloaded.

Inverted C-matrix. Scalar dX.C-1.dX is the reciprocal square of stretch.

Inverted B-1 matrix. Scalars dx.B.dx are therefore the square of stretches.

Given a material line vector dx in the deformed configuration its length (square of length) before deformation is calculated as dx.B-1.dx. Principal components of this tensor are therefore inverted squares of stretches.

k kij

i j

x xCX X

1 jiij

k k

XXCx x

jiij

k k

xxBX X

1 k kij

i j

X XBx x

Solids: Kinematics-deformationInterpretation: There are obviously four possible combinations

Example: elongation I.MHMT3

Simple extension of an elastic rod from an incompressible material

X1

X2

X3

333

222

111

XxXxXx

x1

x2

x3

1 2 3

2 22 3 1

1

1/

Incompressibility (constant volume)

21

11

1

1/ 0 00 00 0

k kij

i j

X XBx x

Cauchy deformation tensor B-1 equals Finger deformation tensor C-1

21

1

1

0 00 1/ 00 0 1/

jiij

k k

xxB

X X

Left Cauchy Green deformation tensor C equals right Cauchy Green tensor B

1 jiij

k k

XXCx x

k kij

i j

x xC

X X

Solids: Kinematics - strainsMHMT3

Without deformation the C-tensor is simply the identity tensor (all stretches are 1). Strain tensors are measures of deviation between the stretch tensors (C or U) and the unit tensor (strains are zero for rigid body motion). The most frequently used measure of strain is the Green-Lagrange strain tensor, defined as

)(21

CE

or the Hencky strain (called also logarithmic stretches or true strain)

lnE U

1 ( )2

k kij ij

i j

x xEX X

Remark: There are many other definitions of strains, Biot, Seyth… but the Green-Lagrange one is probably the most important.

Remark: The words “strain” and “deformation” are more or less equivalent. I used the “strain tensor” and “deformation tensor” to distinguish between the kinematic tensors that are reduced to the zero or to the identity tensor at rigid body motion.

MHMT3

The Green Lagrange strains can be expressed also in terms of displacements of material points

b

A

B Xd

dx F dX

a

( )Ad X

( )Ad X dX

XdXdXdXdddxdXxd

2 2| | | | ( )( ) ( ) 2ji i i k ki j i k i i j i ij j i

j k j i j i

dd d d d ddx dX dX dX dX dX dX dX dX dX E dX dX

X X X X X X

Extension of material fibre (difference of squares of final and initial lengths) is

1 ( )2

ji k kij

j i j i

dd d dEX X X X

giving the final expression of the Green Lagrange strain tensor in terms displacements

Remark: symbol is usually used in literature for displacement. The reason why the symbol is used here is to avoid conflict with velocities .

u

d

u

Solids: Kinematics - strains

1 ( )2

k kij ij

i j

x xE

X X

this is the same as

MHMT3

For one dimensional case |dx|=l, |dX|=l02 2 2( )( ) 2

2

dx dX dx dX dx dX EdXdx dX dx dX dx dXEdX dX dX

Small deformation

In relatively stiff solids (for example steel) the deformations, displacements and gradient of displacement are small quantities, therefore the last term in the Green Lagrange tensor (product of two small quantities) can be neglected, giving the tensor of small deformations

1 ( )2

jiij

j i

ddX X

Solids: Kinematics - strains

Example: elongation II.MHMT3

Simple extension of an elastic rod made from an incompressible material

X1

X2

X31 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

( 1) ( 1) ( 1)

x X d Xx X d Xx X d X

x1

x2

x3

1321

21

1

1

1 0 01 1( ) 0 1/ 1 02 2

0 0 1/ 1

j i k kij

i j i j

d d d dEX X X X

Green Lagrange strain (1 is the principal stretch in axial direction)

Small deformations1

1

1

1 0 01 ( ) 0 (1/ ) 1 02

0 0 (1/ ) 1

j iij

i j

d dX X

displacements

incompressibility constrain

Example – small deformationMHMT3

Let us consider a solid rod rotated along its axis by the angle

X1

X2 1 cosX r

2 sinX r 1 cos( )x r

2 sin( )x r 1 (cos( ) cos )d r

2 (sin( ) sin )d r

Reference configuration Current configuration (after rotation) Displacements

Determine xx component of the small deformation tensor

1

1 2

(cos( ) cos ) (cos cos sin sin cos ) (cos 1) sind r r

X X

1 1 111

1 1 1

1 ( ) cos 1 02

d d dX X X

2 21 1 1 2 211

1 1 1 1 1

1 1( ) cos 1 (cos 2cos 1 sin ) 02 2

d d d d dEX X X X X

Determine xx component of the Green Lagrange tensor

2

2 1

(sin( ) sin ) (sin cos cos sin sin ) (cos 1) sind r r

X X

This term should be zero because it is a rigid body motion

(deformation is small only for small rotation angle )

This term is zero for arbitrary large rotation

This example demonstrates limited applicability of small deformation tensors

Solids: StressesMHMT3

Tensor of stresses [N/m2] describes a vector of force F [N] acting to a small surface ([m2] cross-section of material). But which cross-section? The cross-section defined in the reference configuration (A) or in the deformed body (a)?

x1

x2

x3

FX2X3

F

The stress tensor reflecting the actual load of material is related to the deformed cross section A (true stress, Cauchy stress, denoted by ).

Other stress tensors are more or less artificial constructs related to the reference configuration (cross section A0), for example the Kirchhoff tensor S

0 0jiij mn im jn mn

m n

xV x VS F F S

V X X V

The reason why and when the Kirchhoff stress can be useful is associated with energetic considerations, see next slide…

0

jiij mn

m n

XXVSV x x

ratio of volumes (I am not quite sure, Valenta reports

reciprocal value)

X1

A0

A

V0V

V-volume in deformed configuration

V0-volume in reference configuration

3210

)det( FVV

Solids: Deformation energyMHMT3

The deformation tensors enable calculation of deformation energy (reversibly accumulated in a deformed body) according to different constitutive models. Knowing the deformation energy W as a function of stretches (or deformations) it is possible to calculate the stress tensors as partial derivatives of W, e.g.

[stress] [displacement] = [deformation energy change], e.g. ijij

WSE

W [J/m3] is deformation energy related to unit volume of sample in the reference configuration, Sij are components of Kirchhoff stresses and Eij are components of Green Lagrange strain. Kirchhoff stresses and Green Lagrange strains are the so called conjugated tensors (they are both related to the reference configuration).

Proof will be presented on the next slide…

Solids: Deformation energyMHMT3

It will be later derived that the rate of dissipation of mechanical energy to heat in a unit volume [W/m3] is the scalar product of Cauchy stress tensor and the rate of deformation

ijijDtDW

The rate of deformation can be expressed in terms of material derivative of the Green Lagrange strain

2 ( ) ( )ij m m m m m m m k m k k l l k k l

i j i j i j k i j k i j k l i j

DE x x u x x u u x x u x x u u x xDDt Dt X X X X X X x X X x X X x x X X

j

l

i

kkl

ij

Xx

Xx

DtDE

mn m nij

i j

DE X XDt x x

ji

ij pqp q

xxS

X X

Substituting for the rate of deformation and the Cauchy stress

j jmn i mn m n i m n mn mnpq pq mn

mn p q i j p i q j

x xDE x DE X X x X X DE DEDW S S SE Dt X X Dt x x X x X x Dt Dt

In case of solids the mechanical work is not dissipated to heat, but to the internal (deformation) energy increase. However the energy conversion term remains.

…..completes the proof that mnmn

WSE

pm qn

mm u

DtDx

2kl

Solids: Deformation energyMHMT3

Let us consider the simplest linear elastic and isotropic material described by the famous Hook’s law, when the deformation energy can be expressed in terms of the Green Lagrange tensor

)21

()1(2

2mmijij EEEEW

( )1 1 2ij ij mm ij

ij

W ES E EE

wherefrom the component of Kirchhoff stresses can be calculated by partial derivatives

E-Young’s modulus of elasticity [Pa] -Poisson’s constant

This is the Hook’s law for perfectly elastic linear material

MHMT3

The Hook’s law is an example of

CONSTITUTIVE EQUATION (relation between kinematics and stresses)

which is a linear one (in 1D loads and for small deformations the stress is proportional to deformation =E).

More complicated nonlinear or anisotropic materials are characterized by more complicated expression of deformation energy and by using different deformation tensors and their invariants (look at the frequently used Mooney Rivlin model as an example in next slides).

Kirchhoff stresses for isotropic materials can be expressed in rather general form

Solids: Deformation energy

2( )W I W II W IIISI II IIIC C C

where I,II,III are three invariants of the Cauchy Green tensor

23

22

21)( CtrI

2 2 2 2 2 21 2 1 3 2 3

1 ( )2

II tr C C

1 2 3det( )III C

MHMT3 Solids: Deformation energyThere is a large group of simple constitutive equations based upon assumption that the deformation energy depends only upon the first invariant of the Cauchy Green tensor 2

322

21 I

Then the Cauchy stresses can be expressed asm

j

m

iij X

xXx

IW

VV

02

this is Finger deformation tensor

Neo-Hook (linear model similar to Hook, but not the same, has only 1 parameter). Suitable only for description at small stretches (for rubber < 50%)

)3(1 IcW

2 31 2 3( 3) ( 3) ( 3)W c I c I c I

1 maxmax

3( 3) ln(1 )3

IW c II

5

2 21 max

3i i

i ii

IW c

Yeoh model (third order polynomial)

Gent model characterised by a limited extensibility of material fibers (Imax is the first invariant corresponding to stretches giving infinite deformation energy)

Arruda Boyce model having 7 parameters (max is a limiting stretch). This is the “8 springs” model based upon idea of 8 springs connected in a cube.

Example: Hook (1/3)MHMT3

Homogeneous extension of an elastic rod

X1

X2

X3

x1

x2

x3

21

22

23

1 0 01 0 1 02

0 0 1ijE

Hook’s law ( )1 1 2ij ij mm ij

ij

W ES E EE

2 2 21 2 3

1 ( 3)2mmE

2 2 2 21 1 2 3

2 2 2 22 1 2 3

2 2 2 23 1 2 3

1 ( 3) 0 01 2

0 1 ( 3) 02(1 ) 1 2

0 0 1 ( 3)1 2

ijES

verify that without deformation (i=1) all

stresses are zero

Example: Hook (2/3)MHMT3

The previous result holds for compressible Hook material and large deformations. The stretches are not related by any constraint (for example by the incompressibility constraint).

Let us consider the isotropic loading (uniform expansion/compression) with stretches

321

Kirchhoff stresses are in this case)21(2)1( 2

332211

ESSSS

Cauchy stresses are in this case2

11 22 33( 1)

2(1 2 )Ep

30 1/VV

Xx

because 0 jiij mn

m n

xV xS

V X X

The result indicates that the compressibility of a Hook material is determined by the ratio

3

3 2

1 1 3(1 2 ) 3(1 2 )(1 ) 2

VV p p E E

…and it can be seen that =1/2 represents an incompressible material

Example: Hook (3/3)MHMT3

Hook’s law is usually presented in the simple form

EThis statement is correct for the case of small deformations and unidirectional load in the x-direction (S22=S33=0) , when

21 11 2

1)-(-1-1- 21

23

22

2 2 2 211 1 1 2 3

2 21 1 1

[ 1 ( 1 1 1)]2(1 ) 1 2

[( 1)(1 (1 2 ))] ( 1)2(1 ) 1 2 2

ES

E E E

this follows from requirements S22=S33=0

MHMT3 Example: Mooney Rivlin (1/3)In many cases the principal directions of deformation and stress tensors are known, examples are a pressurized thick wall tube with axial load and uniform elongation, stretched rod or plate….

X1

X2

X3

1

1

x1x3

2x2

13

22

1133

In this case the tensors written in the coordinate system of principal axes are represented by diagonal matrices,

Green deformation Green Lagrange strains 2

122

23

0 00 00 0

k kij

i j

x xC

X X

21

22

23

1 0 01 1( ) 0 1 02 2

0 0 1

k kij ij

i j

x xEX X

Stretches and the Green Lagrange strains are obviously related by

333

3

222

2

111

1 1 1 1

EEE

MHMT3 Example: Mooney Rivlin (2/3)Mooney Rivlin model defines the deformation energy in terms of two invariants of the Green deformation tensor, or in terms of principal stretches as

)3()3( 23

21

23

22

22

212

23

22

211 ccW

Kirchhoff stresses corresponding to stretches are

111

11 1 11 1 1

1W W WSE E

0

1 2 3

1j ji iij mn mn

m n m n

x xV x xS S

V X X X X

22 332 2 3 3

1 1 W WS S

Cauchy stresses follow from definition

111 11 22 33

2 3 2 3 1 1 3 2 1 2 3

1 1 1 W W WS

Evaluating partial derivatives of the deformation energy gives

2 2111 1 2 2 3

2 3

2 ( ( ))c c

2 2333 1 2 2 1

1 2

2 ( ( ))c c

….

The next slide demonstrates how to obtain the same result in a simpler way…

Ou…!!!! non zero stresses are predicted at the

reference configuration. Where is the error?

MHMT3 Example: Mooney Rivlin (3/3)

d1

11

Work done by Cauchy stresses can be expressed directly by stretches

11 2 3 1 22 1 3 2 33 2 1 3

1 2 31 2 3

dW d d dW W WdW d d d

dW is total differential and by comparing coefficients of d the Cauchy stresses can be expressed

11 22 332 3 1 1 3 2 1 2 3

1 1 1 W W W

Consider unit cube (dimensions 1 x 1 x 1 in the reference configuration) deformed to the brick with sides 1,2,3. These sides are principal directions and only the normal (Cauchy) stresses 1=11, 2=22, 3=33 act on the faces. Infinitesimally small motion of faces d1,d2,d3 requires the work dW (calculated as the sum of works done by forces 1 23, 2 13, 3 21).

MHMT3 Solids: Experimental methods

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

Static test

Creep test

Relaxation test

MHMT3 Solids: Experimental methods

Inflation testsTubular samples inflated by inner overpressure.

Internal pressure loadAxial loadTorsion

Confocal probe

Laser scanner

CCD cameras of correlation system

Q-450

Pressure transducer

Pressurized sample (latex

tube)

Axial loading (weight)

Elastic Solids (summary)MHMT3

1. Relationships between coordinates of material points at reference (X) and loaded (x) configuration must be defined. For example in the finite element method the reference body is a cube and the loaded body is a deformed hexagonal element with sides defined by an isoparametric transformation.

2. Function x(X) enables to calculate components of the deformation gradient F and the Cauchy Green deformation tensor (by multiplication C=FTF) at arbitrary point x,y,z.

3. In terms of the Cauchy Green deformation or strain tensor the density of deformation energy W(C) can be expressed.

4. Components of stress tensors are evaluated as partial derivatives of deformation energy with respect to corresponding components of strain tensor.

Anyway, what is typical for solid mechanics, the constitutive equations are expressed in terms of deformation energy W. This is possible because the deformation of elastic solids is a reversible process, time has absolutely no effect upon the stress reactions, and it has a sense to speak about potential of stresses. Nothing like this can be said about fluids, where viscous stresses depend upon the rate of deformation, and viscous friction makes the process irreversible. Therefore the constitutive equations of purely viscous fluids cannot be based upon the deformation energy.

Viscoelastic FLUIDs

Macke

MHMT3

Exhibit features of fluids and solids simultaneously. Deformation energy is partly dissipated to heat (irreversibly) and partly stored (reversibly). Both the deformation and the rate of deformation should be considered.

Viscoelastic FLUIDsMHMT3

The simplest idea of viscoelastic fluids is based upon the spring+dashpot models

Maxwell fluids are represented by a serial connection of spring and dashpot

Voight elastoviscous materials are represented by a parallel connection of spring and dashpot

These materials are more like fluids, because at constant force F no finite (equilibrium) deformation is achieved. Only during the time changes a part of mechanical work is converted to deformation energy, later on all mechanical work is irreversibly degraded to heat.

These materials are more like elastic solids. At constant force F finite (equilibrium) deformation is achieved but not immediately. A sudden change of deformation results to infinite force response.

Viscoelastic FLUIDs MaxwellMHMT3

Viscoelastic fluids of the Maxwell body (serially connected spring and dashpot) can be described by ordinary differential equations (k-stiffness of spring, z attenuation)

x1

xz

k

F

1 1

11

( )

( )

d x x dx dx FF zdt dt dt z

dxdF dx FF kx k kdt dt dt z

giving after elimination of x1

This 1D mechanical model is the basis of the Maxwell model

2dF dxFdt dt

/z k 2 z

2t

where is a relaxation time, the time required for the stress to relax to (1/e) value of the initial stress jump (due to sudden extension).

Viscoelastic FLUIDs MaxwellMHMT3

x1

xz

k

F

The material objectivity is satisfied either by the model

( ) 2ij ij m mk mj mi ij ij

k i j

u uut x x x

( ) 2ij ij m mk mj mi ij ij

k i j

u uut x x x

There exist many objections about the previous generalisation of the Maxwell model, first of all against the way how the time derivative of the stress tensor was evaluated.

or by the upper convected Maxwell (UCM)

Time derivative does not fulfill the principle of material objectivity, which means that the time derivative depends upon to motion of observer (material objectivity requires, that even the time derivatives should satisfy the tensor transformation rules [[A’]]=[[R]].[[A]].[[R]]T ).

Remark: Both lower and upper convective Maxwell models are acceptable from the point of view of mathematical requirements, but they are not the same. The UCM model prevails in practice, at least in papers on polymer melts rheology.

Viscoelastic FLUIDs MaxwellMHMT3

Example: Fiber spinning (steady elongation flow of a polymer extruded from a dye). Axial velocity w is determined by the radius of fiber R

Normal stresses for the upper convected Maxwell

L

z

R12

u wr z

2 0zz zz rr rrF p pR

( 2 ) 2

( )

zzzz zz

rrrr rr

w wwz z z

w wwz z z

0u u wr r z

This is continuity equation relating radial u and axial w velocity (cylindrical coordinate system)

and this is a simplified solution based upon assumption of linear radial velocity profile

: m mmj mi zz zz

i j

u u w wi j zx x z z

: 2rr rr rr rru u u wi j rr r r z

This is the system of two ODE for the two unknown normal stresses zz and rr. Given axial force F it is possible to calculate axial profile of thickness R(z)

2 zz rrFR

200 )(RRww

see also the paper Tembely M.et al, Journal of Rheology vol.56 (2012),pp.159-183, fiber spinning using Oldroyd-B and the structural FENE CR rheological model (you will see close similarity of equations presented here).

F

Viscoelastic FLUIDs OldroydMHMT3

Viscoelastic fluids of the Voigt body combined with a serial dashpot can be described by ordinary differential equations (k-stiffness of spring, z1,z2-attenuations)

x1

xz2

z1k

F

21

2

111

11

2

112

)(

dtxdz

dtdxk

dtdF

dtdxzkxF

zF

dtdx

dtdx

dtxxdzF

giving after elimination of x1

2

2

122

1 )1(dtxdz

dtdxkF

zk

zz

dtdF

or 2

1 2 22 ( )dF dx d xFdt dt dt

for new coefficientskzz 21

1

12

zk

2

2z

Viscoelastic FLUIDs OldroydMHMT3

Replacing viscous stress for F and the rate of deformation for dx/dt

)1(2 21 tt

This equation reduces to Newtonian fluid as soon as the time constants 1=2=0 (1 , relaxation time, describes exponential decrease of stress after the flow is stopped).

Oldroyd suggested generalization of this model by replacement the common time derivative by the convected derivative in the same way like in the previously discussed Maxwell model

mij

mmj

i

m

k

ijk

ijijij

xu

xu

xu

ttt

Material derivative D/Dt

x1

xz2

z1k

covariant convected time derivative

Remark: Introducing convected derivatives makes the Oldroyd’s model nonlinear (due to products of velocity times gradient of stresses and stressed times velocity gradients). Theory is rather complicated, the original Oldroyds model suggested even more complex Jaumann derivatives with vorticity terms, reflecting not only a parallel convection of a moving fluid particle, but also its rotation.

Viscoelastic FLUID integral modelsMHMT3

Viscous fluids: stress depends only upon the rate of deformation at the current time t

Thixotropic fluids: stress depends also only upon the current rate of deformation but viscosity is affected by the deformation history

Viscoelastic fluids: stress at time t is a weighted sum of stresses corresponding to the history of deformation rates. Stress tensors calculated at previous positions of particle must be recalculated to the present position.

particle at time t

particle at a past time t’

x1

x2

x3

)()(2)( ttt ijij

( ') 2 ( ') ( ')ij ijt t t

( ) ( )t t

( ') ( ')t t

Let us assume a fluid particle moving along a streamline:rate of strain (e.g. rate of elongation in 1D case)

Viscoelastic FLUID BoltzmannMHMT3

( )( )( ) 2 ( ') ( ') '

( ') ( ')

tji

ij mnm n

x tx tt M t t t dt

x t x t

memory function

strain rate

Integral viscoelastic models are based upon the Boltzmann superposition principle.

The idea can be expressed for 1D case (a dashpot, taking into account only scalar stress and rate of strain) in form of integral

( ) ( ') ( ') 't

t M t t t dt

The monotonically decreasing memory function describes the rate of relaxation (rearrangement of fibers, entanglements…) and can be identified from the relaxation unidirectional experiments.

The unidirectional model can be extended to 3D for example as

N

i

tt

i

i ieattM1

/)'()'(

i-relaxation times, ai-relaxation moduli

Viscoelastic SOLID BKZMHMT3

1 1

1

1( , , ') ( , , ')2( ( ') ( ')) '

tC CC C

t tCC

I I t t I I t tC t C t dt

I I

Nowadays the more common integral methods are oriented to the hyperelastic rather than the fluid-like constitutive equations.

The Kaye-BKZ equation introduces the potential function which is an analogy of elastic potential (deformation energy W) depending upon the invariants of the Finger tensor C-1 expressing deformation at present time t relative to the configuration at past time t’

1 ( )( ') ( ') ( ) ( ) ( ) ( ') ( ')

jk k iij ij

i j k k

x tx t x t x tC C

x t x t x t x t

where IC-1 and IC are first invariants (traces) of the Finger and Cauchy tensors respectively.

1 1

1

1( , ) ( , )2 ( ')( ( ') ( ')) '

tC CC C

t tCC

I I I IM t t C t C t dt

I I

There exist several modifications of the BKZ model, for example

with the extracted memory function or just simply the model without the second term (Cauchy Green tensor), preserving only the Finger tensor in the integrand (Wagner model)

Bernstein, B., Kearsley, E.A., Zapas, L.J.: Trans. Soc. Rheol. 7, p.391/410 (1963)

Wagner, M.H.: Rheol. Acta 15, p.136/142, (1976)

1( ') ( ') '1 ( 3)( 3)

t

tM t t C t dtI II

Bernstein, Kearsley, Zapas

Viscoelastic SOLID HaslachMHMT3

Differential models (Maxwell, Oldroyd) are suitable for viscoelastic liquids, integral models for liquids as well as for viscoelastic solids (BKZ). However the integral models are rather complicated (need many experimentally determined parameters).

Haslach suggested a method based upon a perfect knowledge of equilibrium state (for example a relationship between stress and strain at a steady state). Time evolution of stresses from a non-equilibrium state towards the equilibrium in selected in such a way that the dissipated energy is maximised .

*(,)

()

Do you remember the Lagrange method, looking for the minimum of total energy?

()*(,)= - Deformation

energy

Equilibrium deformation corresponds to the minimum of total energy.

Curve of equilibrium states

Trajectory on the * surface starting from a non-equilibrium state (,) and maximising dissipated energy

It is quite difficult for me to understand the statement about the maximisation of dissipated energy. There are many other trajectories between the starting and ending points when the dissipated energy will be greater

Viscoelastic SOLID HaslachMHMT3

Principle of the suggested method is based upon definition of affinity X(,) * ( )X

which represents a measure of distance between the non-equilibrium point (,) and the equilibrium curve, when ( ) 0e

ee

X

The evolution proces (trajectory of (t),(t)) is described as the motion on the surface * towards equilibrium in the direction of gradient * (gradient relaxation method)

*dX kdt X

*X d k

dt X

2( )( )d Xkdt

22

2( )( )d kdt

….giving the final result

Viscoelastic SOLID HaslachMHMT3

The nonlinear maximum dissipation evolution is described by the system of first-order nonlinear ordinary differential equations

nn

nn

n

n

n

k

dtd

dtddtd

.........

...

...2

2

11

2

2

2

1

2

2

2

22

2

12

21

2

21

2

21

2

2

1

corresponding for example to a creeping test, when stresses are prescribed and deformations are calculated. Deformation energy is expressed in terms of deformations and the parameter k is relaxation modulus.

The role of stresses and deformations can be exchanged (describing a relaxation test) and the Haslach principle can be applied in fact for almost any pair of conjugated variables (pressure/volume, Kirchhoff stress/Green deformation,…). The deformation energy can be described by any hyperelastic model.

Viscoelastic SOLID HaslachMHMT3

Haslach presents application of the Mooney Rivlin model (energy expressed by principal stretches 1, 2) for description of biaxial creeping test of a rubber sheet

22 2

112

1 2 1 1

2 22

2222 1 2

ddt kddt

Henry W. Haslach Jr.: Maximum Dissipation Non-Equilibrium Thermodynamics and its Geometric Structure. Springer N.Y. 2011

The values of the Mooney-Rivlin constants, C1 = 2.240×105 Pa and C2 = 8.512× 103. Biaxial creep test with unequal loads, an initially unloaded rubber sheet is stressed by s1 = 9.456 × 105 Pa and s2 = 7.359 ×105 Pa with various relaxation moduli, k = 5.0 ×107, 2.0 ×107, and 1.0×107 Pa/s respectively, for the initial conditions λ1 = λ2 = 1.0 and s1 = s2 = 0 at t = 0. Fig shows that the principal stretches relax asymptotically to the equilibrium values of λ1 = 2.0 and λ2 = 1.5, corresponding to the perturbed stresses as calculated from the Mooney-Rivlin equation. Furthermore, the larger the relaxation modulus, k, the faster the relaxation to equilibrium.

This example is presented without permission-it is a copy

from the book

Viscoelastic EFFECTsMHMT3

Kaye effectBarus effect (die swell)

Weissenberg effect (material climbing up on the rotating rod)

H

R1R2

Rotatingdisk

Point nearthe fixed disk

Point near therotating disk

Filament pushedtowards the center bythe stretched filament

Stretched filament

Polymer melt flows against the centrifugal forces towards the rotation axis so that you will have a better idea

about what is going on, imagine that instead of a homogeneous fluid

there are entangled spaghetti fibers

Experiments: solid and fluidsMHMT3

Sinusoidaly applied stress and measured strain (not the 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

Characteristic features of elastic, viscoelastic and viscous liquids are best seen using oscillating rheometer (usually cone and plate configuration)

Weissenberg rheogoniometer from Wikipedia

EXAMMHMT3

Constitutive equations

What is important (at least for exam)MHMT3

You should know Helmholtz decomposition of the velocity gradient tensor into spin and rate of deformation tensors

))((21))((

21 TT uuuuu

p

Decomposition of stress tensor

Fluids

2 ( )3u

Newtonian fluid

What is important (at least for exam)MHMT3

Non Newtonian Fluids

Invariants 3 3

1 1

2 : 2 2ij ji ij jii j

1

2 2 :n

K

Power law liquids

2 :

yp

Bingham liquids

What is important (at least for exam)MHMT3

Solids

j

iij X

xF

dx F dX R U dX

1 ( )2

k kij ij

i j

x xE

X X

( )1 1 2ij ij mm ijES E E

Deformation gradient

Decomposition of deformation to stretch and rotation tensors

Cauchy Green deformation tensor

Hook’s law in terms of Kirchhoff stresses and Cauchy Green deformation tensor

(what is it x and X?)

(what is it R and U?)

(what is it E and ?)

What is important (at least for exam)MHMT3

Viscoelastic fluidsMaxwell model

2t

(draw a combination of dashpots and springs corresponding to this model)