Mech ANL 16.0 L05 Viscoelasticity

2015 ANSYS, Inc. May 27, 2015 1

ANSYS Mechanical Advanced Nonlinear Materials

16.0 Release

Lecture 5: Viscoelasticity

2015 ANSYS, Inc. May 27, 2015 2

Lecture Overview

This lecture will discuss the viscoelastic material model for used in modeling such materials as glass (amorphous solid) and amorphous polymers.

In this Lecture, we will cover the following topics:

A. Background on Viscoelastic Theory

B. Prony Series Function

C. TRS Behavior

D.Shift Functions

E. Examples

F. Defining Material Properties

G.Analysis Settings for Viscoelasticity Models

H.Workshop

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A. Background on Viscoelastic Theory

For some amorphous polymers, there is a change in behavior with respect to temperature.

Below the glass transition temperature, the material may behave like an elastic solid.

Above the glass transition temperature, the material response is similar to a rubbery solid.

At higher temperatures, the behavior is similar to a viscous liquid.

For the temperatures above the glass transition temperature, the response is a combination of an elastic solid and viscous liquid (referred to above as a rubbery solid). This behavior is characteristic of viscoelasticity.

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... Background on Viscoelastic Theory

There are two aspects common to viscoelasticity, which involve time- and temperature-dependency:

Comprised of elastic and anelastic response for deviatoric and/or volumetric strains.

The elastic portion is recoverable and is instantaneous

The viscous portion is non-recoverable and occurs over time

Possible temperature-dependency introduced via

Temperature-dependent relaxation constants

Thermorheologically simple (TRS) assumption

Or have no temperature-dependency at all

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Viscoelasticity describes material response which contains an elastic and viscous part

The elastic response is instantaneous Viscous response occurs over time (anelastic)

The rate effect is such that there is limiting behavior for fast and slow loading

As strain rate decreases, the bulk/shear moduli also decreases

For high strain rates, the elastic response is the limiting behavior

For low strain rates, the viscous response is the limiting behavior

e

s

. e

s

e

. e

. e 0


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Creep

Under constant applied stress, strain increases monotonically.

Cases of linear and exponential creep shown on right

Recovery

If the constant applied load is removed, a portion of the viscoelastic strains will recover (dotted line).

Stress Relaxation

Under constant applied strain, stress decreases asymptotically.

t

e0

e

e

t

s

s0

s


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In viscoelasticity, the constitutive relation is dependent on the stress-strain history, so a hereditary integral is used: As seen above, the deviatoric and volumetric terms are separated.

The relaxation functions G(t) and K(t) are described by Prony series (discussed next)

Accounting for temperature effects can be done by either temperature-dependent Prony constants for G(t) and K(t) or by assuming thermorheologically simple

behavior (discussed later)

t

vol

t

dtttKdtttGt0

''

0

''2 e Ie


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Viscoelasticity models

Maxwell model consists a spring and a viscous dashpot (damper) in series

Kelvin (Voigt) model consists a spring and a dashpot in parallel

Standard linear solid model combines two springs and a dashpot

In WB, we use a Prony Series representation (Generalized Maxwell Model)

The number of dashpots is equal to the number of terms in the Prony series representing the stress response.


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B. Prony Series

Where tiG is the relaxation time for each Prony component Gi. G is the long-term modulus (t=).

Instead of inputting values as shown in the first equation on right, we introduce relative moduli ai

G=Gi/Go.

N values of aiG and ti

G are input for shear (and/or bulk) moduli.

Go is the instantaneous modulus (t=0) whereas G is the long-term modulus (t=).

M

i

t

K

i

K

N

i

t

G

i

G

o

ii

N

i

t

i

Ki

Gi

Gi

eKtK

eGtG

G

G

eGGtG

1

0

1

0

1

t

t

t

aa

aa

a

As with other material behavior, volumetric and deviatoric terms are separated. Similar behavior can be defined for bulk modulus with a separate set of M values of relative moduli ai

K and relaxation times tiK.

Relative moduli ai and relaxation times ti can be input as temperature-dependent constants

Prony Series: Shear modulus G(t) and bulk modulus K(t) are functions of time.

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... Prony Series

As shown from the equations on the previous slide, the relative shear (and/or bulk) modulus decay exponentially. At t=ti, the relative modulus will be at 37% of its value.

The sum of ai should be less than or equal to 1. If the sum of ai is equal to 1, that means that G(t=)=0.

Go Go

G (50%Go)

G (20%Go)

1 pair: a1=0.5 and t1=20 2 pair: a1=0.5, t1=20 and a1=0.3, t1=70.

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Thermorheologically Simple Behavior

As mentioned earlier, viscoelastic materials are often time- and temperature-dependent. Both dependencies may need to be accounted for.

Thermorheologically simple (TRS) behavior means that time & temperature are the same phenomenon. This means that the viscoelastic response vs. logarithmic time function translates with change in temperature.

Another way of stating the above is that the material response to a load at a high temperature over a short duration is the same as the response at a lower temperature over a longer duration.

C. TRS Behavior

ln(t)

G

Shifting of Relaxation Modulus with

Change in Temperature

T2 T1

T0

T0 < T1 < T2

G(0)

G()

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... TRS Behavior

Thermorheologically Simple Behavior (continued)

The TRS assumption allows for a relationship with time and temperature dependency. This adequately describes many amorphous polymers.

The short-term G(t=0) and long-term G(t=) moduli will remain the same, regardless of the temperature (i.e., the upper and lower limits on the previous graph will remain the same for any translational shift).

This allows for the definition of viscoelastic behavior at one temperature yet captures the response at other temperatures.

Using the concepts of reduced time and the shift function (discussed next), the viscoelastic response is shifted to account for behavior at another temperature.

Depending on the material, different shift functions are used.

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D. Shift Functions

Mathematically, the aforementioned TRS behavior is expressed by combining the effect of time and temperature into a variable called reduced time x (a.k.a. pseudo time).

The shift function, A(T), describes the shifting of the response curve (i.e., the relationship between time and temperature) based on a reference temperature Tr .

There are two predefined shift functions which can be used in ANSYS, William-Landel-Ferry (WLF) and Tool-Narayanaswamy (TN) shift functions. A user-defined shift function may also be specified. All are specified via TB,SHIFT with Tr and C1 (and C2) as supplied constants. Tr is in absolute temperature units

r

r

r

TTC

TTC

WLF

TTC

TN

t

tTA

etTA

tTAdt

d

dttTAt

2

1

1

10'

11

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'

'

0

''

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Numerical integration is performed with respect to time.

Hence, the Prony representation is rewritten with pseudo time , which includes the effect of time and temperature.

It is assumed that temperature changes DT vary linearly in any increment of time Dt. (i.e., T is a linear function of t in a substep)

This means that the relaxation times of all Prony components must also obey the relationship involving the shift function A(T). Hence, temperature changes change the relaxation times according to A(T).

Using the concept of reduced time, isothermal equations can now be used to describe non-isothermal processes.

rGi

r

G

i

r

N

i

G

i

G

TTAT

T

TTAdt

d

eGGGi

,

,'

1

0

t

t

aa t

... Shift Functions

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Below is an example of how temperature affects the response of the system (change in shear modulus plotted). As temperature increases, one can see that the relaxation times also decrease. (The curve would look as if it shifted to the left if the response were plotted on ln(t) instead of t.)

Go

G (20%Go)

Example of TRS Behavior (T=20, 50, 100)

T

... Shift Functions

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Some comments on the WLF and TN shift functions:

Although the constants C1 (and C2) are usually evaluated at the glass transition temperature Tg, the user is not restricted to use this. The

constants C1 and C2 may be evaluated at any reference temperature Tr.

The TN shift function, as implemented is essentially the Arrhenius equation.

... Shift Functions

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The TN shift function is usually used in the glass industry. Because of volumetric growth of glass encountered near its glassy state, the concept of fictive temperature is used instead of reduced time.

This extension of the Tool-Narayanaswamy shift function to include a fictive temperature is defined as follows:

Where:

FrT

X

T

X

TC

TN etTA

11

'1

parameter material 10Xre temperatufictive

TF

... Shift Functions

2015 ANSYS, Inc. May 27, 2015 18

The fictive temperature is given by:

Where:

The superscript o represents values from the previous time step.

timesrelaxation re temperatu

increment time

D

fi

t

t

)(

)(

oFfi

oF

ofifi

fiTt

TtTTT

D

D

t

t

... Shift Functions

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The fictive temperature model also modifies the volumetric thermal strain model and gives the incremental thermal strain as:

where the glass and liquid coefficients of thermal expansion are given by:

The total thermal strain is given by the sum over time of the incremental thermal strains:

Where:

The superscript o represents values from the previous time step.

FFgFlgT TTTTT DDD aaae )(

44

33

221

)( TTTTT ggggoggaaaaaa

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221

)( TTTTT lllloll aaaaaa

Dt

TT ee

... Shift Functions

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Various materials may exhibit viscoelastic response

Polymers

Usually described by WLF shift function

Elastomers (rubber industry)

Underfill, mold compound (electronics packaging)

Glass

Described by TN shift function

Metals

Usually, metal anelastic response is negligible and not considered

Other

Wood, concrete

... Shift Functions

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Animation of hyperelasticity with

viscoelasticity. No temperature-

dependency specified

(isothermal). True stress vs.

strain shown in XY plot on top.

Note hysteresis during unloading.

E. Example: Bushing

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Animation of hyperelasticity

and viscoelasticity. No

temperature-dependency

specified (isothermal).

Reaction force from rigid

target surface shows

relaxation

... Example: Pinched Cylinder

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F. Defining Viscoelastic Material Data

WB Engineering Data allows for direct specification of viscoelastic material parameters.

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Procedure for defining material data is similar to other material options

Insert model option from Tool Box

Highlight the Tabular data Icon and insert Prony Series values in Table to the far right

Defining Viscoelastic Material Data

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After defining Prony constants, any one of three shift function options can be then be added as applicable


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Click to edit Master text styles

Viscoelastic Test Data can be read into Engineering Data and the corresponding Prony material properties calculated via Curve Fitting Tool.


1

Input relaxation test data

2

Specify Prony Series

3

Execute Curve fit

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To be fully defined, viscoelastic material model must also include either an Isotropic linear elastic material (EX & NUXY) or nonlinear hyperelastic material.

Elastic input values can be temperature-dependent


Shear and volumetric response can be input by specifying relative moduli and

relaxation times.

Note that shear and volumetric response do not have to have same number

of Prony constants

User does not have to input both shear and volumetric response. Often

volumetric relaxation is negligible.

Up to 100 temperature-dependent sets of 100 pairs of constants can be

input for deviatoric and volumetric response.

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G. Analysis Settings for Viscoelasticity

Viscoelasticity is similar to creep, but part of the deformation is removed when the loading is taken off.

CUTCONTROL will not take viscoelastic strains into consideration, so user must verify that time

step is small enough in transition region. This

means that temperature changes should be kept

small over a given substep.

Large Deflection = ON is manditory if combined with hyperelasticity

For large models with long run times and potential convergence trouble, consider setting up a Restart Control strategy in the event that adjustment to time step range or convergence criteria is necessary

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References for Further Reading

Some general references on viscoelasticity:

1. Viscoelasticy, Wilhelm Flugge, Blaisdell Publishing, 1967.

2. Creep and Relaxation of Nonlinear Viscoelastic Materials, William N. Findley, James

S. Lai and Kasif Onaran, Dover, originally published in 1976.

3. Glass Science and Technology, Vol 3 Viscosity and Relaxation, Edited by D.R.Uhlmann

and N.J.Kreidl, Academic Press, 1986.

4. Relaxation in Glass and Composites, George W. Scherer, John Wiley & Sons, 1986.

5. The Phenomenological Theory of Linear Viscoelastic Behavior, Nicholas W. Tschoegl,

Springer-Verlag, about 1992.

6. Viscoelastic Solids, Roderic S. Lake, CRC Press, 1999.

7. ANSYS Theory Manual, Chapter 4.9Viscoelasticity

2015 ANSYS, Inc. May 27, 2015 30

H. Workshop Exercise

Please refer to your Workshop Supplement:

Workshop 5A: Viscoelasticity

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Mech ANL 16.0 L05 Viscoelasticity