mech anl 16.0 l06 advanced models

2011 ANSYS, Inc. May 27, 2015 1 Release 14.0

ANSYS Mechanical Advanced Nonlinear Materials

16.0 Release

Lecture 6: Advanced Models

2015 ANSYS, Inc. May 27, 2015 2

Advanced Material Models

In this lecture we will discuss more advanced nonlinear material options

A. Mullins Effect

B. Anisotropic Hyperelasticity

C. Bergstrom-Boyce Hyperelasticity

D. Shape Memory Alloy

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A. Mullins Effect

Mullins effect is a stretch softening phenomenon observed in elastomeric materials undergoing cyclic loading.

When an elastomer is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less then that on the initial loading for stretches up to the maximum stretch achieved on the initial loading.

During reloading, as the reloading strain approaches the maximum strain seen in its prior strain history, the stress-strain behavior begins to stiffen and rejoin the reference virgin curve; upon reaching the reference virgin curve, the stress-strain behavior follows that of the virgin stress-strain behavior.

2015 ANSYS, Inc. May 27, 2015 4

Mullins Effect Phenomenon typically observed in compliant filled polymers.

Characterized by a decrease in material stiffness during loading

Readily observed during cyclic loading as the material response along the unloading path differs noticeably from the response that along the loading path.

Although the details about the mechanisms responsible for the Mullins effect have not yet been settled, they might include:

Debonding of the polymer from the filler particles

Separation of particle clusters

Rearrangement of the polymer chains and particles.

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The Ogden Roxburgh pseudo-elastic model of the Mullins effect is a modification of the standard thermodynamic formulation for hyperelastic* materials and is given by:

Where:

is the virgin strain energy potential without Mullins effect

is an evolving scalar damage variable

is the damage function

* The virgin material is modeled using one of the available hyperelastic potentials, and the Mullins effect modifications to the constitutive response are proportional to the maximum load in the material history.

)()(),( ijij FWoFW

)( ijO FW

)(

Mullins Effect

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The arbitrary limits are imposed with defined as the state of the material without any changes due to the Mullins effect.

Then along with equilibrium, the damage function is defined by:

Which implicitly defines the Ogden Roxburgh parameter

)()(

0)1(

ijO FW

0.10 1

Mullins Effect

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The 2nd Piola-Kirchoff stress tensor is:

The modified Ogden-Roxburgh damage function available in ANSYS has the following functional form of the damage variable

Where: r, m and b are user defined material damage parameters

is the maximum virgin potential over the time

interval

ij

O

ij C

W

C

WSij

22

m

Om

Lm

WWerf

r b

11

Mullins Effect

0,0 tt )(max tWW om

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The parameters used in the Ogden-Roxburgh damage can be defined in Engineering Data provided a hyperelastic material model is first defined.

Supports all hyperelastic models except foam

Mullins Effect

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Effect can be plotted via Chart Tool

Mullins Effect

LS01

LS02

LS03

LS04

LS05

LS06

LS07

LS08

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References on Mullins Theory:

1. Section 4.8 of ANSYS 14.0 Theory Manual

2. Ogden & Roxburgh; Pseudo-elastic model for Mullins effect in filled rubber, 1999

3. H.J.Qi,MC Boyce; Constitutive Model for stretch-induced softening of the stress-strain behavior of elastomeric materials, 2004

Mullins Effect

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Workshop Exercise

Please refer to your Workshop Supplement:

Workshop 6A: Mullins Effect

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B. Anisotropic Hyperelasticity

Anisotropic Hyperelasticity can be used to model materials that exhibit direction-dependent large elastic strains, such as biomaterials or reinforced elastomer composites.

One way to view anisotropic hyperelasticity is that there may be fibers (or reinforcements) in an elastomer-like matrix.

Up to two preferred fiber directions A and B, with corresponding material parameters, can be specified.

A B

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Anisotropic Hyperelasticity

Recall that there are several isotropic hyperelastic constitutive models available in ANSYS. Many of these (e.g., Mooney-Rivlin, Yeoh, Arruda-Boyce, Gent, Blatz-Ko) are based on the first three strain invariants, as shown on the left

The Cauchy-Green tensor C is used here, although the invariants can also be expressed as a function of the principal stretch ratios li.

The second strain invariant is neglected in some hyperelastic strain energy functions, such as Arruda-Boyce, Gent, and Yeoh

The third strain invariant provides a measure of the volume change J of the element. If the material is fully incompressible, I3=J

2=1.

2

3

22

2

1

det

trtr2

1

tr

JI

I

I

C

CC

C

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Two material directions (vectors A and B) in the undeformed configuration characterize the anisotropy of the material.

In order to represent anisotropic behavior, an additional six strain invariants are required I4-I9.

The ninth strain invariant does not depend on the deformation

29

8

2

7

6

2

5

4

BA

CBABA

BCB

CBB

ACA

CAA

I

I

I

I

I

I

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In the case of fibers in a matrix that produce anisotropy in two directions A and B, one can ascribe some meaning to the various strain invariants:

The first two strain invariants describe the behavior of the matrix material

The third strain invariant is related to the degree of incompressibility of the material

The fourth and sixth strain invariants represent the fiber characteristics

The fifth and seventh strain invariants are associated with fiber-matrix interactions

The eighth strain invariant can be thought of as being related to fiber-fiber interaction

29

8

2

7

6

2

5

4

BA

CBABA

BCB

CBB

ACA

CAA

I

I

I

I

I

I

2

3

22

2

1

det

trtr2

1

tr

JI

I

I

C

CC

C

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The strain energy density function for anisotropic hyperelasticity can be decomposed into two parts deviatoric Wd and volumetric Wv. Moreover, the deviatoric term can be separated into both isotropic and anisotropic parts:

The volumetric term Wv is a familiar equation arising in other nearly-/fully-incompressible strain energy density functions and is a function of J (third strain invariant) only:

vanisodisod

vdAHYPER

WWW

WWW

,,


211

Jd

Wv

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The anisotropic hyperelastic material model can be defined via APDL commands only (TB,AHYPER, TBTEMP and TBDATA):

Can also be combined with viscoelasticity (TB,PRONY)

Viscoelastic behavior is assumed to be isotropic

The volumetric term Wv is a familiar equation arising in other nearly-/fully-incompressible strain energy density functions and is a function of J (third strain invariant) only

The material compressibility parameter d is input via:

TB,AHYPER,,,1,PVOL and TBDATA,1,d

One can estimate d=2/ko, where ko is the initial bulk modulus of the material.

For fully-incompressible behavior, d=0.


211

Jd

Wv

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Two expresssions are available for characterizing the isochoric part of the strain energy potential

Polynomial (Defined with TB,AHYPER,,,Poly):

Exponential: (Defined with TB,AHYPER,,,Expo):


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Note: The first two terms of the expressions on previous slide represent the deviatoric component related to isotropic behavior Wd,iso This is very similar to the general polynomial form without cij cross-terms:

The six constants ai and bi are via TB,AHYPER,,,POLY or TB,AHYPER,,,EXPO and TBDATA,1,a1,a2,a3,b1,b2,b3

If bi=0, the isotropic term becomes the 3rd order Yeoh model. If only a1 is present, it is like the neo-Hookean model.

The remaining terms of the expressions on previous slide represent the deviatoric component related to the anisotropic behavior, Wd,aniso using the fourth through ninth strain invariants.

3

1

2

3

1

1, 33j

j

j

i

i

iisod IbIaW


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The A material direction constants (AX, AY, AZ) are input via TB,AHYPER,,,3,AVEC and TBDATA,1,AX,AY,AZ

The B material direction constants (BX, BY, BZ) are input via TB,AHYPER,,,3,BVEC and TBDATA,1,BX,BY,BZ

The material anisotropy is defined with these two vectors A and B, not with the element coordinate system ESYS

Vectors A and B can have arbitrary directions and need not be orthogonal

The magnitude of the vectors |A| and |B| will be scaled internally to be equal to 1.

For the case when the hyperelastic material is orthotropic: Constants go are not required because I8=I9=0

If the hyperelastic material is transversely isotropic: Constants em, fn, and go are not required because I6=I7=I8=I9=0

Constants (BX, BY, BZ) are not required

The anisotropic hyperelastic constants may also be temperature-dependent

Use the TBTEMP command to define temperature-dependent data


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Sample input script for Anisotropic Hyperelasticity: To combine with viscoelasticity, add prony series definition:

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C. Bergstrom-Boyce Hyperelasticity When rubber material is compressed for extended period of time, sometimes it loses its ability to return to its undeformed state. This loss of elasticity may reduce the efficiency of an elastomeric gasket, seal or cushioning pad to perform over its operating life. The resulting permanent set that a gasket/seals may cause a leak or reduce cushioning effect of a pad.

Bergstrom-Boyce material model can be used for predicting permanent set. The time dependent material properties (C5, C7,C8) of BB model can be adjusted such that the viscoelastic strains can be treated as Pseudo plastic strains.

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1-0.8-0.6-0.4-0.20True Strain

Tru

e S

tre

ss

Permanent set

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... Bergstrom-Boyce Hyperelasticity

The Bergstrom-Boyce material model is a phenomenological-based, highly nonlinear material model used to model typical elastomers and biological materials.

It allows for a nonlinear stress-strain relationship, creep, and rate-dependence.

It assumes an inelastic response only for shear distortional behavior. The response for volumetric is still purely elastic

The model is based on a spring (A) in parallel with a spring and damper (B) in series.

All components (springs and damper) are highly nonlinear.

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The stress state in A can be found in the tensor form of the deformation gradient tensor (F = dxi / dXj) and material parameters, as follows:

Bergstrom-Boyce Hyperelasticity

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L-1(X) is the inverse Langevin function given by:

This approximation will differ from the polynomial approximation used for the Arruda-Boyce model.


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The stress in the viscoelastic component of the material (B) is a function of the deformation and the rate of deformation.

- Of the total deformation in B, a portion takes place in the elastic component while the rest of the deformation takes place in the viscous component.

- Because the stress in the elastic portion is equal to the stress plastic portion, the total stress can be written merely as a function of the elastic deformation

- All variables in this equation are analogous to the variables for s A

The stress tensor from component B is added to the stress tensor from component A to find the total stress


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The Bergstrom-Boyce (or BB) material model can be defined via APDL commands only (TB,AHYPER, TBTEMP and TBDATA):

Issue the TBDATA data table command to input the constant values in the order shown:


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Sample input script for BB model

BB cannot be combined with any other material models


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Permanent set in Elastomers General Tips:

1.Smaller the value of C5, the more significant is the hysteresis and hence higher the Pseudo plastic strains.

2.As the value of C6 increases, the amount of hysteresis decreases and hence less Pseudo plastic strains.

3.As the value of C7 increases, hysteresis also increases and thus high Pseudo plastic strains.

-2-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1.00-0.80-0.60-0.40-0.200.00

Ture

Str

ess

(M

Pa)

True Strain

C1_b = 5

C1_b = 1

C1_b = 0.5

C1_b = 0.1

C1_b = 0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6

Pse

ud

o P

last

ic S

trai

n

C1_b (material constant C5)

Sensitivity with respect to material constant C5

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Permanent set in Elastomers -1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1-0.8-0.6-0.4-0.20

Tru

e S

tre

ss (

MP

a)

True Strain

C = -2.5

C = -2

C = -1.5

C = -1

C = -0.8

C = -0.6

C = -0.4

C = -0.2 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

-3-2.5-2-1.5-1-0.50

Pse

ud

o P

last

ic S

trai

n

C (Material Constant C6)

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1-0.8-0.6-0.4-0.20

Tru

e S

tre

ss (

Mp

a)

True Strain

m = 2

m = 2.5

m = 3

m = 3.5

m = 4

m = 4.5-0.01

0.01

0.03

0.05

0.07

0.09

0.11

2 2.5 3 3.5 4 4.5 5

Pse

ud

o P

last

ic S

trai

n

m (Material Constant C7)



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References on Bergstrom Boyce Model:

1. Section 4.7 of ANSYS 14.0 Theory Manual

2. J.S. Bergstrom and M.C. Boyce. Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers. Journal of the Mechanics and Physics of Solids. Vol. 46. 931-954. 1998.

3. J.S. Bergstrom and M.C. Boyce. Large Strain Time-Dependent Behavior of Filled Elastomers,. Mechanics of Materials. Vol. 32. 627-644. 2000.

4. H. Dal and M. Kaliske. Bergstrom-Boyce Model for Nonlinear Finite Rubber Viscoelasticity: Theoretical Aspects and Algorithmic Treatment for the FE Method. Computational Mechanics. Vol. 44. 809-823. 2009.

Permanent set in Elastomers

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Workshop Exercise Please refer to your Workshop Supplement:

Workshop 6B: Hysteresis under Uniaxial tension loading

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Workshop Exercise Please refer to your Workshop Supplement:

Workshop 6C: Permanent Set in O-ring

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D. Shape Memory Alloy (SMA) Shape memory alloy (SMA) "remembers" its original, cold-forged shape, returning to the pre-deformed shape when heated.

- Can undergo large deformation without showing residual strains .

It has many applications in industries including medical and aerospace.

Nitinol (Nickel-Titanium alloy) is a type of shape memory alloy (SMA) used in eyeglass frames, antennas, couplings, actuators, medical retrieval devices and inserts

www.jmmedical.com

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Two phases, austenite and martensite, are present in Nitinol:

Austenite is usually stable at low stress values and high temperatures

Martensite is generally stable at high stress values and low temperatures

Martensite plates are formed within the austenite phase. The formation of these martensite plates generally do not involve dislocation motion or diffusional flow

Consider the case of an elevated temperature where only austenite exists at the stress-free state. If a material is loaded, the higher stress induces a phase change to

martensite (path ABC). Unloading will result in a reverse transformation from

martensite back to austenite with hysteresis (path CDA). This is the superelastic effect.

... Shape Memory Alloy (SMA)

2015 ANSYS, Inc. May 27, 2015 36

... Shape Memory Alloy (SMA)

Two SMA simulation options available:

Superelasticity - Phase transformation is based on stress only. Although constants can be temperature-dependent, superelastic effects are usually considered in the context of an isothermal process.

Shape memory effect - original shape restored after a thermal cycle. Also due to a phase transformation between martensite and austenite.

Compression

Remove loading elastic recovery

Heating residual strain recovery

Initial spacer Implanted in the spine

Complete implant

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The SMA Superelasticity model makes the following assumptions:

Phase transformation is based on stress only. Although constants can be temperature-dependent, superelastic effects are usually considered in the context of

an isothermal process.

Accounts for austenite to martensite (AS) and martensite to austenite (SA) phase transformations.

Isotropic, rate-independent, and without inelastic strains.

The elastic modulus, Poissons ratio, and coefficient of thermal expansion are assumed to be the same for the fully transformed austenite and martensite phases.

... SMA Superelasticity

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The SMA Superelasticity material model keeps track of the

fraction of austenite xA and martensite xS present.

The sum should equal 1 (i.e., 100%)

Superscripts AS designate the austenite-to-martensite

transformation (and SA the reverse), the fraction rates can be

expressed as:

The first two equations reflect that fraction rate of one phase is based on the fraction rate during either transformation process

(SA or AS).

Remaining equations indicate that a fraction rate of one phase must equal the other.

0

1

AS

AS

xx

xx

SA

S

SA

A

AS

S

AS

A

SA

A

AS

AA

SA

S

AS

SS

xx

xx

xxx

xxx


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Although the transformation is assumed to be fully recoverable, the SMA

superelasticity equations are developed similar to plasticity models

The transformation strains are considered separately, analogous to how inelastic strains are considered in plasticity.

The transformation function is defined as follows:

where q is the deviatoric stress, p is the hydrostatic pressure, and a is

a material parameter.

This transformation function is used for both transformation processes (SA or AS), so no superscript will be used.

Note the similarity with Drucker-Prager

pqF a3

eqvDP pqF sb 3


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The evolution of the martensite phase is expressed as:

for AS transformation with HAS defined as:

where sASs and sAS

f are both material parameters.

An analogous relationship exists for SA transformation:

as

xx

1

1AS

f

S

ASAS

SF

FH

otherwise 0

0

11 if 1F

F

H

AS

f

AS

sAS

asas

otherwise 0

0

11 if 1

1

F

F

H

F

FH

SA

s

SA

fSA

SA

f

S

SASA

S

asas

asxx


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Stress-strain relationship of SMA Superelasticity :

The first slope (green) is the 100% austenite phase and

is described by elastic modulus

Transformation starts at a stress level sASs and ends at s

ASf when

100% is martensite.

The last slope (purple) is the 100% martensite phase

and is also the elastic modulus


eL

sASf

sASs

e

s

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The unloading response is similar:

At 100% martensite, the transformation starts when the stress goes below sSAs.

The transformation back to 100% austenite is complete at sSAf.

All strains are recovered for this isothermal process.

The material parameter eL describes the maximum amount of transformation strain.

For Nitinol, this is typically between 0.07 and 0.10 (7-10%).

The material parameter a discussed earlier affects the material response in tension and compression.

If tensile and compressive behaviors are the same, a=0.

For Nitinol, this is usually taken to be around 0-0.1.

e

s


sSAs

sSAf

eL

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The SMA Superelasticity option is available from the Engineering Data


2015 ANSYS, Inc. May 27, 2015 44

Youngs modulus and Poissons ratio are

required for this material model

As noted earlier, this input describes the mechanical behavior of the austenite and

martensite phases

The SMA parameters can then be input

The first four constants describe the starting and final stress values of transformation

The epsilon value is the maximum transformation strain

The alpha value affects the compressibility of transformation strain (i.e., degree of which transformation strains are dependent on hydrostatic pressure)

This parameter also produces different response in tension and compression


2015 ANSYS, Inc. May 27, 2015 45

The SMA - Shape memory effect is based on a 3-D thermo-mechanical model for stress-induced solid phase transformations.

The governing equations are derived from an expression for free energy potential defined as:

... SMA Shape Memory Effect

2015 ANSYS, Inc. May 27, 2015 46

Where:


normrmation on transfo constraint the

satisfy tointroducedfunction indicator ) '('I

re temperatu theoffunction increasing

ly montonical postive a,To)-(T (T)

Straintion Transforma Deviatoric '

Straintion Transforma Total

Strain Total

tensorstiffness elastic Material

tr

tr

tr

tr

M

e

b

e

e

e

e

D

ation transformphase during material theof

hardening the torelatedparameter material h

observed is

martensite twinningno which eTemperatur To

eTemperatur T

Parameter Material

b

2015 ANSYS, Inc. May 27, 2015 47

Taking the derivative of the free potential energy wrt deviatoric transformation strain, we arrive at an expression of the transformation stress Xtr:

Where:


2015 ANSYS, Inc. May 27, 2015 48

Numerous experimental tests show an asymmetric behavior of SMA in

tension and compression, and suggest describing SMA as an isotropic

material with a Prager-Lode-type limit surface.

Accordingly, the following yield criteria is assumed:

Where J2 and J3 are the second and third invariants of transformation

stress, m is a material parameter and R is the elastic domain radius.


2015 ANSYS, Inc. May 27, 2015 49

The evolution of transformation strain is defined as:

Where: x is an internal variable (transformation strain multiplier).

x and F(Xtr) must satisfy the classical Kuhn-Tucker conditions

as follows:


2015 ANSYS, Inc. May 27, 2015 50

The elastic stiffness tensor is a function of the transformation strain

defined as:

Where: DA is the elastic stiffness tensor of Austenite phase

DS is elastic stiffness of Martinsite phase

When the material is in its austenite phase, D = DA

When the material undergoes full transformation (martensite phase), D = DS.


Stresses, strains, and the transformation strains are then related as

follows:

2015 ANSYS, Inc. May 27, 2015 51


Graphical illustration of Shape Memory Effect model:

The austenite phase is associated

with the horizontal region abcd.

Mixtures of phases are related to the

surface cdef.

The martensite phase is represented

by the horizontal region efgh.

Point c corresponds to the

nucleation of the martensite phase.

Phase transformations take place

only along line cf .

2015 ANSYS, Inc. May 27, 2015 52


The shape memory effect option is defined by seven constants that

establish the stress-strain behavior of material in loading and unloading

cycles for the uniaxial stress-state and thermal loading.

SMA cannot be combined with other materials for the same material.

h To R b eL Em m

mech anl 16.0 l06 advanced models

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