Kim Walter Lee

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Available online at www.sciencedirect.com

International Journal of Mechanical Sciences 44 (2002) 22852315

Modeling mechanical response of intumescent mat materialat room temperature

J.S. Kim1, M.E. Walter, J.K. Lee

Mechanical Engineering Department, The Ohio State University, 2075 Robinson Laboratory,

206 W. 18th Ave., Columbus, OH 43210, USAReceived 25 October 2001; received in revised form 23 October 2002; accepted 3 November 2002

Abstract

Intumescent mat material is widely used to support ceramic substrates in catalytic converters and behaves

very much like hyper-foam material under compressive loading. Experiments show that compressive loading

curves depend on the ram speed and the number of cycles. The unloading curves show dierent slopes and

paths that depend less on the ram speed and number of cycles. The slopes of the unloading curves decrease

as the plastic strain increases; this is referred to as softening in this study. The eects of rate, softening, and

plastic deformation must be considered to model the mechanical response of intumescent mat material. Finite

deformation theory is applied with a multiplicative decomposition of the deformation gradient tensor. Thedeveloped theory is implemented as an implicit nite element algorithm in ABAQUS TM=STANDARD. The

necessary material parameters are extracted from experiments. Numerical simulations show good agreement

with experiments.

? 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Energy absorbing support structures are often composed of compressible hyper-foam materials

that exhibit viscous reaction and high capacity to retain elastic strain energy for wide ranges ofdeformation. A hyper-foam material can be dened as a highly porous material that can undergo

large deformations. One application of a hyper-foam support material is intumescent or swelling

mat in catalytic converters. The intumescent mat is placed between the catalytic converter substrate

and the outer can. The intumescent mat has very little tensile strength and is primarily subjected to

compression. While the outer can of a catalytic converter expands and shrinks during the assembly

Corresponding author. 2075 Robinson Laboratory, 206 W. 18th Ave., Columbus, OH 43210, USA.

E-mail address: [email protected] (M.E. Walter).1 Present address: ArvinMeritor, 950W 450S Bldg. 2, Columbus, Indiana, 47201, USA.

0020-7403/02/$ - see front matter? 2002 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0020-7403(02)00176-5
mailto:[email protected]:[email protected]


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2286 J.S. Kim et al. / International Journal of Mechanical Sciences 44 (2002) 2285 2315

Nomenclature

h softening parameterF yield function

density

Helmholtz free energy

W strain energy function

J determinant of the deformation gradient tensor Fp principal stretch

Xi position vector

Ei unit direction vector in the reference conguration

F deformation gradient tensor

C right CauchyGreen metric tensor

e Eulerian strain tensorE Lagrangian strain tensor

U right stretch tensor

Cauchy stress

Kirchho stress

y yield stress

S 2nd PiolaKirchho stress

D rate of deformation gradient tensor

L velocity gradient tensor

E(4) spatial elasticity tensor without rate eects

E(4)

total spatial elasticity tensorD(4) material elasticity tensor

T(4) 4th order transformation tensor between D

e and De

T(4)em 4th order transformation tensor between D

e and Dm

N principal direction vector in the reference conguration

n principal direction vector in the current conguration

PK2 2nd PiolaKirchho stress

operation and under service conditions, the compressed mat supports the substrate. In this application,

predicting mat holding force forboth mechanical loadingandunloading is very important in catalyticconverter design [14].

The intumescent mat material contains aluminosilicate glass bers, vermiculite mineral particles,

and an organic binder. The vermiculite is a micaceous hydrated magnesium aluminum-silicate mineral

which, when heated, looses water and increases the mat layer thickness. At temperatures between 400

and 600C, the increase in thickness can be as high as 10 times. Fig. 1 shows a material structure

and Table 1 shows its composition based on mass fraction. Although the mechanical behavior at

elevated temperature is quite dierent from that at room temperature, the process of assembling the

converter part takes place at room temperature, and as a rst step, it is worth developing a room

temperature material response model.


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J.S. Kim et al. / International Journal of Mechanical Sciences 44 (2002) 2285 2315 2287

VermiculitePlatelet

VermiculitePlatelet

AluminosilicateFibers

After Heating

Lateral (x)

Thickness (y)

AluminosilicateFibers

Before Heating

(a)

(b)

Fig. 1. A schematic side view (a) of intumescent mat before heating and after heating. A top view (b) (view from the

thickness direction) of a 29 mm diameter mat specimen, where black specks are vermiculite particles.

Table 1

Mass based composition and density of intumescent mat

Vermiculite Aluminosilicate Organic binder Density (Kg=m3)

40 50% 310% 31 43% 632

A great deal of progress has been made on constitutive model development for foam materials

e.g., [510]. Simo [11]and Papoulia et al. [12] have proposed visco-hyperelastic models to accountfor viscous eects, and rate dependent plasticity formulations are employed to represent plastic

deformations [1316]. Plastic strains are obtained incrementally by satisfying assumed consistency

conditions. However, most of these models assume constant unloading slopes and will not provide

accurate results when applied to intumescent mat materials under cyclic loading conditions.

Fig. 2, a schematic of a quasi-static cyclic compression experiment for intumescent mat, shows a

continuous change in the slope of the unloading and reloading curves. The unloading curve shows

more softening (a smaller tangent slope) at a higher compressive strain. Although this softening is

often referred to as material damage, in order to distinguish between similar softening of the long

term elastic response, this paper will use the term softening to describe the observed decrease in


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0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

v=1.6mm/s

v=0.4mm/s

v=0.005mm/s

Static test

Compressivestress(MPa)

Compressive true strain

T1

T1+T

2

Fig. 3. Stress versus strain curves to 60% compression for dierent compression speeds. T1 is the time to reach 60%

compression. T2 is the hold time at the maximum strain. T1+T2 is the time at which displacement unloading begins.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6



1st cycle loading

2nd cycle loading

3rd cycle loading

Unloading

Fig. 4. Experimental cyclic compression results for 40% compression (v= 1:6 mm=s).

after the 2nd cycle. The peak stresses from the cyclic 40% displacement control experiments are

plotted in Fig. 5. Signicant stress reduction takes place from the 1st to 2nd cycle and only gradual

stress reductions continue after the 3rd cycle.


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0

0.2

0.4

0.6

0.8

1

1.2

2 4 6 8 10

v=1.6mm/sv=0.4mm/s

Peakcompressivestress(MPa)

Number of cycles

Fig. 5. Peak stress variations versus number of cycles for two dierent compression speeds (40% compression).

0

0.5

1

1.5

2

2.5

3

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Monotonic loadingCyclic loading

Compressivestress(M

Pa)


Fig. 6. Comparison of monotonic static loading and cyclic loading from load control compression experiments.

Fig. 6shows monotonic and cyclic static stress versus strain curves from the constant load exper-

iments. During the monotonic loading process, load is applied continuously without unloading. The

cyclic static loading experiment is performed to investigate the plastic behavior. Load is applied to a

certain stress level, and after removing the load, the material is then reloaded to a higher stress level.

The direction of loading and unloading is shown in Fig. 6 with arrows. The results show that the

unloading curves do not follow the monotonic loading curve in the range of strains we are interested


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0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

30% compression

40% compression50% compression



Fig. 7. First cycle compression results for dierent maximum compressive strains ( v= 1:6 mm=s).

in. However, unloading and reloading curves are nearly the same. Note that in Fig. 6, the reloading

curve at higher compressive strain is slightly softer than that at lower strain. This softening eect is

discussed in Section 6.1. All reloading curves appear to rejoin the monotonic loading curve.

The displacement control experiment results in Fig. 7show the 1st cycle of loading and unloading

curves for three dierent maximum compressive strains. All loading curves follow the same loadingpath. Fig. 8 shows a comparison between the 1st and 2nd cycle loading curves when dierent

compressive strain is applied. The 1st loading path is the same regardless of the maximum applied

strain. The path of the 2nd loading is very dependent on the maximum strain reached during the

1st loading. Also, note that the peak stress at the maximum strain level is much less for the second

loading. These dierences between the 1st and 2nd loading cycles suggest that, for a given maximum

strain level, the rate dependencies during the 1st loading are dierent from those during subsequent

loadings. The dierences may be due to damage and plastic deformation incurred during the 1st

loading cycle. From these experiments, it is concluded that intumescent mat is a highly nonlinear

material with distinct loading and unloading characteristics that depend on the maximum displacement

or load, the cycle number, and the rate of loading.

3. Model development

Based on experimental results, the four dierent congurations shown in Fig. 9will be used to ac-

count for rate eects, material softening, and plastic strain. For a given mechanical deformation Fm, it

is assumed that the material reference conguration 0 rst changes to the intermediate conguration

1 due to plastic deformation FP. When the material softens, then the intermediate conguration 1

changes to another intermediate conguration 2 by the softening deformation Fh. In this case, the


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0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

1st loading (40%)1st loading (50%)

1st loading (60%)2nd loading (40%)2nd loading (50%)2nd loading (60%)



Fig. 8. Comparison of the 1st and the 2nd loading curves for dierent maximum compressive strains (v= 1:6 mm=s).

Fh

Fm

Current

Configuration

Reference

Configuration 0

0 2

Position vector X

Position vector X

0

Intermediate

Configuration 1

1

Fe

Fp

F'

Position vector X

Virtual Intermediate

Configuration 2

Position vector X

Fig. 9. Congurational changes due to plastic strain and material softening.


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intermediate conguration 1 is a state after plastic deformation and the intermediate conguration

2 is a state after material softening, that is, dx = FFhFp dX0 = Fm dX0. A visco-hyperelastic model

is developed based on the congurations and 1. The material softening model is based on the

congurations , 1, and 2. The elasto-plastic model is based on the congurations ; 0 and 1.Then, through the 4 congurations shown in Fig. 9, each model is combined to form a complete

visco-hyper-elasto-plastic model with material softening.

3.1. Hyperelastic model development

Hyperelastic models have been in existence since the 1940s and are well documented [68,10,

1922]. For the elastic part of the current development, we follow the work reported by Ogden [20]

and by Simo and Taylor [21], in which the strain energy is expressed in terms of principal stretches.

From the virtual work principal, the internal energy variation is expressed as WdV1=

S Ee dV1=

ee dV; (1)

where W is a strain energy function, S is the 2nd PiolaKirchho stress (PK2 stress), is the

Kirchho stress, Ee is the Lagrangian elastic strain tensor, ee is the Eulerian elastic strain tensor, V1is the reference volume, and V is the current volume. From the relation W =S Ee in Eq. (1),

the PK2 stress can be expressed as

S=

3i=1

1

ei

@W

@ei(N1i N

1i); (2)

where ei is the principal stretch associated with the unit vector N1i in the reference conguration

and represents the tensor outer product. The PK2 stress can be related to the Cauchy stress bythe elastic deformation gradient tensor Fe by

= 1

JeFeSFe

T

; (3)

where Je is the determinant of Fe. The PK2 stress rate can be related to the Lagrangian elastic

strain rate by S = D(4) Ee. Hence, the fourth order material elasticity tensor D(4) can be obtained by

dierentiating Eq. (2) with respect to time.

3.2. Visco-elastic model development

Classical linear viscoelastic models have been incorporated into non-linear large deformation mod-els [11,12,23,24]. The visco-elastic stress response can be viewed as a superposition of the long-term

elastic response and a Prony series (see Papoulia et al. [12] for details):

= +

N=1

H; (4)

where the long-term elastic response can be derived from the virtual work principle with an appro-

priately dened strain energy function W:

= 1

JeFe

@W

@Ee Fe

T

: (5)


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The Prony series terms are dened as

H= t

0

ae1=(tt

) d

dt 1

Je

Fe@W

@Ee

FeT dt: (6)

The constants with subscript are material constants. Through the above equations, the stress

response is divided into static and transient terms. Depending on the complexity of the energy

function, the time integral in Eq. (6) may not be analytically tractable. In general, a numerical

integration scheme is required to implement these equations in the nite element method.

Time t in Eq. (6) can be divided into n + 1 time steps so that tn+1 =tn + t. Then, based on

the known stress and history at time step n, stress at time step n + 1 can be computed by assuming

that is linear during the time increment t:

n+1 =n+1 +

n+1vis =

n+1 +

N

=1

Hn+1 (7)

with

Hn+1 =et= Hn+

(n+1 n)

t a(1 e

t= ): (8)

Since the deformation gradient tensor is known at all times, n+1 can be determined from Eq. (5).

All other terms in Eq. (8)are material constants or are known at time step n and therefore Hn+1 can

be calculated. The Euler forward integration scheme requires only information at step n to calculate

the stress at step n+ 1. This is a signicant advantage in terms of saving computer memory and

computing time.

The spatial elasticity tensorE(4) that includes visco-elastic terms is obtained as

E(4) =visE(4) ; (9)

where

vis= 1 +

N=1

1

ta(1 e

t= ): (10)

The fourth order spatial elasticity tensor E(4) maps the elastic rate of stretch tensor to the Jaumann

rate of Cauchy stress, which can be obtained by taking the Jaumann rate of Eq. (5). This procedure

is described in Appendix A, and the result is as follows:

E(4) = I+ 1

JeP(4) +H(4): (11)

The components of H(4) and P(4) are related to the Cauchy stress, elastic deformation gradient Fe

and material form of the elasticity tensorD(4), in the following manner (see Appendix Afor details):

H(4)ijab=

12

(iabj +ia bj+ ibaj +ib aj); (12)

P(4)ijkm= F

eiPF

ejQF

ekRF

emSD

(4)

PQRS: (13)


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4. Softening model development for the elastic unloading function

4.1. Assumptions for the elastic unloading function

The experimental results in Fig. 6 show that the unloading stress response softens as compressive

strain increases. Constitutive modeling must account for dierent unloading functions for dierent

compressive strains. Ordinarily this would require many material constants, however, for intumescent

mat, experimental observations indicate that the unloading stress responses are self-similar.

Let the stress be a function of strain at an arbitrary base state,

=f(U): (14)

The similarity can be treated by scaling the strain by a factor h, while keeping stress constant:

=f(hU) =f(U): (15)

The scaling factor h is called the softening parameter and is assumed to be a function of the 3 rd

component of the right CauchyGreen metric tensor, i.e., h = h(Cm33). Physically, Cm33 is the thickness

direction stretch ratio which can be used as a thickness strain measure. The stress-strain curve

obtained in this manner is called the master curve.

4.2. Material softening in large deformation theory

Let us imagine two dierent bodies that deform independently to exactly the same shape and size.

Furthermore, assume that one body deforms to the nal state by the deformation gradient tensor Fe

and the other body by F. If the two dierent bodies generate the same stress eld, then F and

Fe are called equivalent deformation gradient tensors. Material softening phenomena for unloadingfunctions can be understood by introducing the three dierent congurations and the respective

deformation gradient tensors shown in Fig. 9. It is assumed that the reference conguration is a

state before material softening shown as 1 in Fig. 9 and the intermediate conguration is a state

after material softening shown as 2 in Fig. 9. The transition from the reference conguration to the

intermediate conguration takes place without changing mechanical properties. Stress is calculated

from F associated with the transition from the intermediate to the current conguration. Similar

methods are used in large deformation plasticity theory.

From the congurations shown in Fig. 9, the deformation gradient tensor Fe can be dened as

follows:

Fe =FFh: (16)

It is necessary to relate Fe and F to determine material properties at dierent states. This relationship

is established for large deformation theory with concepts similar to those used with the innitesimal

elastic functions in Eq. (15). In large deformation theory, suppose that the right CauchyGreen metric

tensor C is used to represent the stress function. The metric tensors, Ce and C, can be expressed

through the spectral theorem:

Ce = (e1)2N11 N

11+ (

e2)

2N12 N12+ (

e3)

2N13 N13 ; (17)

C = (1)2N1 N

1+ (

2)

2N2 N2+ (

3)

2N3 N3: (18)


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Since, in Eq. (15), the material softening occurs in the principal strain direction and since the

direction does not change after material softening, it is assumed that the same physical hypothesis

can be applied to material softening in large deformation. In this case, the following relations for

principal stretches and directions between C and Ce are assumed,

i= (ei )

h (19a)

and

N1i =Ni : (19b)

Therefore, Eqs. (19a) and (19b) lead to the metric tensor relation, C = (Ce)h where Ce is raised

the power h (the softening parameter).

By dierentiating Eq. (16) with respect to time, the relationship between velocity gradients L

and Le can be obtained:

L =Le G(4)L L

e; (20)

for which G(4)L is dened with constant h as follows (see Appendix B for details):

G(4)L =

3i=1

(1 h)niiii+

ij=12;13;23

ei

j i

ej

(ei )2 (ej )

2

eii

(nijji+ nijij) +ej

j(njiji+ njiij)

: (21)

The rate of deformation tensor D is the symmetric part of L and is obtained as follows:

D =De G(4)D D

e or D =T(4)ee De; (22)

where T(4)ee is a transformation tensor.

4.3. A thermodynamics approach to hyperelastic modeling with softening

Suppose that a material point in the intermediate conguration is characterized by the Helmholtz

free energy , Cauchy stress tensor , softening parameter h, and a right CauchyGreen metric

tensor C:

= (C; h); (23a)

=

(C

; h): (23b)Applying the energy conservation equation and Legendre transformation on the 2nd law of thermo-

dynamics leads to the following equation in the absence of temperature [28]:

Le 2F @

@CFT L

@

@hh0; (24)

where is density. With velocity gradients obtained by dierentiating Eq. (16), the rst term in

Eq. (24) is rewritten as

Le = L + FLhF1: (25)


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With Eq. (25) and Lh, the inequality in Eq. (24) can be rewritten as follows (see Appendix C for

details):

h 2F @

@CFT L

@

@h + 1

h2

3i=1

ln(i)ni ni

h 0: (26)

To ensure that the inequality in Eq. (26) is satised for all mechanical processes, the following

relationship must hold:

= 2hF @

@CFT: (27)

Since the terms in the second parenthesis in Eq. (27) are independent ofh, following relationship

holds:

@ @h

= 1h2

3i=1

ln(i)ni ni: (28)

For convenience, the Helmholtz free energy , dened as energy per unit mass, can be converted

to energy per unit reference volume or strain energy W:

W =R2 ; (29)

where R2 is the density in the equivalent intermediate conguration 2. If the strain energy function

W in Eq. (29) is dened as

W

=

W

h (30)then the Cauchy stress can be written as follows:

= 2 1

JF

@ W

@CFT: (31)

Therefore, when there is material softening, Eq. (31) can be used for calculating the stress for a

given strain energy function W.

5. Visco-hyper-elasto-plastic model development

In this section, based on the above theory for modeling visco-hyperelastic materials with soften-

ing, the procedure for developing the visco-hyper-elasto-plastic model is presented. The theoretical

development starts with a rate independent plastic model and then is extended to address rate eects

during the loading and unloading process.

5.1. Assumptions for the rate independent plastic model and its material behavior

As was previously mentioned, with the assumption that the lateral deformation is insignicant

compared to the deformation in the thickness direction, a 1-D plasticity model is plausible. Additional


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Shi

T

Y

m

Compressivestress

Shift so that = y

Trial stress

Yield stress = y

p

1

2

3

4

y>

Compressive strain

Fig. 10. Softening of the elastic unloading response.

assumptions include:

yielding only takes under compressive stress in the thickness direction,

the yield (ow) stress y is a function of mechanical strain, and

the long-term elastic response (or unloading) is a function of elastic strain.

In 1-D plasticity, the yield function F is simply dened as follows:

F=e3 e3 e3 ye3= 0 or e3 e3= e3 ye3; (32)

where e3 is a unit vector in the current thickness direction and is based on the deformation gradient

tensor.

Fig. 10is a schematic of the mat materials 1-D plasticity response during loading and unloading.

The thin solid line represents the plastic (yield stress) response during loading. Experimentally it has

been observed that the softening parameter h decreases as strain increases. Therefore, in Fig. 10, as

the compressive strain increases, the elastic unloading response (thin dotted line) begins to soften

and becomes the thick dotted line. For a given strain m, the trial stress (Point 4) is obtained from the

softened elastic curve (thick dotted line), and the yield stress (Point 1) is determined from the yieldstress curve. The trial stress is a stress predictor used in the numerical method for the plastic strain

calculation. Its value is evaluated from the elastic unloading curve for the given current deformation

based on the between plastic strain at the previous step and is subsequently compared with the

yield condition. Until the trial stress satises the yield condition, the plastic strain is modied and

updated from the previous step. Since the magnitude of the trial stress is higher than the magnitude

of the yield stress, the material has yielded and the trial stress must be modied to satisfy the yield

condition. Satisfying the yield condition requires shifting the thick dotted line to the thick solid line

by the plastic strain magnitude. If the loading is removed from Point 1, the unloading will follow

the thicksolid line. Point 2 is the permanent plastic strain when the load is completely removed.


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5.2. The plastic deformation gradient tensor and yield conditions

In large deformation theory, the deformation gradient tensor can be multiplicatively decomposed

into elastic and plastic parts as follows [25,26]:

Fm =FeFp; (33)

where Fe is due to elastic, and Fp is due to plastic deformation. For analytical convenience, the

material rigid body rotation is assumed to come only from the elastic deformation gradient tensor,

and plastic deformation does not contribute any material rotation during the deformation. Then, the

plastic deformation gradient tensor can be simplied to Up, a right stretch tensor:

Fp =RpUp =Up: (34)

Since plastic deformation is assumed to take place only in the material thickness direction, the right

stretch tensor contains only one component and can be written as:

Up =E1 E1+ E2 E2+ pE3 E3; (35)

where E1, E2 and E3 are the material coordinate directions and p is a plastic stretch in the thickness

direction E3.

During the plastic straining, p begins to decrease and the plastic deformation gradient tensor will

satisfy the yield condition (Eq. (32)). Therefore, the plastic loading and unloading conditions are

dened through p as follows:

Loading condition

p 0 and p 1:

Unloading condition

p = 0:

5.3. Denitions of loading and unloading functions

Fig. 9 shows the four dierent congurations that are needed to dene the loading and unloading

with material softening and plastic strain. For a given deformation Fm, due to plastic deformation,

the material reference conguration 0 rst changes to the intermediate conguration 1. When

the material softens, then the intermediate conguration 1 changes to another conguration 2.

Therefore, the unloading stress can be calculated from F. The elastic unloading stress is dened

between the current conguration and the intermediate conguration 2. Cauchy stress is obtained

from the unloading potential WU as follows:

= 1

J

3i=1

i@ WU(

1;

2;

3)

@i(ni ni): (36)

Recall that any point on the loading stress curve is regarded as the yield stress y, which is a

function of mechanical strain. Therefore the yield stress is derived from the loading potential WLfor a given deformation gradient tensor Fm as follows:

y= 1

Jm

3i=1

mi@WL(

m1;

m2;

m3)

@mi(ni ni): (37)


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5.4. Elastic rate of deformation tensors

The relation between the total mechanical and elastic rate of deformation tensors (Lm and Le) can

be obtained by dierentiating Eq. (33) with respect to time:

Le =Lm p

pML; (38)

where

ML=

3i=1

3j=1

eiej

(N1i E3)(N1

j E3)ni nj: (39)

The rate of deformation gradient tensor De, which is the symmetric part of Le, can be obtained as

follows:

De =Dm p

pMD; (40)

where MD is the symmetric part ofML. The unknown rate of plastic stretch p can be obtained by

taking a time derivative of the yield function. This time derivative results in the following equation:

e3 e3

e3 ye3= B1p +B2= 0; (41)

where the constants B1 and B2 are given by (see Appendix D for details):

B1= 1

p

(T(4)ee MD I)(e3 e3)

1

Je3 P

(4)T(4)ee MDe3

2e3 (ML G(4)L ML)e3

; (42)

B2= 4e3 ( y)Dme3 (T

(4)ee D

m I)(e3 e3) +

1

Je3 P

(4)T(4)ee Dme3

+ (Dm I)(e3 ye3) 1

Jme3 P

(4)Y D

me3 2e3 G(4)L D

me3: (43)

In Eqs. (42) and (43), P(4) is dened as P(4)ijkm=F

eiPF

ejQF

ekRF

emSD

(4)PQRS and D

(4) is dened in the

equation S = D(4) Ee

. Also, P(4)Y is dened as P

(4)Yijkm

=FmiPFm

jQFmkRF

mmSD

(4)YPQRS

and D(4)Y is dened in the

equation Y=D(4)Y E

m. Y is the second PiolaKirchho stress for the yield stress. Therefore, p can

be substituted into Eq. (40) and De is obtained with respect to Dm:

De =

I(4)

1

B1pB2

Dm =T(4)emD

m; (44)

where the fourth order transformation tensor T(4)em is introduced for convenience.


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5.5. Including rate eects for loading processes

From the experimental results, it is seen that there are two kinds of rate eects in intumescent

mat material. One is a rate eect during the loading process and the other is a rate eect duringthe unloading process. To simplify the formulation of the rate dependency for loading processes, the

stress response during the loading process is assumed to be composed of a static yield stress due to

plastic strain and a rate dependent stress due to elastic strain. Thus, two rate eects are incorporated:

one associated with elastic unloading and a dierent one associated with the elastic portion of the

loading.

The viscous property constants associated with the unloading function are determined by whether

or not plastic deformation has occurred. The existence of plastic deformation is determined by

evaluating the static yield condition in Eq. (32). The viscoelastic material properties for the unloading

function obtained from the loading process are used when there is plastic deformation. When there

is no plastic deformation, then the viscoelastic material properties are switched to properties obtainedfrom the unloading function. The nal stress is obtained by adding the rate dependent stress to the

static stress. Considering that the mat material has virtually no tensile strength and is used almost

exclusively in compression, it is assumed that no rate eects are present if the static stress in the

thickness direction becomes slightly positive.

Referring to Fig. 9, the Jaumman rate of Cauchy stress

from the 2 to congurations can be

written as follows:

=E(4)De; (45)

where De is the rate of deformation gradient tensor and E(4) is the spatial elasticity tensor. It is

necessary to convert the spatial elasticity tensor dened for the to 2 congurations to one denedfor the to 1 congurations. By using the transformation tensor dened in Eq. (22), Eq. (45) can

be written in terms of De:

=E(4)T(4)ee De: (46)

If the spatial elasticity tensor is dened for the and 0 congurations, the following relation is

obtained through Eq. (44):

=E(4)T(4)ee T(4)emD

m : (47)

For the proposed visco-elasticity model,

=E(4)

totalDm and E

(4)total =visE

(4)T(4)ee T(4)em : (48)

The viscous term vis is dened in Eq. (10). If there is no plastic strain, then p is zero, and in Eq.

(48), T(4)em becomes the Identity tensor.

5.6. Numerical procedures for the plastic stretch ratio and the stress update

When the deformation gradient tensor is determined at each time increment, it is necessary to

decompose the strain into elastic strain and plastic strain by satisfying the yield function. Suppose


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that the plastic stretch ratio pn

has been obtained after a time step n and that pn+1

is the increment

at time step n+ 1. Then, to satisfy the yield criterion at time step n+ 1:

F(pn

+ pn+1

) = 0 (49)and

pn+1

=pn

+ pn+1

: (50)

Performing a Taylor series expansion of Eq. (49) and neglecting higher order terms results in the

following equation:

pn+1

= F(p

n

)

(@F(pn))=@p

; (51)

which is not a linear equation and will require linearization and iteration.

The elastic trial stress at time step n+1 can be calculated from the elastic deformation gradient

tensor by using the previous plastic deformation gradient tensor at time step n. The procedure issummarized as follows:

(i) assume the elastic deformation gradient tensor to be Fen+1

=Fmn+1

Fp1n

withp at the previous

time step n,

(ii) calculate the elastic trial stress by Eq. (36),

(iii) calculate the yield stress y by Eq. (37), and

(iv) check the yield condition by Eq. (32).

If yield occurs,

(i) update Fp

by the NewtonRaphson method until the yield condition Eq. (32) is satised,(ii) calculate by Eq. (36),

(iii) calculate pvis by using Eq. (8) with a=a

p and =

p, and

(iv) write the stress as = + pvis.

If no yield occurs,

(i) calculate evis by using Eq. (8) with a=ae and =

e and

(ii) write the stress as = + evis.

6. Model parameters and comparisons with experiments

The Ogden strain energy function employed by [27] is used as a base function for both loading

and unloading. The loading function, WL in Eq. (37) and the unloading function, WU in Eq. (36)

are therefore dened as follows:

WL(1; 2; 3) =2L

2L

L1 +

L2 +

L3 3 +

1

L(JLL 1)

; (52)

WU(1; 2; 3) =2U

2U

U1 +

U2 +

U3 3 +

1

U(JU U 1)

; (53)


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Z

X

Displacement loading

with specified speed

Contact between

mat and upper die

Rigid plate

Fig. 11. A schematic diagram of the cyclic compression simulation.

where J is the Jacobean of the deformation gradient tensor, are the principal stretches, and , ,

and are material parameters to be determined from experiments.

The constitutive model is coded in the ABAQUSTM user subroutine called UMAT. A single 8

node solid element is used for comparison with the monotonic and cyclic compression experiments.

The size of the element is 10 mm 10 mm 10 mm. The bottom surface is xed in the normal

direction, lateral surfaces are free, and compressive (normal) displacements are applied to the top

surface nodes of the element. Dierent compression speeds are used to compare with experimental

results.

During cyclic displacement control compression, if the mat response is very slow compared to the

machine response and/or if there is plastic deformation during the loading process, the specimen may

loose contact with the upper loading rod during unloading. This eect is simulated by employinga contact surface between the loading rod and top surface of the mat element. Fig. 11 shows a

schematic diagram for the cyclic compression simulation. Cyclic displacement is applied to the rigid

plate in the Z direction.

6.1. Determination of material parameters

In the proposed visco-hyper-elasto-plastic model, two dierent loading and unloading functions

are dened. As was explained in Section 5.7, the loading function has only a static term and the

unloading function has a static and a transient term. There are two material parameters in the static

term and two material parameters to dene the transient term.The loading function material parameters are determined from the static stress versus strain curve

shown in Fig. 3. The material parameters for the loading function (L and L) are obtained from

non-linear curve tting of the loading curve.

For the unloading function, as shown in Fig. 6,each static curve has a dierent strain at zero stress

and a dierent slope. From the idea of material similarity for each unloading function, all unloading

curves in Fig. 6 can be represented with one master curve by changing the softening parameter

h and by shifting the strain at zero stress to zero strain. Fig. 12 shows the master curve and the

unloading curves from dierent strains. As shown in Table 2, the softening parameter h decreases

with increasing strain. From the master curve shown in Fig. 12, the static material parameter U


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0

0.5

1

1.5

2

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35



Master curve

Unloading curve at 1.1MPa (h=1.0)Unloading curve at 1.6MPa (h=0.98)

Unloading curve at 2.22MPa (h=0.92)

Fig. 12. The master curve and unloading test data at dierent compressive stresses with material softening removed.

Table 2

The softening parameter h and the zero strain point for

unloading curves associated with three dierent maximum

applied mechanical strains

Mechanical strain 0.8 0.9 0.99Softening parameter h 1 0.98 0.92

Zero strain 0.532 0.6 0.65

and U are determined for the unloading function. In the elastic region, transient material parameters

ae and e are obtained from both the relaxation curve at the end of the 1st loading and the rate

dependent 2nd loading curves shown in Fig. 8. For the plastic deformation regime, transient material

parameters ap and

p are obtained from the transient stressstrain curves in Fig. 6 with the static

loading function.

6.2. Simulation results

After obtaining the necessary material parameters, simulations are performed to compare with

several cycles of the compression experiment results. Fig. 13 shows a comparison of the rate eects

during the 1st loading process up to 60% compression. It is seen that in all strain ranges except the

small strain region, there is good agreement between the experiments and simulations at each com-

pression speed. In the small strain region, the simulations show higher stress than the experimental

results, especially as ram speed increases. It appears that the mat material may have dierent rate

eects at dierent compressive strain levels.


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0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

Analysis(static)

Analysis(0.005mm/s)Analysis(0.4mm/s)

Analysis(1.6mm/s)

Test(static)

Test(0.005mm/s)

Test(0.4mm/s)

Test(1.6mm/s)



Simulation (static)

Simulation (0.005mm/s)Simulation (0.4mm/s)Simulation (1.6mm/s)

Experiment (static)

Experiment (0.005mm/s)



Fig. 13. Comparison of simulation results with experimental data at dierent compression speeds (60% compression).

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

AnalysisTest

Compressivestress(MP

a)


SimulationExperiment

Fig. 14. Comparison of the simulation and experimental static cyclic loading and unloading curves.

Fig. 14 shows a comparison between the static cyclic compression experiment presented in

Fig. 6 and the simulation. The simulations show good agreement for the dierent unloading

curves. Fig. 15 compares two cycles from the experimental and simulation results when 60% com-

pression is applied with 1:6 mm=s ram speed. The experimental loading and unloading curves for


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0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

1st cycle test2nd cycle test



1st cycle simulation

2nd cycle experiment1st cycle experiment2nd cycle simulation

Fig. 15. Comparison of simulations with the 60% cyclic, displacement control compression experiment for 2 cycles

(v= 1:6 mm=s).

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2nd cycle test

Compressivestress(MPa

)


1st cycle simulation

2nd cycle simulation

2nd cycle experiment

1st cycle experiment

Fig. 16. Comparison of two cycles of the 50% cyclic compression simulations and experiments for 2 cycles (v=1:6 mm=s).

the 1st and 2nd cycle are in good agreement with the simulation results. Fig. 16 shows a simula-

tion result for 50% compression with 1:6 mm=s ram speed. During the 1st loading, stress from the

simulation is higher than the experimentally determined stress. However, since the operating strain

range for this material is generally higher than 50%, the results are considered to be satisfactory for

the 1st and 2nd cycles.


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0

2

4

6

8

0 50 100 150 200 250 300

ExperimentSimulation


Time (seconds)

Fig. 17. Comparison of two cycles of stress versus time from the 60% compression simulation and experiments

(v= 1:6 mm=s).

Fig. 17 shows two cycles of stress variation with respect to time for 60% compression with

1:6 mm=s ram speed. The gure shows the stress jump during loading and the stress relaxation

during unloading. The peak value at each cycle and the stress relaxation with respect to time are

very well represented by the simulation. However, stress relaxation results from the 2nd cycle showsome discrepancies between experiments and simulations. Fig.18 shows comparisons of peak stresses

versus number of cycles from compression experiments with 50% and 60% strain applied under

1:6 mm=s ram speeds. For both cases, good agreement is seen through the 2nd cycle. Simulation

results show a constant stress level after the 3rd cycle, but experimental results show continued

stress decay with 60% compression data decaying substantially more than for 50% compression.

This may be due to cyclic material damage after plastic deformation. Therefore, incorporation of

cyclic material damage is believed to be necessary for better simulation of longer term cycling.

7. Concluding remarks

This paper presented detailed development of a constitutive model for intumescent mat material.

After starting with hyperelasticity theory, the model incorporated rate eects, plasticity, and material

softening during cyclic loading processes. The model is limited to the room temperature response of

materials that have large lateral to thickness dimension ratios and that are deforming primarily in the

thickness direction. The results of several experiments on intumescent mat were reviewed. A new

material softening model was proposed from the assumption of material similarity. The softening

parameter was determined as a function of the strain in the thickness direction. For the 1-D plastic

model, a new plastic deformation gradient tensor and yield criteria in the thickness direction were


25/31


0

2

4

6

8

2 4 6 8 10

Experiment (60%)Experiment (50%)Simulation (60%)Simulation (50%)

Peakcompressivestress(MPa

)

Number of cycles

Fig. 18. Comparison of simulated and experimental peak stress versus number of cycles ( v= 1:6 mm=s).

proposed. Also, the plastic yield stress function was established on the basis of strain. Two dierent

rate eects in the loading and unloading regions were implemented by employing dierent viscosity

parameters for the plastic and elastic unloading regions.

The complete spatial elasticity tensor in the Eulerian frame was derived for implicit nite elementcoding and the NewtonRaphson method was applied to obtain plastic strain. The theory was coded

in ABAQUSTM implicit using the user subroutine UMAT. Experimental results and simulations are

compared to each other for several experimental cases. For up to two cycles, satisfactory agreement

was achieved with the new model.

Further study is still required to accurately predict loading and unloading processes for many

cycles. It is believed that cyclic damage eects after the 2nd loading and dierent viscous eects at

dierent compressive strain levels should be considered in the future. In addition, future versions of

the model should attempt to incorporate elevated temperature material responses.

Although the new constitutive model developed in this paper has been applied to modeling intu-

mescent mat materials, the same modeling approach could be applied to rubber materials to accountfor Mullins eect (i.e., material damage due to cyclic loading). In addition, the softening model can

be applied to any anisotropic material to account for material property changes due to temperature.

Therefore, numerous applications of the new constitutive model are envisioned.

Acknowledgements

This work was nancially supported by ArvinMeritor Co. We gratefully acknowledge Bill Taylor,

Joe Fuehne, and Han-Jun Kim from ArvinMeritor for supporting this project and for their valuable


26/31


comments. We would also like to thank John TenEyck from Unifrax Corp. for supplying specimens

and for insightful discussions.

Appendix A. The Jaumann rate of Kirchho stress

The 2nd Piola Kirchho stress(PK2) can be related to the true stress by

=Je=FeSFeT

: (A.1)

The Jaumann rate of Kirchho stress can be obtained by dierentiating the Kirchho stress with

respect to time and is described with the rate of stretch tensor De:

=FeSFe

T

+ De +De: (A.2)

By using the following relations:

Ee =FeT

DeFe; (A.3)

Je =Jediv(C) =Je(De I); (A.4)

when S is represented as S=D(4) Ee, the Jaumann rate of Cauchy stress is obtained as

= (De I)+

1

Je(FeD(4)(Fe

T

DeFe)FeT

) + De +De: (A.5)

In tensor component form, Eq. (A.5) can be rewritten as follows.

ij = D

emmij+

1

Je(FeiPF

ejQF

ekRF

emSD

(4)

PQRSDekm) +ikD

ekj+ D

eikkj: (A.6)

New 4th order tensors H(4) and P(4) are dened as follows:

H(4)ijab=

12

(iabj+ iabj+ ibaj+ ibaj) (A.7)

P(4)ijkm= F

eiPF

ejQF

ekRF

emSD

(4)

PQRS: (A.8)

The Eulerian tangential stiness tensor E

(4)

dened in the relation

=E

(4)

D

e

can be written as

E(4) = I+ 1

JeP(4) +H(4): (A.9)

Appendix B. Intermediate derivations for Eqs. (20)(21)

Time dierentiation of Eq. (16) is written as follows:

L =Le FLhF1; (B.1)


27/31


where Fh, F and Fe are dened as:

Fh =e1

1

N11 N11+

e2

2

N12 N12+

e3

3

N13 N13; (B.2)

F =1n1 N11+

2n2 N

12+

3n3 N

13 ; (B.3)

Fe =e1n1 N11+

e2n2 N

12+

e3n3 N

13: (B.4)

Time dierentiation of Fh in Eq. (B.2) is rewritten as follows:

Fh =

3i=1

eii

N1i N

1i +

eii

N1i N1i +

eii

N1i N1i

: (B.5)

In Eq. (B.5), (ei =i) and N

1i are obtained as

eii

=

(1 h)

iei

(N1i N1i)

Ee eii

(ln ei )h (B.6)

N1i =

3j=1=i

2

(ei )2 (ej )

2(N1j N

1j )

EeN1i : (B.7)

Lh is obtained as follows from Lh = FhFh1

:

Lh

=

3i=1

(1 h)

(ei )2 NiiiiE

e

+

ij=12;13;23

ei

j i

ej

(ei )2 (ej )2(Njiji+ Njiij)

ei j +

(Nijji+ Nijij)

iej

Ee

3i=1

(ln ei )hN1i N

1i; (B.8)

where Ee =FeT

DeFe.

Therefore, FLhF1 in Eq. (B.1) is can be written as

FLhF1 =

3

i=1(1 h)niiiiD

e

3

i=1(ln ei )h(ni ni)

+

ij=12;13;23

ei

j i

ej

(ei )2 (ej )

2

eii

(nijji+ nijij) +ej

j(njiji+ njiij)

De: (B.9)

Appendix C. Intermediate derivations for Eqs. (24)--(26)

From Eq. (B.1), Le is written as follows:

Le =L +FLhF1: (C.1)


28/31


Eq. (B.6) can be rewritten with respect to E:

eii

=1

h 1 ei

(i)3 (N

1i N

1i) E

1

h

eii (ln

ei )h; (C.2)

where E

=FTDF. Therefore, from Eq. (C.1), Lh in Eq. (B.8) can be presented as

Lh =

3i=1

1

h 1

1

(i)2

Niiii E

+

ij=12;13;23

ei

j i

ej

(ei )2 (ej )

2

(Njiji+ Njiij)

ei

j

+(Nijji+ Nijij)

ie

j

Ee

3i=1

1

h2(ln i)hN

1i N

1i ; (C.3)

FLhF1 in Eq. (C.1) is obtained as follows:

FLhF1 =

3i=1

1

h 1

niiiiL

1

h2

3i=1

(ln ei )h(ni ni)

+

ij=12;13;23

ei

j i

ej

(ei )2 (ej )

2

eii

(nijji+ nijij) +ej

j(njiji+ njiij)

De: (C.4)

Also,

FLhF1 =

1

h 1

L

1

h2

3i=1

ln(i)ni nih: (C.5)

In this way Eq. (26) is obtained from Eqs. (C.5) and (24).

Appendix D. Intermediate derivations for Eqs. (41)--(43)

The rst half of rst term in Eq. (41) can be written as

e3 e3= e3 e3+ e3 e3+e3 e3: (D.1)

When E3 and e3 are related by the equation FmE3=re3, where r is stretch ratio, e3 is obtained as

follows:

e3=Lme3 [(e3 e3) D

m]e3: (D.2)

When is dened as = 1=JFSF

T, can be written as

= (D I)+L

+ L

T + 1

JP(4)D: (D.3)


29/31


Therefore, the 1st, 2nd and 3rd terms in Eq. (D.1) are given by

e3 e3= Lme3 e3 [(e3 e3) D

m](e3 e3); (D.4)

e3 e3= e3 Lme3 [(e3 e3) D

m](e3 e3); (D.5)

and

e3 e3= (T(4)ee D

e I)(e3 e3) + 2e3 L

e3+

1

Je3 P

(4)T(4)ee Dee3; (D.6)

and therefore Eq. (D.1) is written as follows:

e3 e3= 2e3 Lme3 2[(e3 e3) D

m](e3 e3) (T(4)ee D

e I)(e3 e3)

+ 2e3 Le3+ 1

J e3 P(4)T(4)ee Dee3: (D.7)

In the same way, the 2nd term in Eq. (41) is written as follows:

e3 ye3= 2e3 yLme3 2[(e3 e3) D

m](e3 ye3) (Dm I)(e3 ye3)

+ 2e3 Lmye3+

1

Jme3 P

(4)Y D

me3: (D.8)

In Eq. (D.7), P(4) is dened as P(4)ijkm=F

eiPF

ejQF

ekRF

emSD

(4)PQRS and D

(4) is dened in the equation

S = D(4) Ee

. Also, P(4)Y is dened as P

(4)Yijkm

=FmiPFm

jQFmkRF

mmSD

(4)YPQRS

and D(4)Y is dened in the equation

Y=D(4)Y Em. Yis the 2nd Piola Kirchho stress for the yield stress. Next, with the following relations:

Le =Lm

pML and L

=Le G(4)L L

e; (D.9a)

De =Dm

pMD and D

=T(4)ee De; (D.9b)

Eq. (41) can be expressed as follows:

e3

(

y)e3

=p

p

(T(4)ee MD I)(e3 e3)

1

Je3 P

(4)T(4)ee MDe3 2e3 (ML G(4)L ML)e3

+ 4e3 ( y)Dme3 (T

(4)ee D

m I)(e3 e3) +

1

Je3 P

(4)T(4)ee Dme3

+ (Dm I)(e3 ye3) 1

Jme3 P

(4)Y D

me3 2e3 G(4)L D

me3: (D.10)

Therefore the constants B1 and B2 are obtained as in Eqs. (43) and (44).


30/31


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