Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history

*Corresponding author.1Presently Research Scholar, deputed from CVRDE Avadi, Madras-600 054.

Finite Elements in Analysis and Design 31 (1999) 281—294

Estimation of temperature in rubber-like materials usingnon-linear finite element analysis based on strain history

S. Sridhar*,1, N. Siva Prasad, K.N. SeetharamuDepartment of Mechanical Engineering, Indian Institute of Technology, Madras-600036, India

Abstract

A finite element procedure for hyper-elastic materials such as rubber has been developed to estimate the temperaturerise during cyclic loading. The irreversible mechanical work developed in rubber has been used to determine the heatgeneration rate for carrying out thermal analysis. The evaluation of the heat energy is dependent on the strains. The finiteelement analysis assumes Green—Lagrangian strain displacement relations, Mooney—Rivlin strain energy density func-tion for constitutive relationship, incremental equilibrium equations, and Total Lagrangian approach and the stress andstrain of the rubber-like materials are evaluated using a degenerated shell element with assumed strain field technique,considering both material and geometric non-linearities. A transient heat conduction analysis has been carried out toestimate the temperature rise for different time steps in rubber-like materials using Galerkin’s formulations. A numericalexample is presented and the computed temperature values for various load steps agree closely with the experimentalresults reported in the literature. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Rubber-like materials; Strain energy; Assumed strain field; Non-linear finite elements; Temperature rise inrubber; Strain history

1. Introduction

A number of methods [1—6] have been proposed to evaluate the stresses and deformations inrubber-like material using non-linear finite element formulations. Very little work has beenreported for the estimation of temperature build-up in rubber during cyclic loadings [7—9]. Inmany applications, rubber parts are subjected to cyclic loading and it is essential to predict thestresses and temperatures to avoid premature failure and to estimate the service life of rubber parts.

0168-874X/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reservedPII: S 0 1 6 8 - 8 7 4 X ( 9 8 ) 0 0 0 6 4 - X

Since rubber-like materials are constitutively non-linear and undergo large deformations,it is necessary to consider the material and geometric non-linearities in the finite elementformulations. Material constitutive relations can be obtained by employing the strainenergy density function [10—12]. Mooney—Rivlin form of strain energy density is widelyused, because of its closer approximation to the actual behaviour. The incompressibility con-straint of rubber can be imposed in the Mooney—Rivlin form using the Lagrangian multipliermethod.

The degeneration concept of formulating the general shell element has been adopted [13] toanalyse non-linear media due to its simplicity and efficiency. The degenerated shell element for thinshells exhibits locking behaviour and hence reduced and selective integration is used [14—17].Locking and mechanisms produce spurious energy modes, when coarse mesh is employed. Toavoid the above problems, an assumed strain-field technique is adopted, based on the enhancedinterpolation technique [18,19].

In this paper, a finite element formulation for hyper-elastic materials is presented. Materialnon-linearity has been considered by adopting the Mooney—Rivlin strain energy density function.This accounts for the incompressible behaviour of different types of rubber materials. Theincompressibility is accounted for by an additional constraint equation which has been incorpor-ated by the Lagrangian multiplier method. For geometric non-linear formulation, a Total Lagran-gian approach is chosen and a nine-noded degenerated shell element using Green—Lagrangianstrain—displacement is considered to account for large deformations. To overcome locking prob-lems, assumed strain field technique is employed. The displacement convergence criteria is used inthe formulations.

The temperatures are evaluated based on the strain energy in the domain using thestress and strain values obtained from the analysis. The correctness of the estimated temper-atures is predominantly governed by the accuracy of calculated strain and stress values. Theenergy loss which is converted into heat is represented by the strain energy imposed on therubber material, called as damping. The energy dissipated in rubber material is obtainedas a function of strain amplitude and temperature-dependent loss modulus. From the energydissipated,the heat generation rate is calculated taking the volume of the domain and the fre-quency into consideration. A transient heat conduction analysis is then carried out using theheat generation rate and the material properties of rubber-like density and specific heat toestimate the temperature build-up for various time steps when rubber is subjected to cyclicloading.

The performance of this model is validated by comparing the numerical values with the field dataavailable in the literature. This methodology can be employed for any rubber-like material and thetemperature rise can be estimated when the loads are acting with different frequencies andamplitudes.

2. Finite element formulation

The following assumptions have been made in the finite element formulations for the degen-erated shell element.

282 S. Sridhar et al. /Finite Elements in Analysis and Design 31 (1999) 281—294

Fig. 1. (a) Nodal, global and curvilinear coordinate systems. (b) Local coordinate systems.

2.1. Assumptions

(i) Normals to mid-surface always remain straight.(ii) The normal stresses in the thickness direction is constrained to zero.(iii) Transverse shear deformation is taken into consideration (Mindlin theory).

The degrees of freedom at a nodal point are displacements u,v,w in the global directions x,y,z andtwo normal rotations b

1and b

2as shown in Fig. 1a and b.

2.2. Co-ordinate systems

The four co-ordinate systems employed in degenerated shell element shown if Fig. 1a and b aregiven below.

2.2.1. Global Cartesian co-ordinate system: (x,y,z)Using this co-ordinate system, the geometry of the structure is defined. Nodal co-ordinates and

displacements, as well as the global stiffness matrix and applied force vector are referred to thisco-ordinate system.

2.2.2. Nodal co-ordinate systemA nodal co-ordinate system is defined at each nodal point with origin at the reference surface

(midsurface). The vector »1k

is constructed from the nodal co-ordinates of top and bottom surfaces

S. Sridhar et al. /Finite Elements in Analysis and Design 31 (1999) 281—294 283

at node k,

»3k"X501

k!X"05

k, (1)

where

xk"[x

kyk

zk]T. (2)

The vector »1k

is perpendicular to »3k

and parallel to the global xz-plane.The vector »

2kis perpendicular to the plane defined by »

-kand »

3k.

2.2.3. Curvilinear co-ordinate system — (m,g,f)In this system m and g are two curvilinear co-ordinates in the middle plane of the shell element

and f is a linear co-ordinate in the thickness direction. It is assumed that m, g and f vary between!1 and #1 on the representative faces of the element. The f direction is only approximatelyperpendicular to the shell mid-surface.

2.2.4. Local co-ordinate system: (x@,y@,z@)This Cartesian system is defined at the sampling points wherein stresses and strains are to be

calculated. The direction x@3

is perpendicular to the surface f"constant. The direction x’ istangential to the m direction at the sampling point. The direction x@

2is defined as the cross product

of the other two directions. This local co-ordinate system varies along the thickness for any‘normal’ with this variation depending on the shell curvature and variable thickness.

The direction cosine matrix [h] which relates the transformations between the local and theglobal co-ordinate system is defined by

[h]"[xA yA zA], (3)

where the components of the above matrix are unit vectors along the respective axes in theco-ordinate system.

2.3. Assumed strain field

In general, degenerated shell elements exhibit locking. To overcome this difficulty, assumedstrainfield technique is employed. The membrane and shear strain terms and evaluated usingassumed strain field, by which locking is eliminated.

Fig. 2 shows a typical element, where a,b,c,d,e and f represent appropriate sampling points.The sampling points are used to evaluate the conventional strain displacement matrix termsto serve as a basis for calculating the assumed strains at the nine Gaussian points. Let R

1—R

6be

the assumed strain interpolation functions at the points a to f, respectively. Let A[1] to A[6] bethe conventional strain displacement matrix terms at the six sampling points. Now for gettingthe assumed strain displacement matrix term at a typical Gaussian point (m*,g*) the followingexpression is used:

AM (m*,g*)"6+k/1

Rk(m*,g*)A[k]. (4)


Fig. 2. Assumed strain field.

2.4. Linear strain terms using assumed strain field

2.4.1. Membrane strainThe flexural strains can be decomposed into membrane and bending strain terms. Here the

membranae strain terms are evaluated in local co-ordinate system using the expressions

eNmx{x{

"

3+i/1

2+j/1

Pj(m)Q

i(g)eij

mx{x{(in the x direction), (5)

eNmy{y{

"

3+i/1

2+j/1

Pj(g)Q

i(m)eij

my{y{(in the y direction), (6)

12eNmx{y{

"

12

3+i/1

2+j/1

Pj(m)Q

i(g) eij

mx{y{(in the x direction), (7)

12eNmx{y{

"

12

3+i/1

2+j/1

Pj(g)Q

i(m)eij

mx{y{(in the y direction), (8)

where

P1(z)"

z2bA

zb#1B, P

2(z)"1!A

zbB

2, P

3(z)"

z2bA

zb!1B,

Q1(z)"

12A1#

zaB, Q

2(z)"

12A1!

zaB

and the terms eN ijmx{x{

,eN ijmy{y{

,eN ijmx{y{

are the membrane strains evaluated from the displacement field,a"3'M!1/2N and b"1 as shown in Fig. 2.


2.4.2. Transverse shear termsThe transverse shear strains are evaluated in the natural co-ordinate system using the assumed

strain field given by the expressions

cN mf"3+i/1

2+j/1

Pj(m)Q

i(g)cijmg, (in the m direction), (9)

cN gf"3+i/1

2+j/1

Pj(g)Q

i(m)cijmg, (in the g direction). (10)

The higher-order strain terms using assumed strain field can be evaluated employing the aboveprocedure.

3. Material non-linearity

3.1. Basic concepts

Rubber is a hyper-elastic material, which indicates that the stresses are derivable from thestrain-energy function. It further implies that the stresses induced in the component are notdependent on the manner in which it has been loaded. Also the component recovers to its originalshape on removal of applied load. Auxiliary stresses and strains have to be defined to account forthe continuous change of geometry even for normal loading condition so that at any time thevirtual work can be evaluated with reference to a known configuration. The strain energy fora hyper-elastic material is defined as the energy stored per unit original volume and will be denotedby the symbol ‘¼’. It is a function of invariants (I

1,I2,I3) and the Cauchy deformation tensor (C)

and may therefore be written as

¼"¼(I1,I2,I3). (11)

For an incompressible material such as rubber, the strain-energy function should be independent ofI3

according to [10].Therefore Eq. (11), for an incompressible material can be written as a power series in I

1and

I2

only as

¼"

=+l/0

=+

m/0

Clm

(I1!3)l(I

2!3)m with C

00"0. (12)

The Mooney—Rivlin law is the widely used form of the above equation which is given as

¼"C1(I

1!3)#C

2(I

2!3). (13)

The Mooney—Rivlin law is adopted in the present analysis. The values of the material constants(C

1,C

2) can be experimentally found for different types of rubber materials.

The constrained functional is added to the strain energy function [20] to account for theincompressibility and hence

¼M "¼(eij)!j(JI

3!1), (14)


where eij

is the total strain tensor and j is a Lagrangian multiplier. Hence the stress tensor isevaluated taking into account that I

3"1 at each point, as

Sij"

L¼MLe

ij

"

L¼Le

ij

!

12

jLI

3Le

ij

. (15)

3.2. Definition of stresses

For degenerated shell structures it is assumed that the stress normal to the thickness canbe neglected and the constrained functional and the stresses can be written in a more convenientway. It is assumed that S

ij"S

jiand e

ij"e

jiand the strain invariants (I

1,I2,I3) can be written

as [21]

I1"3#2J

1, (16)

I2"3#4J

1#4J

2, (17)

I3"1#2J

1#4J

2#8J

3, (18)

where

J1"e

11#e

22#e

33, (19)

J2"e

11e22#e

22e33#e

11e33!1

2(e212#e2

13#e2

23), (20)

J3"e

11e22

e33#1

4e12

e13

e23!1

4(e11

e223#e

22e213#e

33e212

). (21)

Hence constrained functional (14) can be written as

¼M "(2C1#4C

2)J

1#4C

2J2!jM IM , (22)

jM "2j, (23)

IM"J1#J

2#4J

3. (24)

By assuming S33

as zero the Lagrangian Multiplier jM at each point can be calculated as

jM "(2C

1#4C

2)(LJ

1/Le

33)#4C

2(LJ

2/Le

33)

(LJ1/Le

33)#2(LJ

2/Le

33)#4(LJ

3/Le

33). (25)

Since e33

is dependent on other strains, it can be evaluated from incompressibility constraintequation which is given as

I3!1"0. (26)

Therefore, from Eq. (15) the stresses in rubber material is defined as

Sij"(2C

1#4C

2)LJ

1Le

ij

#4C2

LJ2

Leij

! jMLIMLe

ij

. (27)


4. Geometric non-linearity

4.1. Basic concepts

ui"

n+k/1

Nkdik, (28)

which N,

are the shape functions of the degenerated shell element, and dik

are the element nodaldisplacements.

Fig. 1a and b represents a degenerated shell element with different co-ordinate systems.The total strain

ei"e

iL#e

iNL, (29)

where eiL

are the linear strains and eiNL

are the non-linear strains.

eiL"B

ijLdj, (30)

eiNL"B

ijNLdj. (31)

It may be noted that for the degenerated shell elementi"1,2,5 and j"1,2,45, BijL

is the non-linear strain displacement matrix, B

ijNLthe non-linear strain displacement matrix and d

+the nodal

displacements.

4.2. Definition of strains (e)

The strain components are defined in the local co-ordinate system. This is also the mostconvenient way of expressing the stress components and their resultants for the shell analysis. Thefive significant strain terms are

e@x

e@y

cx{y{

cx{z{

c@y{z

"

u@x@

vy@

u@y@

#

v@x@

u@z@

#

w@x@

vz@

#

wy@

#

12GA

u@x@B

2#A

v@x@B

2#A

w@x@B

2

H12GA

u@y@B

2#A

v@y@B

2#A

w@y@B

2

HAu@x@B A

u@y@B#A

v@x@B A

v@y@B#A

w@x@B A

w@y@B

Au@x@B A

u@z@B#A

v@x@B A

v@z@B#A

w@x@B A

w@z@B

Au@y@B A

u@z@B#A

v@y@B A

v@z@B#A

w@y@B A

w@z@B

, (32)

where u@,v@,w@ are the displacement components in the local co-ordinate set X@i.

The displacement gradients in the x@ and y@ directions are evaluated in their respective directionsin the local Cartesian co-ordinate system.


Fig. 3. Damping loop of rubber.

The strain obtained from the above finite element formulations have been used to estimate thetemperatures in the rubber material when it is subjected to cyclic loading. The methodology forestimating the temperature has been explained below.

5. Estimation of temperature

5.1. Evaluation of energy dissipation using strain amplitude

A part of the total strain energy is converted into absolute damping as shown in Fig. 3.‘Damping’ is the difference between deforming work and elastic recovery [21]. The energy loss iscaused by internal friction and is determined by the loading and unloading phases of a processwhich is shown in Fig. 3. Damping due to surface friction between the rubber and a mating surfacemay be present. But there is also internal friction within the elastic body itself, called materialdamping. In both the cases the work of damping is changed into heat, which thus accounts for theenergy dissipated in the cyclic loading. In Fig. 3 the area of the loop º

1!º

2is a measure of

damping and is equal to the energy loss per oscillation and is called absolute damping. The areaº

1is the total strain energy. The energy dissipated per unit volume per loading cycle in the

material E is calculated by

E"PG11e20, (33)

where e0

is the strain amplitude experienced during cyclic loading and G11 is the loss modulus.


This energy loss E is converted into heat generation QQ .

QQ "El, (34)

where l is the frequency of loading.

5.2. Galerkin+s approach for transient analysis

The temperature rise was estimated due to the dissipated strain energy. For the transientanalysis, the temperature distribution was assumed as two-dimensional and there is no variationin the temperature profile along the length. These assumptions were made as the load onan average is treated as uniform at all cross sections considering the entire cycle in calculatingthe temperature rise. The two-dimensional heat conduction problem was solved using finiteelement method to find out equilibrium temperatures assuming a constant heat input equivalentto the strain energy.

The governing equation is given by

k+ 2¹#QQ "oc¹t

, (35)

where k is the thermal conductivity, QQ the internal heat generation, o the density of rubber, c thespecific heat of rubber, ¹ the temperatures and t the time.

The approximate solution to this governing differential equation is assumed to be

¹*(x,y)"n+i/1

Ni(x,y)¹

i, (36)

where Niis the shape function (interpolating function at a particular point), ¹* the temperature at

the particular point and ¹ithe temperature at the nodes.

Galerkin’s method is followed in the present analysis. The residue R(x,y) due to the approxima-tion in ¹ is given as

!oc¹*t

#k+2¹*#QQ "R(x,y). (37)

Weighting this residue with respect to shape function,

PPNiG!oc¹*t

#k+ 2¹*#QQ H dx dy"0. (38)

The following boundary conditions are applied(i) ¹ ¹

b— at specified nodes.

(ii) Heat loss due to convection on surface

!k¹n

"h(¹*!¹f), (39)

where h is the convective heat transfer coefficient and ¹f

the ambient temperature. The initialcondition of the rubber component is assumed as room temperature. Now assuming a three noded


Fig. 4. Track pad and related track parts.

triangular element, the final form of the Eq. (38) can be represented as

[C]M¹Q N#[K]M¹N"M f N, (40)

where

[C]"PP ocNiN

jdx dy"Thermal capacitance, (41)

[K]"PA

[B]T[D][B]t dA#Pl

h[N]T[N]t dl"Thermal ‘stiffness’, (42)

M f N"PA

QQ [N]Tt dA!Pl

q[N]Tt dl#Plh

h¹f[N]Tt dl"Heat load vector (43)

q-heat flux.In the above equations, the terms [B] and [D] are the convectional temperature gradient matrix

and material constants matrix respectively for three noded triangle element.Hence the temperature build-up in the rubber material can be found using the Eq. (40).

6. Numerical example

A tank track pad subjected to cyclic loading given in [7] has been studied using the aboveformulations and the temperature build-up has been evaluated for various time steps. The profilegeometry of the track and rubber pad is shown in Fig. 4. The bottom of the track is fitted withrubber pad which is in contact with ground. The weight of the tank is distributed to the tracks andpads through road wheels. The rubber and experiences loading and unloading as the track moveswith the rotation of wheels to complete a cycle during the movement of the tank. The system isrestrained in the lateral surfaces. The following data have been considered:

Type of rubber Styrene Butadiene rubberLoad per unit area 3400 kN/m2


Fig. 5. Loss modulus vs. temperature.

Fig. 6. Temperature—time response of a pad.

Frequency 4.14HzDensity 1160 kg/m3Specific Heat 1420 J/kg°CPoisson’s ratio 0.48Constant C

1870 kPa

Constant c2

185 kPaThermal conductivity of the rubber 0.32 W/m°CHeat transfer coefficient 17 W/m2°C

The loss modulus varies with temperature [7] as shown in Fig. 5. At each time step, temperaturesare calculated and loss modulus is updated.


In the present case, study, the frequency of Eq. (34) is arrived at using formula l"nw/t

c, where

nw

is the no. of wheels loading the pad in one cycle and tcis the time taken for the pad to complete

one cycle.The results from the present analysis are shown in Fig. 6. The temperatures are estimated for

various time steps with the speed of the tank set out 30 kmph and are compared with the Ref. [8].From the figure it can be seen that the results are in close agreement.

7. Conclusion

In this paper, a finite element procedure for hyper-elastic materials such as rubber has beendeveloped to estimate the rise in temperature due to cyclic loading. The rise in temperature ismainly dependent on the dissipation of energy in the material, caused due to the strains. Hence theaccuracy of estimation of temperature rise will depend upon the accuracy of calculating of strains.To achieve this, finite element formulations using a nine-noded degenerated shell element withboth geometric and material non-linearities have been presented considering the behaviour ofrubber. To eliminate the locking phenomena, assumed strain-field technique has been employed.This suits well for the estimation of strains. For the thermal analysis, a two-dimensional analysis issufficient due to its accuracy in evaluating the temperatures from the strain energy obtained in theprevious step. A transient heat conduction analysis has been carried out to estimate the temper-ature rise for different time steps in rubber-like materials by taking heat energy as input, usingGalerkin’s formulations. A numerical example is presented to compare the computed temperatureswith these reported in the literature. There is a close agreement between the present method and theexperimental results. Hence the method presented in this paper is reliable in estimating thetemperature rise in rubber-like materials when they are subjected to cyclic loading.

References

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Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history