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

Finite Elements in Analysis and Design 31 (1998) 85—98

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

S. Sridhar1, N. Siva Prasad*, K.N. SeetharamuDepartment of mechanical Engineering, Indian Institute of Technology, Chennai 600 036, 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 functionfor 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. ( 1998 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 deformationsin rubber-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].In many applications, rubber parts are subjected to cyclic loading and it is essential to predictthe stresses and temperatures to avoid premature failure and to estimate the service life of rubberparts.

Since rubber-like materials are constitutively non-linear and undergo large deformations, it isnecessary to consider the material and geometric non-linearities in the finite element formulations.

*Corresponding author.1Presently research scholar, deputed from CVRDE Avadi, Chennai 600 054, India.

0168-874X/98/$ — see front matter ( 1998 Elsevier Science B.V. All rights reservedPII: S 0 1 6 8 - 8 7 4 X ( 9 8 ) 0 0 0 5 1 - 1

Material constitutive relations can be obtained by employing the strain energy density function[10—12]. Mooney—Rivlin form of strain energy density function is widely used, because of its closerapproximation to the actual behaviour. The incompressibility constraint of rubber can be imposedin the Mooney—Rivlin form using the Lagrangian multiplier method.

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 the stress andstrain values obtained from the analysis. The correctness of the estimated temperatures is predomi-nantly governed by the accuracy of calculated strain and stress values. The energy loss which isconverted into heat is represented by the strain energy imposed on the rubber material, calleddamping. The energy dissipated in rubber material is obtained as a function of strain amplitudeand temperature-dependent loss modulus. From the energy dissipated, the heat generation rate iscalculated taking the volume of the domain and the frequency into consideration. A transient heatconduction analysis is then carried out using the heat generation rate and the material properties ofrubber-like density and specific heat to estimate the temperature build-up for various time stepswhen rubber is subjected to cyclic loading.

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 and amplitudes.

2. Finite element formulation

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

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).

86 S. Sridhar et al. /Finite Elements in Analysis and Design 31 (1998) 85—98

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

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

1and b

2as shown in Fig. 1a and b.

2.2. Coordinate systems

The four coordinate systems employed in degenerated shell element shown in Fig. 1a and b aregiven below.

2.2.1. Global Cartesian coordinate system (x, y, z)Using this coordinate system, the geometry of the structure is defined. Nodal coordinates and

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

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

(mid-surface). The vector V1k

is constructed from the nodal coordinates of top and bottom surfaces

S. Sridhar et al. /Finite Elements in Analysis and Design 31 (1998) 85—98 87

at node k,

V3k"X 501

k!X "05

k, (1)

where

xk"[x

kykzk]T. (2)

The vector V1k

is perpendicular to V3k

and parallel to the global xz-plane. The vector V2k

isperpendicular to the plane defined by V

1kand V

3k.

2.2.3. Curvilinear coordinate system (n, g, f)In this system m and g are two curvilinear coordinates in the middle plane of the shell element

and f is a linear coordinate 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 coordinate 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. Thedirection x@ is tangential to the m direction at the sampling point. The direction x@

2is defined

as the cross product of the other two directions. This local coordinate system varies along thethickness for any “normal” with this variation depending on the shell curvature and variablethickness.

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

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

where the components of the above matrix are unit vectors along the respective axes in thecoordinate system.

2.3. Assumed strain field

In general, degenerated shell elements exhibit locking. To overcome this difficulty, assumedstrain field technique is employed. The membrane and shear strain terms are 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. Thesampling points are used to evaluate the conventional strain—displacement matrix terms to serve asa basis for calculating the assumed strains at the nine Gaussian points. Let R

1—R

6be assumed

strain interpolation functions at the points a—f, respectively. Let A [1] to A [6] be the conventionalstrain—displacement matrix terms at the six sampling points. Now, for getting the assumedstrain—displacement matrix term at a typical Gaussian point (m*, g*) the following expression isused:

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

membrane strain terms are evaluated in local coordinate 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)

12

eNmx{y{

"

12

3+i/1

2+j/1

pj(m)Q

i(g) eij

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

12

eNmx{y{

"

12

3+i/1

2+j/1

pj(g)Q

i(m)eij

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

where

P1(z)"

z2b A

zb#1B, P

2(z)"1!A

zbB

2, P

3(z)"

z2b A

zb!1B,

Q1(z)"

12 A1#

zaB, Q

2(z)"

12 A1!

zaB,

and the terms eN ijmx{x{

, eN ijmy{y{

, eN ijmx{y{

are the membrane strains evaluated from the displacement field,a"3~1@2 and b"1 as shown in Fig. 2.


2.4.2. Transverse shear termsThe transverse shear strains are evaluated in the natural coordinate 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 the strainenergy function. It further implies that the stresses induced in the component are not dependent onthe manner in which it has been loaded. Also the component recovers to its original shape onremoval of applied load. Auxiliary stresses and strains have to be defined to account for thecontinuous change of geometry even for normal loading condition so that at any time the virtualwork can be evaluated with reference to a known configuration. The strain energy for a hyper-elastic material is defined as the energy stored per unit original volume and will be denoted by thesymbol “¼”. It is a function of invariants (I

1, I

2, I

3) and the Cauchy deformation tensor (C) and

may therefore be written as

¼"¼(I1, I

2, I

3). (11)

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

according to Ref. [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)


eij

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

3"l 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 can beneglected and the constrained functional and the stresses can be written in a more convenient way.It is assumed that S

ij"S

jiand e

ij"e

jiand the strain invariants (I

1, I

2, I

3) 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

4(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 Eq. (14) can be written as

¼M "(2C1#4C

2)J

1#4C

2J2!jM IM , (22)

jM "2j, (23)

IM"J1#2J

2#4J

3. (24)

By assuming S33

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

jM "(2C

1#4C

2)LJ

1Le

33

#4C2

LJ2

Le33

LJ1

Le33

#2LJ

2Le

33

#4LJ

3Le

33

. (25)

Since e33

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

I3!I"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)

Nk

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 coordinate 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 element

i"1,2, 5 and j"1,2, 45,

where BijL

is the linear strain—displacement matrix, BijNL

the non-linear strain—displacement matrix,and d

jthe nodal displacements.

4.2. Definition of strains (e)

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

ex{

ey{

cx{y{

cx{z{

cy{z{

"

Lu@Lx@

Lv@Ly@

Lu@Ly@

#

Lv@Lx@

Lu@Lz@

#

Lw@Lx@

Lv@Lz@

#

Lw@Ly@

#

12G A

Lu@Lx@B

2#A

Lv@Lx@B

2#A

Lw@Lx@B

2

H12G A

Lu@Ly@B

2#A

Lv@Ly@B

2#A

Lw@Ly@B

2

HALu@Lx@BA

Lu@Ly@B#A

Lv@Lx@BA

Lv@Ly@B#A

Lw@Lx@BA

Lw@Ly@B

ALu@Lx@BA

Lu@Lz@B#A

Lv@Lx@BA

Lv@Lz@B#A

Lw@Lx@BA

Lw@Lz@B

ALu@Ly@BA

Lu@Lz@B#A

Lv@Ly@BA

Lv@Lz@B#A

Lw@Ly@BA

Lw@Lz@B

, (32)

where u@, v@, w@ are the displacement components in the local coordinate set x@i.

The displacement gradients in the x@- and y@-directions are evaluated in their respective direc-tions in the local Cartesian coordinate system.


The strains 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"P*G11*e20, (34)

where e0is the strain amplitude experienced during cyclic loading and G11 is the loss modulus. This

energy loss E is converted into heat generation QQ .

QQ "E*v, (35)

where v is the frequency of loading.

Fig. 3. Damping loop of rubber.


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 variation inthe temperature profile along the length. These assumptions were made as the load on an average istreated as uniform at all cross-sections considering the entire cycle in calculating the temperaturerise. The two-dimensional heat conduction problem was solved using finite element method to findout equilibrium temperatures assuming a constant heat input equivalent to the strain energy.

The governing equation is given by

k+2¹#QQ "ocL¹Lt

, (36)

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, (37)

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 approximation in ¹ is given as

!ocL¹*Lt

#k+2¹*#QQ "R(x, y). (38)

Weighting this residue with respect to shape function,

PPNiG!ocL¹*Lt

#k+2¹*#QQ H dx dy"0. (39)

The following boundary conditions are applied:(i) ¹"¹

"at specified nodes

(ii) Heat loss due to convection on surface

!kL¹Ln

"h(¹*!¹f), (40)

where h is the convective heat transfer coefficient, and ¹f

the ambient temperature.The initial condition of the rubber component is assumed as room temperature. Now assuming

a three-noded triangular element,The final form of the Eq. (39) can be represented as

[C]M¹Q N#[K]M¹N"M f N, (41)


where

[C]"PPocNiN

jdx dy is the thermal capacitance, (42)

[k]"PA

[B]T[D][B]t dA#P1

h[N]T[N]t dl is the thermal “stiffness”, (43)

Mf N"PA

QQ [N]Tt dA!Plq

q[N]Tt dl#Plh

h¹f[N]Tt dl is the heat load vector, and (44)

q is the heat flux.

In the above equations, the term [B] and [D] are the conventional temperature gradient matrixand material constants matrix, respectively, for a three-noded triangle element.

Hence, the temperature build-up in the rubber material can be found using Eq. (41).

6. Numerical example

A tank track pad subjected to cyclic loading given in Ref. [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 witha rubber pad which is in contact with the ground. The weight of the tank is distributed to the tracksand pads through road wheels. The rubber pad experiences loading and unloading as the trackmoves with the rotation of wheels to complete a cycle during the movement of the tank. The systemis restrained in the lateral surfaces.

The following data have been considered:

Type of rubber Styrene Butadiene rubberLoad per unit area 3400 kN/m2Frequency 4.0 HzDensity 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, temper-atures are calculated and loss modulus is updated.

In the present case study, the frequency of Eq. (35) is arrived at using the formula v"nw/t

c,

where n8

is the no. of wheels loading the pad in one cycle and t#

the time taken for the pad tocomplete one cycle.


Fig. 4. Track pad and related track parts.

Fig. 5. Loss modulus vs. temperature.

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


The results from the present analysis are shown in Fig. 6. The temperatures are estimated forvarious time steps with the speed of the tank 30 kmph and are compared with the Ref. [8]. Fromthe 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,the accuracy of estimation of temperature rise will depend upon the accuracy of calculating ofstrains. To achieve this, finite element formulations using a nine-noded degenerated shellelement with both geometric and material non-linearities have been presented considering thebehaviour of rubber. To eliminate the locking phenomena, assumed strain field technique has beenemployed. This suits well for the estimation of strains. For the thermal analysis, a two-dimensionalanalysis is sufficient due to its accuracy in evaluating the temperatures from the strainenergy obtained in the previous step. A transient heat conduction analysis has been carried out toestimate the temperature rise for different time steps in rubber-like materials by taking heat energyas input, using Galerkin’s formulations. A numerical example is presented to compare thecomputed temperatures with those reported in the literature. There is a close agreement betweenthe present method and the experimental results. Hence, the method presented in this paper isreliable in estimating the temperature rise in rubber-like materials when they are subjected to cyclicloading.

References

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