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Computational Mechanicshttps://doi.org/10.1007/s00466-020-01862-w
ORIG INAL PAPER
A thermo-mechanical material model for rubber curing and tiremanufacturing simulation
Thomas Berger1 ·Michael Kaliske1
Received: 25 November 2019 / Accepted: 5 May 2020© The Author(s) 2020
AbstractIn this contribution, the phase change of unvulcanized to vulcanized rubber is described by a thermo-mechanical materialmodelwithin the finite elementmethod (FEM) framework. Before the vulcanization process (curing), rubber exhibits an elasto-visco-plastic behaviour with significant irreversible deformations without a distinct yield surface. After exposing rubber tohigh temperature, the molecular chains build-up crosslinks among each other and its mechanical behaviour changes to stifferviscoelastic material. The proposed model assumes, that both phases are present during the vulcanization process. The ratiochanges from the uncured phase at the beginning to the cured phase according to the current state of cure. A constitutive curingformulation is introduced into the model, to capture the shape change during the vulcanization and to ensure, that the secondlaw of thermodynamics is fulfilled. A multiplicative split of the deformation gradient is assumed to describe incompressiblematerial. Thermal expansion due to the change of temperature is taken into account in the volumetric part, as well as shrinkageduring the vulcanization process. In the isochoric part, the phase change from elasto-visco-plastic to viscoelastic materialis described by micro-macro transition based on the micro-sphere model. The consistent formulation of the material modeland its tangent are important for a successful implementation into a three-dimensional finite strain FEM framework. Thecapabilities of the model are shown by the simulation of an axisymmetric tire production process starting at the green tireinserted into the heating press up to a post-cure inflation step.
Keywords Rubber curing · Uncured rubber · Tire production · Finite element
1 Introduction
Rubber is used in a large variety of applications in industry.With certain fillers like carbon black or sulfur, its prop-erties can be tuned for every specific application. In mostcases, rubber is used in its vulcanized state above the glasstransition temperature, where rubber has an elastic andincompressible behaviour. By far, the most important appli-cation for rubberlike elastomers are car tires. To predictthe performance of a tire, engineers have created a lot ofmaterial models describing its behaviour at specific load-ing including temperature- and time-dependencies.However,in recent years, the development of more powerful simula-tion techniques and computer hardware led to the interest
B Michael [email protected]
1 Institut für Statik und Dynamik der Tragwerke (ISD),Technische Universität Dresden, FakultätBauingenieurwesen, 01062 Dresden, Germany
in simulating the production process of the product and thelow crosslinked uncured rubber. Before rubber is exposedto the vulcanization temperature and the crosslinks of themolecular chains have not been built up, rubber is in itsuncured phase and has a soft and fluid like behaviour. In[1], a strain energy density function for uncrosslinked buta-diene rubber is proposed, where a polynomial strain energydensity function is fitted to shear and uniaxial experiments.The distinct plastic and viscous flow of the uncured mate-rial are not addressed in this formulation. A more advancedstep based on various experiments is presented in [2], whereuniaxial tension, relaxation and compression tests have beencarried out. They showed that plastic flowoccured even at lowstrain without a distinct yield surface. Also, rate-dependentbehaviour could be identified by a hysteresis at cyclic load-ing. They used a generalized Maxwell model to describingthe viscoelastic behaviour ignoring the plastic flow. A devel-opment for the representation of uncured rubber has beendone by [3], where the plastic flow and the viscoelastic partare decoupled from each other and considered separately.
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Computational Mechanics
The model is capable to represent the short-term mould-ing effects and they used an endochronic plastic evolutionlaw. For a more complex description, micro-macro transi-tion via the micro-sphere model [4] has been used, whichsplits the three-dimensional deformation into a finite num-ber of one-dimensional stretches. Thismaterialmodel is usedin the contribution at hand to represent the uncured rubberbefore vulcanization. The phase change during vulcanizationis addressed by few researchers. In [5], a small-strain modelis given,where the increased stiffness is introduced by addingadditional springs to the model. This represents the additionof crosslinks during curing, while the stress response of thenewly added springs is only affecting the current and futurestrain increment and not the previous strains. A continuousmodel could be achieved by defining the elastic modulusas definite time integrals. They extended the model to finitestrains [6] and to viscoelasticity [7]. While curing shrink-age is modelled by an additional element in the volumetricpart of the strain energy, the shape change during vulcan-ization is not addressed in the approach. In [8], a so-calledpseudoplastic strain is introduced that takes into account thepermanent strain during vulcanization. They introduced thatthe behaviour of rubber is highly dependent on the amountof crosslinks in the rubber. For predicting the crosslinks, twochemical reactions are proposed, first the raw material isreacting with a vulcanization agent and the resulting productis further reacting into a fully crosslinked material.
In [8], the amount of crosslinks is determined by a mov-ing die rheometer, where the reaction moment of the rubberduring vulcanization is measured and related to the state ofcure. Various researcher have been focused on describing theamount of cure over time at different temperatures. A widelyused model is presented in [9], where the isothermal curingkinetics are described. They evaluated a differential scanningcalorimeter to determine the amount of crosslinks over timeat constant temperature. They used the measured current andtotal heat input in the sample to experimentally observe thecuring reaction. A so-called induction period has been intro-duced in [10], whichmodels the temperature-dependent timethat elapsed before vulcanization starts. The model in [9] isdeveloped further for non-isothermal cases and with a tem-perature dependent power term in the equation, see [11,12].These models are then applied to a tire cross-section in themould, where only the heat transfer problem and the vulcan-ization process are addressed. Therefore, regions of over-or undercure could be identified neglecting the mechanicalprocess of in-moulding and the phase change due to vulcan-ization. In [13], the stream of hot steam into the bladder hasbeen depicted by computational fluid dynamics (CFD) sim-ulations, resulting in a precise prediction of the temperatureof the bladder that heats the tire. However, the fully thermo-mechanical simulation of a tire production process from theuncured green tire that is pressed into a mould, exposed to
heat to enforce vulcanization and phase change of the tire toobtain the final cured tire, has not been addressed so far.
In this contribution, a thermo-mechanical consistent frame-work is presented for the phase change of rubber from theuncured to the cured phase. It is assumed that the behaviouris mainly dependent on the amount of crosslinks inside andthat both phases contribute, dependent on the current state ofcure, to the overall representation of the material. The phasechange and the different geometries of the rubber before andafter vulcanization are addressed, while always fulfilling thesecond law of thermodynamics. To ensure that no additionalenergy is evolving in the model during vulcanization, a so-called curing strain is introduced. This permanent strain isirreversible and will lead to a new equilibrium configurationafter curing.
The paper is structured as follows. In Sect. 2, an evolutionlaw for the state of cure for a non-isothermal curing processis introduced based on the model from [9]. The equationsof motion in a continuum mechanical setting are presentedin Sect. 3, as well as the fundamental ideas of the materialmodel for the vulcanization process. A micro-sphere basedapproach and its implementation are discussed in Sect. 4.Its capabilities and the consistent derivation of the evolutionlaws are presented in Sect. 5, where the model is first appliedto simple numerical examples. The production process of atire from green tire to cured tire is shown and carried out bya finite element simulation.
2 State of curing
Vulcanization of rubber is assumed as crosslinking of the freemolecular chains with each other. In this section, a temper-ature dependent function is introduced, which describes theamount of crosslinks c in the rubber, assuming no crosslinksc = 0 for completely uncured rubber and amaximumamountof crosslinks c = 1 for fully cured rubber. Breaking ofcrosslinks is not considered and only monotonic increasingof the state of cure (SOC) is modelled. Above a certain tem-perature, at around 403K, the curing process starts and theamount of crosslinks increases. The vulcanization process ishighly temperature dependent. For larger temperatures, theprocess starts earlier and is faster than at lower temperatures.
Experimentally, this phenomenon can be captured via amoving die rheometer (MDR) test, see [8]. A rubber sampleis placed in the machine, where a rotational displacement isapplied at the top surface and the reaction moment is mea-sured at the bottom of the sample. During this test, the sampleis heated and vulcanization takes place. During vulcaniza-tion, the material becomes stiffer and the reaction momentincreases and reaches a maximum at the end of vulcaniza-tion. Via post-processing, the state of cure is computed asratio of the moment at the current time step divided by the
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maximum moment. Thus, it is possible to measure the timewhen vulcanization starts and also the velocity of the pro-cess at different temperatures. The rubber sample is smalland thin enough to assume constant and homogeneous tem-perature over the whole body. During the experiment, thetemperature of the sample will evolve over a certain time.It is assumed, that due to the small sample the rubber willheat quickly and will reach the temperature of the climatechamber immediately.
In [9], an empirical function for an isothermal curingprocess, that takes into account the rate- and temperature-dependent behaviour, is presented. This model represents thecuring process very well for an isothermal state. During themoulding and heating of a tire in the press via surface heatexchange, the temperature increases slowly and the processcannot be assumed isothermal. Therefore, a model is intro-duced, where the curing rate is computed by the current stateof cure
c = k cmtn(1 − ctn
)n. (1)
The temperature dependency of the curing evolution is rep-resented by an Arrhenius-type equation
k = k0 exp
(− E
R θ
). (2)
The gas konstant R is used and the parameters k0 and Ehave to be determined via an optimization from experimentalresults. In [11,12], a temperature dependent power term isproposed. The different rate of curing at the beginning of thevulcanization process is modelled via a linear temperaturedependency of the power term
m = m1 θ + m2. (3)
The new state of cure is then obtained by an update formulawith a predefined timestep Δt
ctn+1 = ctn + cΔt . (4)
Obviously, the rate of cure, Eq. (1), is zero when the state ofcure is equal to one, which means, the rubber is fully cured,but also when it is equal to zero. Therefore, assuming nocrosslinks at the beginning would lead to no evolution ofthe state of cure. It is assumed, that at the beginning of thevulcanization process, a small initial amount of crosslinks isalready present
c0 = 0.0002. (5)
This value represents a state of cure of 0.02% at the begin-ning, which is small enough to ensure, that the unvulcanized
(a)
00.10.20.30.40.50.60.70.80.91
0 500 1000 1500 2000 2500
c[−]
time [s]
sim 140° Csim 150° Csim 160° Cexp 160° C
(b)
00.10.20.30.40.50.60.70.80.91
0 500 1000 1500 2000 2500
c[−]
time [s]
sim 140° Csim 150° Csim 160° Cexp 160° C
Fig. 1 Numerically obtained state of cure over time compared to theexperimental data determined by a moving die rheometer at 160◦C fora a tire tread compound and b a rubber containing the tire belts
rubber model can still be applied and is large enough to startthe self-induced curing process. In Fig. 1, the simulated stateof cure over time is plotted versus the experimental resultsfrom an MDR test for two tire compounds. The curing timeand process can be captured nicely at the tested temperaturewith the proposed function and the behaviour at lower tem-perature is according to the results from the literature (Tables1 and 2). To show the capabilities at various temperatures, themodel is applied and compared to experimental results from[10]. The parameters are identified via optimization algo-rithm and are shown in Table 3. The state of cure is plottedover time for 4 different temperatures in Fig. 2. The tem-perature dependent curing model captures well the differentcuring kinematics at the tested temperatures.
3 Thermo-mechanically consistentvulcanization framework
During the production process of a tire, both phases ofthe rubber are important. The uncured phase applies duringmoulding in the press and the cured phase in the post curinginflation. Therefore, both phases should be considered ade-quately. A thermo-mechanical consistent formulation has to
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Table 1 Curing parameters used in the simulation for a tire tread com-pound
Gas constant R[
kg m2
s2 mol K
]8.3144598
Activation energy E[
Jgmol
]98,050.5806
Thermal parameter k0 [−] 1.473792 × 1010
Power term 1 n [−] 1.57
Power term 2 m1[ 1K
]0.0
Power term 3 m2 [−] 0.94
Table 2 Curing parameters used in the simulation for a rubber contain-ing the tire belts
Gas constant R[
kg m2
s2 mol K
]8.3144598
Activation energy E[
Jgmol
]104,450.5806
Thermal parameter k0 [−] 2.473792 × 1010
Power term 1 n [−] 1.1
Power term 2 m1[ 1K
]0.0
Power term 3 m2 [−] 0.5
Table 3 Curing parameters used in the simulation for a rubber contain-ing the tire belts
Gas constant R[
kg m2
s2 mol K
]8.3144598
Activation energy E[
Jgmol
]107,050.5806
Thermal parameter k0 [−] 36.673792 · 1010Power term 1 n [−] 1.6
Power term 2 m1[ 1K
]0.005
Power term 3 m2 [−] −1.21575
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 500 1000 1500 2000 2500 3000
c[−]
time [s]
sim 110° Csim 130° Csim 140° Csim 150° C
exp 110° Cexp 130° Cexp 140° Cexp 150° C
Fig. 2 Numerically obtained state of cure over time compared to exper-imental data, provided in [10]
be found to assemble both phases into one model in orderto be able to describe intermediate transition states as well.The main idea for the proposed material formulation is touse already developed and proven material descriptions andconnect them in a way, to fulfil the second law of thermody-namics and that a permanent curing strain is present.
3.1 Fundamental equations of motion
An initially undeformed bodyB0 is defined as reference con-figuration at time t0 = 0. The mapping ϕ(X, t) links everymaterial point from its initial positionX to its spatial positionx at time t
x = ϕ(X, t). (6)
The current deformation of one material point at time t > t0can be identified as difference of the current position and itsreference position
u = x − X. (7)
The deformation gradient F = Gradϕ(X, t) maps aninfinitesimal line element from the reference configurationto the spatial configuration
dx = F dX. (8)
The gradient operator Grad[·] denotes the spatial derivativeswith respect to the reference configuration and grad[·] withrespect to the current configuration. The volume change V/V0
of an infinitesimal small volume element dV0 is representedby the determinant of the deformation gradient
J = detF = V
V0. (9)
To ensure that no self-penetration results from the deforma-tion ϕ, the Jacobianmust always be positive J > 0. The rightand left Cauchy Green tensor are derived as
C = FT F, b = FFT (10)
by the deformation gradient. In rubberlike materials, incom-pressible behaviour is found. To enforce this feature, amultiplicative split of the deformation gradient into volu-metric and isochoric parts
F = Fvol F (11)
is achieved by
Fvol = J 1/31 (12)
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τvolτvol
U(J) αv αc
Fig. 3 Rheological representation of the volumetric material model:incompressible behaviour, thermal expansion and curing shrinkage areaddressed in the volumetric part
and
F = J−1/3 F. (13)
3.2 Free energy function and Kirchhoff stress
The free energy function of the model is split additively intoa volumetric part and an isochoric part
Ψ (F, θ, c, I) = U (J , θ, c) + Ψ (F, θ, c, I). (14)
The volumetric part U (J , θ, c) is a penalty term, whichenforces the incompressibility of the model. The volumetrictemperature expansion is associated with for the volumet-ric part as well as the curing shrinkage. The volumetric freeenergy function
U (J , θ, c) = κ(J − ln(J ) − 1)
+ κ αv (θ0 − θ) ln(J ) + κ c αc ln(J ) (15)
is represented by the rheological model depicted in Fig. 3.The bulkmodulus κ should be large enough to ensure incom-pressible behaviour. The volumetric thermal expansion ischaracterized by the parameter αv , which defines the vol-umetric change stemming from a temperature change of 1K.The last term of Eq. (15) describes the shrinkage due tovulcanization. The state of cure is monotonically increas-ing from c0 to 1, therefore, the maximum curing shrinkageis defined by αc.
It is assumed that during vulcanization some molecu-lar chains are entangled and build connections with otherswhile others can move freely. Therefore, the overall mate-rial response is represented by both phases depending on thecurrent state of cure c. The isochoric free energy function issplit further
Ψ (F, c, θ, I) = (1 − c) Ψ u(F, θ, Iu)+ cΨ c(Fm, θ, Ic), (16)
with a set of internal variables Iu and Ic, into the uncured andcured rubber part, respectively. In the cured rubbermodel, thedeformation gradient is decomposed further
F = Fm Fc. (17)
The permanent set due to curing is captured by the curingdeformation gradient Fc, with the evolution law presented inthe following section. In the undeformed and uncured stateno curing strain is present, Fc = 1.
The associated Kirchhoff stresses are as well additivelydecomposed into volumetric and deviatoric contributions
τ := τ vol + P : τ . (18)
In Eq. (18), the fourth-order deviatoric projection tensor isused
Pabcd = 1
2
[δac δbd + δad δbc
]− 1
3δabcd . (19)
The volumetric stresses are the negative pressure derivedfrom the penalty term of the free energy function
τ vol = J∂U (J , θ, c)
∂ J1. (20)
The deviatoric stresses are obtained from the isochoric partof the free energy function
τ = 2∂Ψ (F, θ, c, I)
∂g. (21)
The covariant reference metric tensorG and the current met-ric tensor g are introduced. These mappings are identifiedby the Kronecker delta in the Cartesian coordinate system,G = δAB and g = δab. These metrics are necessary forthe identification of the co- and contravariant objects in theLagrangian and Eulerian manifolds. According to Eq. (16),the isochoric stresses are further decomposed depending onthe state of cure
τ = (1 − c) τ u + c τ c. (22)
From Eq. (22), it can be seen that for a fully cured material,i.e. c = 1, the isochoric stresses are only derived from thecured free energy function Ψ c.
3.3 Evolution law of the curing strain
The basic underlying rheology of the framework is shown inFig. 4, where the uncured and cured material are representedby so-called black boxes. The main feature of the vulcan-ization is its change of the mechanical behaviour and that anew equilibrium shape is present. As discussed previously,when the state of curing increases, the ratio of the stresses andfree energy change from the uncured model to the cured one.For rubber-like materials, the curing process increases thestiffness of the material. For a thermo-mechanical consistentformulation, the free energy should not increase during the
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Computational Mechanics
Ψu
Ψc
(1 − c)
cdc
ττ
F
Fm Fc
Fig. 4 Thermo-mechanical framework of the proposed vulcanizationmodel, where any reasonable material model can be inserted for theuncured and cured rubber
vulcanization process. Therefore, a symbolic curing elementdc is included in the lower branch, which also representsthe shape change of the material during curing, see Fig. 4.Instead of defining an evolution law directly for the curingelement, an evolution law is derived from the conservationof energy. In other words, the partial derivative of the freeenergy function with respect to the state of cure always has
to be zero ∂Ψ∂c := 0. Evaluating this statement with the free
energy function Eq. (16) and the chain rule leads to
∂Ψ
∂c= Ψ c − Ψ u + c
∂Ψ c
∂bm
∂bm∂c
= 0. (23)
The evolution law for the curing strain dc can be obtainedby rearranging Eq. (23) and multiplying with the curing rateEq. (1) from the previous section
dc = −∂bm∂t
= −∂bm∂c
∂c
∂t= − (
Ψ u − Ψ c) cc
(∂Ψ c
∂bm
)−1
.
(24)
For a free energy function based on the first invariant, forexample Yeoh’s model introduced in [14], the first part iseasily obtained as a scalar multiplied by the identity tensor
∂Ψ c
∂bm= ∂Ψ c
∂ I 11 (25)
using the relation ∂ IA∂A = 1 from [15]. To project the curing
strain into the stress direction, the projection tensor
Np = τ c
|τ c| (26)
is introduced with |τ c| = √τ c : τ c. The final constitutive
evolution law for the curing strain is
−dc = 1
2Lv(bm)b
−1m = (
Ψ u − Ψ c) cc
(∂Ψ c
∂ I 1
)−1
Np.
(27)
The evolution law is introduced in a similar manner than in[16] using the Lie time derivative. The always positive curingrate and the assumption, that the free energy of the cured rub-ber is larger than the uncured, ensure, that the curing strainis monotonically increasing. For a fully cured material, thecuring rate is zero and the curing strain is not increasing fur-ther. If the free energy of the cured rubber is significantlylarger than the uncured, almost all of the deformation beforethe vulcanization process is transferred to the curing strain,resulting in a new equilibrium configuration. Obviously, Eq.(24) is not defined for a state of cure equal to zero. As statedin Eq. (5), a small amount of crosslinks is present even foruncured rubber and an initial state of cure c0 > 0 is intro-duced. The proposed framework for the representation ofthe vulcanization process can be used for any given mate-rial model of uncured and cured rubber. Parameters for bothphases can be identified independently by experiments thatare done before and after the vulcanization process.
3.4 Thermo-mechanical consistency
It is assumed, that all energy during the phase change is usedfor the evolution of the permanent set. Therefore, the heatgeneration due to rubber curing, for example investigated in[17], is neglected in this contribution. For a constant temper-ature, the second axiom of thermodynamics, the so-calleddissipation inequality, reads
D = τ : d − Ψ ≥ 0. (28)
The time derivative of the free energy function can beobtained by the product rule
Ψ = (1 − c)∂Ψ u
∂bb : d + c
∂Ψ c
∂bm: bm + ∂Ψ
∂c: c. (29)
Using the definition of the material time derivative with the
found evolution law bm = l bm + bm lT + Lc(bm), Eq. (28)is
D =[
τ − (1 − c)∂Ψ u
∂bb − c
∂Ψ c
∂bmbm
]
: d
− c∂Ψ c
∂bm:(1
2Lv(bm)b
−1m
)− ∂Ψ
∂c: c ≥ 0. (30)
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Computational Mechanics
The equation should be valid for every displacement rate and,therefore, the part in brackets is the definition of the stress, asstated in Eq. (22). The last part is equal to zero via definitionof the evolution law. The remaining part is
D = −c∂Ψc
∂ I1(Ψu − Ψc)
c
c
(∂Ψc
∂ I1
)−1
1 : Np ≥ 0 (31)
with Eq. (25). Due to the definition of the projection tensorand its trace 1 : Np = tr
(Np
) = 1, the dissipation inequal-ity is fulfilled only if the free energy of the cured rubber islarger than of the uncured rubber, which is one of the mainassumptions of the curing model. The model is based on theassumption of not changing free energy during curing, stillan always positive internal work is present while the internalvariable Fc is evolving. Due to the definition of the mono-tonic increasing state of cure c > 0, the dissipation is alwayspositive and the process is irreversible.
4 Vulcanizationmodel based on themicro-spheremodel
The linearization of the proposed evolution law is rather dif-ficult and can lead to a complex and numerically expensiveimplementation. Therefore, a micro-macro transition basedon the micro-sphere model is proposed and disscused in thissection. A three-dimensional deformation is represented bya number of uniaxial stretches pointing to the surface of adeformed micro-sphere. This approach simplifies the defor-mation state and even a more complex formulation of thematerial model can be used.
4.1 Kinematic description
The macroscopic deformation is linked to a single point ofthe micro-sphere via the Lagrangian unit orientation vector
r = cosϕ sinϑe1 + sinϕ cosϑ e2 + cosϑ e3. (32)
This vector with |r| = 1 and the Cartesian basis vectors e1,e2 and e3 points to the surface of the sphere, as illustrated inFig. 5. The spatial orientation vector of the deformed sphereis introduced as the mapping with the isochoric deformationgradient
t = F r. (33)
The stretch of such a material line element in the orientationdirection yields
λ = √t · t. (34)
Fig. 5 Micro-sphere model in spatial coordinates e1, e2 and e3
Homogenization over the unit sphere is done, where the ori-entation vectors are pointing to an infinitesimal small areadA = sinϑ dϕ dϑ . The area of a part of the micro-sphere isexpressed in polar coordinates
A(ϕ, ϑ) =∫ ϕ
0
∫ ϑ
0sinϑ dϕ dϑ. (35)
With the polar coordinates in the range of ϕ ∈ [0, 2π ] andϑ ∈ [0, π ], the total area of the sphere is |S| = 4π. Theaveraging operator is defined as
〈·〉 = 1
|S|∫
S(·) dA. (36)
Assuming affinity between themacroscopic andmicroscopicstretches, λ and λ, respectively, it holds
λ = λ. (37)
Therefore, the macroscopic free energy function can be iden-tified as
Ψ (F, θ, c, I) = 〈ψ(λ, θ, c,P)〉. (38)
The microscopic free energy is now a function of a one-dimensional stretch and a set of internal variables P . Thetemperature and the state of cure are constant over themicro-sphere, therefore, for every unit direction the sametemperature and state of curing are used for the microscopicfree energy function.
A discrete averaging operator is introduced for the numer-ical implementation of the model. The integral in Eq. (36) isreplaced by a number of unit direction vectors and the accom-panying weighting factors
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τ τ
(1 − c)
c
ψep
ψph
εp
ψve
εv
ψh εh
ψce
ψc1 εc1
εc
λλe λc
λce λci
Fig. 6 Combined material model
〈(·)〉 = 1
|S|∫
S(·) dA ≈
m∑
i=1
(·)iwi . (39)
A set of 21 directional vectors and corresponding weightingfactors are taken from [18,19] using the symmetry of thesphere. These values fulfil the isotropy condition and a stressfree state in the reference configuration.
4.2 Rheological model and free energy function
The proposed combined rheologicalmodel is shown in Fig. 6.For uncured rubber, the visco-elasto-plasticmaterialmodel in[3] is used and, for cured rubber, the viscoelastic generalizedMaxwell model is applied. Similar to the framework, seeFig. 4, a curing element is introduced in the cured rubberbranch. The curing strain has to be computed in every unitdirection of the micro-sphere model and saved as an internalvariable Pi . The microscopic free energy function is similarto Eq. (16) represented by a ratio of an uncured and a curedfree energy function
ψ(λ, θ,P) = fEQ(θ)[(1 − c) ψu(λ,Pu) + cψc(λ,Pc)
].
(40)
Anentropy elastic thermo-mechanical free energy is employedwith the temperature scaling function fEQ , which is adoptedfrom [16]. During the production process of a tire, the shortterm moulding process is not much effected by changingtemperature in the rubber. The external heat sources fromthe press will have a larger influence than the internal heatsources from the viscous dissipation of the material. Thetemperature-dependent mechanical behaviour affects the tireproduction process, as the hotter material has a lower vis-
cosity and flows into all edges of the mould. However, inthis contribution, an isothermal free energy function withfEQ = 1 is assumed for simplification. Note, that the tem-perature still has an influence on the evolution of cure and onthe volumetric expansion in Eq. (15).
4.2.1 Uncured rubber model
The model used for the uncured rubber is taken from [3]and is presented in short. As shown in [2], rubber before thevulcanization process shows very different behaviour thanafter curing. Without crosslinks of the molecular chains, theuncured rubber exhibits softer behaviour. The stress-strainresponse is highly nonlinear with rate-dependency of the ini-tial loading and kinematic hardening. A hysteresis can beobserved during cyclic loading and plastic flow is presenteven at small strains. A distinct yield surface cannot beidentified as well as a ground state elastic response. In theused material model, see Fig. 7, the rate-independent elasto–plastic representation is seperated from the rate-dependentviscoelastic part and it is assumed that these two features existindependently from each other. The plastic flow is defined asrate-independent with a so-called endochronic plasticity law.For quasi-static loading, the stress–strain response showskinematic hardening while plastic strain evolves, which ismodelled by an additional spring element in parallel to themodified endochronic dashpot. In the rate-dependent branch,a nonlinear spring is coupled to a single dashpot in par-allel to a Maxwell element. The single dashpot describesrate-dependency of the initial loading. The kinematic hard-ening part shows also rate-dependency, that the stress-strainresponse yields a larger slope for faster loading rates. Thisfeature is captured by the additional Maxwell element in par-allel to the dashpot, consisting of a nonlinear spring and anonlinear dashpot. The according free energy function of theuncured rubber model is split into rate-independent and rate-dependent parts
ψu(λ,Pu) = ψ p(λ,P p) + ψv(λ,Pv). (41)
A multiplicative split of the total stretch in the orientationdirection is introduced for the rate-independent branch
λ = λep λp. (42)
The logarithmic strains ε := ln(λ) are introduced and addi-tively decomposed into
ε = εep + εp. (43)
The rate-independent branch for the uncured rubber modelconsists of two storage functions for the elastic response andthe post-yield hardening part
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Computational Mechanics
ψ p(λ,P p) = ψep(λep) + ψph(λp). (44)
Generic power type expressions are used for the free energyfunctions
ψep(λep) = μep
[λ
δepep
δep− 1
δep− ln(λep)
]
, (45)
ψph(λp) = μph
δph
[λp − 1
]δph , (46)
where μep > 0, μph > 0, δep ≥ 2 and δph ≥ 2 are materialparameters. The free energy functions are always positive forevery stretch and also the conditionψ(λ = 1) = 0 is fulfilled.The stresses are defined as the derivatives with respect to thelogaritmic strains and stretches
σp = ∂ψep
∂εand τ p = ∂ψep
∂λ= σp
λ. (47)
The evolution of the internal variable for the plastic strain isdefined with the power-type evolution equation
εp = γp(βp)βp, (48)
with γp = zηp
∣∣∣βp
∣∣∣mp
and the thermodynamical driving force
βp = −∂ψ p
∂εp. (49)
The evolution of the arclength z = |ε| ensures a smoothhysteresis and an always positive plastic strain, which is notaffected by the loading rate. For more information about thespecific plasticity approach, the reader is referred to [3,20].
Similar to the rate-independent branch, the total stretch isdecomposed into elastic and viscoelastic stretches
λ = λve λv, (50)
where the viscoelastic stretch λv is further decomposed into
λv = λeh λh . (51)
The according logarithmic strains are additively decomposedas
ε = εve + εv (52)
and
εv = εeh + εh . (53)
τ τ
ψep
ψh
εp
ψve
εv
εh
ψph
λpλep
λ
λ
λv
λhλeh
λve
Fig. 7 Uncured rubber material model and definition of the stretches,introduced in [3]
The free energy function of the rate-dependent branch isdefined in a similar manner with two storage functions
ψv(λ,Pv) = ψve(λve) + ψh(λeh). (54)
Generic power-type expressions, which are always positiveand zero for no deformation, are employed
ψve(λve) = μve
[λ
δveve
δve− 1
δve− ln(λve)
]
, (55)
ψh(λeh) = μh
δh[λeh − 1]δh . (56)
The viscous strains evolve defined by the following evolutionlaws
εv = γv(βv)βv with γv(βv) = 1
ηv
∣∣∣βv
∣∣∣mv
, (57)
εh = γh(βh)βh with γh(βh) = 1
ηh|βh |mh . (58)
The thermodynamical driving forces are introduced as thederivatives with respect to the logarithmic strains
βv = −∂ψv
∂εv
and βh = −∂ψh
∂εh. (59)
4.2.2 Cured rubber model
For the cured rubber material, the generalized Maxwellmodel is used, that captures well the viscoelastic proper-ties of the cured rubber. The single nonlinear spring depicts
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Computational Mechanics
long term elasticity that can be observed if the rubber isstretched and held until relaxation is finished.Afinite numberof Maxwell branches can be connected in parallel to modela wide range of frequencies. In this contribution, only onebranch is considered to show the basic capabilities. How-ever, an extension of the model is possible without mucheffort. The overall stretch is again decomposed into curingand elastic stretch and their logarithmic strains
λ = λe λc, ε = εe + εc. (60)
For the Maxwell branch, the elastic stretch is further decom-posed into elastic and inelastic stretches
λe = λce λci , εe = εce + εci . (61)
The free energy functions for the cured rubber branch is
ψc(λe,Pc) = ψce(λe) + ψci (λce). (62)
For both springs, the same power type function is usedmulti-plied by a scaling factor γ∞ and γ1 for the long term responseand the viscous spring, respectively,
ψce = γ∞ μc
[λ
δce
δc− 1
δc− ln(λe)
]
(63)
and
ψci = γ1 μc
[λ
δcce
δc− 1
δc− ln(λce)
]
. (64)
Similar to the viscoelastic branch of the uncured rubbermodel, the evolution law of the dashpot is defined by the
thermodynamically driving force βv = − ∂ψc
∂εci
εci = γci (βv)βv with γci = 1
ηci
∣∣∣βv
∣∣∣mci
. (65)
For the evolution of the curing strain, no evolution law isdefined directly. As stated in Eq. (23), the free energy shouldremain constant during vulcanization. In the micro-sphereapproach, this remains true for every orientation direction r
∂ψ
∂c= 0. (66)
This leads to the formulation of the evolution law of thecuring strain
εc = ψu − ψc
c ∂ψc∂εc
c. (67)
The derivative of the cured free energy with respect to thecuring strain is obtained by the chain rule
∂ψc
∂εc= −γ∞ μc
[λδce − 1
] − γ1 μc[λδcce − 1
]. (68)
The free energy of the viscoelastic branch depends on theoverall stretch and the curing strain. For a more efficientimplementation of Eq. (66), the total derivative dψc
dεchas to
be used. The total derivative of the viscoelastic stretch withrespect to the curing strain reads
dλcedεc
= ∂λce
∂λc
∂λc
∂εc+ ∂λce
∂λci
∂λci
∂εci
∂εci
∂βv
∂βv
∂λce
dλcedεc
. (69)
Rearranging Eqs. (68) and (69) leads to the total derivativethat is inserted into the evolution law of the curing strain, Eq.(67),
dψc
dεc= −γ∞ μc
[λδce − 1
]
+ γ1 μcλ
δcece − 1
λce
∂λce∂λc
∂λc∂εc
1 − ∂λce∂λci
∂λci∂εci
∂εci
∂βv
∂βv
∂λce
. (70)
The rather simple chosen material model for the cured rub-ber, captures its basic features sufficiently. However, thereare different models like a non-linear viscoelastic model pre-sented in [21] or [22]. It is necessary to derive the free energyfunction with respect to the curing stretch and any model canbe inserted. Due to the fact that the curing strain is onlyaffecting the cured material model, any model can be imple-mented for the uncured rubber without further derivations.In [4], a non-affine micro-sphere model for rubber is intro-duced, where a stretch fluctuation field on the micro-sphereis determined by a principle of minimum averaged free ener-gies. This approach could also be used for modelling therubber curing process, as the micro-stretches λ for cured anduncured rubber are equal. However, even the evolution law ofthe simple affine visco-elastic model, Eq. (67), is quite com-plex. Increasing the complexitiy of the cured rubber modelwould increase the complexitity of the evolution law further.
4.3 Algorithmic stresses and tangents
In a fully coupled thermo-mechanical simulation, the tem-perature of the new timestep Δtn+1 is computed based onthe heat transfer equation for every integration point, simul-taneously. The current state of cure is obtained by Eq. (4), thestate of cure at the previous timestep and the current temper-ature. The newly obtained curing rate, Eq. (1), will be usedto update the curing strain. The new state of cure will lead toa different ratio of uncured to cured rubber, combined with
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Computational Mechanics
the updated curing strain. The stresses and tangent modulihave to be derived for a new time step tn+1, where the stressand internal state variables from the previous time step tn areknown. The history variables are updated with the flow rulein the Eulerian setting.
The volumetric stress is the hydrostatic pressure, derivedfrom the volumetric free energy function, Eq. (15),
τ vol = p1 (71)
with
p = JU ′ = κ [J − 1 + αv(θ0 − θ) + c αc] . (72)
The volumetric tangent terms are derived further and take theform
Cvol = 4∂2ggU = (p + s)g−1 ⊗ g−1 − 2pI, (73)
using the second order derivative ofU , the fourth order identytensor Iabcd = [
δacδbd + δadδbc]/2 and
s = J 2U ′′ = κ [1 − αv(θ0 − θ) − c αc] . (74)
From Eq. (21) and the averaging operator, Eq. (36), the iso-choric stresses
τ = 〈τλ−1t ⊗ t〉 (75)
are obtained with the identity 2∂gλ = λ−1t ⊗ t. The furtherderivative of the stresses with respect to the metric and usingthe expression ∂g(t⊗ t) = 0 leads to the algorithmic tangent
Calgo = 〈(d − τλ−1)λ−2t ⊗ t ⊗ t ⊗ t〉. (76)
The scalar stresses in the microscopic setting are derivedfrom the free energy function, Eq. (40), as
τ = ∂ψ
∂λ= (1−c)τu+c τc = (1−c)(τp+τv)+c (τce+τc1).
(77)
Analogously, the scalar tangent terms are derived
d = dτ
dλ= (1−c)du+c dc = (1−c)(dp+dv)+c (dce+dc1).
(78)
The microscopic stresses and tangents of the uncured andcured rubber phase are determined in the next sub-sections.The isochoric tangent term is then obtained as
Ciso = P :[Calgo + 2
3(τ : g)I
− 2
3
(τ ⊗ g−1 + g−1 ⊗ τ
)]: PT . (79)
The total tangent is the sum of volumetric and isochorictangent leading to a consistent linearization of thestresses
C = Cvol + Ciso. (80)
4.3.1 Uncured rubber phase
In this section, the scalar stresses and tangents of the uncuredrubber model are presented in short, for more details, thereader is referred to [3]. In direction r, the logarithmicmicro-stress of the elasto-plastic branch is derived as
σp = ∂ψep
∂εep= μep
(λ
δepep − 1
). (81)
The scalar stresses are computed by the relation ∂εep∂λ
= λ−1
and the chain rule
τp = σp
λ= μep
λδepep
λ. (82)
Before computing the stresses, the plastic flow εp in directionr needs to be known. The updated logarithmic plastic strainis derived from its value at the previous time step tn and theevolution law of the plastic flow Eq. (48)
εtn+1p = εtnp + εp|tn Δt . (83)
This steps leads to the nonlinear residual equation
rp = εtn+1p − εtnp − γp(βp)βpΔt = 0, (84)
that is solved by a local Newton iteration. In contrast to [3],the scalar tangent term is derived directly and is not obtainedwith the implicit function theorem
dp = dτpdλ
= μep
λ
dλδepep
dλ− τp
λ. (85)
The derivative of the elastic stretch is challenging, becauseit is also dependent on the backstress βp, which is depen-dent on the elastic stretch itself. That relation leads to theexpression
dλδepep
dλ=∂λ
δepep
∂λ+ ∂λ
δepep
∂εp
∂εp
∂ z
∂ z
∂λ+ ∂λ
δepep
∂εp
∂εp
∂βp[
∂σp
∂λδepep
dλδepep
dλ− ∂βph
∂λp
(∂λp
∂λ+ λp
∂λδepep
dλδepep
dλ
)]
,
(86)
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Computational Mechanics
with the definition of the backstress βp = σp − βph
and the stress in the kinematic hardening branch βph =μph
[λp − 1
]δph−1λp. Due to the self dependency of the
elasto-plastic stretch λep, its derivative with respect to theoverall stretch is on the left and the right hand side of Eq.(86). Rearranging leads to the expression to calculate thetangent
dλδepep
dλ=
∂λδepep
∂λ+ ∂λ
δepep
∂εp
∂εp∂ z
∂ z∂λ
− ∂λδepep
∂εp
∂εp
∂βp
∂βh∂λp
∂λp∂λ
1 − ∂λδepep
∂εp
∂εp
∂βp
[∂σp
∂λδepep
− ∂βh∂λp
∂λp
∂λδepep
] . (87)
The evaluation of all the derivatives and the final form of thederivative can be found in the Appendix. The viscoelasticstresses
σv = ∂ψve
∂εve= μve
(λδve
ve − 1)
(88)
and the stress in the hardening branch
βh = μh [λeh − 1]δh−1 λeh (89)
are derived in a similarway.Due to the viscoelastic kinematichardening branch, two internal iterations have to be done forthe viscous logarithmic strain
rv = εtn+1v − εtnv − γv(βv)βvΔt = 0 (90)
and an inner iteration for the hardening strain
rh = εtn+1h − ε
tnh − γh(βh)βhΔt = 0. (91)
With the two strains at hand, the microscopic stress in direc-tion of r reads
τv = σv
λ=
μve
(λ
δveve − 1
)
λ. (92)
For the viscoelastic part, the tangent is even more com-plex, due to the second dashpot in the Maxwell element.Therefore, the total derivative of the hardening stretch λhalso depends on the viscous stretch λve and both depend onthe backstress βv and βh , respectively. First, the derivativeof the elastic stretch of the Maxwell element is computedas
dλehdλ
= ∂λeh
∂λ+ ∂λeh
∂εh
∂εh
∂βh
∂βh
∂λeh
dλehdλ
+ ∂λeh
∂λδveve
dλδveve
dλ. (93)
The total derivative in Eq. (93) will now be transferred to theright hand side, and also the total derivative of λ
δveve will be
separated
dλehdλ
=∂λeh∂λ
1 − ∂λeh∂εh
∂εh∂βh
∂βh∂λeh
+∂λeh
∂λδveve
1 − ∂λeh∂εh
∂εh∂βh
∂βh∂λeh
dλδveve
dλ(94)
= a1 + a2dλδve
ve
dλ. (95)
The abbreviations a1 and a2 are introduced for simplificationof the notation. The derivative of the stretch of the singlespring is computed accordingly
dλδveve
dλ=∂λ
δveve
∂λ
+ ∂λδveve
∂εv
∂εv
∂βv
[∂σv
∂λδveve
dλδveve
dλ− ∂βh
∂λeh
dλehdλ
]
. (96)
Inserting Eq. (93) into Eq. (96) leads to the total derivative ofthe viscoelastic stretch, which is needed for the algorithmictangent
dλδveve
dλ=
∂λδveve
∂λ− ∂λ
δveve
∂εv
∂εv
∂βv
∂βh∂λeh
a1
1 − ∂λδveve
∂εv
∂εv
∂βv
[∂σv
∂λδveve
− ∂βh∂λeh
a2] . (97)
The derivation of all parts can be found in the Appendix.Finally, the tangent for the viscoelastic part is obtainedby
dv = dτv
dλ= μve
λ
dλδveve
dλ− τv
λ. (98)
4.3.2 Cured rubber phase
For cured rubber, the internal iteration in the Maxwell ele-ment is solved in a similar way as Eq. (91). After computingthe inelastic strains of the cured and uncured rubber phase,the free energy of all springs is known. After a time step Δthas elapsed and a large enough temperature is present, thestate of cure increases according to Eq. (4). Thus, the ratio ofuncured to cured free energy changes and to fulfil Eq. (23),an additional Newton iteration for the curing strain has to becarried out. Analogously to Eqs. (83) and (84), the residualequation for the logarithmic curing strain is
rc = εtn+1c − εtnc − εcΔt = 0, (99)
which cannot be solved analytically for εtn+1c . Linerization is
neccesary to obtain the new curing strain
Lin rc = rc|εtnc + Kc Δεk+1c = 0 (100)
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Computational Mechanics
with
Kc = drcdεc
|εkc . (101)
The curing strain of the last time step is set as initial valuefor the update procedure ε0c = ε
tnc . The curing strain of the
next iteration step k + 1 is computed as
εk+1c = εkc + Δεk+1
c (102)
with
Δεk+1c = −K−1
c rc|εkc . (103)
If a certain tolerance |rc| < T OL is reached, the updateprocedure is terminated and the curing strain has been found.For evaluating the tangent term Kc, the evolution law of thecuring rate is noted in short as
εc = ψu − ψc
c dψcdεc
c = e1e2
c
c. (104)
The tangent term of Eq. (101) using the quotient rule reads
Kc = 1 −de1dεc
e2 − e1de2dεc
(e2)2c
c. (105)
In the current framework, the free energy of the uncured rub-ber model does not depend on the curing strain εc, therefore,the derivative of the numerator is already known from Eq.(70). The quite complex derivation of the complete tangenttermKc is shown in the Appendix. In Table 4, the structure ofall internal iterations is shown for updating the logarithmicstrains. After all the iterations have been done, the internalstretches fulfilling the evolution laws are knownand the stressof the cured rubber phase is computed as
τc = τce + τci (106)
with
τce = γ∞ μcλ
δce − 1
λand τci = γ1 μc
λδcce − 1
λ. (107)
The scalar tangent terms are derived consequently for theelastic and viscoelastic branch separately
dc = dτcdλ
= dce + dci . (108)
The microscopic stress of the elastic branch is derived fromEq. (107a)
dce = dτcedλ
= γ∞ μc δcλ
δc−1e
λ
dλedλ
− τce
λ. (109)
The elastic stretch depends on the the total stretch, the curingstrain and also on the viscoelastic strain
dλcedλ
= ∂λce
∂εci
∂εci
∂βi
∂βi
∂λce
dλcedλ
+ ∂λce
∂λe
dλedλ
. (110)
Using the evolution law of the curing strain, Eq. (70), oneobtains the final derivative of the elastic strain with respectto the overall stretch
dλedλ
= z1z2
. (111)
For notation, the numerator z1 and the denominator z2 arewritten separately,
z1 = ∂λe
∂λ+ ∂λe
∂εc
cΔt
c ∂ψc∂εc
[∂ψu
∂λ− ∂ψci
∂λce
∂λce∂λ
1 − ∂λce∂εci
∂εci∂βi
∂βi∂λce
]
(112)
and
z2 = 1 + cΔt(c ∂ψc
∂εc
)2
[(∂ψce
∂λe+ ∂ψci
∂λce
∂λce∂λe
1 − ∂λce∂εci
∂εci∂βi
∂βi∂λce
)
c∂ψc
∂εc
+ (ψu − ψc) c∂2ψc
∂εc ∂εc
∂εc
∂λe
]. (113)
The tangent term of the inelastic part of the cured rubbermodel is
dci = γ1 μc
λδc λδc−1
cedλcedλ
− τci
λ. (114)
The total derivative of the elastic stretch of the Maxwellbranch is already given in Eq. (110) using the result fromEq. (113). The complete workflow of the material model isshown in Table 4. First, the new state of cure is computedwith the current temperature. Then, all internal iterations inthe uncured and cured rubber phase are carried out. With thefound internal strains, the free energies of the uncured andcured rubber are known and the iteration to find the curingstrain is carried out. After every residual reached the toler-ance, the stresses and tangent moduli are computed.
5 Numerical examples
5.1 Vulcanization of a single material point
In this section, the correct derivation of the vulcanization andthe conservation of the free energy are addressed. Therefore,
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Computational Mechanics
Table 4 Simulation workflow for the thermo-mechanical curing model
Compute current state of cure, based on the updated temperature
etargnirucetupmoC1.1 c = k cmtn
(1 − ctn)n
erucfoetatsetupmoC2.1 ctn+1 = ctn + c Δt
Compute all internal variables εc, εp, εv and εh
seulavlaitiniteS1.2 kc = 0, ε0c = εtnc
DO WHILE TOL < [rc]3.1 Set initial values kp = 0, ε0p = εtn
p
DO WHILE TOL < |rp|3.2 Residual equation rp = εp − εtn
p − γp (βp )βp Δt = 0niLnoitaziraeniL3.3 rp = rp|εtn
p+ ∂rp
∂εp|εkp
pΔt = 0
3.4 Compute Kp = ∂rp
∂εp|εkp
p
3.5 Solve Δεkp+1p = −K−1
p rp|εkpp
3.5 Update εkp+1p = ε
kpp + Δε
kp+1p , kp = kp + 1
4.1 Set initial values kv = 0, ε0v = εtnv
DO WHILE TOL < |rv|5.1 Set initial values kh = 0, ε0h = εtn
hDO WHILE TOL < |rh|
5.2 Residual equation rh = εh − εtn
h − γh(β
h)β
hΔt = 0
niLnoitaziraeniL3.5 rh = rh|εtnh
+ ∂rh
∂εh|ε
khh
Δt = 0
5.4 Compute Kh = ∂rh
∂εh|ε
khh
5.5 Solve Δεkh+1h = −K−1
h rh|ε
khh
5.6 Update εkh+1h = εkh
h + Δεkh+1h , kh = kh + 1
4.2 Residual equation rv = εv − εtnv − γv (βv )βv Δt = 0
niLnoitaziraeniL3.4 rv = rv|εtnv
+ ∂rv
∂εv|εkv
vΔt = 0
4.4 Compute Kv = ∂rv
∂εv|εkv
v
4.5 Solve Δεkv+1v = −K−1
v rv|εkvv
4.6 Update εkv+1v = εkv
v + Δεkv+1v , kv = kv + 1
2.2 Residual equation rc = εc − εtnc − εcΔt = 0
niLnoitaziraeniL3.2 rc = rc|εtnc
+ ∂rc
∂εc|εkc
cΔt = 0
2.4 Compute Kc = ∂rc
∂εc|εkc
c
2.5 Solve Δεkc+1c = −K−1
c rc|εkcc
2.6 Update εkc+1c = εkc
c + Δεkc+1c , kc = kc + 1
Compute algorithmic stress and tangent
ssertsetupmoC1.6 τ = (c − 1) τu + c τc
tnegnatetupmoC2.6 d = (c − 1) du + c dc
noitisnartorcam-orciM3.6 τ = 〈τλ−1t ⊗ t〉noitisnartorcam-orciM4.6 Calgo = 〈(d − τλ−1)λ−2t ⊗ t ⊗ t ⊗ t〉
.
.
.
the deformation of a single material point is prescribed ana-lytically by
F =⎡
⎢⎣
λ 0 00 1√
λ0
0 0 1√λ
⎤
⎥⎦ . (115)
The material parameters for the uncured rubber are takenfrom [21] and the material parameters for the cured rubberare arbitrarily chosen. The parameters are given in Tables
1, 6 and 7. Similar to the forming and building process ofa tire, the deformation is applied first and kept constant asit is the case in the moulding press. Next, the temperatureis increased and held for a longer time, thus, the vulcan-ization process is started. In Fig. 8, the prescribed stretchand temperature versus time are plotted. In this example, thedeformation is kept constant for a long time, thus, the freeenergy during vulcanization is studied without viscoelasticeffects of the uncured rubber. After 200 s, the final deforma-tion of λ = 2.5 is applied and the viscoelastic branch of the
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Computational Mechanics
11.21.41.61.82
2.22.42.6
0 500 1000 1500 2000 2500 300020
40
60
80
100
120
140
160
λ[−
]
θ[°
C]
time [s]
λ θ
Fig. 8 Deformation and temperature driven curing process of amaterialpoint. The applied stretch and the prescribed temperature plotted versustime
uncured rubber relaxes. The nonlinear effects of the elasto-plastic and viscoelastic branch of the uncured rubber modelcause a nonlinear increase of the free energy function. Due tothe relatively slow loading rate, the viscoelastic effects are notsignificant in this example in contrast to a faster tiremouldingprocess. The stored energy reduces until it is fully relaxed andis only represented by the time-independent elasto-plasticbranch. After 1500 s, the temperature is increased and thestate of cure evolves over time according to Eq. (4). Due tothe evolution of the curing variable, the representation of thefree energy changes but its amount remains constant
ψ = (1 − c)ψu + cψc. (116)
Cured rubber usually shows much stiffer behaviour than theuncured rubber and, to conserve the free energy during thistransformation, curing strain evolves according to Eq. (67).As seen in Fig. 9, the curing strain of the first unit directionvector r1 = e1, which is equal to the uniaxial loading direc-tion, evolves very fast. Due to the exponential free energyfunction of the cured rubber, the curing strain has to increaserapidly in comparison to the slowevolutionof the state of cureto ensure that the free energy is not increasing. The portionof the cured rubber of the overall free energy cψc is increas-ing according to the state of cure, as seen in Fig. 9. The finalvalue of the curing strain in loading direction is λc = 2.32,which means that nearly 93% of the applied stretch is trans-formed into a permanent set, caused by the vulcanizationprocess. In Table 5, the convergence behaviour of the inter-nal iteration for the curing strain is shown. After a maximumnumber of 2 iteration steps, the residual equation is fulfilled,which verifies a correct derivation and implementation ofthe tangent term Eq. (101). In Fig. 10, the micro mechanicalstress response of the first unit direction vector is shown. Atthe beginning of the simulation, the overall stress response isrepresented solely by the uncured rubbermodel.After 1500 s,
0
0.5
1
1.5
2
2.5
0 500 1000 1500 2000 2500 30000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
[−]
energy
[N/m
m2]
time [s]
cεc
ψc ψc
Fig. 9 Resultant free energy and the evolution of the curing strain plot-ted versus time, the state of cure and the cured part of the free energyfunction evolves accordingly over time
the ratio of uncured to cured rubber is changed and the stressis increasing according to the state of cure.
5.2 Rubber block simulation
After demonstrating the correct implementation on an inte-gration point level, the model will be applied to a simplegeometry. A fully coupled thermo-mechanical simulationwill be carried out with a commercial finite element code,extended by user subroutines tomodel thematerial behaviourand evolution of cure. A tension test of a simple rubber blockmodel, see Fig. 11a, is investigated, to show the capabilitiesof themodel on a structural scale. The samematerial and cur-ing parameters are chosen as in the first example, depictedin Tables 1, 6 and 7. The bottom nodes are fully constrainedand the top nodes are driven vertically up to 100% structuraldeformation while constrained in the other two directions.In Fig. 11b, the deformed configuration is plotted with thedisplacements in the loading direction. After the total dis-placement has been applied, the temperature at the top andbottom will be increased up to 160◦C and, via conductivity,the temperature of the block will increase non homogeneous.Therefore, the state of cure will evolve non homogeneouslyand the block will be represented by the uncured and by thecured rubber model, simultaneously. After the block is fullycured, the boundary conditions are removed and the blockis deforming to its new equilibrium configuration, Fig. 11c.Approximately 90% of the deformation is transferred to apermanent set and the shape of the block has significantlychanged. In Fig. 12 the reaction force in loading directionis plotted versus time. The uncured rubber is softer than thecured rubber and, therefore, the reaction force is increasingwhile the state of cure is evolving.
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Computational Mechanics
Table 5 Residuals of theinternal iteration for the curingstrain
time 500s 1000s 1500s 1600s 2000s 2500s
kc = 0 4.722E−009 7.308E−009 4.722E−009 1.974E−003 5.326E−005 7.765E−007
kc = 1 3.093E−022 3.275E−022 3.068E−022 4.521E−009 1.895E−011 1.551E−016
kc = 2 1.214E−017 5.216E−017
00.10.20.30.40.50.60.70.80.91
500 1000 1500 2000 2500 3000
τ[M
Pa]
time [s]
τ(1 − c) τu
c τc
Fig. 10 Uncured and cured stress contributions plotted versus time aswell as the overall stress in loading direction
5.3 Tire production simulation
In tire building, all tread and belt compounds are assembledtogether on a drum. The carcass, innerliner and sidewall rub-ber compounds are wrapped around the steel bead and, then,stuck togetherwith the other rubber parts of the belt and tread.The formed green tire on the drum starts to rotate and rollerspress all parts together, so that all trapped air within the rub-ber compounds is squeezed out. At this point, the green tire ismeasured and a finite element model is created, see Fig. 13.
The fully coupled thermo-mechanical simulation is car-ried out based on a commercial finite element code, that isextended by user specific subroutines to include the mate-rial model and the evolution of cure. The subroutines arewritten in Fortran. Due to the very stiff behaviour of themould, it ismodelled by rigid surfaces. The simulation is thencarried out in an axisymmetric setting, neglecting the treadpattern and only takes into account circumferential grooves.Therefore, the number of degrees of freedom is decreasedand the simulation is completed faster in comparison to a fullthree-dimensionalmodel. Taking into account lateral groovesof the tread pattern leads to large mesh distortions in thisarea, which have to be addressed by mesh smoothing algo-rithms like an Arbitrary Lagrangian-Eulerian framework or ameshless discretization often used in impact simulations. Thewhole production process simulation at hand takes around3.5hours on an INTEL® Core™i7 CPU @ 3.60GHz and 16GBRAM. The results are shown in the following section step bystep.
(a) ux[mm]
10
0
(b)
ux[mm]
10
10
0
(c)
ux[mm]
0
Fig. 11 Displacement of the block model in loading direction ux at thea undeformed configuration, b deformed state and c after vulcanizationin the new equilibrium configuration
5.3.1 Moulding simulation
The short-term in-moulding process is highly dependent onthe uncured phase of the rubber. Due to the large inner pres-sure, that is used to press the green tire into the mould, largestrains are present especially in the tread area. Therefore, acorrect derivation and implementation of the material is cru-cial to find a converged state. First, the green tire is placedinside the opened mould, Fig. 14a, contact between the elas-
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Computational Mechanics
Table 6 Mechanical properties of uncured rubber material used in thesimulation
Bulk modulus κ [MPa] 100
Elasto–plastic modulus μep [MPa] 0.09
Elasto–plastic power term δep [−] 6.2
Plastic hardening modulus μph [MPa] 0.4
Plastic hardening power term δph [−] 6.2
Plastic flow parameter ηp [MPa] 0.2
Plastic flow power term mp [−] 1.0
Viscoelastic modulus μve [MPa] 0.26
Viscoelastic power term δve [−] 5.2
Visco hardening modulus μh [MPa] 0.65
Visco hardening power term δh [−] 8.2
Viscous flow parameter ηv [MPa · s] 5.0
Viscous flow power term mv [−] 2.0
Visco hardening flow parameter ηh [MPa · s] 600.0
Visco hardening flow power term mh [−] 1.0
Table 7 Mechanical properties of cured rubber material used in thesimulation
Bulk modulus κ [MPa] 100
Elastic modulus μc [MPa] 6.35
Elastic power term δc [−] 4.54
Long term scaling factor γ∞ [−] 1.0
First branch scaling factor γ1 [−] 0.4
Viscous flow parameter ηv1 [MPa · s] 5.0
Viscous flow power term mv [−] 1.0
02468
1012141618
0 200 400 600 800 1000
RF
x[M
Pa]
time [s]
Fig. 12 Reaction force in loading direction RFx versus time
tic bladder and the green tire is established by applying alow inner pressure. Due to the low stiffness of the uncurednylon cords, the tire will expand until the side of the mouldis closed, Fig. 14b. Up to this point, the tire center is heldby Dirichlet boundary conditions to keep it inside the mould.After bladder and tire are in contact, this boundary conditionis not active any more. The green tire is now only held bycontact with bladder and mould. The mould is then closed upto its final position, Fig. 14c, still with the low inner pressure.Establishing contact in the tread area is difficult, therefore,
Point A
Point B
Point C
Fig. 13 Discretized green tire and its material sections
the tire has to be discretiziesed very fine in this region. Thetire will change its shape significantly, where the material isflowing into the mould to form the circumferential grooves.The final position of the tire, Fig. 14e, is achieved after theinner pressure of 25 bar is applied.While kept in this position,the viscoelastic branch of the uncured rubber relaxes and thematerial flows further into the edges of the mould. Valida-tion of the in-moulding process is very difficult, because thepress is a closed system and it is not possible to observe thetire during the moulding explicitly. In [23], temperature sen-sors are placed on the surface of the tire and, when contactis established between the tire and the hot mould, an instantincrease of the temperature could be observed. The numer-ical results correspond to the experimental results, the firstpart that is in contact with the mould is the bead area. Afterthat the sidewall near the belt edges establishes contact. Atlast, the area between bead and sidewall is in contact withthe mould.
5.3.2 Vulcanization simulation
By the model presented in Sect. 2, the description ofunvulcanized rubber, exposed to large temperature, can beassociated with the change of its mechanical behaviour. Thisprocess will take place at different times and at different ratesdepending on the mixture of the rubber compound. For heat-ing the tire, the surface temperature of the moulding pressand the air inside the bladder are increased and due to heatexchange over the tire surface, it is heated from the outside.For all rubber compounds, the same thermal properties, pre-sented in Table 8, are taken from the literature. It is assumed,that the thermal parameters during vulcanization are con-stant. The temperature distribution of the cross-section overtime is shown in Fig. 15a–h. The surface area in contactwith the mould is heated directly and, therefore, faster thanthe inner surface in contact with the bladder. After 700s, Fig. 15d, the sidewall has reached the final temperatureof 160◦C while the tread area is significantly colder. Thistemperature difference in the tire cross-section is anotherreason for the non-uniform state of cure in the tire, shown in
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Computational Mechanics
Fig. 14 Single steps of the in-moulding simulation: a inserting of thegreen tire into the mould, b inflation of the bladder and closing of thesides of the press, c fully closedmoulding press with low inner pressure,d increased inner pressure and e finally applied inner pressure of 25 bar
Fig. 16a–h. The parameters for the curing rate, Eq. (4), arefitted for every compound of the tire based on experimentaldata. The fast heating of the sidewall will cause the mate-rial to vulcanize there first while most of the other regionsare still uncured, see Fig. 16c. The rubber, that contains thebelts, shows vulcanization behaviour that starts early but at
Table 8 Thermal properties of the rubber material used in the simula-tion
Density ρ[kgm3
]1100
Thermal conductivity kv
[ Wm K
]0.14
Specific heat cs[
Jkg K
]2010
Heat exchange parameter h[
Wm2 K
]100
slower rate, which is observed in the curing process as well.After 1500 s, the SOC in the belt rubber starts to evolve whilethe tread material has not started to vulcanize, Fig. 16d. Thefaster curing process of the tread compound will overtake thebelt rubber and after 2100 s, the tread is nearly fully cured,while the belt rubber is only half way cured, Fig. 16g. In thecurrent model, the feature of overcuring is not represented,where the stiffness of the rubber will decrease again due tochemical alterations and breaking of established crosslinks.
In Fig. 17a, the temperature evolution of 3 different pointsare plotted. The temperature in the relatively small sidewallis increasing very fast in comparison to the thicker parts in thebelt and subtread area. Point B at a circumferential groove isheated faster, due to the closer distance to the heating press. InFig. 17b, the different curing kinematics for different rubbercompounds canbe seen.Aspresented inFig. 1, the belt rubberstarts the vulcanization process faster than the tread rubber.But the tread compound will reach the fully cured state fasterand so, after 2500 s, both compounds are nearly at the sametime fully cured. The curing kinematics of the sidewall andtread compound are similar, but due to the faster temperatureevolution, the sidewall area is much faster fully cured.
5.3.3 Post curing inflation
After the tire has been fully cured, it is taken out of themould and cooled down to room temperature again. Duringthis step, the different heat expansion parameters of rubberand reinforcements lead to stress inside the tire. To min-imize the pre-stress, tire manufacturers add an additionalstep in the production process, so-called post curing infla-tion (PCI). The tire is fixed on a rim and an inner pressure isapplied while the tire is cooled down. Due to the constraintsby the inner pressure, no free thermal shrinkage occurs andthe PCI tire has a wider section width than a tire without postcuring inflation [24]. Modelling this phenomenon requires arealistic model of the complex mechanics and phase transi-tions of the nylon cords that undergo large shrinkage duringvulcanization and cooling. However, the tire after releas-ing from the moulding press has now a different shape thanthe green tire. In Fig. 18, the new equilibrium shape of thecured tire is compared to the stress-free reference config-uration of the green tire. The tread grooves of the cured
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Computational Mechanics
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Fig. 15 Contour plots of the temperature in the tire cross-section at a 100 s, b 300 s, c 500 s, d 700 s, e 900 s, f 1100 s, g 1300 s and h 1500 s
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Computational Mechanics
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Fig. 16 Contour plots of the current state of cure in the tire cross-section at a 900 s, b 1100 s, c 1300 s, d 1500 s, e 1700 s, f 1900 s, g 2100 s andh 2300 s
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Computational Mechanics
(a)
20
40
60
80
100
120
140
160
0 500 1000 1500 2000 2500 3000
θ[°
C]
time [s]
Point APoint BPoint C
(b)
00.10.20.30.40.50.60.70.80.91
0 500 1000 1500 2000 2500 3000
c[−]
time [s]
Point APoint BPoint C
Fig. 17 a Temperature and b state of cure of the 3 points of interest areplotted versus time: at the sidewall (Point A), at the belt material underthe groove (Point B) and at the subtread (Point C)
Fig. 18 Comparison of the shape between the uncured green tire andthe fully cured tire. (Color figure online)
tire are sharper than in the uncured tire and represent thenegative geometry of the mould. The width of the tire isdecreased after the curing process. The shape of the curedtire could be achieved without changing the reference con-figuration.
6 Conclusion
Understanding the production process of a tire is neccessaryto optimize the final product. The finite element method is apowerful tool for analysing the process and to identify areasof improvement in the curing process. Therefore, vulcan-ization of rubber is described by a model valid for rubbercompounds of the tire. It could be observed that different
rubber mixtures will vulcanize in a different way and, conse-quently, the tire curing process will be non-uniform over thetire cross-section. These results can then be used to optimizethe tire curing process and to obtain a perfectly cured tire ina minimum time.
A thermo-mechanically consistent framework for thephase transition of the vulcanization of rubber has been pre-sented, which includes the evolution of a permanent curingstrain, the change of the mechanical behaviour and cur-ing shrinkage. The presented framework is split into anuncured and a cured phase, therefore, the models for bothphases can be chosen and identified independently from eachother. The material model is implemented in a finite ele-ment framework and is used to simulate the tire productionprocess. The procedure can be modelled in an axisymmet-ric setting from moulding of the green tire, heating of thetire inside the press and non-uniform vulcanization to thefinal cured tire. The production process is finished afterthe tire is released from the mould and a changed shapeof the cured tire has been achieved. In the current axisym-metric model, lateral grooves are neglected. The shape ofthese is most important for the behaviour of the tire at wetsurfaces and influences the noise of the running tire signifi-cantly.
The uncured rubber material flows into the small cavitiesin the mould due to high inner pressure from the bladder.In a pure Lagrangian setting, this causes large mesh distor-tions that will lead to numerical difficulties and an unstablesimulation. Overcoming these challenges, mesh smoothingtechniques, like the Arbitrary Lagrangian-Eulerian (ALE)formulation or a meshless representation have to be appliedfor modelling the fully three-dimensional production pro-cess with lateral grooves. The shrinkage behaviour of thenylon reinforcement cords, due to curing and cooling, has tobe addressed to model the influence of the post-cure infla-tion.
Acknowledgements OpenAccess funding provided by Projekt DEAL.The generous support of this research by Hankook Tire, Daejeon, SouthKorea, is gratefully acknowledged.
Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indi-cate if changes were made. The images or other third party materialin this article are included in the article’s Creative Commons licence,unless indicated otherwise in a credit line to the material. If materialis not included in the article’s Creative Commons licence and yourintended use is not permitted by statutory regulation or exceeds thepermitted use, youwill need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Computational Mechanics
Appendix
In order to provide all information required for the materialmodelling, additional remarks are given subsequently.
Appendix A Elastic-plastic Tangent
The total derivative for the rate-independent tangent requiresthe total derivative of the elasto-plastic stretches with respectto the overall stretch
dλδepep
dλ= z1
z2. (117)
The numerator is
z1 = δep λδep−1ep
(1
λp− λep
ηp
∣∣∣βp
∣∣∣mp
Δt∂ z
∂λ+ (mp + 1)γpΔt
∂βph
∂λp
)
(118)
and the denominator
z2 = 1 + δep λδepep (mp + 1)γp Δt
(
μep + ∂βph
λp
λp
δep λδepep
)
,
(119)
with the derivative of the evolution of the arclength
∂ z
∂λ= ln(λ) − ln(λtn )
ln(λ) − ln(λtn )
1
Δt λ(120)
and the derivative of the backstress
∂βph
∂λp= μph
[(λp − 1)δph−1 + (δph − 1)(λp − 1)δph−2λp
].
(121)
Appendix BViscoelastic Tangent
The rate-dependent branch of the uncured rubber model ismore complex due to the viscoelastic hardening branch andthe derivative of the elastic stretch with respect to the overallstretch is
dλδveve
dλ= δve
λδveveλ
+ δve λδveve (mv + 1) γv Δt ∂βh
∂λeha1
1 + δve λδveve (mv + 1) γv Δt
[μve − ∂βh
∂λeha2
]
(122)
with the help of the substitution
a1 =1
λh λve
1 + λeh (mh + 1) γh Δt ∂βh∂λeh
(123)
and
a2 =− λeh
δve λδveve
1 + λeh (mh + 1) γh Δt ∂βh∂λeh
. (124)
The used derivative of the backstress is consequently
∂βh
∂λeh= μh
[(λeh − 1)δh−1 + (δh − 1) (λeh − 1)δh−2 λeh
].
(125)
Appendix C Evolution Law for the Curing Strain
The tangent for the local Newton iteration, Eq. (101), isdiscussed in detail. Substitution of the curing strain rateεc = e1
e2cc and the quotient rule lead to
Kc = 1 −de1dεc
e2 − e1de2dεc
e22
c
c. (126)
The derivative of the numerator has been already found, Eq(70),
de1dεc
= −dψc
dεc. (127)
The derivative of the denominator is obtained by the deriva-
tive of e2 = −γ∞ μc
[λ
δce − 1
]+ b1
b2, where the numerator
and denominator of the viscoelastic part are derived seperatlyby the quotient rule
de2dεc
= −γ∞ μc δcλδcedλcedεc
+∂b1∂εc
b2 − b1∂b2∂εc
b2(128)
with
b1 = γ1 μc[λδcce − 1
]and
∂b1∂εc
= γ1 μc δc λδc−1ce
dλcedεc
.
(129)
The total derivative of the viscoelastic stretch is already foundin Eq. (69), which leads to
dλcedεc
= − λce
1 + (mv + 1) γv Δt γ1 μc λδcce δc
. (130)
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Computational Mechanics
The derivative of the denominator of Eq. (128) is finally
db2dεc
=(mv + 1)Δtγ1 μc δc
⎡
⎢⎣mv
ηv
∣∣∣βv
∣∣∣mv
βv
γ1 μc δc λδccedλcedεc
λδcce + γv δc λδc−1
cedλcedεc
⎤
⎥⎦ .
(131)
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