Microstructural model for the time-dependent …Microstructural model for the time-dependent...

Microstructural model for the time-dependent thermo-mechanical analysis of cast ironsCitation for published version (APA):Pina, J. C., Kouznetsova, V., van Maris, M. P. F. H. L., & Geers, M. G. D. (2015). Microstructural model for thetime-dependent thermo-mechanical analysis of cast irons. GAMM-Mitteilungen, 38(2), 248-267.https://doi.org/10.1002/gamm.201510014

DOI:10.1002/gamm.201510014

Document status and date:Published: 01/01/2015

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https://research.tue.nl/en/publications/microstructural-model-for-the-timedependent-thermomechanical-analysis-of-cast-irons(e8204093-4855-4dfc-8577-8943573dd5a3).html

GAMM-Mitteilungen, 15 March 2015

Microstructural model for the time-dependent thermo-mechanical analysis of cast irons

J.C. Pina1,2, V.G. Kouznetsova2,∗, M.P.F.H.L. van Maris2, andM.G.D. Geers2

1 Materials innovation institute (M2i), P.O. Box 5008 2600 GA, Delft, The Netherlands2 Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB,

Eindhoven, The Netherlands

Received XXXX, revised XXXX, accepted XXXXPublished online XXXX

Key words Cast iron, thermo-mechanical, thermo-viscoplasticity, stress relaxation

In this paper, a multiscale modelling approach is followed for the modelling of time and tem-perature dependent behaviour of compacted graphite cast iron (CGI) material. Cast irons areoften used in heavy duty machinery parts subjected to elevated temperatures for prolongedperiods of time. This, in combination with its complex heterogeneous microstructure, playsthe crucial role in determining the life time of the material. In this work a 2D microstructuralmodel of CGI is developed. The geometry is based on the micrographs of the material. Thepearlitic matrix is modelled with the temperature dependent elasto-visco-plastic model cali-brated on the pearlitic steel experiments. The graphite particles are modelled as anisotropicelastic. The results of the simulations of tensile and stress relaxation tests at different temper-atures between20C and500C show that the macroscopic mechanical behaviour of the ma-terial deteriorates rapidly above350C. At the microstructural scale, the anisotropic graphiteparticles act as stress concentrators promoting the formation of strain percolation paths, thatbecome more critical at higher temperatures.

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1 Introduction

Nowadays, several industrial applications rely on components operating at high temperaturesover long periods of time. These applications, require materials which can endure the de-manding mechanical conditions as well as the often extreme thermal and environmental loads.Cast iron is a typical example of such a material, which has a good compromise between itsmechanical and thermal properties. Therefore, it is extensively used for high temperaturethermo-mechanical applications. The most prominent examples hereof can be found in theautomotive industry, e.g. in cylinder heads and blocks, exhaust manifolds and disk breaks.

The good combination of mechanical and thermal properties of cast iron ensues from itsmicrostructure, composed of graphite inclusions embeddedin a ferrite/pearlite matrix. Themorphology of the graphite inclusions plays a fundamental role in defining the material prop-erties [1,2] and is frequently used to identify the different cast irons as:Nodular or SpheroidalGraphite Iron (SGI); Compacted or Vermicular Graphite Iron (CGI)andFlake or Lamellar

∗ Corresponding author E-mail:[email protected], Phone: +31 40 247 5885, Fax: +31 40 244 7355


2 J.C. Pina et al.: Microstructural model of a cast iron

Graphite Iron (FGI). In particular, in CGI and FGI the graphite particles form a 3D intercon-nected network that, combined with graphite’s high thermalconductivity, yields thermal prop-erties superior to those of steels. Nevertheless, the presence of the graphite inclusions provesto be detrimental for the mechanical properties, because they introduce stress concentrationsites promoting damage initiation during the early stages of the material’s lifetime [3–9].

At elevated temperatures, the response of the material becomes more sensitive to temper-ature and time. This is reflected in thermal softening and higher creep or stress relaxationsrates [2, 10–21]. Combined with cast iron’s highly heterogeneous microstructure, this maysignificantly limit the thermo-mechanical life performance of the material by acceleratingdamage initiation and propagation. Therefore, it is imperative to understand the elevated tem-perature behaviour of cast iron at different length and timescales in order to provide accuratelifetime predictions of the material and design engineering components accordingly.

In recent years, a few phenomenological models for thermo-mechanical analyses of castirons have been developed [22–26]. The main limiting requirement of these models is thatthey rely on a large amount of experimental data. They do not directly account for the in-fluence of microstructural features on the material thermo-mechanical response (i.e. mor-phology and anisotropy of the graphite inclusions, matrix time and temperature dependentthermo-mechanical behaviour, interaction between the phases and interface behaviour).

In a different approach, microstructural modelling has been used in the past to study themechanical response of cast irons: SGI [3, 27–33], CGI [34–38] and FGI [39]. In additionto the graphite morphology, typical for each cast iron type,other microstructural informationhas been included in these models:

• graphite anisotropy, to evaluate its influence on the local and global mechanical andthermo-mechanical response of FGI and CGI [38,39],

• matrix time-dependent response for high strain rate applications (e.g. machining) [31–33,36,37,40],

• graphite/matrix interface [39,40] and damage [36,37,40].

In spite of the vast amount of work done, hardly any attentionwas given to the combinedeffect of time and temperature on themicrostructuralmechanical response of cast iron. There-fore, a microstructural model for the elevated temperaturemechanical analyses of cast irons ishere proposed. The model is aimed at transient conditions such as the ones present in enginesduring their operating life in a temperature range between20 C and500 C. The point ofdeparture for the present analyses is the cast iron microstructural models presented in [39]and [38], where the temperature dependent thermal and mechanical properties of the graphiteand matrix are included, as well as the morphology and anisotropy of the graphite inclusions.In this work, the time and temperature dependent behaviour of the pearlitic matrix is addedby means of the thermo-viscoplastic model described in [41]. The developed microstructuralmodel is dedicated here to CGI but can be adapted to FGI or SGI in a straightforward mannerby introducing the appropriate morphology of the graphite inclusions.

The goal of this paper is to provide insight into the time-dependent response of CGI fortemperatures up to500 C. The role played by the microstructure and temperature on thebehaviour of the material at both the macro and micro scales is investigated. In order to as-sess the validity of the model parameters identified on a pearlitic steel for the pearlitic matrix,


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micro-indentation tests have been performed. The results from the elevated temperature anal-yses indicate that the sensitivity of the material to temperature rapidly increases for temper-atures above400 C. The higher temperature sensitivity is driven by the thermal recovery inthe pearlitic matrix. The combination of time, temperatureand microstructural heterogeneityleads to higher local plastic deformation and earlier damage initiation at the microstructurallevel.

This paper is organized as follows. In the next section the CGI microstructural model isintroduced. The constitutive behaviour of graphite is discussed in Section 2.1. In Section 2.2,the micro-indentation tests on C75 pearlitic steel and the CGI pearlite matrix are presentedtogether with the pearlite model parameters. The thermo-viscoplastic model for the pearliticmatrix is described in Section 2.3. Then, in Section 3, the elevated temperature analyses onCGI are discussed starting with the tensile tests in Section3.1, followed by the numericalstress relaxation tests in Section 3.2. The main conclusions are summarized in Section 4.

2 CGI microstructural model

The cast iron investigated in this paper is CGI with a carbon content between 3.5 and 3.8 wt%and a graphite volume fraction between 10 and 12% [42]. The matrix is composed of pearlite(95%) and ferrite (5%). In CGI, the graphite particles predominantly reveal a vermicularmorphology, even though nodular particles are always present [43]. Due to the complex andhighly heterogeneous 3D morphology of the vermicular graphite particles, it is challengingand computationally demanding to incorporate all relevantmicrostructural features present inCGI in a descriptive model.

Here, as in [38] a Finite Element Method (FEM) based computational homogenization ap-proach [44] is used to create a microstructural model for CGI. The Representative VolumeElement (RVE) geometry is extracted from a 2D scanning electron microscope (SEM) micro-graph. A 2D finite element mesh of this image is obtained by converting each pixel into afinite element. The different phases (matrix or graphite) are identified based on the pixel greyscale level. The main advantage of this method is that the 2D shape of the graphite particles iscaptured with the experimental resolution. One drawback ofthis 2D model is that it disregardsthe inherent 3D nature of the CGI microstructure. Nevertheless, these simplifications make itpossible to incorporate microstructural details in the microstructure for a RVE size that wouldbe computationally intractable in 3D, especially in the non-linear regime considered here. TheCGI microstructure and the corresponding FEM model are shown in Figure 1. The size of theRVE is defined such that it includes a sufficient number of graphite particles with a graphitearea fraction of10.2% (typical for CGI), whereas the size is not too large to compromise thecomputations.

2.1 Graphite

In graphite, anisotropy dominates both the mechanical and thermal properties. This anisotropyis related to graphite’s layered structure with strong covalent bonds within the basal planesand weak van der Waals bonds between them, which makes graphite a transversely isotropicmaterial (Figure 1.c) [2,5,45–49].

In previous work [38, 39], the pronounced effect of graphite’s anisotropy on the local andglobal response of cast iron has been studied in detail, bothnumerically and experimentally.



a) SEM micrograph

y

x

graphiteisotropic

matrix

graphiteanisotropic

b) FEM model c) Graphite crystalline structure

C

A

E= 36 GPac

E= 1020 GPaa

Fig. 1 CGI microstructure and corresponding FEM model.

Graphite’s anisotropy within cast irons is related to the crystalline structure of the graphiteparticle which in turn is defined by the particle principal growth mechanism [2,45,46,48,49].The latter also controls the morphology of the particle. In nodular particles, the principalgrowth direction is the “c” crystallographic direction, which leads to a “onion” type crystallinestructure. On the other hand, lamellar particles grow primarily in the “a” crystallographicdirection leading to a crystalline structure in which the graphite platelets are mostly orientedparallel to the “a” direction [2,45–47].

In CGI, the predominantly vermicular graphite particles have a growth mechanism thatalternates between the “a” and “c” directions. This leads toparticles in which some parts re-semble the graphite crystalline structure of nodular graphite, while in other parts it resemblesthe crystalline structure of lamellar graphite [2, 45, 46, 49]. The nodular structure generallyoccurs close to the nucleus of the particle, whereas the lamellar type structure generally ap-pears in the elongated “worm” shaped branches of the particle. In these parts, a more clearlyoriented structure is present with stacked layers of graphite platelets running parallel to thegraphite growth direction. It can therefore be reasonably assumed that the graphite in the elon-gated “worm” like branches of vermicular particles, behaves as transversely anisotropic, withits principal direction (i.e. “a” direction) oriented parallel to the particle growth direction (Fig-ure 1.c). The corresponding elastic constants for graphiteare given in Table 1. The remainingparticles, will be assumed isotropic. The elastic constants for isotropic graphite are taken byaveraging of the in-plane and out-of-plane properties of graphite, i.e.E11 = E22 = 528 GPaandν12 = ν23 = ν31 = 0.0875.

Table 1 Graphite elastic constants: in-plane (directions 1 & 2) andout-of-plane (direction 3)Young’s moduli, Poisson’s ratios and shear moduli [50].

E11 = E22 [GPa] E33 [GPa] G23 = G31 [GPa] ν12 ν23 = ν311020.4 36.364 0.280 0.163 0.012

2.2 Matrix

For the pearlite matrix, the thermo-viscoplastic model presented in [41] is used to describe thetime-dependent behaviour of CGI. Because of the complexityof performing an independentmodel parameter identification process for pearlite withinCGI, the material parameters asidentified in [41] from elevated temperature tests (tensileand creep tests) on C75 pearlitic



steel (0.708 wt% percent of carbon) will be used here. To assess whether the pearlitic steelused for the model parameter identification is a valid substitute for the pearlite present in CGI,room temperature micro-indentation tests were performed on both materials. The results fromthe tests are shown in Table 2, indicating that there is an adequate agrement between bothmaterials in terms of the Young’s modulus and the hardness.

Table 2 Material properties obtained by micro-indentation tests on pearlite in CGI and C75pearlitic steel

Pearlite (CGI) C75Young’s modulus [GPa] 184.0± 14.6 190.3± 10.7

Hardness [GPa] 3.66± 0.54 3.16± 0.18

The pearlite thermo-viscoplastic model makes use of twelvematerial parameters: two elas-tic constants,E andν, the thermal expansion coefficientα, and nine parameters describing theviscoplasticity. The values of the Young’s modulus for different temperatures used here aregiven in Table 3 [51] and those of the thermal expansion coefficient in Table 4 [52]. Poisson’sratio is assumed to be temperature independent and equal to0.29, based on experimental evi-dence showing its minor temperature sensitivity in the considered temperature range [53,54].

Table 3 Elastic modulus of steels for different temperatures [51]

T [ C] 20 204 427 537 649E [MPa] 207 186 155 134 124

Table 4 Thermal expansion coefficient of pearlitic steel for different temperatureranges [52]

T [ C] 0-100 0-200 0-300 0-400α [µm/m C] 11.6 12.6 13.3 14.0

The remaining parameters, except for the melting temperature Tm and activation energyQ, were determined from experimental data on C75 pearlitic steel (from monotonic tensiletests at different temperatures and creep tests at420 C). Details are given in [41]. The strainrate sensitivity and thermal function parameters are givenin Table 5, and those correspondingto the yield stress, hardening and recovery functions are given in Table 6. The significance ofthe different parameters is discussed in the following subsection.

Table 5 Strain rate sensitivity and thermal function parameters

Q [kJ/mol] Tm [ C] A [s −1] C [MPa] n350 1371 1.26 · 1011 91.20 3.82

2.3 Pearlite thermo-viscoplastic model

The model developed by Pinaet al. [41] is used here to account for the time and temperaturedependent behaviour of the pearlitic matrix. The model is based on earlier work by Freed andWalker [55]. The viscoplastic strain rate is based on the Zener-Hollomon relation [56,57], in



Table 6 Yield stress, hardening and recovery parameters.

τy(T ) [MPa]δ D0 [MPa] hD [MPa] m

20 C 350 C 420 C 500 C

0.075 9.12 3133 0.38 324 287 275 240

which the viscoplastic strain rate is obtained as the product of two functions: an Arrheniustype thermal function, and a function known as the Zener parameter. The thermal functionaccounts for the sensitivity of the viscoplastic strain rate to temperature, whereas the Zenerparameter characterizes the strain rate sensitivity. Thisprovides a simple way to incorporatethe influence of temperature on the viscoplastic strain rate. The evolution of the internalvariables due to isotropic hardening, as well as dynamic andthermal recovery are also takeninto account.

2.3.1 Kinematics

In the present model the kinematics is described in terms of the right Cauchy-Green straintensorC = FT

· F, whereF = (∇0x)T is the deformation gradient tensor with∇0 thegradient with respect to the reference configuration, andx the position vector in the currentdeformed configuration.

The material velocity is taken into account by the spatial velocity gradient tensorL andits symmetric and skew-symmetric parts, i.e., the rate-of-deformation tensorD and the spintensorΩ, respectively:

L = F · F −1 = D +Ω, (1)

where the superimposed dot denotes the material time derivative.The rate of the right Cauchy-Green deformation tensor is determined by taking the material

time derivative ofC, that is:

C = FT · F + FT· F = 2FT · D · F. (2)

The thermal and mechanical contributions to the thermo-mechanical deformation process aredefined through the multiplicative decomposition of the deformation gradient tensorF into athermal partFth and a mechanical partFM [58]:

F = Fth · FM . (3)

The tensorFM expresses the mapping from the reference configurationΩ0 to the intermediatemechanical configurationΩM , andFth maps the deformation fromΩM to the current config-urationΩ. The mechanical partFM is further decomposed into an elastic and a viscoplasticpart via:

FM = Fe · Fvp. (4)

The plastic flow is taken volume-preserving, i.e.Jvp = det(Fvp) = 1 [59].



Considering equations (3) and (4), the expression for the spatial velocity gradient tensorLin terms of its thermal, elastic and viscoplastic parts can be obtained as an additive decompo-sition:

L = Le + Lvp + Lth = (De +Ωe) + (Dvp +Ωvp) + (Dth +Ωth), (5)

whereLe, Lvp andLth are given by:

Le = Fth · Fe · Fe−1 · Fth

−1, (6a)

Lvp = Fth · Fe · Fvp · Fvp−1 · Fe

−1 · Fth−1, (6b)

Lth = Fth · Fth−1. (6c)

Finally, an expression for the additive split ofC in its thermal, elastic and viscoplastic partscan be obtained by introducing equations (3) and (4) into (2). This yields:

C = Ce + Cvp + Cth, (7)

with

Ce = FT ·

[

Fth · Fe · Fe−1 · Fth

−1 + Fth−T · Fe

−T · FeT· Fth

T]

· F

= 2FT· De · F, (8a)

Cvp = FT ·

[

Fth · Fe · Fvp · F −1 + F −T · FvpT· Fe

T · FthT]

· F

= 2FT· Dvp · F, (8b)

Cth = FT·

[

Fth · Fth−1 + Fth

−T· Fth

T]

· F = 2FT· Dth · F. (8c)

whereDe, Dvp andDth are respectively the elastic, viscoplastic and thermal rate-of-deformationtensors. The constitutive relations forDvp andDth will be defined in the following sections.

2.3.2 Thermal deformation

The thermal effects are incorporated in the model via the thermal part of the deformationgradient tensorFth, which is assumed to be purely volumetric [58,60]. For mechanically andthermally isotropic materialsFth, can be defined as:

Fth = φ(T )1/3I . (9)

The scalar functionφ(T ) describes the volumetric deformation induced by a temperaturechange(T − T0) relative to the reference temperatureT0. The simplest form forφ(T ) is[58,60]:

φ(T ) = 1 + α(T − T0), (10)

whereα is the volumetric thermal expansion coefficient which, in general, can be temperaturedependent.



Furthermore, from equations (6c) and (9) the symmetric and skew symmetric parts ofLth

can be obtained as:

Dth =1

2

[

Lth + LthT]

=1

3

φ(T )

φ(T )I = Lth, (11a)

Ωth =1

2

[

Lth − LthT]

= 0, (11b)

2.3.3 Elastic model

In the constitutive framework applied in this paper, the stress-strain relation is given by a smalldeformation elastic model which relates the rate of the 2nd Piola-Kirchhoff stress tensorS tothe rate of the elastic right Cauchy-Green strain tensorCe. The 2nd Piola-Kirchhoff stresstensorS is defined as:

S= JF −1· σ · F −T = F −1

· τ · F −T , (12)

whereσ andτ = Jσ are the Cauchy and the Kirchhoff stress tensors, respectively. Theconventional constitutive relation betweenS andCe is then expressed as:

S= 4Celas : Ce, (13)

where4Celas is the fourth-order elasticity tensor, for an isotropic material given by [61]:

4Celas = [µ− λln(Je)]C −1

e · 4IRT · C −1e + 1

2λC −1

e ⊗ C −1e , (14)

with 4IRT the right transpose of the fourth-order identity tensor defined as4IRT : A = A T .In (14),λ andµ are the elastic Lame’s constants.

2.3.4 Thermo-viscoplastic model

A classical expression is used to describe the viscoplasticflow, given byDvp, which assumesthat the viscoplastic deformation accumulates in the direction of the deviatoric stressτ d,whereby its magnitude is determined by the rate of the viscoplastic multiplierγvp:

Dvp = γvpN, (15)

with N the flow direction, defined in terms of the deviatoric part of the Kirchhoff stress tensoras:

N =3

2

τd

τeq, τeq =

√

(

3

2τd : τ d

)

. (16)

The definition of the plastic flow given by equation (15) is completed by postulating a zeroviscoplastic spin [59]:

Ωvp = 0. (17)

Next, γvp is defined by a Zener-Hollomon decomposition [55–57]:

γvp = θ(T )Z(ϕ,D), (18)



whereθ(T ) > 0 is an Arrhenius type thermal function andZ(ϕ,D) > 0 is the Zener param-eter; withD > 0 the drag strength, accounting for the isotropic hardening.

Viscoplastic deformation will take place provided that:

ϕ(τeq , τy) = τeq − τy(T ) ≥ 0. (19)

whereτy(T ) is the yield stress, which here depends on the temperature only, since hardeningis alredy accounted for in (18).

The Zener parameter is defined as [41]:

Z(ϕ,D) = A sinhn⟨ ϕ

D

⟩

= A sinhn⟨

τeq − τy(T )

D

⟩

, (20)

whereA > 0 andn > 0 are temperature independent rate sensitivity parameters.The expression forθ(T ) is given by [41]:

θ(T ) =

exp[

−QR T

]

if Tt ≤ T < Tm,

exp[

−QR Tt

(

ln(

Tt

T

)

+ 1)

]

if Tc < T ≤ Tt,

exp[

−QR Tt

(

ln(

Tt

Tc

)

+ 1)]

if 0 < T ≤ Tc.

(21)

whereQ is the activation energy for creep,R = 8.314 J/molK is the universal gas con-stant,Tm is the melting temperature andTt = Tm/2 andTc = 350 C are the transitiontemperatures.

Finally, the evolution equation for the drag strengthD is given by [55]:

D = h(D) [γvp − θ(T )r(D)] = θ(T )h(D) [Z(ϕ,D)− r(D)] , (22)

with the initial valueD = D0 the minimum or annealed drag strength. Equation (22) de-scribes the evolution of the drag strength as the result of the interaction of strain hardening,dynamic recovery viah(D), and thermal recovery viar(D). The corresponding expressionsfor h(D) andr(D) are:

h(D) = hD

[

(D −D0)/δ C

sinh[(D −D0)/δ C]

]m

= hD

[

ϕD

sinh[ϕD]

]m

, (23)

r(D) = A sinhn

[

(D −D0)

δ C

]

= A sinhn [ϕD] , (24)

with

ϕD =(D −D0)

δ C, (25)

whereC is the power-law breakdown strength,hD is the isotropic hardening modulus,δ andm are material parameters related to thermal and dynamic recovery.



2.3.5 Non-linear system of equations for the viscoplastic behaviour

To summarize, the viscoplastic constitutive model is described by the following set of equa-tions:

S= 4Celas :

[

C− 2FT · θ(T )Z(ϕ,D)N · F − 2FT · Dth · F]

, (26a)

D = θ(T )h(D) [Z(ϕ,D)− r(D)] . (26b)

where (26a) is obtained by substituting equations (7), (8b), (15) and (8c) in equation (13),with Dth given by (11a), and using expression (18) forγvp.

For the numerical solution of boundary value problems involving this constitutive model,a finite element formulation has been developed and implemented.

3 Numerical simulations on CGI RVE

In this section, the CGI RVE is used to investigate effect of time and temperature on themechanical response of CGI. First, monotonic tensile testsat 20, 350, 400, 450 and500 C areperformed to assess the influence of temperature. Then, the time-dependency is investigatedby means of stress relaxation tests, which are carried out atthe same temperatures as used inthe tensile tests.

3.1 Elevated temperature uniaxial tensile response

The tensile tests are performed by prescribing a macroscopic strain of 2.5 % to the RVE inthe horizontal direction (see Figure 1). The deformation isapplied in 50 seconds, i.e. with astrain rate of5 · 10−4 s −1. The periodic boundary conditions have been prescribed in such away that the overall stress state remains uniaxial, see Appendix A for the details.

The resulting macroscopic stress-strain response for the different temperatures is presentedin Figure 2. From the results shown in Figure 2.a, it is obvious that the mechanical perfor-mance degrades with increasing temperature. Figure 2.b, reveals that also the hardening ofthe material due to plastic deformation decreases with temperature.

The thermal softening observed in Figure 2.a, can also be noticed when comparing the ma-trix response at20 C and500 C. The drop in stress hardening with temperature can be tracedback to the decrease of drag strength with increasing temperature, following Equation (22),as shown in Figure 3. At higher temperatures, a higher activation energy promotes thermalrecovery which counteracts the effect of strain hardening due to plastic deformation. This isillustrated in Figure 4 where the contributions to the drag strength from plastic deformation,Dγvp

= h(D)∆tγvp, and thermal recovery,Dr = −h(D)∆tθ(T )r(D), are separately pre-sented. Figures 4.a and 4.b, reveal that the higher activation energy at500 C not only enablesthermal recovery but also enhances plastic flow, leading to ahigher contribution ofDγvp

tothe drag strength. Nevertheless, with the pearlite parameters used, the increase inDγvp

at500 C is counteracted by the thermal recovery such that the overall drag strengthD is lowerthan at20 C.

The high microstructural heterogeneity of cast irons (graphite morphology and propertiesof the phases), combined with the temperature-dependency,have an important effect on thedistribution of the local microstructural fields. As shown in Figures 5.a and 5.b, high stress andstrain concentrations arise at various locations surrounding the graphite particles. The most



0 100 200 300 400 500350

400

450

500

550

Temperature [ºC]S

tres

s h

ard

enin

g [

MP

a]

T = 20 ºC − Stress hardening = 535 MPa





a) b)

0 0.005 0.01 0.015 0.02 0.0250

100

200

300

400

500

600

700

800

900

Strain

Str

ess

[MP

a]

T = 20 ºC

T = 350 ºC

T = 400 ºC

T = 450 ºC

T = 500 ºC

Fig. 2 Influence of temperature on the uniaxial tensile response ofCGI as predicted by RVE simula-tions: a) homogenized macroscopic stress-strain response; b) stress hardening as a function of tempera-ture.

a) T = 20 ºC b) T = 500 ºC

65.00

57.00

48.98

40.97

32.96

24.95

16.93

8.920

0.912

D [MPa] Drag strength

Fig. 3 Uniaxial tensile test, drag strengthD in the matrix phase at the end of macroscopic loading: a)at20

C; b) at500

C.

critical areas, are those in the vicinity of anisotropic graphite particles whose long axis is ori-ented at45 with respect to the loading direction, especially in regions where several particlesare clustered together (see Figure 1). The anisotropic graphite particles oriented transverselyto the loading direction show little resistance to deformation upon loading. Consequently,high strains are observed within these graphite particles (Figure 6), and in the matrix adja-cent to the tip of the particles, which behaves as a notch in the matrix. At this spots, the hightstrain areas extend far into the matrix leading to the formation of strain percolation paths (Fig-ures 5.c and 5.d). At higher temperatures, this effect is enhanced even further as demonstratedin Figures 5.b and 5.d. At500 C, the stress fields are only half the magnitude of those at



Dr= -h(D) ∆t θ(T) r(D)

c) T = 20 ºC d) T = 500 ºC

a) T = 20 ºC b) T = 500 ºC

10.00

8.75

7.50

6.25

5.00

3.75

2.50

1.25

0.00

Dγvp

[10-2 MPa].

0.00

-1.25

-2.50

-3.75

-5.00

-6.25

-7.50

-8.75

-10.00

Dr[10-2 MPa]

Dγvp

= h(D) ∆t γvp

..

Fig. 4 Uniaxial tensile test, contributions to the drag strengthD from plastic deformation (top) andthermal recovery (bottom) at the end of macroscopic loading: left) at20

C; right) at500

C.

20 C, yet they entail higher and more extensive plastic strains.Therefore, it can be expectedthat damage will initiate earlier and propagate faster at higher temperatures.

3.2 Stress relaxation

The stress relaxation simulations are carried out at the same temperatures as the uniaxial ten-sile tests described in the previous section: 20, 350, 400, 450 and500C. The boundary con-ditions are prescribed such that an overall uniaxial stressstate is reproduced (see Appendix A).Two load steps are considered:i) pre-load: a macroscopic strain of 1% is prescribed on theRVE in the horizontal direction (see Figure 1) in 20 seconds,i.e. at a strain rate of5 ·10−4s−1;ii) stress relaxation: once the prescribed macroscopic strainis reached, the RVE is held in itsdeformed state for 2 hours allowing the stress to relax.

The results are presented in Figure 7, where the stress relaxation is defined as the drop instress between the peak stress reached at the end of the pre-loadτt=20s, and the stress at the



von Mises equivalent stress

Equivalent viscoplastic strain

c) T = 20 ºC d) T = 500 ºC

15.00

13.10

11.20

9.40

7.50

5.60

3.70

1.90

0.00

εvp

[%]

a) T = 20 ºC b) T = 500 ºC

950

844

738

631

525

419

313

206

100

vm[MPa]

Fig. 5 Uniaxial tensile test, equivalent von Mises stresses (top)and equivalent viscoplastic strain (bot-tom) in the matrix phase at the end of macroscopic loading: a)and c) at20

C; b) and d) at500

C.

end of the relaxation periodτt=2h:

Sr =

[

τt=2h − τt=20s

τt=2h

]

∗ 100%. (27)

Figure 7 gives a clear indication of the effect of the temperature on the time-dependent be-haviour of CGI. Below350C, there is a negligible difference in the relaxation response ofthe material. From this point onwards, increasing the temperature leads to a sharp increase inthe amount of stress relaxation.

The observed behaviour is driven by the elevated temperature response of the pearlitic ma-trix, captured by the model described in Section 2.3, and calibrated on the experimental datafor pearlitic steel [41]. Experiments indicated that pearlitic phases present a low sensitivityto temperature below350C, while above this temperature, the resistance of the material tocreep deteriorates drastically which explains the response observed in Figure 7.

In general terms, stress relaxation is driven by the time- and temperature-dependent plasticdeformation of the material. As discussed in the previous section, the resistance to plastic



ε11

strain component

a) T = 20 ºC b) T = 500 ºC

17.00

14.80

12.60

10.40

8.30

6.10

3.90

1.70

0.00

ε11

[%]

Fig. 6 Uniaxial tensile test,ε11 strain component in the graphite particles: a) at20

C; b) at500

C.

0 100 200 300 400 50012

13

14

15

16

17

18

19

Temperature [ºC]

Str

ess

rela

xati

on

[%

]

T = 20 ºC − Sr = 12.51 %

T = 350 ºC − Sr = 12.68 %

T = 400 ºC − Sr = 13.82 %

T = 450 ºC − Sr = 15.55 %

T = 500 ºC − Sr = 18.6 %

0 0.5 1 1.5 24

6

8

10

12

14

16

18

20

time [hours]

Str

ess

rela

xati

on

[%

]

T = 20 ºC − Sr =12.51 %

T = 350 ºC − Sr =12.68 %

T = 400 ºC − Sr =13.82 %

T = 450 ºC − Sr =15.55 %

T = 500 ºC − Sr =18.6 %

a) b)

Fig. 7 Stress relaxation of CGI as predicted by RVE simulations: a)stress relaxation percentage vs.time for the different temperatures; b) stress relaxation percentage as a function of temperature.

deformation is described in the model by the drag strength. At lower temperatures, stressescan exceed the drag strength and enable plastic deformationand hence stress relaxation. Athigher temperatures, the stresses are lower; yet, a higher stress relaxation is observed, whichcan be explained by the thermal recovery. As a consequence ofa higher activation energy,thermal recovery becomes more pronounced at higher temperatures. It counteracts the strainhardening effect of the drag strength enabling plastic deformation. Hence, in spite of the lowerstresses, a higher stress relaxation takes place at higher temperature.

To illustrate this situation, Figures 8 and 9 show the incremental viscoplastic strain∆tγvpand the contributions toD from the plastic deformation and thermal recovery in the matrix.Figure 8 reveals a marked difference in the matrix incremental viscoplastic strains∆tγvp, atthe end of stress relaxation at20C and500 C. Moreover, the difference in∆tγvp can be



a) T = 20 ºC b) T = 500 ºC

1.50

1.31

1.13

0.94

0.75

0.56

0.38

0.19

0.00

∆t γvp

[10-4]

∆t γvp

.

Fig. 8 Stress relaxation test, incremental viscoplastic strain∆tγvp in the matrix phase at the end ofstress relaxation: a) response at20

C; b) response at500

C.

traced back to the the drag strength and thermal recovery. The higher thermal recovery at500C (Figure 9) induces a lower drag strength. This results in a higher amount of plasticdeformation and hence more stress relaxation.

The combined effect of microstructural heterogeneity and temperature can be also appreci-ated in Figures 8 and 9. The presence of the graphite particles introduces stress concentrationsin their neighbourhood in the microstructure. At500C the contribution to the drag strengthD from thermal recovery is twice as high as the contribution from plastic deformation. There-fore, over time the drag strength decreases resulting in higher plastic strain rates and theformation of strain percolation paths (Figure 8). It is expected that this triggers damage at themicrostructural level.

4 Conclusions

The aim of this paper was to investigate the elevated temperature mechanical behaviour of castirons. For this purpose, a new microstructural based model for elevated temperature mechan-ical analyses of CGI has been developed. The model incorporates the time and temperaturedependent response of the matrix and the morphology of the graphite inclusions along withthe intrinsic mechanical anisotropy of graphite. The behaviour of the pearlitic matrix is mod-elled through a thermo-viscoplastic model [41] implemented in a finite element framework.To validate the representativeness of the material parameters identified on C75 pearlitic steelfor the behaviour of the pearlitic matrix, micro-indentation tests have been performed on thepearlitic matrix of CGI as well as on a C75 pearlitic steel. The results, in terms of Young’smodulus and hardness, revealed an adequate agreement between the two materials.

From the numerical results of the elevated temperature tests on CGI it can be concludedthat:

• The effect of temperature on the mechanical response of CGI is dominant for tempera-tures above350C. This is confirmed by the tensile tests as well as the stress relaxation



Dr= h(D) ∆t θ(T) r(D)

c) T = 20 ºC d) T = 500 ºC

a) T = 20 ºC b) T = 500 ºC

2.50

2.19

1.88

1.56

1.25

0.94

0.63

0.31

0.00

Dγ

vp

[10-4 MPa].

0.00

-0.63

-1.25

-1.88

-2.50

-3.13

-3.75

-4.38

-5.00

Dr[10-4 MPa]

Dγvp

= h(D) ∆t γvp

..

Fig. 9 Stress relaxation test, contributions to the drag strengthD from plastic deformation (top) andthermal recovery (bottom) at the end of stress relaxation: left) response at20

C; right) response at500

C.

tests. In the former, a sharp decrease in the hardening rate with temperature has beenobserved at the macroscopic level. In a similar fashion, a fast increase in the resultingstress relaxation is perceived for temperatures exceeding350C.

• For higher temperatures, the amount of thermal recovery overcomes the contributionof plastic deformation to hardening, leading to a reductionof the drag strength. This,directly influences either the hardening or the relaxation of the material at the local andglobal level.

• At lower temperatures, the effect of thermal recovery is negligible and the only driver forstress relaxation is dynamic recovery.

• At the microstructural level, the highly heterogeneous microstructure of cast irons leadsto stress concentrations in the areas surrounding the graphite particles. As the tempera-ture is increased, the combined effect of time, temperatureand heterogeneity promotes



local plastic deformation at the microstructural scale. This may accelerate the damageinitiation and evolution, therefore reducing the time to failure of the material.

Acknowledgements This research was carried out under the project number MC2.06270 in the frame-work of the Research Program of the Materials innovation institute (M2i) (www.m2i.nl).

A Computational homogenization

Within the framework of the computational homogenization adopted for the microstructuralmodelling [44], the microstructural displacement fields~um( ~Xm) of a given point with the po-sition vector~Xm in the reference configuration can be described through a macroscopic con-tribution (resulting from the macroscopic deformation gradient tensorFM ) and a microstruc-tural fluctuation field~uf( ~Xm) that represents the local variations with respect to the averagemacroscopic fields. Expressing the displacement~um( ~Xm) within the microscopic volumerelative to some arbitrarily chosen pointk yields

~um( ~Xm) = ~uk( ~Xk) + (FM − I ) · ( ~Xm − ~Xk) + ~uf ( ~Xm), (28)

where the subscriptsm, M andf refer to the micro and macro scale quantities and the micro-scopic fluctuation field, respectively.

In computational homogenization framework, the macro-to-micro scale transition is achievedby imposing that the macroscopic deformation gradient tensor should be equal to the volumeaverage of its microscopic counterpart, i.e.:

FM =1

V0

∫

V0

Fm dV0, (29)

From the expression of the microscopic displacement field (28), the microscopic deforma-tion gradient tensor can be obtained and introduced in equation (29) leading to the followingboundary integral:

∫

Γ0

~uf ⊗ ~n0 dΓ0 = 0, (30)

where~n0 is the normal to the RVE boundaryΓ0 in the reference configuration.This condition can be fulfilled by means of different boundary conditions, such as uni-

form displacement, uniform traction and periodic boundaryconditions. In this work, periodicboundary conditions are used since they yield better approximation to the apparent macro-scopic properties [44]. In the case of a 2D RVE, as used in thiswork (Figure 10), the periodicboundary conditions lead to the following displacement constraint relations [44]:

~u+m = ~u−

m + (FM − I) · ( ~X+m − ~X−

m), (31)

where superscript “+” refers to the top and right part of the boundary, while “−” refers to thebottom and left part (see Figure 10).

Furthermore, the macroscopic gradient tensorFM is prescribed through the RVE cornernodes 1,2 and 4 in the form of displacements as:

~up = (FM − I) · ~Xp, p = 1, 2, 4 (32)



21

x

y

V

Γleft

4 3

Γright

Γtop

Γbottom

Fig. 10 Schematic representation of a 2D unit cell showing the boundary conditions with prescribeddisplacements at corner nodes for a 2D macroscopic uniaxialtensile stress condition.

where~up are the prescribed microscopic displacements and~Xp are the position vectors of therespective corner nodes.

The micro-to-macro scale transition is based on the Hill-Mandel principle of macro-homo-geneity. Based on this principle it can be shown that the relation between the macroscopic andmicroscopic stresses is expressed as [44]:

PM =1

V0

∫

V0

Pm dV0 =1

V0

∑

p=1,2,4

~fp ⊗ ~Xp, (33)

whereP represents the first Piola-Kirchhoff stress tensor and~fp are the reaction forces at theprescribed corner nodes. To derive the second equation in (33), the equilibrium conditionon the RVE has been used, in combination with the divergence theorem and the periodicboundary conditions (31), see [44] for the details.

From (33) it is clear that to obtain a homogenized uniaxial stress state in the horizontaldirection, for the RVE illustrated in Figure 10, the following boundary conditions can beapplied on the corner nodes:

~u1 = ~0, u2x = u∗, u2

y = 0, ~f4 = ~0, (34)

with u∗ the prescribed horizontal displacement. All the other nodes on the RVE boundary arelinked through the periodic boundary conditions.

References

[1] K. B. Rundman, Cast Irons, in: Encyclopedia of Materials: Science and Technology, editedby K. H. J. Buschow, R. W. Cahn, M. C. Flemings, B. Ilschner, E.J. Kramer, S. Mahajan, andP. Veyssiere (Elsevier, Oxford, 2001), pp. 1003 – 1010.



[2] J. R. Davis (editor), ASM Specialty Handbook: Cast Irons(ASM International, 1996).[3] M. J. Dong, C. Prioul, and D. Francois, Metallurgical and Materials Transactions A28(11), 2245–

2254 (1997).[4] C. A. Cooper, R. Elliot, and R. J. Young, Acta Materialia50, 4037–4046 (2002).[5] P. Q. Dai, Z. R. He, W. Q. Wu, and Z. Y. Mao, Materials Science and Technology18(9), 1052–

1056 (2002).[6] P. Dierickx, C. Verdu, and A. Reynaud, Scripta Materialia 34(2), 261–268 (1996).[7] F. Iacovello, O. Di Bartolomeo, V. Di Cocco, and V. Piacente, Materials Science and Engineering

A 478, 181–186 (2008).[8] Z. R. He, G. X. Lin, and D. Ji, Materials Science and Engineering A234–236, 161–164 (1997).[9] H. Pirgazi, S. Ghodrat, and L. A. I. Kestens, Materials Characterization90, 13–20 (2014).

[10] C. G. Chao, T. S. Lui, and M. H. Hon, Metallurgical Transactions A19(5), 1213–1219 (1988).[11] C. G. Chao, T. S. Lui, and M. H. Hon, Metallurgical Transactions A20(3), 431–436 (1989).[12] C. G. Chao, T. S. Lui, and M. H. Hon, Journal of Materials Science24, 2610–2614 (1989).[13] R. B. Gundlach, AFS International Cast Metal Journal4(3), 11–20 (1979).[14] R. B. Gundlach, Transactions of the American Foundrymen’s Society87, 551–560 (1980).[15] R. B. Gundlach, Transactions of the American Foundrymen’s Society91, 389–422 (1983).[16] J. R. Kattus and B. McPherson, Report on Properties of Cast Iron at Elevated Temperatures -

ASTM Special Technical Publication No. 248 (American Society for Testing Materials, Philadel-phia, 1959).

[17] T. S. Lui and C. G. Chao, Journal of Materials Science24, 2503–2507 (1989).[18] P. Rodriguez and S. L. Mannan, Sadhana20(1), 123–164 (1995).[19] K. Roehrig, Transactions of the American Foundrymen’sSociety86, 75–88 (1978).[20] G. Winter, M. Riedler, R. Minichmayr, and W. Eichlseder, Cast iron versus aluminium cylinder

head materials: their properties concerning to thermo-mechanical fatigue, in: 22nd DANUBIA–ADRIA Symposium on Experimental Methods in Solid Mechanics, (Monticelli Terme/Parma–Italy, 2005).

[21] K. R. Ziegler and J. F. Wallace, Transactions of the American Foundrymen’s Society92, 735–748(1984).

[22] T. Seifert and H. Riedel, International Journal of Fatigue32(8), 1358–1367 (2010).[23] T. Seifert, G. Maier, A. Uihlein, K. H. Lang, and H. Riedel, International Journal of Fatigue

32(8), 1368–1377 (2010).[24] F. Szmytka, L. Remy, H. Maitournam, A. Koster, and M. Bourgeois, International Journal of

Plasticity26(6), 905–924 (2010).[25] F. Szmytka, M. H. Maitournam, and L. Remy, Computers & Structures119, 155–165 (2013).[26] F. Szmytka, P. Michaud, L. Remy, and A. Koster, International Journal of Fatigue55, 136–146

(2013).[27] L. Collini and G. Nicoletto, The Journal of Strain Analysis for Engineering Design40(2), 107–

116 (2005).[28] N. Bonora and A. Ruggiero, International Journal of Solids and Structures42, 1401–1424 (2005).[29] M. Kuna and D. Z. Sun, Journal De Physique IV6, 113–122 (1996).[30] D. Steglich and W. Brocks, Computational Materials Science9(1), 7–17 (1997).[31] L. Chuzhoy, R. E. DeVor, S. Kapoor, and D. Bammann, Journal of Manufacturing Science and

Engineering124(2), 162–169 (2002).[32] L. Chuzhoy, R. E. DeVor, D. J. Bammann, S. G. Kapoor, and A. J. Beaudoin, Journal of Manu-

facturing Science and Engineering125(2), 181–191 (2003).[33] L. Chuzhoy, R. E. DeVor, and S. G. Kapoor, Journal of Manufacturing Science and Engineering

125(2), 192–201 (2003).[34] N. K. Fukumasu, P. L. Pelegrino, G. Cueva, R. M. Souza, and A. Sinatora, Wear259, 1400–1407

(2005).[35] W. M. Mohammed, E. G. Ng, and M. A. Elbestawi, The International Journal of Advanced Man-

ufacturing Technology56(1-4), 233–244 (2011).



[36] G. Ljustina, R. Larsson, and M. Fagerstrom, Finite Elements in Analysis and Design80, 1–10(2014).

[37] G. Ljustina, M. Fagerstrom, and R. Larsson, European Journal of Mechanics-A/Solids37, 57–68(2013).

[38] J. C. Pina, S. Shafqat, V. G. Kouznetsova, H. J. P. M., andM. G. D. Geers(2015), submitted.[39] J. C. Pina, V. G. Kouznetsova, and M. G. D. Geers, Inernational Journal of Solids and Structures

(2015), In press.[40] W. M. Mohammed, E. G. Ng, and M. A. Elbestawi, The International Journal of Advanced Man-

ufacturing Technology56(1), 233–244 (2011).[41] J. C. Pina, V. G. Kouznetsova, and M. G. D. Geers, Mechanics of Time-Dependent Materials

18(3), 611–631 (2014).[42] S. Shao, S. Dawson, and M. Lampic, Materialwissenschaft und Werkstofftechnik29(8), 397–411

(1998).[43] S. C. Lee and Y. B. Chang, Metallurgical Transactions A22(11), 2645–2653 (1991).[44] V. G. Kouznetsova, W. A. M. Brekelmans, and F. P. T. Baaijens, Computational Mechanics27(1),

37–48 (2001).[45] A. Hatton, M. Engstler, P. Leidbenguth, and F. Mucklich, Advanced Engineering Materials13(3),

136–144 (2010).[46] D. Holmgren, R. Kallbom, and I. L. Svensson, Metallurgical and Materials Transactions A38(2),

268–275 (2007).[47] B. Miao, D. O. North Wood, W. Bian, K. Fang, and M. H. Fan, Journal of Materials Science

29(1), 255–261 (1994).[48] M. P. Shebatinov, Metal Science and Heat Treatment16(4), 288–293 (1974).[49] A. Velichko, A. Wiegmann, and F. Mucklich, Acta Materialia 57(17), 5023–5035 (2009).[50] O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence,and T. Weng, Journal of Applied Physics

41(8), 3373–3382 (1970).[51] G. E. Dieter, Mechanical Metallurgy (McGraw-Hill Education, UK, 1989).[52] AISI 1070 Steel, Hot Rolled, 19-32 mm (0.75-1.25 in) round, www.matweb.com, Accessed:

January 2014.[53] E. Garofalo, P. R. Malenock, and G. V. Smith, Determination of Elastic Constants, Special Tech-

nical Publication, American Society for Testing and Materials p. 10 (1952).[54] G. M. E. Cooke, Fire Safety Journal13(1), 45–54 (1988).[55] A. D. Freed and K. P. Walker, International Journal of Plasticity9, 213–242 (1993).[56] C. Zener and J. H. Hollomon, Transactions of ASM33, 163–235 (1944).[57] C. Zener and J. H. Hollomon, Journal of Applied Physics15(1), 22–32 (1944).[58] S. Hartmann, Fakultat fur Mathematik/Informatik und Maschinenbau, Technische Universitat

ClausthalFac3-12-02, 1–13 (2012).[59] E. A. de Souza Neto, D. Peric, and D. R. J. Owen, Computational Methods for Plasticity: Theory

and Applications (John Wiley & Sons, 2008).[60] A. Lion, International Journal of Plasticity16(5), 469–494 (2000).[61] R. L. J. M. Ubachs, P. J. G. Schreurs, and M. G. D. Geers, International Journal of Solids and

Structures42, 2533–2558 (2005).


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