Computational Materials Science 64 (2012) 306–311

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Mesoscale analysis of deformation and fracture in coated materials

R.R. Balokhonov a,⇑, V.A. Romanova a, S. Schmauder b, E. Schwab a

a Institute of Strength Physics and Materials Science, SB, RAS, 634021 Tomsk, Russiab Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre, University of Stuttgart, Germany

a r t i c l e i n f o

Article history:Received 1 November 2011Received in revised form 19 March 2012Accepted 4 April 2012Available online 6 May 2012

Keywords:Computational mechanicsBoundary-value problemCoated materialsStrain rateLocalized plastic flow and fracture

0927-0256/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.commatsci.2012.04.013

⇑ Corresponding author. Tel.: +7 3822 286937; fax:E-mail address: rusy@ispms.tsc.ru (R.R. Balokhono

a b s t r a c t

The deformation and fracture of a coated material are simulated. A dynamic boundary-value problem in aplane strain formulation is solved numerically by the finite-difference method. To simulate the mechan-ical response of the steel substrate use was made of the relaxation constitutive equation based on micro-scopic dislocation mechanisms. A fracture criterion takes into account crack origination and growth inthe elastic-brittle coating. Numerical experiments were conducted for varying strain rate of tensionand compression. Macroscopic behavior of the coated material is shown to be controlled by interrelatedprocesses of localized plastic flow in the substrate and cracking of the coating that strongly depends on anexternal strain rate.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Hard and superhard coating depositions, including nanostruc-tured coatings, and investigations of their mechanical, physicaland chemical properties have been the object of a great numberof papers, see, e.g., a review [1]. Particular emphasis has been placedon the microstructure, strength, stiffness, heat and corrosion resis-tance of coatings. Another important problem is investigations ofthe properties of a coated material as a whole. For instance, it isappropriate to perform microstructure-based numerical simula-tions of the ability of a material with a coating possessing certainphysical–mechanical properties to resist external loading.

Nowadays there are a large number of studies addressing mul-tidimensional numerical simulation and modeling of materialswhere explicit consideration is given to microstructure, for in-stance [2–7]. This involves development of constitutive modelsfor components of a composite.

Special attention is given to interfaces. The majority of publica-tions devoted to the interfacial problems considers the interfacialfracture, decohesion and debonding, see, e.g. [8–10]. Nevertheless,a comprehensive study of the phenomena related to the irregularinterface geometry effects has not been provided so far.

The main purpose of the paper is to investigate numerically themechanisms of deformation and fracture in a coated material withan irregular interface between the coating and the substrate undertension and compression at different strain rates.

ll rights reserved.

+7 3822 492576.v).

2. Formulation of the problem

The dynamic boundary-value problem is solved. Presented inFig. 1 is a coated material mesovolume (250 � 180 lm). This is alateral face of a flat experimental test piece. Since the size of thespecimen in the z-direction is about 2 cm, mechanical behaviorof the mesovolume is simulated within the plane strain formula-tion. The coating-substrate interface geometry corresponds to theconfiguration found experimentally and is accounted for explicitlyin the calculations (Fig. 1).

2.1. Basic equations, plastic flow in the substrate and fracture in thecoating

The total system of equations includes expressions for the strainrate tensor, the mass conservation law and the equations ofmotion:

_eij ¼12ð _ui;j þ _uj;iÞ; _V=V ¼ _ekk; rij;j ¼ q€ui; ð1Þ

where ui is the displacement vector, eij and rij are the strain andstress tensors, V is the specific volume and q is the current density,the upper dot and comma in the notations stand for the time andspace derivatives, respectively. The stress tensor is the sum ofspherical and deviatoric parts

_rij ¼ K _ekkdij þ _Sij; _Sij ¼ 2l _eij �13

_ekkdij � _epij

� �; ð2Þ

XB1 B3

B4

STE250 steel

Y

B2 FeB100 μm Brittle boron coating

Plastic steel substrate

(b) (a)

50 μm

Fig. 1. Microstructures of the coated material: (a) used in calculations and (b) experimental [12].

R.R. Balokhonov et al. / Computational Materials Science 64 (2012) 306–311 307

where K and l are the bulk and shear moduli, Sij is the deviatoricstress tensor, _ep

ij is the plastic strain rate tensor, and dij is theKronecker delta.

A physically-based constitutive model is used to describeplasticity of the steel [11]

_epij ¼

32

_epeq

reqSij;

_epeq ¼ _ep

r exp �G0

kT1� req � raðep

eqÞ~r

� �w" #z( )

;

T ¼ T0 þZ eeq

0

bq0Cv

reqdepeq;

where req and epeq are the equivalent stress and plastic strain, G0 is

the energy that a dislocation must have to overcome its short-rangebarrier solely by its thermal activation, ~r is the stress above whichthe barrier is crossed by a dislocation without any assistance fromthermal activation, _ep

r is the reference value of the plastic strain rate,T0 is the initial temperature, Cv is the heat capacity, b is the fractionof plastic work which is converted into heat, b ffi 1, z = 2/3 and w = 2for many metals [11]. ra ep

eq

� �¼ rs � ðrs � r0Þ exp �ep

eq=epr

� �is the

athemal part of the yield stress. This yield function describes isotro-pic strain hardening, where rs is the saturation stress, r0 is the yieldpoint, ep

r is the plastic strain reference value.To describe fracture of the boride coating use was made of the

Huber’s type criterion. The criterion is thought to poorly describefracture of brittle materials. In this work we show that when it isapplied to coated materials, if real geometry of interfaces is takeninto consideration that results in appearance of localized regions oftensile stresses under any type of external loading, the maximumdistortion energy criterion works fairly well and provides a correctdescription of crack propagation directions at tension and com-pression. Earlier the same conclusion was made for the fractureof reinforcing particles in metal-matrix composites [6]. We havemodified the criterion to account for the difference in strength val-ues of the tensile and compressive regions:

req ¼ Cten; if ekk > 0 and req ¼ Ccom; if ekk < 0 ð4Þ

where Ccom and Cten are the values of the tensile and compressivestrengths.

According to the criterion, Eq. (4), fracture occurs in the local re-gions undergoing bulk tension. The following fracture conditionsare prescribed for any local region of the coating: if the bulk defor-mation ekk assumes a positive value and req reaches its critical va-lue Ccom, then all components of the deviatoric stress tensor in thisregion are taken to be zero, and in the case of ekk < 0 and req P Cten,pressure P is equal to zero as well.

The boundary conditions on surfaces B1 and B3 simulate uniax-ial tension/compression along the X-direction, while on surfaces B2

and B4 they correspond to the free surface and symmetry condi-tions (Fig. 1).

_uxðx; y; tÞ ¼ const ¼ �v for t P 0; ðx; yÞ 2 B1;

_uxðx; y; tÞ ¼ const ¼ v for t P 0; ðx; yÞ 2 B3;

rijðx; y; tÞ � nj ¼ 0 for t P 0; ðx; yÞ 2 B2;

_uyðx; y; tÞ ¼ 0 for t P 0; ðx; yÞ 2 B4;

rxyðx; y; tÞ ¼ 0 fort P 0; ðx; yÞ 2 B1 [ B3 [ B4:

ð5Þ

Here t is the time, v is the constant loading velocity and nj is the nor-mal to surface B2.

2.2. Description of the numerical procedure

The system (1)–(5) in the plane strain formulation is solvednumerically by the finite-difference (FD) method [13]. In contrastto the finite-element (FE) method, where the solution of the sys-tem is approximated, the FD method approximates the derivativesentering this system.

The region presented in Fig. 1a is represented as a mesh contain-ing N uniform rectangular cells N = Nx � Ny. The mesh is ‘‘frozen’’into the material and is deformed together with it. The system ofequations for this mesh is replaced by a FD analog. Use is made ofan explicit conditionally stable scheme of the second order of accu-racy. For the time step, it is necessary that Courant criterion be sat-isfied as follows: Dt ¼ kC

hminCl

, where hmin is the minimum step of themesh, Cl is the longitudinal velocity of sound, and 0 < kC < 1 is theCourant ratio. The stability condition implies that an elastic wavewithin one time step would not cover the distance longer thanthe minimum mesh step.

The values of stress rij, strain eij, and density q are computed inthe cell centers (points I, II, III, IV in Fig. 2), while those of displace-ments ui and velocities _ui correspond to the mesh nodes (points 1,2 . . .. in Fig. 2). Let us use the following definition of partialderivatives:

@F@x¼ lim

D!0

RB Fð�n ��iÞdS

D;

@F@y¼ lim

D!0

RB Fð�n ��jÞdS

D;

�n ¼ @x@n

�iþ @y@n

�j ¼ @y@S

�i� @x@S

�j

where B is the boundary of region D, S is the arc length, �n is the nor-mal vector, and �t is the tangent vector (Fig. 2).

Applying Eq. (6) to the rectangular regions bounded by dashedlines (Fig. 2), we obtain for the stress derivatives correspondingto, e.g., node k

RC rijð�n ��iÞds ¼

RC rij

@y@S ds, from which ðrij;xÞk ¼

1D ðrI

ijðy2 � y1Þ þ rIIijðy3 � y2Þ þ rIII

ij ðy4 � y3Þ þ rIVij ðy1 � y4ÞÞ, andR

C rij ð�n � �jÞds ¼R

C rij@x@S ds, whence

ðrij;yÞk ¼1DðrI

ijðx2 � x1Þ þ rIIijðx3 � x2Þ þ rIII

ij ðx4 � x3Þ þ rIVij ðx1 � x4ÞÞ:

IV III

I II

m

x

3

4

2

1 k

Boron coating

5 8

7 6

y

D D

D

t

n

Steel substrate

Fig. 2. Schematic representation for approximating the space derivatives.

308 R.R. Balokhonov et al. / Computational Materials Science 64 (2012) 306–311

Derivatives ui,j, in their turn, correspond to the cell centers andare calculated from the values of ui in the surrounding nodes. Inparticular, for cell m we have

ðui;xÞm ¼1

2D

u5i þ u6

i

� �ðy6 � y5Þ þ ðu6

i þ u7i Þðy7 � y6Þþ

þðu7i þ u8

i Þðy8 � y7Þ þ ðu8 þ u5i Þðy5 � y8Þ

!;

ðui;yÞm ¼1

2D

u5i þ u6

i

� �ðx6 � x5Þ þ ðu6

i þ u7i Þðx7 � x6Þþ

þ u7i þ u8

i

� �ðx8 � x7Þ þ ðu8 þ u5

i Þðx5 � x8Þ

!;

where D is the area of the respective quadrangle. The computationis performed in time steps, moving from one layer n to another n + 1

_ui ¼unþ1

i � uni

Dt:

In modeling multi-phase materials the interface between thestructure components goes across the computational mesh nodes(Fig. 2, thick solid line), with the constitutive models for the coat-ing and substrate materials prescribed on either side of this inter-face. The continuity of displacements and normal stresses at thisinterface is, therefore, preserved and the conditions of an idealmechanical contact are satisfied.

3. Computational results

Diffusion borating technique is used to deposit FeB coating on alow-carbon mild steel substrate [12,14]. Experimental stress–strain curves for this type of steel under compression at differentstrain rates are presented in Fig. 3a by symbols. Solid lines corre-spond to the thermomechanical model predictions. Parameters ofthe model given by Eq. (3) were derived for this steel during thecomputations.

Graphically shown in Fig. 3b is the macroscopic response of themicrostructure investigated under different external loading condi-tions. According to the calculated dependences, the coated materialshows higher tolerance to compressive stresses than to externaltensile loading, that is typical for composite materials and, as thecalculations show, is associated with basic differences of fractureprocesses in the coating under tension and compression.

Fig. 4 shows the distribution of the stress tensor components un-der external compressive loading for the cases of needle-like andplane ‘‘coating-substrate’’ interfaces. For the first case the teeth isseen to experience tensile stresses in compression. The samepattern as in Fig. 4 is obtained in the case of external tension, thedifference being in the sign of the stress tensor components. Thus,the local tensile stresses develop in different places under tensionand compression. This fact is responsible for the difference infracture processes under tension and compression (Fig. 5).

In both tension and compression cracks originate in the localregions experiencing tensile loading. Under compression, theregions are situated at the sides of the boron teeth (regionsmarked by + in Fig. 4). Cracks successively nucleate on boridetooth sides and propagate along the axis of compression (Fig. 5,states 2–7). No formation of the main longitudinal crack is, how-ever, observed. The upper coating layer maintains its stressedstate and resists to the load, while multiple cracking of a boridetooth unloads the coated material in the intermediate sublayer(Fig. 5, state 7). The stress–strain curve in this case exhibits localdrops of the averaged stress, whose general level, however,continues to increase, and no catastrophic loss of strength isobserved (see Fig. 3b).

A different fracture pattern is found under external tension ofthe coated material. The crack nucleates in the local region ofhighest concentration of tensile stress, which is situated at aboron tooth base, and propagates in the boride coating towardsthe free surface of the specimen (Fig. 5, states 8–13). This unloadsthe coated material along the direction of applied tension. Adescending section (points 8–13 in Fig. 3) in the stress–strain curveappears.

Experimental evidences [14–16] for coated materials and forparticle-reinforced metal matrix composites show that cracks inthe coating and in the reinforcing particles under compression arelargely oriented along the direction of compression (see Fig. 6b asan example).

Numerical tests for the compression and tension of the micro-structure presented in Fig. 1 were carried out by varying the strainrates (Fig. 7). The calculations show a difference between the coatedmaterial responses to different types of loading: tension andcompression.

The following conclusions can be made. First, under a hightensile strain rate the fracture process intensifies in comparisonwith that observed under quasistatic loading (Figs. 5 and 8).Under quasistatic loading only one crack is propagated, whereasat high strain rate there arises a multiple cracking of thecoating. This is due to the fact that the stress under high strainrate increases very rapidly, and release waves from the firstcrack formation are not in time for unloading the nearby stressconcentration regions.

The second conclusion is that the higher the strain rate of com-pression, the less intensive the cracking of the coating (Fig. 9) and,as a result, the higher the dynamic strength of the coated material(Fig. 7). This effect is due to pronounced plasticity in the steel sub-strate. The higher the strain rate, the higher the yield stress in thesteel substrate, and hence, the lower the difference in mechanicalproperties between the substrate and coating materials. This re-sults in the lower stress concentrations near the interface andthe later cracking of the coating.

xxσ

0

-8

-2

-4

-6

0

yyσ

4

-4

2

0

-2

xyσ

1

-1

0.5

0

-0.5

Serrated interface

+ + + + + + +– – – – – – –

Plane interface

– Local compressive regions+ Local regions experiencing tensile loading

Fig. 4. Distribution of stress tensor components (�102 MPa) for needle-like (a) and plane ‘‘coating-substrate’’ interfaces (b) that corresponds to the point (1) in Fig. 3.

0

200

400

600

Quasistatic11025010008000

Experiment Calculation Strain rate, s-1

ε, %

<σ>, MPa

-0,3 -0,150 5 10 0,15

200

250

300

350

400

450

<σ>, MPa

ε, %

1 4

3

2 5

6

7

8

9

10

11

12

13

Compression

Tension

(b)(a)

Fig. 3. Dynamic properties of STE250 steel used as a substrate (a) and homogenized stress–strain curves for the coated steel under quasistatic tension and compression (b).

0

0

0.0025

0.005

0.0075

0.01

0

4

6

8

2 0

5

7.5

10

2.5

52

10

7

8 12

Fig. 5. Distributions of the equivalent stress (req, �102 MPa) for states (2–12) presented in Fig. 3.

Compression TensionTensile regions

Compressive region

7 12

Experiment

0

2

1

(a)

Equivalent plastic strain

Equivalent plastic strain

peqε ,%

(b)Fig. 6. Calculated results for states (7) and (12) presented in Figs. 3 and 5 (a) and microscopic section of a TiN coating deposited on stainless steel after nanoindentation [15](b).

R.R. Balokhonov et al. / Computational Materials Science 64 (2012) 306–311 309

0 0.564

66

68

70

<σ>, MPa

14 ,10 −× sε

ε=−0.3, %

ε=−0.32, %

ε=−0.34, %

1 1.5 2-0,4 -0,3 -0,2 0,20

40

50

60

70

<σ>, MPa

ε, %

103

3×103

8×103

12×103

24×103

1

2

5

4

3

6

(a) (b)

Fig. 7. Homogenized stresses vs. (a) specimen strain and (b) strain rates for fixed strains.

0

250

500

750

MPaeqσ

31 5

Fig. 8. Cracking of the coating at a strain rate of 24 � 103 s�1. Distribution of equivalent stresses and fractured regions corresponding to the states (1–6) shown in Fig. 7.

210×eqσMPa

0

4

8

12

0

0.2

0.4

0.6

peqε , %

0.8

(a) (b) (c) Fig. 9. Distributions of equivalent stresses and equivalent plastic strains (fractured regions in the coating are marked by black color) for different strain rates of compression:(a) 3 � 103 s�1, (b) 8 � 103 s�1, (c) 24 � 103 s�1. Total strain is – 0.37% (see Fig. 7).

310 R.R. Balokhonov et al. / Computational Materials Science 64 (2012) 306–311

4. Conclusions

The results of numerical simulations have shown that even inthe case of simple uniaxial compression of the material with a bor-ide coating, the boride teeth undergo tensile stresses, and localconcentrations of tensile and compressive stresses are developedthroughout the steel–boride interface. Note that in the volumeoccupied, these local regions of tension are comparable with thoseexperiencing compressive stresses, while tensile stresses in theirabsolute value can compare with the value of external compressive

loading. Formation of the regions experiencing tensile loading isassociated with the curvature of interfaces and their complicatedgeometry.

The computational investigations agree well with the experi-ments and show the following:

1. Cracks originate and propagate predominantly in the regions oflocal tension under any type of external loading. In the course ofexternal tension and compression, cracks originate in differentzones and propagate in different directions.

R.R. Balokhonov et al. / Computational Materials Science 64 (2012) 306–311 311

2. The serrated shape of the material–coating interface retardspropagation of the main longitudinal crack and prevents thecoating from peeling under compression of the specimen.

3. The onset of fracture decreases exponentially with the strainrate of compression, and only slightly depends on the strain rateof tension due to reduced plasticity. Moreover, under tensionthe fracture intensifies with the strain rate increasing.

Acknowledgments

The support of the President of Russian Federation (project MD-202.2011.8), Russian Foundation for Basic Research (project 12-01-00436-a) and Deutsche Forschungsgemeinschaft (projects SCHM746/84-1, SCHM 746/103-1) is gratefully acknowledged.

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