modeling of mechanical damage detection in cfrps via ......modeling of mechanical damage detection...

COMPOSITES

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Composites Science and Technology 67 (2007) 1518–1529

SCIENCE ANDTECHNOLOGY

Modeling of mechanical damage detection in CFRPs viaelectrical resistance

Z.H. Xia a,*, W.A. Curtin b

a Department of Mechanical Engineering, The University of Akron, Akron, OH44236, USAb Division of Engineering, Brown University, Providence, RI 02912, USA

Received 23 February 2006; received in revised form 10 July 2006; accepted 18 July 2006Available online 15 September 2006

Abstract

Carbon fiber reinforced polymer composites (CFRPs) are inherently multifunctional materials that, in addition to their primary func-tion as a structural material, allow for the sensing and monitoring of in situ damage nucleation and evolution by the measurement of thematerial electrical resistance. Here an analytic model is developed for the transverse (perpendicular to the fibers) electrical resistance ofpristine and damaged unidirectional composites, complementing earlier work on the longitudinal resistance. The ratio of transverse tolongitudinal resistance for undamaged materials provides a direct measure of the internal density of fiber–fiber electrical contacts, akey material parameter in linking to the response of damaged materials. Under uniaxial loading with evolving fiber breakage, the nor-malized transverse resistance versus strain is predicted to have exactly the same form as that for the longitudinal resistance. Numericalstudies show this agreement for uniform fiber–fiber contact distributions but, for random contact distributions, the longitudinal resistanceis larger than predicted while the transverse resistance is smaller; these differences are shown to arise as a result of the statistically-pref-erential breaking of longer fiber segments. Analysis of multiple numerical simulations shows that variations in the electrical resistance arenot directly correlated with variations in the stress–strain response. Thus, statistical methods are required to relate resistance to strain ordamage. The Weibull modulus of the resistance change increases with increasing applied strain, with values exceeding 10 and 20 for thetransverse and longitudinal resistance, respectively, demonstrating increasing reliability at higher damage levels and good correlation ofaverage resistance change to applied strain. The present study shows that both longitudinal and transverse resistance changes are sensitiveto damage in a predictable manner and can be used together to improve the reliability of damage assessment during loading of CFRPs.� 2006 Elsevier Ltd. All rights reserved.

Keywords: B. Modeling; A. PMCs; B. Electrical properties; B. Mechanical properties

1. Introduction

On-board damage assessment and life-prediction, orprognosis, is a key technology for extending the practicalmission life of air vehicles and other structural components.In carbon fiber reinforced polymer (CFRP) composites,used for many important structures such as helicopterrotors, fan blades, and pressure vessels, the mechanicaldeformation and electrical resistance are coupled, suggest-ing the possibility of real-time sensing that is safer and

0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.07.017

* Corresponding author. Tel.: +1 401 863 1477; fax: +1 401 863 9025.E-mail address: [email protected] (Z.H. Xia).

cheaper than conventional methods. The fact that the elec-trical resistance of a carbon fiber composite changes with‘‘damage’’ has been established by several experimentalstudies in the last decade [1–4], leading to some empiricalcorrelations. However, the precise nature of the actual dam-age states giving rise to the resistance change has not beenwell understood, making the empirical correlations interest-ing but not directly coupled to predictions of remainingstrength or life. To make the electrical response predictivefor damage assessment, modeling efforts at various scalesare desirable. Usually, composite components are complex,consisting of multiple plies, and have a variety of damagemodes. Here, we focus on individual unidirectional plies

mailto:[email protected]

NT

Nth

NT

Nth

E

NT

Nth

NT

Nth

E

Fig. 1. Schematic of voltage lead geometries and sample dimensions for(a) square and (b) hexagonal fiber arrays.

Z.H. Xia, W.A. Curtin / Composites Science and Technology 67 (2007) 1518–1529 1519

and establish the relationship between evolving fiber dam-age and the anisotropic electrical resistance. Other impor-tant damage modes, such as delamination and transversecracking in multi-ply specimens, are important but requirecontinuum models and are not the focus of this work. How-ever, the present work develops the single-ply constitutiverelations for resistance versus stress, including evolvingfiber damage, that can be used in computational studiesof resistance in damaged multi-ply laminates and actualcomponents.

Many models have been developed to characterize com-posite mechanical behavior, and in particular strength, dueto evolving fiber damage under loadings [5–9]. Differentcomputational techniques and algorithms have been devel-oped to simulate the failure process of the composites atmultiple length scales [10,11]. The mechanics modeling isthus well-established. The associated electrical responseof the composite has received much less attention. Todor-oki et al. developed Response Surface models to predictdelamination damage from measurements of electricalresistance [12]. Direct electro-mechanical modeling of com-posites under loading was performed by Park et al., whoderived an analytical model for the longitudinal resistancechange during loading that explained many features seen intheir experimental data [13,14]. Based on an analogy to themechanical ineffective length, i.e., the length over which abroken fiber recovers stress, Park et al. proposed the exis-tence of a ‘‘characteristic electrical ineffective length’’,which is the length over which a broken fiber recovers cur-rent-carrying capability and is nominally the typical dis-tance between electrical contacts of nearby fibers. Xiaet al. [15] built on these ideas to create coupled electricaland mechanical models to describe the longitudinal electri-cal resistance change due to fiber breaks in continuous fiberreinforced composites under longitudinal tensile loading.

An important aspect of the electrical response is theanisotropic nature of the conductivity or resistivity. Inthe longitudinal direction, individual fibers carry currentover long distances without the need for any fiber–fibercontacts. In the transverse direction, current is only trans-mitted through the existence of such fiber–fiber contacts.The difference in longitudinal and transverse resistancescontains information about the fiber–fiber contacts, andhence the ‘‘electrical ineffective length’’, which is thenrelated to the quantitative coupling of electrical resistanceto mechanical damage. The longitudinal and transverseresistances have been measured in many CFRP systems[16–19], but this information has not been related to dam-age. The purpose of the present work is to analyze thetransverse resistance, to relate it to the longitudinal resistiv-ity and the electrical ineffective length, and then to predicthow the transverse resistance evolves with damage. Wederive analytic models that are compared to numerical sim-ulations, which generally support the analytic models butwith some modifications. The overall results suggest thatusing both longitudinal and transverse resistance togethercan provide insight into the underlying damage evolution

and ultimately lead to a prognosis capability for predictingimpending failure in CFRP composite components.

The remainder of this paper is organized as follows. InSection 2, we present an analytical model and a numericalmicromechanics model for the determination of resistancesof undamaged composites. In Section 3, we develop ananalytical model for the transverse resistance associatedwith fiber breakage, and compare the analytical solutionswith the numerical results. The statistical variation in mea-sured resistance, and its correlation with mechanical dam-age and applied strain, is also discussed. In Section 4, wesummarize our results and discuss future directions forthe application of these models in the prediction of remain-ing strength and life of composites and in the creation ofmulti-ply and component-scale models for damage detec-tion using electrical resistance.

2. Transverse resistance for undamaged composites

We envision an array of nominally aligned carbon fibersin a matrix material, with fiber volume fraction Vf and fiberdiameter d. The overall dimensions of the sample are longi-tudinal length L, transverse length LT, and through-thick-ness length Lth. There are N fibers, with NT in thetransverse direction and Nth in the thickness direction, withN = NTNth, NT / LT, and Nth / Lth. Schematic viewsdown the longitudinal axis of composites with square andhexagonal fiber arrays are shown in Fig. 1.

In carbon fiber reinforced polymer composites, undulat-ing fibers lead to electrical contacts between fibers [19]. Dueto these fiber contacts, current can flow in the transversedirection through the fiber network, as schematicallyshown in Fig. 2a. Fiber contacts do not affect the currentflow in the longitudinal direction in an undamaged com-posite (Fig. 2b) although the contacts do cause the redistri-bution of current flow in a damaged composite. As a result,the CFRP composite has a highly anisotropic resistance.The average length of the fiber segments between contactsis defined as

dce ¼LN2N c

; ð1Þ

Fig. 2. Schematics showing boundary conditions and current paths for (a) transverse resistance and (b) longitudinal resistance.

1520 Z.H. Xia, W.A. Curtin / Composites Science and Technology 67 (2007) 1518–1529

where Nc is the total number of contact points in the com-posite and the factor 2 is introduced because each contactpoint connects two fibers. dce is also the ‘‘characteristicelectric ineffective length’’, i.e., the average length overwhich a broken fiber regains current carrying capabilitydue to the contacts. dce is the key material parameter forboth transverse resistance and the sensitivity of both trans-verse and longitudinal resistance to mechanical damage.We consider dce to be fixed during mechanical deformationof the material.

We now consider the anisotropic composite resistance.The longitudinal resistance of the undamaged unidirec-tional composite is readily calculated. Since the currentdoes not need to flow through the contact points, each fibercan be considered as an independent parallel resistor. Thelongitudinal resistance RL is thus

RL ¼qLN¼ q

pd2L4V fLTLth

� �: ð2Þ

where q is the fiber resistivity (resistance per unit length).For modeling the transverse resistance, we assume for sim-plicity that the contact points are distributed uniformly, as

Fig. 3. Schematic of fiber contacts in a composite: a fiber contact network (a)(b), each having an equivalent circuit (c).

shown in Fig. 3a for a square fiber array. Considering eachsegment of fiber between two neighboring contact points asa single resistor, the entire composite is then a resistor net-work. We approximate this resistor network as a set of NT

sheets of identical resistors of resistance Ri in series in thetransverse direction, as shown in Fig. 3b. The transverseresistance of the series of resistors is

RT ¼ N TRi: ð3Þ

Although the fiber segments are physically oriented alongthe longitudinal direction, they are electrically in parallelin the transverse direction, as illustrated in Fig. 3c. Thelength of the resistors in the sheets depends on how the fi-bers contact one another. For a square fiber distribution,each fiber is able to contact four neighboring fibers. We con-struct a unit cell for the electrical circuit, as schematicallyshown in Figs. 4a and c. From Figs. 4a and c, it is evidentthat the relevant resistor segment length is 2dce since the cur-rent does not flow in the through-thickness direction and inthe transverse direction it flows over two fiber contact spac-ings before reaching a contact with a neighboring fiber inthe transverse direction. Fig. 4e shows the equivalent circuit

consisting of a series of resistor sheets composed of parallel fiber segments

Fig. 4. Unit cells for uniform fiber–fiber contact distributions. Top view of possible fiber contact in (a) square and (b) hexagonal array. 3d view of unitcells for (c) square array, (d) hexagonal array (see Fig. 1), and corresponding equivalent circuits of (e, f).


for the unit cell in Fig. 4c, from which the resistance is cal-culated to be 2qdce, consistent with the effective resistorlength of 2dce. For a hexagonal fiber distribution, thereare additional contacts in the diagonal direction in the unitcell, as shown in Fig. 4b. If the fiber contact points are uni-formly distributed, there are two large unit cells in series inthe transverse direction, as shown in Fig. 4d. We have cal-culated the resistance of each unit cell by constructing thecorresponding equivalent resistor network connecting inputand output leads, as shown in Fig. 4f, and find resistances of2.12qdce and 2.54qdce, respectively. Because these two unitcells are in series, the overall effective resistance is the aver-age value 2.34qdce. The effective resistor length, on a per-fi-ber basis, is then 2.34dce. In general, the effective resistanceof a single sheet can be expressed as

Ri ¼ qðbdceÞ=N i; ð4Þwhere the geometry factor b equals 2 and �2.4 for squareand hexagonal fiber distributions, respectively. Further-more, each sheet resistor consists of Ni individual fiberresistors in parallel with Ni = NthL/(bdce). Thus, combiningEqs. (3) and (4) and using Eq. (1), the total transverse resis-tance is

RT ¼qðbdceÞ2

N thLNT ¼ qATðbdceÞ2

LT

LthL; ð5Þ

where AT is a geometric factor depending on the fiber ar-ray, with AT = 1, 2=

ffiffiffi3p

for the square and hexagonal ar-rays, respectively, as shown in Fig. 1. It can be seen fromEq. (5) that the transverse resistance is directly related tothe electrical ineffective length. The ratio of transverse tolongitudinal resistance is

RT

RL

¼ N 2T

bdce

L

� �2

¼ AT4V f

pLT

d

� �2 bdce

L

� �2

; ð6Þ

which depends only on geometrical factors, fiber volumefraction, fiber diameter, and the electrical ineffective length.Since all parameters aside from bdce are measurable a pri-

ori, the value of bdce can be derived directly from measure-ments of the transverse and longitudinal resistances. Eqs.(5) and (6) are the first main results of this paper.

To verify that the analytical solution of Eq. (6), derivedfor an idealized geometry, is appropriate for randomly dis-tributed contact points, the transverse resistance wascalculated numerically using a 3d electrical network modelin which the contact points are distributed at random


throughout the composite. In any one simulation, a speci-fied number of Nc fiber contact points are introduced.Numerically, each contact point is considered as a nodeof a 1D rod element and each rod element is assigned a resis-tance Rk = qdk, where q is the fiber resistivity (ohm/m) anddk is the length of the element (fiber segment) between twocontact points. Kirchoff’s law is applied at each nodal point,and Ohm’s law is applied to each resistor element. Satisfy-ing these two physical laws on all nodes generates a matrixequation in finite-element form and standard finite-elementsolution procedures can be used to calculate the voltage andcurrent distributions throughout the material, and the mea-sured electrical resistance, for any set of boundary condi-tions. We use a preconditioned conjugate gradient methodto solve the problem. Desired boundary conditions areapplied on the model by specifying voltages at the pointsor areas at which leads are attached to the physical speci-men. A ‘‘ballast resistor’’ is added in series with the sampleto simplify calculation of the sample resistance. Given thevalue of the ballast resistor Ro and voltage of the batteryVapp, the resistance of the sample can be determined fromthe voltage drop DV across the resistance Ro asRT = (Vapp/DV � 1)Ro. The sizes of the composite systemsused for all calculations reported here are NT = 80,Nth = 20 for square fiber distribution, NT = 40, Nth = 40for the hexagonal fiber distribution, with lengths varyingfrom L = 2–8 mm.

Fig. 5 shows the ratio of the transverse to longitudinalresistance as a function of the ratio of characteristic electri-cal ineffective length to composite length, bdce/L, in log–logform. At small bdce/L, the resistance ratio scales as

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0.001

ce /L

RT

R/L

L=4 mmL=6 mmL=4 mm (uniform)L=8 mm (uniform)L=2 mm (uniform)AnalyticalRevised analytical Square, L=2 mmSquare, L=4 mm

Hexagonal ( =2.4)

Square ( =2)

0.01 0.1 1

Fig. 5. Ratio of transverse resistance to longitudinal resistance RT/RL

versus normalized fiber contact density parameter bdce/L for undamagedcomposites. Note the power-law behavior with exponent 2 (dashed lines)for small bdce/L, as predicted by the analytical model of Eq. (6). Solid linesshow prediction including a finite-size correction (Eq. (7)). Solid symbols:square fiber distribution with sample size NT = 80, Nth = 20; opensymbols: hexagonal fiber distribution with sample size NT = 40, Nth = 40.

(bdce/L)2, in agreement with Eq. (6), and is in quantitativeagreement with Eq. (6) using the geometry factors b = and2.4 for square and hexagonal fiber distributions, respec-tively. Therefore, measurements of RT and RL can be usedderive the internal ineffective electric length dce for a realis-tic disordered system. The predictions of Eq. (6) do notmatch the numerical results in the regime bdce/L P 0.1; thisis an end effect. The fiber segments at the longitudinal endsof the sample are not involved in the transverse electricalcircuit and thus have zero voltage drop and zero current,as explicitly shown by example in the computed voltagedistribution in Fig. 6. When bdce approaches the compositelength L, the excluded lengths represent a large fraction ofthe composite length and the sample resistance rises rela-tive to the analytic model. To account for the end effect,we simply remove from consideration 1.25 layers of lengthbdce in the network model (0.625 on either side, chosen tobest-fit the data but close to the average distance of thenearest contact to the end of the fiber), effectively shorten-ing the length of the composite from L to L � 1.25(bdce).Eq. (5) then becomes

RT ¼qðbdceÞ2

N thLð1� 1:25bdce=LÞNT ð7Þ

and the ratio RT/RL follows. The prediction using Eq. (7) isalso shown in Fig. 5, and provides an excellent fit to thenumerical results. In most practical situations L� dce

and this correction is not necessary but it is important torecognize it for interpreting numerical simulations, whichmay not always be in this limit.

We now use Eq. (6) to derive the key parameter dce frompublished experimental data on longitudinal and transverseresistance. In the calculation, the geometric factors AT inthe transverse direction and 1/AT in the through-thicknessdirection in Eq. (6) are set equal to unity since thegeometric factor is equal to the ratio of transverse tothrough-thickness fiber–fiber spacings and we assume auniform fiber distribution in the composite. FromEq. (6), the electrical ineffective length is then bdce ¼dL=LT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðRT=RLÞp=4V f

p¼ d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðqT=qLÞp=4V f

p, where d, L, LT

and Vf are geometric parameters and RT/RL or qT/qL (ratioof transverse to longitudinal resistivities) is measured. Thelongitudinal, through-thickness and transverse resistancesof unidirectional composites with different fiber volumefractions were measured by several researchers [16,17].They found that the longitudinal conductivity (1/qL) is pro-portional to the volume fraction, in agreement with Eq. (2).Using their experimental results for longitudinal and trans-verse resistance, we calculate the electric ineffective lengthbdce as shown in Fig. 7. The electrical ineffective lengthdecreases exponentially with an increase in the fiber volumefraction, reflecting the fact that fiber contacts rapidly multi-ply as the fibers become closely packed. Park et al. [14]deduced the electrical ineffective length by considering thechange in longitudinal resistance as a function of strainand fiber damage (see below), and obtained results also

Fig. 6. Voltage distribution for transverse resistance measurement showing longitudinal end effect in which ends of fibers do not carry any current orvoltage drop.

0.01

0.1

1

10

100

0

Fiber volume fraction

(nl

ec)

mm( ,)

Longitudinal [14]Transverse [16]Through-thickness [16]Transverse [17]Through-thickness [17]

0.2 0.60.4 0.8

Fig. 7. Characteristic electrical length versus fiber volume fraction, asdeduced from Eq. (6) using experimental data on longitudinal, transverse,and/or through-thickness electrical resistance.


shown in Fig. 7, again reflecting an exponential form sim-ilar to that derived from Todoroki’s data. From these twoexperiments, typical values of bdce for fiber volume fractionin the range 0.5–0.6 can vary over a wide range, 0.02–2 mm. Such variation is not surprising since the density

of fiber–fiber contacts is likely very sensitive to the methodof material processing.

3. Transverse and longitudinal resistance versus damage for

tension loading

When the composite is subject to tension in the longitu-dinal fiber direction, some fibers will break due to the statis-tical distributions of fiber strength. These broken fibers willthen transfer load to other fibers, possibly inducing them tobreak as well. At some critical level of load, i.e., the tensilestrength, the propagation of fiber breaking will proceedacross the entire composite and the composite will fail.Two types of models, global load sharing (GLS) and localload sharing (LLS), have been developed to predict thecomposite failure process. The GLS model assumes thatthe load lost by a fiber at some axial position, due to break-age and slippage, is transferred equally to all unbroken(elastic) fibers in the cross-sectional plane [20,21]. Thus,the mechanical strains of all intact fibers at any arbitrarycross-section are equal, which permits simplified analyticaltreatment. In reality, the load carried by broken fibers islocally distributed (so-called Local Load Sharing (LLS))[7–9,22–24], leading to highly inhomogeneous internalstresses just prior to failure, with failure occurring fromthe unstable propagation of a critical damage cluster. Animportant feature emerging from LLS models is the


prediction of a size effect in the composite tensile strength.However, the differences in the stress–strain relation andextent of fiber breakage between GLS and LLS modelsare relatively small until just before failure [21], and hencean analytic GLS model is valuable for characterizing dam-age evolution prior to global failure.

The dependence of longitudinal electrical resistance ver-sus mechanical damage (fiber breaking) has been recentlyaddressed within the GLS and LLS frameworks. Usingthe GLS model assumption, Park et al. developed a corre-lation between the mechanical failure and longitudinal elec-trical resistance change [13]. Okabe et al. [9] proposed a 3Dshear lag model for more realistic calculations of the fiberdamage and pointed out that the local strain concentrationaround a cluster of fiber breaks can be neglected except justbefore fracture, and micrographs of damage in CFRPs at75% of fracture strain supported the calculations. Buildingon these results, Xia et al. developed a directly-coupledmechanical/electrical model based on a 3D shear–lagmodel to predict the resistance versus mechanical damage[15]. Here, we develop analytic GLS models for the trans-verse resistance versus mechanical damage under tensionloading. This complements the results obtained by Parket al. for longitudinal loading. In fact, we will show thatthe transverse and longitudinal resistance changes withdamage are identical, within the GLS assumption for thedamage evolution, and for uniform fiber contact spacings.Our analytic models will then be shown to agree well withLLS simulation models up to the point of failure.

The fiber strength distribution is described by the stan-dard Weibull model, where the probability of fracture ofa fiber of length d at applied tensile stress r is

P fðL; rÞ ¼ 1� exp � dL0

rr0

� �m� �; ð8Þ

where r0 is the characteristic strength of the fibers at alength L0 and m is the Weibull modulus describing the var-iability in strength at any fixed length [e.g., 6–11]. Withinthis framework, the number of fiber breaks in one parallelsheet of fiber segments of length d = bdce is

Nb ¼ N i 1� exp � bdce

dc

Eferc

� �m� �� ; ð9Þ

where we have introduced the applied strain e = r/Ef, withEf the fiber Young’s modulus. We have also introduced thecharacteristic mechanical ineffective length dc and the char-acteristic fiber strength rc that emerge naturally in the GLSmodel, given by

dc ¼r0rL1=m

0

s

!m=ðmþ1Þ

; rc ¼rm

0 sL0

r

� �1=ðmþ1Þ

;

where r is the fiber radius and s is the interfacial shear yieldstress. The fiber breakage reduces the number of current-carrying resistors in each parallel sheet and hence, follow-ing from Eq. (4), the transverse resistance becomes

RT ¼q0ðbdceÞN i � N b

NT: ð10Þ

Here q 0 is the resistivity of the piezoresistive fibers in ten-sion, which can be expressed as q 0 = (1 + ae)q where a isthe piezoresistance factor. Substituting Eq. (9) into Eq.(10) leads to

RT ¼ðbdceÞ2N T

N thLð1þ aeÞ exp

bdce

dc

Eferc

� �m� �: ð11Þ

The ratio of the resistance change DRT with applied strainto the zero-stress resistance RT0 is then

DRT

RT0

¼ ð1þ aeÞ expbdce

dc

Eferc

� �m� �� 1: ð12Þ

The transverse resistance change has the same form as thatfor longitudinal resistance derived by Park et al. [13]. InPark’s formula, the factor b is not included and the param-eter dce is used to fit experimental data. Here we see that theadditional factor of b enters, accounting for the differencebetween physical fiber–fiber contacts and effective currentflow paths through the fiber network; b as the same valuefor both longitudinal and transverse resistance. We haveverified (see data below also) that the form of Eq. (12),including the factor b, is relevant for the longitudinal resis-tance change with damage as well. As discussed by Parket al., the effect of fiber damage on the electrical resistanceis much larger than its effect on the mechanical strain be-cause the ‘‘characteristic ineffective electrical length’’ bdce

is typically much larger than the mechanical ineffectivelength dc, enhancing the argument in the exponent of Eq.(12). This is a feature that makes damage detection by elec-trical resistance very attractive in comparison to damagedetection by direct mechanical means. Eq. (12), and theidentical result for the longitudinal resistance change, isthe second main result of this paper.

To assess the validity and accuracy of the analytic resultof Eq. (12), which applies to both longitudinal and trans-verse resistances under tensile loading, we have comparedthe predictions of Eq. (12) to results from numerical simu-lations using a coupled shear–lag mechanical analysis andelectric network resistance analysis described in detail inRef. [15]. The electrical model is identical to that discussedabove, but with information on broken fibers and fiberstresses in the network exported from an accompanyingLocal Load Sharing mechanical simulation model similarto those in the literature [9,15]. The composite system stud-ied here is a carbon fiber reinforced polymer matrix com-posite with hexagonal and square fiber arrays. Themechanical and electrical properties of the fibers andmatrix used in this study are listed in Table 1, and corre-spond to a T700S carbon fiber/epoxy composite. Withthe mechanical and electrical parameters fixed, we varyonly the electrical ineffective length (fiber contact spacing)dce in the simulations, which can vary widely with materialand processing, as indicated in Fig. 7.

Table 1Fiber and matrix material properties used in this work

Property T700S Fiber Epoxy matrix

Fiber radius, r (lm) 3 –Elastic modulus, E (GPa) 230 3.4Poisson’s ratio, m 0.22 0.35Weibull modulus, m (L0 = 25 mm) 4.22 –Weibull strength, r0 (MPa) 2426 –Shear strength, ry (MPa) – 25Piezo factor a of fiber 5.41 or 0


Fig. 8 shows the longitudinal and transverse resistancechange as a function of applied strain for uniform fiber con-tact distributions for the hexagonal fiber array. If the fiber–fiber contacts are distributed uniformly, so that every fiberresistor segment has exactly the length dce, the predictionsof the analytical model match those of the numerical modelvery closely and the longitudinal and transverse resistancesare indeed identical, as predicted. This is also true forsquare fiber arrays (not shown). This demonstrates thatthe electrical GLS model works well for describing thedamage evolution up to the failure point in small sampleswith low fiber moduli. In particular, since both longitudinaland transverse cases are well-predicted, our results demon-strate that clusters of fiber breaks, formed by the local loadtransfer among nearby fibers and occurring generally inplanes perpendicular to the longitudinal axis, do not createa notable additional electrical anisotropy. With increasingclusters of damage, the electrical resistance should becomelarger in the longitudinal direction and have less of an effectin the transverse direction, but such effects are small for thesystem studied here (low fiber Weibull modulus and mod-erate load sharing) up to the point of failure.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

Strain

(

Res

ista

nce

ch

ang

eR/

R0)

Transverse (Random)Transverse (Uniform)

Longitudinal (Random)Longitudinal (Uniform)Analytical

L =1.45

T =0.87

L =1.0T =1.0

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Str

ess

(Mp

a)

curve

0.005 0.01 0.015 0.02

Fig. 8. Transverse and longitudinal resistance changes (DR/R0) versusapplied strain, as predicted by numerical simulations (symbols) andanalytical models (lines; Eqs. (12) and (13)) for uniform fiber–fibercontacts and one case of random fiber–fiber contacts. The correspondingstress–strain curves is also shown. [Hexagonal fiber array b = 2.4,piezoresistance factor = 0, dce = 100 lm, L = 4 mm, 1024 fibers,NT = 32, and Nth = 32].

For the random fiber contact distributions in the hexag-onal fiber array, the analytical result of Eq. (12) does not

agree with the data; the transverse and longitudinal electri-cal responses for one random distribution are shown inFig. 8 for comparison. The longitudinal resistance changesfaster with strain than predicted while the transverse resis-tance changes more slowly than predicted. We will discussthis further below but it is not due to the development ofdamage clusters since the difference does not occur for uni-form fiber contacts. The numerical results can nonethelessbe well-fit to the form of Eq. (12) if the characteristic elec-tric ineffective length bdce is multiplied by an additional fit-ting factor c,

DRT

RT0

¼ ð1þ aeÞ expcTbdce

dc

Eferc

� �m� �� 1;

DRL

RL0

¼ ð1þ aeÞ expcLbdce

dc

Eferc

� �m� �� 1; ð13Þ

i.e., there is an effective ‘‘characteristic ineffective electricallength’’ cbdce that can fit the data, as shown for the oneparticular case in Fig. 8 with cL = 1.45 for the longitudinalresponse and cT = 0.87 for the transverse response.

We have performed 25 simulations of electrical andmechanical response for square fiber arrays having differ-ent random fiber strengths (chosen from the same Weibulldistribution) and different random contact points (with thesame average dce). The results for the longitudinal andtransverse response are shown in Fig. 9. For each particu-lar random distribution, the electrical response is slightlydifferent; note, however, that the sample-to-sample varia-tions in stress–strain behavior are much smaller up to thepoint of failure. The statistical spread in the resistance ver-sus strain among the different samples is thus intrinsic to

0

0.05

0.1

0.15

0.2

0.25

0.3

0

Strain

( e

gnahc ecnatsise

RR

LR/

0L

)

Numerical (Transverse)Numerical (Longitudinal)Analytical

L =1.50 curves

650

1300

1950

0

)aP

M( ssertS

L =1.24

T =0.76

T =1.21

0.005 0.01 0.015 0.02

Fig. 9. Longitudinal and transverse resistance change versus appliedstrain as simulated for 30 different realizations of random fiber–fibercontact distributions, and predictions of the analytical model of Eq. (13)with adjustable c factor. Also shown are the stress–strain curves for the 30samples. [Square fiber array; fiber piezo-resistance factor = 0;dce = 125 lm, L = 2 mm, NT = Nth = 40].


the randomness in both fiber strengths and fiber contactspacings. Each resistance curve can be fit to the analyticalform of Eq. (13) with a particular value of c. The longitu-dinal resistance data is bounded by cL values of 1.24 and1.50 while the transverse resistance data is bounded by cT

values of 0.76 and 1.21. In all cases, the longitudinal resis-tance is larger than the transverse resistance.

Ignoring the statistical variations for the moment, theaverage values of cL and cT for longitudinal and transverseresistances differ. Performing a number of simulations forvarious values of bdce for both hexagonal and square fiberarrays and fitting the resulting resistance versus strain toEq. (13) using c as a fitting parameter yields the resultfor c vs. bdce shown in Fig. 10. The average value of c isessentially independent of the ineffective length bdce andindependent of the fiber arrangement. Furthermore, theaverage value of cT for transverse resistance is �0.95, justslightly less than unity, while the average value of cL forlongitudinal resistance is �1.4. Since the mechanical inef-fective length dc is 683 lm while the present results spanthe range of bdce = 100–1000 lm, the robustness of Eq.(13) with a constant value c is clear.

The differences between analytical and numericalresults, i.e., deviations from c = 1, stem from two factors:(i) longer fiber segments are statistically more likely tobreak, as evident from the length dependence of the Wei-bull model (Eq. (8)) and (ii) the current flow is non-uniformin the composites with random contact distributions and isnot precisely a series array of parallel resistor sheets. Wenow analyze these factors.

In the longitudinal case, the current flow is uniform inthe fibers. But since longer fiber segments are more likelyto break and since longer fibers have proportionally largerresistances, each such fiber break leads to an increase inresistance that is larger than that due to the breaking of

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0

βδ ce

γL,

γT

Longitudinalγ L

Uniform

Hexagonal (β =2.4)

Square (β =2)

Transverseγ T

400 800 1200

Fig. 10. Fitting factors c versus bdce for random fiber–fiber contactdistributions as obtained by fitting simulated resistance versus strain datato the model of Eq. (13).

an average segment length. This point was previously rec-ognized by Park [25] and implies that cL > 1. This effectcannot be accounted for exactly because the macroscopicresistance change due to a broken fiber depends not onlyon the resistance of the broken fiber segment itself (propor-tional to its length) but also on the surrounding resistornetwork configuration, which confers, on average, an addi-tional factor. Fig. 11 shows the longitudinal resistancechange upon breaking a single fiber in a random net-work versus the length ‘ of the broken fiber, as calcu-lated numerically for many different fiber segments in arandom network. The relationship is nearly linear withfiber length, as expect, scaling as DR/RL0 � 3.6825 ·10�6(1 + 1.485‘/dce) with RL0 = 0.1875 X. For randomly-distributed contact points, the distribution of fiber segmentlengths is a Poisson distribution, P ð‘Þ ¼ e�‘=dce=dce. Com-bining this with the probability of failure versus length(Eq. (8)), the average length of a broken fiber segment ‘b

versus applied strain can be computed. At low strains,the average broken length is ‘b ¼ 2dce and at high strainsthe average broken length approaches ‘b ¼ dce. Using thedata in Fig. 11, the effective resistance change due to anaverage broken fiber relative to an average fiber is, at most,cL = DR(2dce)/DR(dce) = 3.97/2.485 = 1.60, which is quiteclose to, but slightly larger than, the average value of�1.4 shown in Fig. 10.

In the transverse case, the lengths of the broken fibersare the same as for the longitudinal case, being governedby the fiber strength statistics and not by the fiber contactsor by the geometry of the electrical measurement. How-ever, the transverse current flow is highly inhomogeneous:longer fibers carry less current than the average and hencebreaking such fibers leads to smaller changes in resistancecompared to the uniform case. This is consistent with theobservation of cT < 1 for the transverse resistance. Pro-

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0

Normalized length of fiber break ( / ce)

(R

esis

tan

ce c

han

ge

R/R

0)

Transverse

Longitudinal

o Transverse X Longitudinal

2 4 6 8 10

Fig. 11. Longitudinal and transverse resistance changes due to a singlebroken fiber segment in a random fiber–fiber contact network versusnormalized broken fiber segment length (dce = 100 lm, hexagonal fiberarray).


ceeding further, Fig. 11 also shows the network transverseresistance change upon breaking a single fiber segment ver-sus the length of the broken segment (resistor) which, onaverage, follows the empirical form DR/RT0 � 1.085 ·10�5[1 � 0.3925 ln(‘/dce)]. The change in network resistanceis highly stochastic, with a weak trend of decreasing resis-tance with increasing resistor length. Shorter fibers havelower resistance and hence carry much more current, pro-viding a much larger resistance change upon being broken.However, Fig. 11 also shows that some resistor segmentscarry little or no current, and thus make no change tothe resistance when broken. The heterogeneity of the cur-rent distribution under application of a transverse voltageis also shown explicitly for one particular case in Fig. 12.Due to the statistically-preferential breaking of longer fibersegments, we can proceed similarly to the longitudinal caseand estimate that that the transverse cT should be no smal-

ler than cT = DR(2dce)/DR(dce) = 0.72, which is indeedslightly smaller than the simulation value of �0.95.

Finally, we return to the statistical distribution of theresistance versus strain shown by the multiple simulationdata in Fig. 9. This issue is important for damage detection,wherein the resistance change is measured and the strain ordamage is then determined. The variability in measuredresistance for nominally identical but statistically differentmaterial samples implies a limit to the accuracy of thestrain determination. The main question is whether ornot the statistical variations of the resistance change arerelated to the sample-to-sample variations in compositestress at a given strain. In other words, is a particular resis-tance change reflective of the particular damage in the com-posite? Because the resistance is a convolution of the

Fig. 12. Current distribution under transverse applied voltage in a slice ofcomposite perpendicular to the fiber axis, for a hexagonal fiber array withrandom fiber–fiber contacts. Hexagonal mesh shows fibers arrangement,with a fiber at each node; see also Fig. 3. The current flowing through eachfiber is highly varying due to the complex random network created by thefiber contacts.

mechanical damage and the randomness of the electricalcontact network, there is not necessarily a direct correla-tion. Fig. 9 shows that the sample-to-sample variations instress vs. strain are much smaller than the variations inresistance versus strain. Therefore, there is no direct sam-ple-to-sample correlation and the electrical resistance isinherently less predictable than the strain, implying anuncertainty in strain prediction using the measuredresistance.

To further study the statistical nature of the compositeresponse, we calculate the Weibull modulus of the resis-tance at different applied strain levels using a two-parame-ter Weibull distribution, as shown in Fig. 13. The Weibullmoduli of both transverse and longitudinal resistancesincrease with the applied strain, indicating increasing reli-ability as the point of failure is approached, which is agood feature for damage detection. Furthermore, the Wei-bull moduli of transverse and longitudinal resistances canexceed 10 at applied strains larger than about 80% and60% of the failure strain, respectively, indicating that thestrain can be deduced from the resistance with high reliabil-ity. There is thus a strong correlation between the averagemechanical damage and average electrical response. Just atcomposite failure, the Weibull modulus of the failure strainis 63 while those for the longitudinal and transverse resis-tance changes are 13.8 (18.3) and 9.4 (11.2) for the squareand hexagonal fiber arrays, respectively. The relatively highWeibull moduli of the resistance changes at failure implythat there are also good correlations between the failurestrain (or strength) and the resistance change and thereforeresistance data can be used to estimate the remainingstrength and life of the composites.

0

5

10

15

20

25

30

35

0 20 40 60 80 100 120

Percentage of average failure strain (%)

Wei

bu

ll m

od

ulu

s

Transverse

δce =50 μm

Weibull modulus offailure strain=63.4

Hexagonal

Longitudinal

100

100

100

50

100

Fig. 13. Weibull moduli of transverse and longitudinal resistance distri-butions versus applied strain. Solid lines and open symbols: square array;dashed lines and filled symbols: hexagonal array.


4. Discussion and conclusions

We have presented coupled electrical–mechanical mod-els to predict the relationship between mechanical damageand electrical response. These models provide a basis forusing measured electrical resistance to estimate the maxi-mum applied loads, whether intentional or inadvertent,and to assess in real time the severity of internal damageat the ply level in a component. Three new geometric/statis-tical material parameters, bdce, cT and cL, need to be deter-mined from experiments. The characteristic ineffectivelength parameter bdce links the fiber–fiber contact densityto the undamaged anisotropic electrical resistance, andthe damaged to undamaged electrical resistance. It dependsslightly on fiber geometry but is easily determined by mea-suring the ratio of transverse to longitudinal resistance ofundamaged composites (see Eq. (6) and Fig. 7). The cparameters are governed by an interplay of geometry andfiber statistics but are essentially constant (Fig. 10). Theycan be deduced by fitting the measured longitudinal andtransverse resistance changes versus strain to Eq. (13), orthe values obtained here can be used a priori. All other truematerial parameters (a,q,dc,rc,m,s) are fiber or matrixproperties that can be obtained a priori by other tests.Eq. (13) thus provide relationships between the evolvingcomposite strain and composite electrical resistancechanges due to internal fiber damage during tensile loadingof uniaxial composites.

According to our analytical and numerical analysis,both longitudinal and transverse resistance change can beused for assessment of fiber damage. The statistical varia-tions in longitudinal resistance are smaller than those ofthe transverse resistance and therefore longitudinal mea-surements give more precise estimates of strain. For exam-ple, in a T700s fiber/epoxy composite with Vf = 0.5,bdce = 0.5 mm, and other properties are listed in Table 1,a measured longitudinal resistance change DR/Ro = 0.1corresponds to a maximum strain attained during the priorload history of �0.0142 ± 0.001 while the same measuredtransverse resistance change corresponds to a maximumstrain of �0.0142 ± 0.002. The residual stiffness of thecomposite is 95 GPa, about 81% of original stiffness, withan uncertainty of ±9.2 GPa from longitudinal and±17.6 GPa from transverse resistance changes and usingthe stress–strain curve of the composite. However, detec-tion using the transverse resistance could be practicallymore attractive since the transverse resistance is one tothree orders of magnitude higher than the longitudinalresistance and thus may permit more-precise measurementsin realistic situations. In fact, using a combination of longi-tudinal and transverse resistance measurements with differ-ent spatial positions of the voltage leads can provideinformation about damage location as well as severity.We will discuss these practical issues in future work.

The models developed here use the Global Load Sharingmodel. As noted earlier, this model provides an upperbound relative to the true damage evolution that occurs

under conditions of Local Load Sharing. The GLS modelalso does not contain the volume dependence (size-scaling)of the strength that is important in large components.However, the GLS model provides a very good representa-tion of the composite damage at lower strains where dam-age detection is desirable. In practical application, wewould identify a maximum safe strain, either experimentalor computed by LLS mechanical models, which wouldaccount for size effects and other non-GLS effects, and thenuse the GLS model to determine the damage and internalstrain from the electrical resistance up to the attainmentof the safe strain level. Another mode of application is tomeasure electrical resistance on small patches or strips ofa composite component and apply the GLS model on thissmaller volume, where the differences with LLS will besmaller.

We have yet to address the critical ‘‘prognosis’’ issuesof time- and cycle-dependent material degradation andthe associated remaining strength and remaining life.However, the mechanical models needed for predictionexist (see, for instance, Refs. [26–30]) and our models hereprovide the direct connection between evolving fiber dam-age and electrical resistance. Thus, we anticipate that thedetection and prediction of time- and cycle-dependentdamage evolution can be obtained building on presentand past work.

The models we have developed here for unidirectionalplies of a composite can also be used as constitutive inputfor large-scale electrical–mechanical analysis of multi-plycomposites or laminates with various loadings or damagestates. The constitutive properties of the composite canbe derived from generalizations of Eqs. (2), (5) and (13).Specifically, the longitudinal, transverse, and through-thickness resistivities of a single ply can be expressed as

qL ¼ qðpd2=V fÞð1þ aeÞ exp ðcLbdce=dcÞðEfe=rcÞmb c;qT ¼ qATðbdceÞ2ð1þ aeÞ exp ðcTbdce=dcÞðEfe=rcÞmb c;qth ¼ qð1=ATÞðbdceÞ2ð1þ aeÞ exp ðcthbdce=dcÞðEfe=rcÞmb c;

ð14Þwhere it is expected that cth � cT. If, due to manufacturingmethods, the fiber–fiber contact distribution is anisotropic,i.e., the number of contacts influencing transverse andthrough-thickness current flow differs, then generalized bfactors may also be necessary (e.g., see Fig. 7). Modelsfor the electrical resistance of multi-ply laminates can beconstructed using Eqs. (14) suitably rotated to match thedesired ply lay-up sequence. Multi-ply structures also thenrequire additional contact resistances between neighboringplies to account for the fiber–fiber contacts that generateinter-ply electrical connectivity. In any case, the above for-mulas link the electrical resistance to fundamental materialproperties and mechanical deformation, and therefore aresuitable constitutive models for large-scale finite elementmodels of laminates or components. We have made pro-gress in this area, as well, on which we will report in futurepublications.


Acknowledgements

W.A.C. and Z.H.X. thank the US Air Force Office ofScientific Research and DARPA DSO for support of thiswork through grant FA9550-04-1-0402 from the Mechan-ics of Multi-functional Materials Program.

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