Modeling and analysis of the electrical resistance measurement of ...

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2514 L. Shen et al. / Composites Science and Technology 67 (2007) 2513–2520measurement of the electrical behavior of composite laminates(both intact and damaged), and the other is the developmentof a mathematical model, which relates themeasured electrical behavior to the mechanical damage.Typical mathematical models include the artificial neuralnetwork [13] and the response surface [16,17] models. Inthese mathematical models first a system of functions, whichrelates the mechanical damages to the measured electricaldata, is assumed with some unknown coefficients. Thesecoefficients are then obtained by matching the well-defineddamage patterns to the measured electrical data. Once theunknown coefficients are determined, the models can predictdamage using the electrical data as input.The current techniques for the measurement of the electricalbehavior of composite laminates are the two-probe andfour-probe methods. The two-probe method is based onthe definition of resistance according to Ohm’s law. In thismethod two electrodes are used to measure the electricalresistance. However, contact resistance may be introduceddue to the imperfect bonding between the electrodes andthe composite laminates. Since the contact resistance isunknown and difficult to determined, and sometimes, canbe even more profound than the real resistance of the sampledue to poor contacts at the probe, in general the two-probemethod is not recommended in self-sensing measurements.Alternatively, the four-probe (resistance) method was proposed,in which a pair of electrodes are used for the currentinput, and another pair of electrodes for the voltage output[6–12]. Based on the measured voltage and current, the resistancebetween the voltage contacts is then derived. However,the validity of this approach to obtain an accurate measurementof the resistance has not been investigated. In the meantime, the potential and voltage change methods were alsointroduced. Both potential and voltage change methodsare based on the principle of the four-probe method withexception that the resistance is not calculated from the measuredvoltage and charged current. Instead, both methodsuse directly the measured potential and voltage change asthe input in the mathematical models for the prediction ofthe mechanical damages.In this study, the basic problems in 2-D were investigatedthrough numerical analysis, i.e., the validity rangeof the four-probe method and the relative magnitude ofthe voltage and resistance fractional changes due to damage.Two beam-type specimens with and without damagewere designed, and numerically simulated using the finiteelement method (FEM). For the two-probe and four-probemethods, various resistances and voltage changes were simulatedand compared. As a result, the validity and theapplicability of the four-probe methods were discussed.2. The two-probe and four-probe methods2.1. The two-probe methodResistance measurement: Fig. 1a schematically shows abody, on which a pair of electrodes with conductive wiresFig. 1. Schematic diagrams of two-probe method; (a) point electrodes forresistance measurement and (b) plate electrodes for conductivitymeasurement.is attached. This is a common practice to measure the electricalresistance between the two electrodes of the bodyusing the two-probe method. The two-probe method isbased on Ohm’s law, i.e., V = IR with V, I and R beingvoltage, current and resistance between the two electrodesrespectively. When connecting an electrical meter to thetwo ends of the conductive wires, a circuit is formed, inwhich a current is produced by the power of the meter.Based on the current and voltage and using Ohm’s law, aresistance is measured. However, the measured resistanceshould be carefully understood. In fact, in addition to thetarget resistance R 0 between the two electrodes of the specimen,the measured resistance R also includes the resistanceR 1 of electrodes with wires, the resistance R 2 of the meter,and the contact resistance R 3 due to the imperfect bondingbetween the electrodes and the body. The measured resistanceR can be approximated as the specimen’s resistanceR 0 , only if the sum of the other three resistances is smallcompared with the target resistance R 0 .Because the conductivities of the wires and electrodessuch as silver paint paste are many orders of magnitudehigher than that of the measured body, the resistance R 1of the wire is relatively small. Also, the meter resistanceR 2 can be controlled to be relatively small by using meterswith higher resolution. But the contact resistance R 3 is notalways controllable. For example, even though a perfectbonding may be achieved by careful treatment and preparation,some imperfect bonding may develop when the systemis subjected to loading during service or experimentalmeasurements. That is why the four-probe methods wereproposed by Wang et al. [6–12]. However, the conditionsunder which the four-probe method can be effectively usedfor the resistance measurement of composite panels have
L. Shen et al. / Composites Science and Technology 67 (2007) 2513–2520 2515not been studied comprehensively, and may result in significanterrors. This will be analyzed later.Conductivity measurement: When measuring the conductivitiesof materials such as composite panels with continuouscarbon fibers and polymer matrix, a rectangular orstrip panel with plate electrodes is commonly used asshown in Fig. 1b. Such a specimen can lead to a uniformcurrent density in the rectangular or strip panel. The silverpaint paste was usually used as electrodes. If it was carefullyprepared, the bonding between the electrodes andthe panel can be assumed perfect. In this case, the measuredresistance R can be approximated as the resistanceR 0 of the panel between two plate electrodes. Then, theconductivity of the composite panel r can be determinedas r = L/(RA), where, L and A are the length and crosssectionalarea of the panel, respectively. Since the compositepanel is orthotropic, the conductivities in the other twodirections will be different and can be measured similarly.2.2. The four-probe methodsResistance method: The initial design of the four-probemethod for a strip specimen by Wang et al. [6] is shownin Fig. 2a. The outer and inner pairs of electrodes wereused as current and voltage contacts, respectively. If thecurrent density between the pair of voltage contacts is uniform,the resistance of the segment between the voltagecontacts can be calculated through Ohm’s law:R ¼ V I ;where, V and I are the voltage and current from the voltageand current contacts, respectively. Because the two quantitiesof voltage and current are independent of the resistancescaused by electrodes, wires and imperfect electrodeabcSSLSilver Paint ElectrodesLA 1 A 2A 3 A 4B 1 B 2Silver Paint Electrodes2A1A2B1B2C1CFig. 2. Schematic diagrams of four-probe method for (a) volumeresistance measurement, (b) surface and oblique resistance measurement,and (c) multi-prove method.2D1D2E1EB 3B 4ð1Þbonding, the measured resistance is accurate. Based onthe measured resistance, the electrical conductivity canthen be obtained. But one should keep in mind that thecondition of uniform current density between the pair ofvoltage contacts must be met Eq. (1) to be valid. Otherwise,significant errors may occur. Therefore, sufficient spacingbetween the two voltage contacts and between the currentand voltage contacts was suggested [6]. However, it appearsthat a detailed investigation on the validity of the fourprobemethod has not been carried out.Furthermore, the four-probe method was used to measurethe resistance change of a strip of composite paneldue to damage such as the delamination induced by successiveindentations or impacts [7–9]. Fig. 2b is a schematicdiagram showing the application of the four-probemethod. The resistances between the electrodes locatedon the same side and different sides of the panel are namedas the surface and oblique resistance, respectively [10,11].In order to measure the change of surface resistance R A2 A 3between electrodes A 2 and A 3 due to the damage, electrodesA 1 and A 4 are used as current contacts, and the voltagebetween electrodes A 2 and A 3 before and after damage ismeasured. Then, the corresponding resistance before andafter damage is obtained using Ohm’s law, leading to theresistance change due to the damage. Similarly, the obliqueresistance change between electrodes A 2 and B 3 due to thedamage is measured with electrodes A 1 and B 4 used as currentcontacts.Potential and voltage change methods: As mentionedabove in the introduction, the potential and voltage changemethods have also been proposed by Wang et al. [11] inaddition to the four-probe resistance method (referred toas the four-probe method throughout the paper). Insteadof calculating resistance through Ohm’s law, the measuredpotential and voltage are directly adopted in sensing themechanical damage of the composite panels. The differencebetween the two methods is that in the potential method,the ground is used as one electrode for voltage contact.So, the measured voltage is the potential of the other electrode,thus the name potential method. However, since theground is also one electrode of the current contacts, thecontact resistance due to the electrode is involved in themeasurement. The voltage change method avoids the contactresistances. But it depends on the structural configurationand relative locations between the current and voltagecontacts. For example, different current contacts will leadto different voltage changes even though the fractionalchange in voltage can avoid the effect due to the currentmagnitude. Also, the voltage change depends on the panelsize even though the local information around a pair ofvoltage contacts holds the same. As a result, the trainedresponse surface or neural network can only be valid forthe specific composite panel used.The four-probe resistance method does not involve theabove mentioned problems caused by the potentialand voltage methods because the resistance changebetween two voltage contacts depends only on the material
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