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

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2516 L. Shen et al. / Composites Science and Technology 67 (2007) 2513–2520properties and geometry structures of the local compositepanel, i.e., the panel material sufficiently far away fromthe voltage contacts has no or little effect on the resistancebetween the two voltage contacts. However, the existingmethod of using Eq. (1) to derive the resistance changebetween two voltage contacts cannot be valid for the highlyorthotropic composite panel. This problem has not beenaddressed and will be investigated using numerical experimentsin this study.Multi-probe method: Another application of the fourprobemethod was carried out by Todoraki et al. [16] whereit was called the multi-probe method, as shown in Fig. 2c.A constant electric current was charged from the left edge(1A and 2A) to the right edge (1E and 2E), and electricpotential changes at each electrode were measured withelectrodes 1E and 2E as the ground. So, the potentialchange in each electrode is actually the voltage change withrespect to electrode 1E or 2E as a fixed contact in a pair ofvoltage contacts.3. Numerical analysis3.1. A specimen without damageA 2-D strip specimen without delaminations is used tocarry out a numerical analysis, as shown in Fig. 3a. Thedimensions are similar to the specimen used by Wanget al. [10,11], i.e., L 0 = 200 mm, L 1 = 10 mm, L 2 =30mm, L 3 = 2 mm and H = 3.2 mm. The electrical conductivityin longitudinal direction is taken as r L =15X 1 mm 1and the electrical conductivity in the through-thicknessdirection r T is used as a variable to carry out a parametricstudy. For a composite panel the through-thickness conductivityr T (in the order of 10 3 X 1 mm 1 ) is usuallymuch smaller compared to the longitudinal conductivityr L . In order to measure the surface and oblique resistancesaccording to the two- and four-probe methods, the electrodesA 0 to A 6 and B 0 to B 6 are mounted on both sidesof the specimen (Fig. 3). The electrode width isL 3 = 2 mm and its conductivity is set at 10 8 X 1 mm 1 sothat the electrode resistances can be assumed negligible.aL 3L 0L 1 L 2A 0 A 1 A 2 A 3 A 4 A 5 A 6B 0 B 1B 2 B 3 B 4 B 5 B 6bL 3L 0L 1 L 2A 0 A 1 A 2 A 3 A 4 A 5 A 6B 0 B 1B 2 B 3 B 4 B 5 B 6Fig. 3. 2-D specimen and electrode locations (a) without delaminationand (b) with delamination.HHThe surface resistances between the pairs of electrodesA 1 –A 5 and A 2 –A 4 and the oblique resistances between electrodepairs A 1 –B 5 and A 2 –B 4 , namely R A1 A 5, R A2 A 4, andR A1 B 5, R A2 B 4, respectively, will be investigated using thetwo- and four-probe methods.According to the two-probe method, a unit current I isapplied as input to an electrode pair and the correspondingvoltage output V is obtained from numerical simulationusing the finite element method. Then, using Ohm’s law,the resistance R between the electrode pair is derived asR = V/I. Because no contact resistance is involved in thenumerical experiment, the corresponding value of resistancecan be used as target resistance to investigate thevalidity of the four-probe method.Based on the four-probe method, a unit current I isapplied as input to the electrode pair A 0 –A 6 or A 1 –A 5 ,and the voltage V between the electrode pair A 1 –A 5 orA 2 –A 4 as output is obtained from numerical experiments.Then, the surface resistance R between the electrode pairsA 1 –A 5 or A 2 –A 4 is derived as R = V/I according to Chung[18]. For the oblique resistance measurements, instead, aunit current I is applied as input to the electrode pair A 0 –B 6 or A 1 –B 5 , and the voltage V between the electrode pairA 1 –B 5 or A 2 –B 4 as output is obtained from numerical experiments.Subsequently, the oblique resistance R between theelectrode pairs A 1 –B 5 or A 2 –B 4 is derived as R = V/I.As mentioned previously, the four-probe method is validfor the resistance measurement if the current densitybetween the voltage contacts is uniform. The numericalexperiment is designed to investigate under what conditionsthis uniformity assumption can be met. For this purpose,various through-thickness conductivities such asfrom 0.001 to 20 X 1 mm 1 are used. It will be seen thatwhen the through-thickness conductivity is not small comparedto the longitudinal one, as is the case for isotropicmaterials, the four-probe method works well. Otherwise,its accuracy is poor.3.2. A specimen with a delaminationFig. 3b shows the same specimen as shown in Fig. 3a butwith a delamination located at its center with a length of25 mm. The longitudinal and through-thickness conductivitiesr L and r T are taken as r L =15X 1 mm 1 andr T = 0.005 X 1 mm 1 , respectively, which are close to thereal values for the composite panel. The surface resistancesR A1 A 5and R A2 A 4, the oblique resistances R A1 B 5and R A2 B 4, andthe through thickness resistance R A3 B 3are numericallyobtained according to the two-probe method. Since nocontact resistance is involved in the numerical calculationby the two-probe method, the resistances obtained by thetwo-probe method are exact and serve to check the accuracyof the four-probe method. The corresponding resistancesare also obtained using the four-probe method,where the outside electrode pairs are used as the currentcontacts for the case of surface/oblique resistance. Forexample, to obtain R A1 A 5or R A2 B 4, the electrode pair A 0 –
L. Shen et al. / Composites Science and Technology 67 (2007) 2513–2520 2517Fig. 4. Finite element mesh for (a) specimen 1 (no delamination), and (b)specimen 2 (with delamination).A 6 or A 1 –B 5 is used as the current contacts and the voltagebetween the electrode pair A 1 –A 5 or A 2 –B 4 needs to beobtained numerically. The resistance change for each casedue to delamination is obtained by comparing it with thecorresponding value of the specimen without delamination.For the case of the through thickness resistance R A3 B 3, twopairs of current contacts are used, i.e., A 2 –B 2 and A 2 –B 4 .As voltage percentage change was used in the literature,another purpose of the numerical analysis is to check if thevoltage and resistance percentage changes due to thedelamination using the four-probe method are close tothe accurate resistance percentage change.3.3. FEM simulationsThe commercially available FEM code, ABAQUS, waschosen for this study. To ensure convergence, very finemeshes (9310 and 9598 linear quadrilateral elements) wereused for specimens 1 and 2, respectively. The meshesaround the electrodes A 3 and B 3 are shown in Fig. 4aand b for specimens 1 and 2.4. Results and discussion4.1. The validity range of the four-probe methodSince the finite element method is used in this study, forthe two-probe method, the numerical calculations do notinvolve any resistances caused by electrodes with wiresR 1 , the meter R 2 , and the contacts R 3 in the calculation.Thus it can be used to represent the exact solution R 0 forthe system. In reality, the measured resistance throughthe two-probe method in general involves the contact resistanceR 3 and other resistances; it can not be used as accurateresistance of the sample, which is also the reason thatthe four-probe method was introduced. To be able to checkthe validity of the four-probe method, the resistanceobtained from the definition, which is the same as the resistancefrom the two-probe method without contact, wire,and meter’s resistances involved, is used as reference(two-probe), then the resistance from the four-probemethod is calculated through the ratio of the measuredvoltage to the input current. The comparison of the resistancesbetween the two is demonstrated below.The first specimen without delamination shown inFig. 3a is designed to check the validity of the four-probemethod numerically. Based on the accurate theoreticalresults, the resistances at various locations are calculatedusing the four-probe method. The surface and obliqueresistances for various through-thickness conductivitiesare listed in Table 1, and also plotted in Figs. 5 and 6.From these comparisons, it is clearly seen that when thethrough-thickness conductivity is not too small comparedto the longitudinal conductivity, the four-probe methodcan work well. For the present case, if the longitudinalconductivity is 15 X 1 mm 1 , then the through-thicknessconductivity should be larger than 1 X 1 mm 1 for thefour-probe method to be applicable. However, for commonlyused composite panels, the through-thickness conductivityis much smaller than the longitudinal one, sayaround 0.005 X 1 mm 1 . Thus, the four-probe methodcannot be effectively used to measure the surface and obliqueresistances for real composite panels. The underlyingreason for this is that when the through-thickness conductivityis small, the voltage in the through-thickness directionexists due to the dominant electrical conduction inthe longitudinal direction. Fig. 7 shows the contour plotof the current density vector, where the current density0.005 A/mm is applied between electrodes A 0 and A 6 . Also,the potential at each electrode and the magnitude of thecurrent density at the middle of two adjacent electrodesare displayed in Fig. 7. Voltage drop in the through-thicknessdirection was also reported based on measurementsTable 1Comparison of the electrical resistances between the accurate and four-probe methods (r L =15X 1 mm 1 )r T (X 1 mm 1 ) R (X) A1A5 R (X) A2A4 R (X) A1B5 R (X) A2B4Two-probe Four-probe Two-probe Four-probe Two-probe Four-probe Two-probe Four-probe0.001 2.123 1.2157 1.693 0.6712 3.211 2.1597 2.992 1.7970.005 1.135 0.5433 0.9005 0.2983 1.218 0.6100 1.093 0.42420.01 0.8156 0.4011 0.6762 0.2186 0.8717 0.4143 0.7537 0.25800.1 0.4002 0.2548 0.2753 0.1293 0.4002 0.2547 0.2761 0.12941 0.2798 0.2474 0.1561 0.1237 0.2798 0.2474 0.1561 0.12375 0.2543 0.2450 0.1318 0.1225 0.2543 0.2450 0.1318 0.122510 0.2486 0.2434 0.1269 0.1217 0.2486 0.2434 0.1269 0.121715 0.2459 0.2424 0.1247 0.1212 0.2459 0.2424 0.1247 0.121220 0.2443 0.2415 0.1235 0.1208 0.2443 0.2415 0.1235 0.1208
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