Theoretical and Experimental Investigation of Magnetostrictive Tagged Composite Beams

Theoretical and Experimental Investigation of Magnetostrictive Tagged Composite Beams by Florin G. Jichi Bachelor of Science Technical University of Timisoara, Romania, 1995 Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mechanical Engineering College of Engineering & Information Technology University of South Carolina 2000 Department of Mechanical Engineering Director of Thesis Department of Mechanical Engineering Second Reader Dean of the Graduate School

ACKNOWLEDGEMENTS As any research endeavor cannot be done alone, but more likely is a team effort, there are many members involved. Therefore, I would like to express my gratitude and deeply thanks to Dr. Victor Giurgiutiu for all his guidance and help throughout my entire time spent here at USC. Without his support, none of what I accomplished would not have been possible. In addition, I would like to sincerely thank Dr Abdel-Moez E. Bayoumi for his time from his busy schedule. My gratitude also goes to, Adrian Cuc, Paulette Goodman, Jingjing Jack Bao, Greg Nall, Radu Pomirleanu, and Andrei Zagrai, my colleagues, who made my time spent here so pleasant and fun. I would like to express my forever gratitude and thanks to my mother, father, sister and brother-in-law for their trust, incommensurable support, love and encouragement.

ABSTRACT Among novel non-destructive evaluation techniques for structural health monitoring, the magnetostrictive-tagged fiber-reinforced composites stand out as especially suitable due to: (a) distributed sensory properties; (b) non-contact damage detection; and (c) straight forward manufacturing implementation. Experimental data and mathematical modeling of a magnetostrictive-tagged fiber reinforced composite specimen under bending (flexural) loading are presented. A brief review of the state of the art identifies previous work on axially loaded magnetostrictive composites, but finds no previous work on bending. Description of bending specimen design and fabrication is followed by the theoretical analysis and by the description of the experimental set-up and equipment used. Several analysis models were used. Test data, with and without magnetic annealing between loading cycles, is presented and results are discussed. Numerical values for the stress and strain versus magnetic flux density coefficients are given for both annealed and non-annealed cases. Piezomagnetic coefficients for the magnetostrictive composite are calculated. The correlation between the results developed in the presented paper for bending and previously published results for axial loading is found to be within 10% after correction factors depending on the quantity of the magnetostrictive material are applied. In conclusion, the usefulness of this method for structural health monitoring and further work are discussed.

TABLE OF CONTENTS ACKNOWLEDGEMENTS...II ABSTRACT... III TABLE OF CONTENTS... IV LIST OF FIGURES... VI LIST OF TABLES...XII 1 INTRODUCTION...1 1.1 PREVIOUS WORK ON MAGNETOSTRICTIVE TAGGED COMPOSITES FOR STRUCTURAL HEALTH MONITORING (TENSILE EXPERIMENTS)...1 1.2 CHARACTERISTICS OF ETREMA S TERFENOL-D MAGNETOSTRICTIVE MATERIAL...5 1.3 PRESENT INVESTIGATION...7 2 ANALYSIS OF A MAGNETOSTRICTIVE COMPOSITE BEAM...8 2.1 STATIC ANALYSIS OF SIMPLY SUPPORTED COMPOSITE BEAM UNDER CENTRAL LOAD...9 2.2 ANALYSIS OF A MAGNETOSTRICTIVE LAMINATED COMPOSITE...10 2.2.1 Micromechanics analysis of the MS tagged composite specimen...10 2.2.2 Lamination Analysis...13 2.3 7-PLY BALANCED-ORTHOTROPIC MODEL...17 2.4 14-PLY CROSS-PLY ANGLE MODEL...20 2.5 28-PLY CROSS-PLY ANGLE MODEL...23 2.6 56-PLY CROSS-PLY ANGLE MODEL...25 2.7 CONVERGENCE ANALYSIS...25 2.7.1 Convergence of the strain...26

2.7.2 Convergence of stress...30 3 MAGNETOSTRICTIVE COMPOSITE BEAM EXPERIMENT...39 3.1 SPECIMEN DIMENSIONAL DESIGN...39 3.2 DESCRIPTION OF THE MS TAGGED COMPOSITE SPECIMEN...43 3.3 SPECIMEN PREPARATION FOR THE EXPERIMENT...44 3.4 EXPERIMENTAL DESIGN...45 3.4.1 Description of the experiment...45 3.4.1.1 Clamping fixture design...48 3.4.1.2 Protective wood fixture design for the strengthen the gaussmeter probe...48 3.4.2 List of the equipment used...49 3.4.3 Equipment Calibration...50 3.4.4 LabView Virtual Instrument Environment...51 4 EXPERIMENTAL PROCEDURE AND RESULTS...54 4.1 MAGNETIC ANNEALING...54 4.2 TESTING PROCEDURE FOR MS TAGGED COMPOSITE BENDING EXPERIMENT...54 4.3 RESULTS WITHOUT MAGNETIC ANNEALING...55 4.3.1 Displacement and strain without magnetic annealing...55 4.3.2 Magnetic flux density without magnetic annealing...59 4.4 RESULTS WITH MAGNETIC ANNEALING BETWEEN LOADING-UNLOADING CYCLES...63 4.4.1 Displacement and strain with magnetic annealing between loading-unloading cycles...63 4.4.2 Magnetic flux density with magnetic annealing between loading-unloading cycles66 4.5 MAGNETIC FLUX DENSITY ANALYSIS...70 4.6 PIEZOMAGNETIC STRESS AND STRAIN COEFFICIENTS...72 4.6.1 Piezomagnetic stress coefficient, e 31 without magnetic annealing...73

4.6.2 Piezomagnetic stress coefficient, e 31, with magnetic annealing...74 4.6.3 Piezomagnetic stress coefficient,e 31,using experimental magnetic flux density without magnetic annealing and the 56-ply strain results model...75 4.6.4 Piezomagnetic stress coefficient, e 31 using experimental magnetic flux density with magnetic annealing between loading-unloading cycles and the 56-ply strain results model...77 4.6.5 Piezomagnetic strain coefficient d 31 using experimental magnetic flux density without magnetic annealing and 56-ply stress model...78 4.6.6 Piezomagnetic strain coefficient,d 31,for annealed specimen experiment and 56-ply stress model...79 5 ANALYSIS AND DISCUSSION...80 5.1 COMPARISON OF PIEZOMAGNETIC COEFFICIENTS FOR MAGNETOSTRICTIVE TAGGED COMPOSITES WITH PREVIOUSLY PUBLISHED DATA...80 5.2 COMPARISON BETWEEN EXPERIMENTAL, DESIGN AND MODEL STRAIN FOR WITH AND WITHOUT MAGNETIC ANNEALING...83 5.3 COMPARISON BETWEEN EXPERIMENTAL AND MODEL PIEZOMAGNETIC STRESS COEFFICIENT WITHOUT MAGNETIC ANNEALING...84 5.4 COMPARISON BETWEEN EXPERIMENTAL AND MODEL PIEZOMAGNETIC STRESS COEFFICIENT WITH MAGNETIC ANNEALING...84 6 CONCLUSIONS...86 7 BIBLIOGRAPHY...89 8 APPENDIX...91

LIST OF FIGURES Figure 1 Magnetic flux in axial and thickness direction during axial loading of magnetostrictive tagged composite specimens (White, 1999)... 3 Figure 2 Transverse (x-axis) magnetic flux density versus load: (a) neat resin;... 4 Figure 3 Stress/magnetic flux density cyclic testing results: (a) without annealing between cycles; (b) with annealing between cycles (Quattrone, Berman, and White, 1998)... 5 Figure 4 Strain versus magnetic strength (Butler, 1988)... 6 Figure 5 Relationships between a) magnetic flux density versus magnetic strength, b) stress versus magnetic flux density, c) stress versus magnetic strength... 6 Figure 6 Schematic of the static system... 9 Figure 7 Schematic of a generic lay-up... 14 Figure 8 Balanced-orthotropic lay-up model used in the analysis... 17 Figure 9 Cross-ply model used in the analysis... 20 Figure 10 Strain distribution of the 7, 14, 28 model with 0 /90 degree ply angle in longitudinal direction... 26 Figure 11 Convergence of the strain result of the models to the experimental data... 28 Figure 12 Strain distribution for the 7, 14, 28 mathematical models with 0 /90 degree ply angle in transversal direction.... 29

Figure 13 Stress thickness distribution of the 7-ply balanced-orthotropic model with 0 degree fiber angle... 30 Figure 14 Stress thickness distribution in longitudinal direction for the 14-ply 0 /90 degree fiber angle model... 31 Figure 15 Thickness distributions of longitudinal stress for 28-ply 0 /90 degree fiber angle model.... 32 Figure 16 Thickness distributions of longitudinal stress for 56-ply 0 /90 degree fiber angle model.... 32 Figure 17 Stress-thickness distribution for the 7, 14, 28-ply models with 0 /90 degree fiber angle... 33 Figure 18 Convergence of results on the specimen surface under maximum load condition (F max = 58.9 N, M max = -88.29 Nm/m) as predicted by various models and measured by experiment for stress results... 35 Figure 19 Stress thickness distribution in longitudinal direction for 14-ply 90 /0 degree fiber angle model... 36 Figure 20 Stress - thickness distribution in longitudinal direction for the 28-ply 90 /0 degree fiber angle model... 36 Figure 21 Stress thickness distribution in longitudinal direction for 56-ply 90 /0 degree fiber angle model... 37 Figure 22 Stress-Thickness Distribution for the 7 ply 0 degree angle model and 14, 28- ply models with 90 /0 degree fiber angle... 38

Figure 23 General view of the experiment set-up... 45 Figure 24 Schematic of the experiment and data flow... 46 Figure 25 Position of the gaussmeter probe, strain gage, and displacement transducer47 Figure 26 Concept design of the protective wood fixture... 48 Figure 27 Overview of the assembly of magnetic sensor and carved wood... 49 Figure 28 Front panel of the program created for the MS composite beam experiment using the LabView virtual instrument... 52 Figure 29 The logic flow of the program created for the MS composite beam experiment using the LabView virtual instrument... 53 Figure 30 Mechanical data resulting from the experiments without magnetic annealing between loading cycles: displacement vs. load... 56 Figure 31 Mechanical data resulting from the experiments without magnetic annealing between loading cycles: strain vs. load... 58 Figure 32 Mechanical data resulting from the experiments without magnetic annealing: strain vs. displacement... 58 Figure 33 Magnetic flux density responses, without magnetic annealing, to bending strain in top surface of MS-tagged composite specimens: superposed 10 consecutive cycles... 60 Figure 34 Mean value and standard deviation of the magnetic field... 61 Figure 35 The resulted magnetic field as function of measured strain... 61 Figure 36 Mean values of magnetic flux density and standard deviation versus strain

for the specimen without annealing... 63 Figure 37 Displacement results versus applied force for annealed specimen... 65 Figure 38 Strain results for the MS composite specimen with magnetic annealing between cycles... 66 Figure 39 Magnetic flux density results for specimen with magnetic annealing between cycles... 68 Figure 40 Mean value and standard deviation of magnetic flux density results vs. load for the specimen with magnetic annealing between cycles... 68 Figure 41 Experimental magnetic flux density results versus strain with magnetic annealing of the specimen between cycles... 69 Figure 42 Mean value and standard deviation of magnetic flux density results vs. strain for the specimen with magnetic annealing between cycles... 69 Figure 43 Magnetic flux density in relationship with model longitudinal stresses on the surface of the composite specimen model... 70 Figure 44 Standard deviation of the stress from the mean value versus magnetic flux density... 71 Figure 45 Magnetic flux density versus strain on the top surface of the model material considered and 0 /90 degree fiber angle without magnetic annealing... 71 Figure 46 Plot of the average magnetic flux density vs. strain showing the trend lines in determination of the piezomagnetic stress coefficient e 31 for non -annealed specimen case 74

Figure 47 Plot of the average magnetic flux density vs. strain showing the trend lines in determination of the piezomagnetic stress coefficient e 31 of magnetic annealing of the specimen... 75 Figure 48 Plot of the average magnetic flux density with no magnetic annealing specimen vs. 56-ply model strain showing the trend lines in determination of the piezomagnetic stress coefficient e 31... 76 Figure 49 Plot of the average magnetic flux density with magnetic annealing vs. 56-ply model strain showing the trend lines in determination of the piezomagnetic stress coefficient e 31... 77 Figure 50 Trend line for calculating the piezomagnetic strain coefficient... 78 Figure 51 Trend line for computing the piezomagnetic strain coefficient d 31 for the case of magnetic annealing of the specimen between cycles... 79

LIST OF TABLES Table 1 Dimensions of the MS tagged composite specimen... 9 Table 2 Mechanical properties considered for the magnetostrictive-tagged woven composite material specimen... 11 Table 3 Lay-up of the composite models for 14, 28 and 56-ply models... 21 Table 4 Comparison of the longitudinal strain on the specimen surface under maximum load condition (F max = 58.9 N, M max = - 88.29 Nm/m) as predicted by various models and measured by experiment... 27 Table 5 Stress results on the surface for 7 balanced-orthotropic, 14, 28-ply cross-ply models 34 Table 6 Stress results on the surface for 7, 14, and 28 ply 0 and 90 /0 cross-ply models 38 Table 7 List of the equipment used in the magnetostrictive composite beam experiment... 50 Table 8 The displacement data without magnetic annealing... 56 Table 9 Strain data without magnetic annealing... 57 Table 10 Magnetic flux density results without magnetic annealing... 59 Table 11 Displacement results with annealing between loading-unloading cycles... 64 Table 12 Strain results with annealing between loading-unloading cycles... 65

Table 13 Experimental magnetic flux density with magnetic annealing between loading-unloading cycles... 67 Table 14 Summary of magnetostrictive coefficient for MS tagged composites, as determined in the present work and by previous investigators... 81 Table 15 Comparison of the design, model and measured strain... 83 1 INTRODUCTION In recent years, numerous applications in civil engineering construction used composite materials. The increase of the usage of the composite materials in civil construction imposed problems of evaluating the in-service composite civil engineering structures. The evaluation of the composite is a wide area in which engineers and researchers have proposed several technologies, for in-service Non-destructive evaluation (NDE). Conventional NDE methods, initially developed for metallic structures, have been shown to be less effective in monitoring composite structures due to the micromechanical complexity of the composite material. New NDE technologies are required. The NDE technology analyzed in this paper is proposed for the inspection of advanced composite by using the magnetostrictive particle tagging technique. Magnetostrictive-tagged composites permit: (a) distributed sensory properties; (b) noncontact damage detection; and (c) straight forward manufacturing implementation. 1.1 PREVIOUS WORK ON MAGNETOSTRICTIVE TAGGED COMPOSITES FOR

STRUCTURAL HEALTH MONITORING (TENSILE EXPERIMENTS) Terfenol-D is a magnetic anisotropy-compensated alloy Tb x Dy 1-x Fe 2 that shows a strong magnetostrictive behavior. The name Terfenol-D represents the composition of the material and the original name of the Navy Laboratory at which the work begun. Ter represents Terbium, Fe from iron, nol Naval Ordnance Laboratory and D Dysprosium. A magnetostrictive (MS) material, such as Terfenol-D, produces magnetic field when subjected to mechanical strain. This phenomenon is known as the converse magnetostrictive effect. White and collaborators (White, Albers and Quattrone, 1996; White and Brouwers, 1998) did extensive work on magnetostrictivetagged composites under axial loading. White (1999) reviewed the magnetostrictive tagging methodology of composites for structural health monitoring and gave an update on recent results. An experiment in which MS-tagged composite specimens were subjected to uniaxial tension in a testing machine was presented. Neat resin specimens tagged 2.24% by volume with magnetostrictive Terfenol-D powder were used. The resulting magnetic field was measured in both the axial and thickness directions (Figure 1). Trovillion et al. (1999) studied the magnetic characteristics of neat resin and glassfiber-reinforced magnetostrictive composites subjected to axial load. The fiber reinforced polymer composite (FRP) specimens consisted of 4 layers of continuous strand glass mat fibers embedded in a polyester resin. Trovillion et al. (1999) studied the magnetic characteristics of neat resin and glass-fiber reinforced magnetostrictive composites subjected to axial load. The fiber reinforced polymer composite (FRP) specimens consisted of 4 layers of continuous strand glass mat fibers embedded in a polyester resin.

Figure 1 Magnetic flux in axial and thickness direction during axial loading of magnetostrictive tagged composite specimens (White, 1999) The top lamina of the composite was impregnated with Terfenol-D powder at a volume fraction of 2.24% for that lamina. The specimens were subjected to uniaxial loading under load control at a rate of 0.02 kn/s. Hall-effect device were used to measure the magnetic field response to subsequent loading and unloading. As seen in Figure 2, both measuring devices gave similar results. Magnetic annealing (i.e. rearranging the magnetic dipoles chains of the magnetostrictive molecules through application of strong magnetic field) was performed by applying a magnetic field through the thickness of the specimen using a pair of 800 Gauss permanent magnets. Nersessian N. and Carman G.P. presented at ASME conference in Orlando results of tests for five different volume fraction composites namely 10 %, 20%, 30 %, 40 % and 50 %. The composite were tested under constant magnetic field with varying the mechanical load and constant mechanical load with varying magnetic field conditions. Results for the constant magnetic filed test indicated that modulus generally increases with volume fraction and increasing H/H max. For low fields, initial dip is noticed in modulus attributed to domains becoming more mobile at lower magnetic field levels. Results presented for the constant load test show a strong dependence of strain output on applied pre-stress.

Magnetic Flux Density (gauss) 2.5 Gauss Probe Hall Effect Chip 2.0 1.5 1.0 0.5 Magnetic Flux Density (gauss) 0.75 0.50 0.25 Gauss Probe Hall Effect Chip (a) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Load (kn) (b) 0.00 0.0 0.5 1.0 1.5 2.0 Load (kn) Figure 2 Transverse (x-axis) magnetic flux density versus load: (a) neat resin; (b) composite sample (Trovillion et al, 1999) Quattrone, Berman, and White (1998) studied the magnetic response repeatability of MS tagged composites under cyclic loading. Terfenol-D active-tagged composites were subjected to uniaxial tension and the magnetic response in the axial direction under repeated loading and unloading was measured. Two types of experiments were performed: without magnetic annealing between loading cycles (Figure 3a) and with annealing (Figure 3b).

Slope 45/10 mg/mpa Slope 140/10 mg/mpa (a) (b) Figure 3 Stress/magnetic flux density cyclic testing results: (a) without annealing between cycles; (b) with annealing between cycles (Quattrone, Berman, and White, 1998) Krishnamurthy, Anjanappa, and Wang (1999) consider health-monitoring detection of delaminations in composite materials using an excitation coil and a sensing coil. The open-circuit voltage induced in the sensing coil is proportional to the stress generated in the magnetostrictive layer by the presence of the delamination. 1.2 CHARACTERISTICS OF ETREMA S TERFENOL-D MAGNETOSTRICTIVE MATERIAL An application manual for designing the magnetostrictive transducers using Etrema Terfenol-D was prepared by J. Butler (1988). A physical description of the magnetostrictive material, with emphasis on Etrema Terfenol-D, was given. The nominal Etrema Terfenol-D strain versus magnetic strength, H, as illustrated in this manual, is shown in Figure 4.

Figure 4 Strain versus magnetic strength (Butler, 1988) The theoretical stress versus magnetic flux density, stress versus magnetic strength and magnetic flux density versus magnetic strength curves are presented in Figure 5. Butler (1988) also provides an analysis of the properties of Etrema Terfenol-D with emphasis on strain, stress and magnetic field relationships. Some design considerations and fundamental concepts for designing magnetostrictive transducers are also presented. Figure 5 Relationships between a) magnetic flux density versus magnetic strength, b) stress versus magnetic flux density, c) stress versus magnetic strength 1.3 PRESENT INVESTIGATION In the present investigation, studies of the theoretical and experimental of stressstrain and magnetic flux density response of a woven tagged composite subjected to bending loading were performed. Several two-dimensional simulation models for strain and stress prediction in the composite were investigated. The analysis was performed under two separate assumptions regarding woven composites modeling: (a) balancedorthotropic equivalent and (b) cross-ply equivalent. The experimental investigation considers the bending moment created by the loading and unloading of a simply supported MS-tagged composite beam using several weights. At each loading and unloading step, the magnetic flux density induced by the converse magnetostrictive effect

was measured. Comparison between the experimental and analysis results are then presented. Based on the comparison of experimental and analysis data, piezomagnetic stress and strain coefficients under bending are calculated. Comparison of these coefficients with the results obtained by other investigators for axial loading is performed. It is shown that, although exact comparison is not fully possible since bending and axial stress distributions are essentially different, the results are similar. The piezomagnetic stress and strain coefficients determined in this paper for MS tagged composite under bending can be used in industry as reference data for design of novel NDE devices based in the magnetostrictive effect in MS tagged composites.

2 ANALYSIS OF A MAGNETOSTRICTIVE COMPOSITE BEAM The analytical work performed on the MS tagged woven composite specimen consisted of: (a) loading analysis; (b) micromechanical analysis; (c) lamination analysis. One balanced-orthotropic and several cross-ply lamination models were used. Converge analysis of the strain and stress predictions versus number of plies in the lamination model were also performed. A magnetostrictive composite stress-strain model has been developed for simulating the behavior of the woven magnetostrictive composite. Two dimensional mathematical model study was performed for the 7 ply balanced-orthotropic model, and 14, 28 and 56-ply multidimensional composite model with 0 /90 and 90 /0 degree. Influence of the Terfenol-D was considered for the first and last ply of the 7-ply model, in first 2 plies and last 2 plies for the 14-ply model, for the first 4 plies and last 4 plies of the 28-ply model and the 8 plies for the 56-ply model. 2.1 STATIC ANALYSIS OF SIMPLY SUPPORTED COMPOSITE BEAM UNDER CENTRAL LOAD To determine the bending moment generated by the centrally-placed load, a static analysis of simply supported composite beam was performed. In this analysis, the dimensions and mechanical properties of the woven composite presented in Table 1 were used. A schematic of the system and how the load is applied is shown in Figure 6. The

loading and unloading of the specimen was assumed to be performed using 2-kg weights. Table 1 Dimensions of the MS tagged composite specimen Characteristics Dimension [mm] Span, L 600 Width, w 100 Thickness, t 6.5 Thickness of the lamina 0.92857 Force given by the bricks Strain Gages 2-Kg bricks Composite Material Plate w L Figure 6 Schematic of the static system The force exerted by the one brick is calculated considering the weight of 2 kg and gravitational acceleration of 9.81 m/s 2. Thus the force of the brick is The moment per unit width exerted by the weights is: F = 2.09.81 = 19.62 N (1) FL M 4w (2) Where L is the actual length of the beam, w- the width of the beam, F is the forced applied by one weight. During the loading cycle, up to three weights were sequentially put on the specimen and then removed. The maximum force value, corresponding to

three weights, was F max = 58.9 N. The corresponding moment per unit width was M max = 88.29 Nm/m. 2.2 ANALYSIS OF A MAGNETOSTRICTIVE LAMINATED COMPOSITE 2.2.1 Micromechanics analysis of the MS tagged composite specimen Since a woven layer has fibers in two directions, warp and fill, the analysis of a woven composite layer cannot be directly performed through the Classical Lamination Theory (CLT), which assumes fibers aligned with just one direction. An equivalence principle needs to be applied. Tsai (1992) suggested that the predictions of elastic constants and strength of woven composite could be made using classical micro and macro mechanics with appropriate empirical correction factors (Tsai, 1992, page 7-14). One approach is to replace a woven composite layer with two equivalent conventional layers representing the weave and the warp of the original fabric. The micro-mechanics stiffness and strength formulas are then applied to the equivalent plies. One shortcoming of this approach is that the order of 0 and 90 plies may strongly affect the results. Another approach is the replace the balanced woven composite layer by an orthotropic layer with averaged properties. For a woven composite having fibers aligned with the loading axes, this hypothesis yields balanced-orthotropic behavior. In our analysis, both approaches were taken. In either case, a conventional micromechanics analysis (Jones, 1999) was first applied to determine the basic MS composite properties from the properties of its constituents, given in Table 2.

Table 2 Mechanical properties considered for the magnetostrictive-tagged woven composite material specimen Mechanical properties Dimension 1 Young s modulus of the fiber, E f 72.4 GPa 1 Young s modulus of the matrix, E m 3.25 GPa 2 Young s modulus for the Terfenol-D, E MS 30 GPa 1 Poisson ratio of the fiber, f 0.2 1 Poisson ratio of the matrix, m 0.3 Fiber weight fraction, w f 0.3 1 Fiber density, f 2.54 g/ml 1 Matrix density, m 1.18 g/ml 3 Terfenol-D density, MS 9.25 10 3 kg/m 3 1 Thermal expansion coefficient for the fiber, α f 5 10-6 m/m per ºC 1 Thermal expansion coefficient for the matrix, α m 3.0 10-6 m/m per ºC Terfenol-D weight fraction 15 Terfenol-D particle size 4 38-40 μm Note: 1 Malik, 1992; 2 Butler, 1988; 3 De Laicheisserie, 1993, Trovillion et al., 1999. The fiber volume fraction v f is calculated as (Malik, 1992, page 81, equation 2.5): v f wf f wf 1 w f m f (3) For the values given in Table 2, the fiber volume fraction values yield v f = 0.16. Since the sum of the volume fractions is equal to 1, the volume fraction for the matrix is v m = 1-v f for the plies which does not include the Terfenol-D, and v m = 1- v f - v MS for the plies that contain MS material. The basic moduli of elasticity in the material axes were calculated following (Jones) 1999 as:

E E v E v E v 1 f f m m MS MS E [(E ) v (E ) v (E ) v ] 1 1 1 1 2 f f m m MS MS (4) The ply Poisson s ratio was calculated as: v v v 12 f f m m MS MS E 2 21 12 E1 (5) Shear modulus for the fiber, matrix and Terfenol-D for each ply were defined as: G f E f m MS Gm GMS 2 1 2, 1 2, 1 f E m E MS (6) Following Jones (1999, page 134), we calculate the shear modulus for the plies as: G 12 Gf GmGMS v G G v G G v G G f m MS m _ MS f MS MS f m (7) Thermal expansion coefficients for the ply were: E v E v E v 1 E v E v E v f1 f f m1 m m MS MS MS f f m m MS MS (8) 1 v 1 v 1 v 2 f f1 f m m1 m MS MS MS 1 12 (9) and 1 [ ] 2 (10)

2.2.2 Lamination Analysis The elastic behavior of multidirectional plies can be described in terms of the stiffness matrix, the compliance matrix, (Jones, 1975). The goal is to determine the strain-stress distribution in the material. The plane stress stiffness matrix [Q] of a orthotropic composite ply is computed as: E1 21E1 0 112 21 112 21 E E 112 21 112 21 0 0 G12 (11) 12 2 2 [Q] 0 Where E 1 and E 2 are given by Equation (4). The composite is assumed to consist of a layup of multiple plies. The lay-up schematic for generic composite is presented in Figure 7 where x-y are the loading axes. In our case, the angle of the plies is either 0 or 90-degree. θ 1 θ k y x θ n z Figure 7 Schematic of a generic lay-up Considering the transformation matrix:

2 2 cos sin 2cossin 2 2 cossin cossin cos sin (12) 2 2 [T] sin cos 2 cos sin The stiffness matrix in loading axes [Q] is: T 1 1 [Q] [T] [Q] [T] (13) Knowing the stiffness matrix, one calculates the extensional stiffness matrix [A], coupling stiffness matrix [B], bending stiffness matrix [D] for the composite laminate: [A] [Q] (z z ) 1 k k1 k k 2 2 [B] [ [Q] k1(z k z k1)] 2 k 1 3 3 [D] [ [Q] k1(z k z k1)] 3 k (14) The variable z k represents the distance from the midplane to the bottom of the kth lamina; z k-1 represents the distance from the midplane to the top of the kth lamina. These distances were calculate using the formula: N layer zk h (k ) 2 (15) In matrix form the general equation is: 0 N A B M B D (16) Where {N} is the load vector, {M} is the moment vector, {ε 0 } is the mid-surface strain vector, and {κ} is the curvature vector. For the design purposes, knowing the loading, the

state of deformation general solution is: 0 1 A B N B D M (17) Next, the effect of thermal conditions is analyzed by considering only the thermal effect, i.e. without loading F=0. The loads derived from the above-mentioned conditions are: N layer [N] [Q] (z z ) T T k1 k1 k k1 k1 k1 N 2 2 k 1 T k1 layer z z [M] [Q] k1( ) T, k1 2 (18) Here, the term k1represents the coefficient of thermal expansion in the loading axes, which is calculated using the transformation matrix and the coefficients of thermal expansion in the material axes: { } [T] { } T k 1 k 1 k1 (19) The loads and moments due to thermal effect: N layer {N} [Q] { } (z z ) T T k1 k1 k k1 k1 k1 (20) Nlayer 2 2 z k z k1 T k1 k1 k1 k1 2 {M} [Q] { } ( ) T (21) The load and moment vector are composed as: N NN MMM T T (22) A general loading vector it is defined as:

N0 N1 N2 {NM} M0 M1 M 2 (23) Solving the state of deformation Equation (17) yields the loading axes strain on the top and bottom of each ply is calculated. { } { } z { } top k 1 0 k 1 { } { } z { } botk1 0 k (24) Using the strains in loading axes, the strains in material axes are calculated for the top and the bottom ply: 1 {} [T] { k 1} T (25) Consequently, the stresses in the longitudinal (L) and transversal (T) directions at the top and the bottom of each ply are: { } [Q] { } topk1 k1 top k 1 (26) { } [Q] { } bot k1 bot k1 k1 (27)

2.3 7-PLY BALANCED-ORTHOTROPIC MODEL Ply 1 Ply k Ply n Z X Y Figure 8 Balanced-orthotropic lay-up model used in the analysis The balanced-orthotropic model, presented in Figure 8, uses the assumption that a balanced woven composite can be represented by a composite having averaged properties. Using the longitudinal and transverse moduli for a uni-directional fiber composite layer we calculated the averaged properties: E1 E 1 2 (EL E T) 2 (28) The MS Terfenol-D volume fraction and Poisson s ratio were v MS = 2.24% and ν MS = 0.3 for the plies with Terfenol-D, and zero for the other plies. Two indexes were defined in order to make a distinction between the laminas with and without Terfenol-D, N layer =7, N layers =6.. Performing the CLT analysis yields the balanced-orthotropic (BO) matrices:

7 6 6.59610 7.29110 0 6 0 0 9.66810 (N/m) (29) 6 7 [A] BO 7.29110 6.59610 0 12 13 3.63810 2.2710 0 0 0 0 (N) (30) 13 13 [B] BO 2.27410 9.0910 0 244.47 25.897 0 [D] BO 25.897 190.99 0 0 0 34.346 (N m) (31) The thermal effect was included for complete analysis procedure but since the cure temperature was the same with environmental temperature, Tk 1 0, then all the stresses due to thermal effects are zero. vector is: Considering only the bending moment, the load vector is zero and the moment M {M vector} 0 0 (Nm/m) (32) The values for this loading vector are: 0 0 0 (N/m) NM 88.29 (N) 0 0 (33)

The strain and curvature vector is determined then using the general state of deformation equation mentioned above. Thus the strain and curvatures are: 0 0 0 0 and 0.366 0.050 0 (1/m) (34) The values for the stresses in longitudinal direction for the top and bottom layer are: 12.72 8.72 5.23 L_top 1.74 1.74 5.23 9.08(MPa) 9.08 5.23 1.74 L _ bot 1.74 5.23 8.72 12.72(MPa) (35) The strains in the longitudinal direction for the top and bottom layer are: 1347 962 577 top 192 192 577 962(με) 962 577 192 bot 192 577 962 1347(με) (36)

2.4 14-PLY CROSS-PLY ANGLE MODEL Ply 1 Ply k Ply n Z X Y Figure 9 Cross-ply model used in the analysis In the cross-ply analysis (Figure 9), each woven composite layer is approximated by a couple of unidirectional cross-ply layers. The cross-ply layer can be either 0 /90 or 90 /0. Since the choice between 0 /90 and 90 /0 directly affects the accuracy of the results on the specimen surfaces, we initially considered both cases in our analysis. Additionally, the number of the layers was gradually increased from 14 to 28 and 56, and a convergence study was performed. During this convergence study, as the number of layers were doubled, the thickness of the layer was correspondingly halved, such that the overall thickness of the specimen was maintained. The difference in 14, 28 and 56-ply mathematical models procedure from the balanced-orthotropic model is that the Young s modulus of elasticity in longitudinal and transversal direction were not averaged The 28- ply and 56-ply models were obtained by subsequent subdivision of the 0/90 layers and application of symmetry principles. The ply angle distributions for the 14, 28 and 56-ply

models are presented in Table 3: Table 3 Lay-up of the composite models for 14, 28 and 56-ply models Number of Layers Lay-up 14 [(0/90) 3 /0] S 28 [(0/90) 7 /90] S 56 [(0/90) 14 ] S Terfenol-D is assumed present in the first two layers and last two layers and the for the Young s modulus values in the longitudinal and transversal directions are: E 1 = 14.96 GPa, E 2 = 3.94 GPa. The Young s modulus of elasticity values for the layers that does not contain magnetostrictive material are: E 1 = 14.31 GPa, E 2 = 3.84 GPa. The A, B and D matrices for the 14-ply model were: 7 6 11 6.610 7.2910 2.5110 6 7 9 [A] 14 7.2910 5.6010 1.8310 11 9 6 2.5110 1.8310 9.67 10 (N/m) (37) 12 13 3.6410 2.27 10 0 0 0 0 (N) (38) 13 13 [B] 14 2.27 10 9.09 10 0 244.47 25.9 0 [D] 14 25.9 190.99 0 (N m) 0 0 34.35 (39) Stress and strain results in longitudinal direction for the 14-ply 0 /90 cross-ply laminate, at the top and bottom of the plies, are:

18.01 3.95 12.32 2.57 7.39 1.28 2.46 L_top 0.00-0.64-4.93-1.92-9.85-3.29-15.44 (MPa), 15.44 3.29 9.85 1.92 4.93 0.64 0.00 L_bot - 2.46-1.28-7.39-2.57-12.32-3.95-18.01 (MPa), 1191 1021 851 680 510 340 170 top 0-170 -340-510 -680-851 -1021 (με), 1021 851 680 510 340 170 0 bot - 170-340 - 510-680 - 851-1021 -1191 (με) (40) For the 90 /0 cross-ply laminate the stress and strain results are: 5.94 19.81 4.14 12.64 2.48 6.32 0.83 L_top 0.00-3.16-1.66-9.48-3.31-16.51-5.10 (MPa), 5.10 16.51 3.31 9.48 1.66 3.16 0.00 L_bot - 0.83-6.32-2.48-12.64-4.14-19.81-5.94 (MPa), 1524 1307 1089 871 653 436 218 top 0-218 - 436-653 - 871-1089 -1307 (με), 1307 1089 871 653 436 218 0 bot - 218-436 - 653-871 -1089-1307 -1524 (με) (41)

2.5 28-PLY CROSS-PLY ANGLE MODEL The 28-ply model is computed in a similar manner with the 14-ply model. The changes occur in the indices, the ply angle, and the thickness of the lamina. The A, B and D matrices were: 7 6 11 6.110 7.2910 1.3810 6 7 9 [A] 14 7.2910 6.110 3.1810 11 9 6 1.3810 2.1810 8.97 10 (N/m) (42) 12 13 13 3.6410 1.2510 1.0510 13 13 13 [B] 14 2.27 10 9.0910 1.0510 13 13 1.0510 1.0510 101.23 (N) (43) 231.19 25.9 0 [D] 14 25.9 204.27 0 (N m) 0 0 30.59 (44) The stress and strain results are presented next:

L_top 19.06 4.53 16.33 3.83 13.03 3.06 10.42 2.38 7.82 1.70 5.21 1.02 2.61 0.34 0.00-1.30-0.68-3.91-1.36-6.52-2.04-9.12-2.72-11.73-3.49-14.97-4.18-17.69 L _ bot 17.69 4.18 14.97 3.49 11.73 2.72 9.12 2.04 6.52 1.36 3.91 0.68 1.30 0.00-0.34-2.61-1.02-5.21-1.70-7.82-2.38-10.42-3.06-13.03-3.83-16.33-4.53-19.06 top 1259 1169 1079 989 899 809 719 629 539 449 259 269 179 89 0-89 -179-269 -359-449 -539-629 -719-809 -899-989 -1079-1169 bot 1169 1079 980 899 809 719 629 539 449 359 269 179 89 0-89 -179-269 -359-449 -539-629 -719-809 -899-989 -1079-1169 -1259 (45)

2.6 56-PLY CROSS-PLY ANGLE MODEL The 56-ply model was obtained by subsequent subdivision of the 0 and 90 layers and application of symmetry principles. The changes appear in setting the indexes and the thickness of the lamina. The A, B and D matrices for the 56-ply model were: 6.2210 7.2910 2.8210 7 6 11 6 7 9 [A] 56 = 7.2910 5.97 10 2.0610 (N/m) 11 9 6 2.8210 2.0610 9.67 10 (46) -13-72.14-3.98 10 0-13 [B] 56= -3.9810 72.14 0 (N) 13 0 0 9.0910 (47) 224.47 25.90 0 [D] 56 = 25.90 211 0 (N m) 0 0 34.35 (48) The maximum strain on the surface of the specimen model was found to be: = 1296 (49) The corresponding maximum stress was: = 19.6 MPa (50) 2.7 CONVERGENCE ANALYSIS The convergence of the strain and stress distribution for various models was studied. Attention was focused on the longitudinal stress and strain under maximum load

conditions (F max = 58.9 N, M max = - 88.29 Nm/m). The strain values predicted on the top and bottom surfaces were also compared with the experimental value measured during the tests described in Chapter 4. 2.7.1 Convergence of the strain Figure 10 shows the distribution of the longitudinal strain for the 7-ply balancedorthotropic model and the 14-ply and 28-ply 0 /90 cross-ply models. (The 56-ply results were not plotted on Figure 10 to avoid cluttering the drawing). The sign convention used in Figure 10 is negative to the left and positive to the right. 14 plies 0/90 deg 28 plies 0/90 deg 7-ply balancedorthotropic model Figure 10 Strain distribution of the 7, 14, 28 model with 0 /90 degree ply angle in longitudinal direction The study shows that the largest strain value was predicted by the 7-ply balancedorthotropic model. This value is also very close to the experimental value (1347 με vs. 1333 με). The convergence of the strain distribution for the cross-ply models is apparent as the models are becoming more refined i.e. strain distribution is getting closer to the experimental data as the number of plies employed in the model is increased. The strain

values predicted by each model, at the surface of the composite, and comparison with experimental data, are presented in Table 4. Table 4 Comparison of the longitudinal strain on the specimen surface under maximum load condition (F max = 58.9 N, M max = - 88.29 Nm/m) as predicted by various models and measured by experiment 7-ply balancedorthotropic model 14-ply 0 /90 model Strain [με] 28-ply 0 /90 model 56-ply 0 /90 model Experimental 1347 1191 1259 1296 1333 The reason for the difference between the balanced-orthotropic model and the other models is that the balanced-orthotropic model considers each of the plies at the same angle. Thus, the strain that could be carried is larger than the more realistic models in which the strains are lower because the angle of ply is alternating from 0 to 90 degree. The results show that the plies with transverse fiber to the longitudinal direction of measuring (90 degree fiber angle) shrink and oppose the longitudinal stress direction. The convergence of the strain data of the models is presented in Figure 11.

1385 1330 Strain [micro strain] 1275 1220 1165 1110 1055 1000 Strain response of the models for surface Experimental Baseline Strain Strain response for quasi-isotropic model 7 14 28 56 Number of the plies Figure 11 Convergence of the strain result of the models to the experimental data The figure shows that the maximum strain given by the 7-ply balanced-orthotropic model, is approximately the same as the experimentally obtained strain. The graph also shows that the strain prediction by the cross-ply models improves with increasing number of layers.

14-ply 0/90 deg model Balanced orthotropic model 28-ply 0/90 deg model Figure 12 Strain distribution for the 7, 14, 28 mathematical models with 0 /90 degree ply angle in transversal direction. The strain distribution for each model in transversal direction is presented in Figure 12.The strains on the left side of the symmetry line represent positive strains and on the right side represent negative strains. This case shows the strain distribution for the 14 and 28 ply models have close results. It is also observed that for the 7-ply model the strains are larger.

2.7.2 Convergence of stress For the 7-ply balanced-orthotropic model, the longitudinal stress distribution is presented in the Figure 11. It can be noticed the stress distribution is rather linear, in accordance with the balanced-orthotropic assumption. The slight discontinuity between the outer first and last layers and the other layers is due to the outer layers being stiffer due to presence of MS tagging in the modified layers, layer 0 and 6. Stresses -Thickness Distribution in Longitudinal Direction for 7 plies Figure 13 Stress thickness distribution of the 7-ply balanced-orthotropic model with 0 degree fiber angle. 15. The stress distribution for the 14-ply 0 /90 cross-ply model is presented in Figure

Stress - Thickness Distribution for 14 plies 0/90 degree fiber angle model Figure 14 Stress thickness distribution in longitudinal direction for the 14-ply 0 /90 degree fiber angle model In this model, the stress distribution is different then the stress distribution for the 7-ply balanced-orthotropic model.it can be noticed the stress in the layers with 90-degree fiber angle is lower than the stress in the layers with 0-degree fiber angle. For the layers that contain the magnetostrictive material an increased stress is noticed. The cause of this is the increased stiffness due to Terfenol-D. Similar model behavior is observed for 28 ply and 56-ply 0 /90 cross-ply models as can be shown in Figure 15 and 16.

Stress - Thickness Distribution for 28 plies 0/90 degree fiber angle model Figure 15 Thickness distributions of longitudinal stress for 28-ply 0 /90 degree fiber angle model. Stress - Thickness Distribution for 7, 14, 28, 56 plies models Figure 16 Thickness distributions of longitudinal stress for 56-ply 0 /90 degree fiber angle model.

The longitudinal stress distribution for the 14, 28-ply 0 /90 cross-ply model is presented in Figure 17. In the cross-ply models, the stress distribution is substantially different then the stress distribution from the 7-ply balanced-orthotropic model. The stresses in the layers with 90-degree fiber angle are substantially smaller. This produce alternating changes from high stress to low stress as the layer stiffness change from the high to low in accordance with the 0 and 90 orientations of the cross-ply laminate. 7 plies balanced-orthotropic model 28 plies 0/90 deg model 14 plies 0/90 deg model Figure 17 Stress-thickness distribution for the 7, 14, 28-ply models with 0 /90 degree fiber angle The 56 ply 0 /90 degree fiber angle model is presented separately since the graph would have been hard to visualize and the stress distribution for each model would have been too hard to follow if they were superimposed on Figure 17. The results reveal that the layers that contain Terfenol-D are stiffer than the other layers. The 7 ply balanced-orthotropic model stress distribution shows that the model does not accurately simulate the real woven composite material because larger stress values were observed than in the cross-ply. This statement is made from a stress distribution

point of view. Depending on the fiber orientation and the stress amplitude, the stress carried is consequently proportional. For the 0 degree fiber orientation the stress is larger. For the fiber oriented to a 90-degree angle to the longitudinal direction, the stress amplitude is small. However, this shows that even the transversal fiber carry load. For analysis purposes, Table 4 presents the largest stresses on the surface for each model. Table 5 Stress results on the surface for 7 balanced-orthotropic, 14, 28-ply crossply models 7 ply balancedorthotropic model [MPa] 14 ply 0 /90 model [MPa] 28 ply 0 /90 model [MPa] 56 ply 0 /90 model [MPa] 12.72 18.01 19.06 19.63 Table 5 shows that the longitudinal stresses obtained by the 14, 28, and 56-ply models are larger than the stresses obtained with the balanced-orthotropic model. These results are due to increased stiffness in the longitudinal direction and less stiffness in transverse direction typical of cross-ply models. In the case of the balanced-orthotropic model the stiffness in both direction are the same, but of lower value than the longitudinal cross-ply stiffness.. Figure 18 presents the results of Table 5 in graphical form. It can be appreciated that the balanced-orthotropic model grossly underestimates the stresses, whereas the cross-ply models shown a definite convergence. It can be estimated that, by further increasing the number of layers in the model, convergence towards an asymptotic value of around 20 MPa would be obtained.

Longitudinal stress [MPa] 25 20 15 10 5 0 balanced-orthotropic model cross-ply models 0 7 14 28 56 Number of the layers Figure 18 Convergence of results on the specimen surface under maximum load condition (F max = 58.9 N, M max = -88.29 Nm/m) as predicted by various models and measured by experiment for stress results Stress analysis for 90 /0 cross-ply models was also performed. As before, the maximum load case (F max = 58.9 N, M max = - 88.29 Nm/m) was considered. The next graphs refer to the mathematical models with 90 /0 degree fiber angle. In Figure 19, 20 and 21 the 14, 28 and 56-ply models with 90 /0 degree fiber angle are presented.

Figure 19 Stress thickness distribution in longitudinal direction for 14-ply 90 /0 degree fiber angle model Figure 20 Stress - thickness distribution in longitudinal direction for the 28-ply 90 /0

degree fiber angle model Stress - Thickness Distribution in Longitudinal Direction for 56 plies 90/0 Degree Fiber Angle Model Figure 21 Stress thickness distribution in longitudinal direction for 56-ply 90 /0 degree fiber angle model In Figure 22 the stress distributions for the 14 and 28 cross-ply models with 90 /0 degree fiber angle are presented. 56-ply model is not included to not complicate the graph and make it hard to visualize the stress thickness distribution. In this graph, the 7-ply balanced-orthotropic model is also represented. The thickness distribution for the balanced-orthotropic model is similar to 0 /90 models stress distribution. Here is visible, in addition, the influence of the Terfenol-D in increasing the stiffness of the material with results in larger stress

7-ply balanced-orthotropic model 14-ply 90/0 deg model 28-0ply90/0 deg model Figure 22 Stress-Thickness Distribution for the 7 ply 0 degree angle model and 14, 28-ply models with 90 /0 degree fiber angle For analysis purposes Table 6 presents the largest stresses on the surface for each model. In this case, the comparison shows that the stress carried in the transverse direction, when the top ply has 90 degree ply angle, is smaller than the stress given by the balanced-orthotropic model. Table 6 Stress results on the surface for 7, 14, and 28 ply 0 and 90 /0 cross-ply models 7 ply balancedorthotropic model [MPa] 14 ply 90 /0 model [MPa] 28 ply 90 /0 model [MPa] 56 ply 90 /0 model [MPa] 12.72 5.94 5.55 5.4

3 MAGNETOSTRICTIVE COMPOSITE BEAM EXPERIMENT The experimental results alluded to the previous section were obtained during a carefully conducted experiments, as described next. 3.1 SPECIMEN DIMENSIONAL DESIGN For pre-design purpose, the mechanical properties of the specimen were evaluated with a very simplified theory that took into account the following data: 1 Volume fraction of the fiber: v f = 0.5 2 Volume fraction of the matrix v m = 1- v f = 0.5. 3 Young s modulus of the E-glass fiber the: Ef 9 72.410 Pa 4 Young s modulus of the resin Em 9 3.2510 Pa. We assumed an average Young s modulus for random fiber composite calculated with (Malik, 1992, equation 3.42, page 130): 3 5 E E E 8 8 11 22 (51) E 11 and E 22 are calculated using the rule of mixture for longitudinal modulus, E 11, and transverse modulus E 22 (Malik, 1992, equations 3.26,3.27, page 123).

E11 Ef vf Em vm (52) E 22 Ef Em E v E v f m m f (53) The result is E 9 17.808 10 Pa. The shear modulus for the composite (Malik, 1992, equation 3.43, page 130) material is assumed to be: 1 1 G E E 8 4 11 22 (54) The result is G 9 6.061 10 Pa. Simple statics was used to determine the bending moment of a simply supported composite beam centrally loaded by a concentrated force. To dimension the specimen for further computations, some mechanical and dimensional quantities were assumed. The equivalent Young s modulus, E, calculated above was considered. The width was measured on the composite specimen, w = 100 mm. The maximum deflection was considered not to exceed the value of q max = 36 mm based work previously done in the area of materials testing. calculated: To determine the thickness of the composite, the length, L to thickness, t, ratio was k 1 3 2wEq max L F k max, t (55) The maximum force was calculated based on the force of one weight and the

number of weights to be used in the experiment. The maximum force to be used in the experiment was considered to be the force exerted by ten weights, F max = 196.2 N. The thickness is then 3F max k t E w a (56) t = 5.051 mm. The length of the specimen was calculated as: L t k (57) with L = 449.9 mm, thus t = 6.5 mm. Furthermore the length of the specimen is adjusted to L = 579.6 mm. Rounding up, the final length for following calculations is L = 600 mm. The design dimensions of the specimen we choose to be: Length 1000 (mm) (58) Width 100 (mm) (59) Thickness 6.75 (mm) (60) The next step was to calculate the moment of inertia for determining the deflection. The moment of inertia was calculated using the well-known formula for a rectangle: w t I 12 3 (61) The result is: I 3 4 2.563 10 mm. Then the deflection is

q F L 3 F E I 24 (62) The result is q F 4.104 8.208 12.312 16.417 20.521[mm] 24.625 28.729 32.833 36.937 for 19.62 39.24 58.86 78.48 F 98.1 (N) 117.72 137.34 156.96 176.58 (63) At this point, the maximum stress and strain were calculated. Because of symmetry, the stress and strain have same absolute value when equally distanced from mid-plane. The values for the strain are: 83.6 167.5 250.8 334.4 417.9(MPa) 501.5 585.1 668.7 752.3 445 889 1334 1778 2223( ) 2668 3112 3557 4002 (64) The maximum stress was max t M 2 I and the maximum strain was max E max. Assuming the flexural strength to be SF 6 15010 Pa and a safety factor of SF=1.5, admissible tensile stress is calculated as

SF a SF (65) The result is a 6 10010 Pa. The admissible strain value is a a E (66) ε a = 5617 με. (67) Comparison of the result with the strain predicted above shows that adequate safety is built into the specimen. These simple calculations were used to ensure that the specimen was adequate for experimentation. 3.2 DESCRIPTION OF THE MS TAGGED COMPOSITE SPECIMEN An MS tagged composite specimen was fabricated at Reichhold Chemicals (Raleigh, N.C.) by binding 7 layers of fiberglass woven roving 36oz./sq.yd. and Atlac 580-05 Urethane-modified Vinyl Ester Resins. The specimen was 1000 mm long, and had a 100 mm by 6.5 mm cross section. MS Terfenol-D tagging powder was used in the two outside layers in the middle 500 mm of the span. Out of 1000 grams of resin, 250 grams had the MS powder. The resin was cured with 1 % MEKP (methyl ethyl ketone peroxide) at room temperature for approximately 90 minutes. The target weight fraction of the glass fibers in the composite was w f = 30 %. The weight fraction of MS Terfenol-D tagging in the resin was 15 %.

3.3 SPECIMEN PREPARATION FOR THE EXPERIMENT The specimen was instrumented with strain gages in the mid-span section on the upper and lower surfaces. The specimen surface was prepared for bonding the strain gages using the procedures given by Measurements Group (Raleigh, N.C.). The bonding area was made planar using abrasive sandpaper. Next, the surface was cleaned using Measurements Group surface conditioner M-Prep Conditioner A (a water based acidic surface cleaner). Measurements Group strain gages types CEA-06-125UT-120 were applied with M-Bond AE-10 Kit gage adhesive. To protect the strain gages and reduce their exposure to the environment, the strain gages were covered with one sided Scotch tape.

3.4 EXPERIMENTAL DESIGN 3.4.1 Description of the experiment Lakeshore Gaussmeter Strain Indicator Magnetostrictive Composite Solartron Displacement Transducer National Instrument Amplifier Figure 23 General view of the experiment set-up The overall configuration of the experiment can be seen in Figure 23. The experimental setup permitted the simultaneous measurement of beam deflection, mechanical strains, and magnetic response of the magnetostrictive (MS) tagged beam. Data flow through the data acquisition and processing modules is presented schematically in Figure 24. The MS-tagged composite beam was supported on concrete blocks (500 mm equivalent span) and loaded gradually with an incremental number of clay weights (2 kgf = 19.6 N each). Strain gages were placed on both the upper and lower surfaces of the specimen at the mid-span and were connected in a half bridge configuration to the strain indicator. The magnetic flux density produced from the magnetostrictive particles was measured by the gaussmeter.

Strain Gages Strain Indicator LVDT Displacement Transducer SCXI Unit DAQ PC Magnetic Sensor Gaussmeter Figure 24 Schematic of the experiment and data flow For correlation purposes, the mid-span displacement was also measured. An LVDT displacement transducer and a non-magnetic (aluminum and brass) clamping fixture were used. Details of the mid-span instrumentation are given in Figure 25. Initial trials showed that the strain gauge and LVDT electromagnetic fields did not influence the Gaussmeter reading of the magnetostrictively induced magnetic field. The MS tagged composite beam, supported on concrete blocks, was loaded gradually with 2-kg weights. In order to avoid excessive strain in the composite material, the number of loading weights was limited to three. The test procedure was as follow: load the specimen, acquire data and repeat until all the three weights were on the specimen. Next, reverse the procedure and perform unloading. The strain gages were connected to a strain indicator, which provided the actual strain value at top and bottom surfaces. At the same time, the displacements were measured with an LVDT displacement transducer. Hence, the relationship between loads, displacements and strains could be established. Simultaneously, the magnetic field developed by the MS

active tagging particles was detected using the gaussmeter. The gaussmeter provided the value of the magnetic flux density read by the probe on the surface of the MS tagged composite material. The data was collected using National Instruments LabView software and associate hardware consisting of a SCXI signal-conditioning module, and the Gateway computer. The information was processed using National Instruments LabView software. The data collection was developed for simultaneous on-line operation with the strain gages, displacement transducer and the gaussmeter. At each loading step, complete data collection was performed. The configuration of the strain gages, displacement transducer and position of the gaussmeter probe are presented in Figure 25. Strain Gage The rod of the Displacement Transducer Gaussmeter Probe Clamping Fixture Figure 25 Position of the gaussmeter probe, strain gage, and displacement transducer 3.4.1.1 Clamping fixture design A clamping device was designed in order to achieve a better position of the

displacement transducer on the magnetostrictive composite material.the fixture is composed of 2 aluminum parts (see Figure 25), which are kept attached to the composite material with a brass bolt and nut. A brass bolt was chosen for use in order to diminish the magnetic influence of all the components to the gaussmeter probe. 3.4.1.2 Protective wood fixture design for the strengthen the gaussmeter probe A special fixture had to be constructed to ensure proper and consistent alignment of the gaussmeter probe with respect to the composite surface. Its design is presented in Figure 26: 2 mm Sensor Tip Sensor Rod Wood Figure 26 Concept design of the protective wood fixture Considering the fragility of the gaussmeter probe and the advice from Lakeshore Cryotonics, a wood fixture for the protection of the probe tip was designed and fabricated. The distance from the tip of the sensor to the upper surface of the specimen was 2 mm. The open area was covered with clear tape. The sensor rod was attached to the wood fixture using Pro Seal Blue RTV Silicone made by Pacer Technology (Rancho Cucamonga, CA.). A view of this apparatus is given in the Figure 27.

Sensor Tip Sensor Rod Machine Carved Wood Figure 27 Overview of the assembly of magnetic sensor and carved wood 3.4.2 List of the equipment used In this experiment, many different pieces of equipment were used to achieve the simultaneous measurements of beam deflection, mechanical strains, and magnetic response of the MS tagging particles. The equipment list is presented in Table 7.