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Hybrid smart composite plate under low velocity impact
Composite Structures 56 (2002) 175–182 www.elsevier.com/locate/compstruct
Hybrid smart composite plate under low velocity impact Jin-Ho Roh, Ji-Hwan Kim
*
School of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul 151-742, Republic of Korea
Abstract This paper presents an approach to the detection of the strains and optimization of shape memory alloys (SMAs) under low velocity impact. It is assumed that the sensor is mounted on the opposite side of the impact process. SMA fibers are embedded within the layers of a composite plate. The first-order shear deformation theory of plates is used, and the finite element method is adopted in the numerical analysis. With different skew angles of the piezoelectric sensor, we can illustrate that the measured charge manifests the strain components induced by the impact. By changing the SMA volume fractions and the temperature increment, we can significantly reduce the deflections. It can be shown that the application of piezoelectric sensors is effective in the monitoring of the transient strain. In addition, the optimization of the distribution of volume fraction of SMA fibers provides further benefits in reducing the deflection of the plate. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Shape memory alloy; Piezoelectric; Impact; Active strain energy tuning; Hybrid smart structure
1. Introduction Intelligent/smart structures have features of integrated sensors and actuators within a host material, and they will have a tremendous effect upon numerous industrial fields. The idea of applying ‘‘smart’’ materials to mechanical and structural systems has been studied by many research workers in various disciplines. Shape memory alloys (SMAs) as the fiber reinforcement or actuator in the composite material can have numerous adaptive capabilities, but their logy behavior is the main reason for the limited applications of SMA. Piezoelectric materials can be used as the sensors and actuators simultaneously due to the piezoelectric effects and fast response of the materials. On the other hand, the displacement is quite small due to the small strain magnitude. Yang [1] discusses smart hybrid composites which incorporate SMA with other structural or functional materials. In addition, he suggests that with the simultaneous use of SMA and a ferroelectric ceramic, the composite material can effectively sense and actuate to reduce the structural vibration without the use of an external control. Owing to their weak impact resistance properties, laminated composite plates are very susceptible to damage when impacted by a solid object. This kind of *
Corresponding author. Tel.: +82-2880-7383; fax: +82-2887-2662. E-mail addresses: [email protected] (J.-H. Roh), jwhkim@ plaza.snu.ac.kr (J.-H. Kim).
damage is internal and it cannot be easily detected from its appearance. Accordingly, numerous experimental and analytical techniques have been developed for composite plates subjected to low velocity impact. Tan and Sun [2] predict the history of a contact force analytically based on the transverse shear deformable plate theory. Subsequently, Sun and Chen [3] investigate the effect of initial stresses on the low velocity dynamic response of the laminates by the finite element method. Smart materials can present an attractive tool for reducing deflections and stresses in the structures subjected to a low velocity impact. In particular, SMAs can generate significant tensile stresses. Reducing the deflection and strain induced by the impact can be achieved as the SMA fibers are embedded within a composite material in a hybrid smart composite. Birman [4] considers an approach to the optimum design of shape memory alloys in hybrid cross-ply composite plates subjected to low velocity impact. The results illustrate that SMA fibers embedded within the layers of a composite plate can significantly enhance its global resistance to low velocity impact. The constitutive relationships and associated micromechanics for a hybrid SMA composite material are considered in a number of studies. Rogers et al. [5] employ a rule of mixtures to develop the micromechanics for SMA composites with graphite/epoxy and nitinol/epoxy plies. Piezoelectric sensors measure the average local strain over the area to which they are bonded, and various investigations are concentrated on the methodology of
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 8 9 - 1
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J.-H. Roh, J.-H. Kim / Composite Structures 56 (2002) 175–182
using piezoelectric devices as the sensors to monitor structural damage. This concept is exploited by Yin et al. [6] in detecting edge delamination for composite plates subjected to remote uniaxial loading. Yin and Shen [7] investigate numerically the feasibility of using piezoelectric sensors to measure strains under low velocity impact in composites. In the present study, low velocity impact for the hybrid smart plate is considered. SMA fibers are embedded within the layers, and the piezoelectric sensors made from PVDF films are mounted on the opposite side of the impact. For the different skew angles, each sensor measures the charge signal induced by the impact. When the fibers are heated, the SMA fibers try to contract to their normal or memorized length and therefore generate a recovery force. Therefore, the resultant in-plane forces will adaptively change the structural response of the plate. This approach is referred to as active strain energy tuning (ASET), which is used as a control technique. The shape memory effect is employed to reduce transverse deflections of the plates and the global stresses of the material. The first-order shear deformation theory of plates is used, and also the finite element formulation presented to model the dynamic response of the hybrid composite plates. It is shown that SMA fibers can reduce deflection, and that the piezoelectric sensors are effective in monitoring the impact-induced strain.
2. Finite element model Fig. 1 shows the hybrid smart composite plate with the piezoelectric film subjected to impact by the mass ms . The SMA fiber is embedded in a composite matrix. Sensors A, B and C are mounted on the opposite side of impact, and located as shown in Fig. 1(a). The piezoelectric film sensor measures the charge due to the impact for the skew angle h as shown in Fig. 1(b). The Mindlin plate theory is used, and the finite element method is adopted. Each sensor made by a piezoelectric film measures the induced charge signal with respect to skew angles.
Fig. 1. Mechanical model for smart composite plate subjected to impact: (a) hybrid composite plate; (b) piezoelectric film.
By using the nine-node isoparametric shape function, the displacements for the element can be written as n oT ð1Þ fqge ¼ u10 ; v10 ; w10 ; h1x ; h1y ; . . . ; u90 ; v90 ; w90 ; h9x ; h9y ; e
where the subscript e denotes the element, u0 ; v0 ; w0 and hx ; hy stand for the mid-plane displacements and rotational angles, respectively. The recovery tensile stress can be evaluated analytically or experimentally. In this study, experimental stress–temperature curves are employed as in [4]. Thermal effects are incorporated by experimental recovery stress–temperature relationships of the SMA fiber. Thus, the constitutive equation of a piezoelectric includes the effect of thermal expansion. The constitutive equations of the hybrid smart plate with SMA fibers and piezoelectric materials can be written as T
frg ¼ ½Qfeg ½e fEg þ frr gjs ½Qfagjc DT ;
ð2Þ
fDg ¼ ½efeg þ ½nfEg þ fPgDT ;
ð3Þ
where ½Q and ½Q indicate the reduced stiffness matrices for a hybrid composite material and the composite medium, respectively. In addition, ½e is the piezoelectric stress/charge coefficient matrix, ½E is the electric field, frr g is the recovery stress, and fag is the thermal expansion coefficient. The volume fractions of SMA fibers and the composite medium are denoted by js and jc , respectively. While ½D stands for the electric displacement, ½n is the permittivity matrix, and fPg is the thermal stress coefficient. The overall properties of the hybrid composite materials are calculated by using micromechanical equations based on Chamis’ multicell model [4]. The governing equations for SMA fiber-reinforced hybrid composite plates subjected to a combined external, thermal and recovery stress load can be derived by the principle of virtual work dU ¼ dV ;
ð4Þ
where U and V are the strain energy and the work of the external force of the system, respectively. We can express the strain energy U as follows: Z 1 T U¼ feg frg dv: ð5Þ 2 v And work of the external force V is composed of V 0 and V 00 . V 0 is due to in-plane forces and the effect of thermal loads which includes the recovery forces of the Nitinol (Nickel–Titanium-Naval Ordnance Laboratory) fibers, and the heating of the Nitinol fibers. V 00 is due to inertia force. dV 0 can be written as follows: Z Z 2 X 2 X 1 h dw Nab ðoa wÞðob wÞ dA; ð6Þ dV 0 ¼ 2 A a¼1 b¼1
J.-H. Roh, J.-H. Kim / Composite Structures 56 (2002) 175–182 r T r T where Nab ¼ Nab þ Nab . In addition, Nab and Nab denote the recovery tension developed in the Nitinol fibers and compressive in-plane thermal loads in the a- and bdirections, respectively. Thermal loads are generated by temperature change DT , and are developed by both the activation and deactivation processes of the Nitinol fibers. Hence, the thermal load N T are given by the relations 2 T3 2 3 Z hk Nxx axx X 4 NyyT 5 ¼ Qk 4 ayy 5DT jc dz; ð7Þ hkþ1 k axy NxyT
where ½Q is the matrix for the composite material, axx ¼ a11 m2 þ a22 n2 , ayy ¼ a22 m2 þ a11 n2 and axy ¼ 2ða11 a22 Þmn. Symbols axx , ayy and axy denote coefficients of thermal expansion in the x-, and y-directions. Symbol 1 denotes the direction parallel to the fibers while symbol 2 stands for the direction normal to them. The recovery tension, N r , is developed in the Nitinol fibers as 2 2 3 2 r 3 Nxx m 4 Nyyr 5 ¼ rr 4 n2 5js h: ð8Þ Nxyr 2mn The angle of the fiber’ location is denoted by h while m and n denote cos h and sin h, respectively. Now, the virtual of inertia force dV 00 is written as Z Z 00 dV ¼ dwhq€ w dA: ð9Þ A
Finally, the equations of motion for the impactor and hybrid composite plate system are obtained as follows: € s þ F ¼ 0; ms w
ð10Þ
½Mf€ qg þ ð½Ks þ ½Kr DT ½KT Þfqg ¼ fFg;
ð11Þ
where ½M and ½Ks are the system mass and stiffness matrices, and ½Kr and ½KT are the geometric stiffness matrices due to recovery stress and thermal stress, respectively. The force vector, fFg ¼ f0; 0; . . . ; F ; . . . ; 0; T 0g , and F is the contact force between the plate and the impactor. The solution for Eqs. (10) and (11) can be deduced by various methods. In this study, Newmark’s constant-acceleration time integration algorithm will be used. Eq. (11) may be written for each time step as iþ1 ½KfqgtþDt
Dt2 i fFgtþDt þ ½Mfbg; ¼ 4
ð12Þ
where ½K ¼
Dt2 ð½Ks þ ½Kr ½KT DT Þ þ ½M 4
and Dt2 f€ qgr : 4 The same procedure can be used also for solving the equation of motion for the impactor. The evaluation of
fbgr ¼ fqgr þ Dtfq_ gr þ
177
the contact force depends on a contact law which relates the contact force with the indentation. The contact law depends on the permanent indentation after the unloading cycles, and the modified version [7] is to be used as follows: Loading
Unloading Reloading
F ¼ ka1:5 ; 0 < a 6 am ; i1 4 pffiffih 2 r ð1 ms Þ =Es þ 1=E2 : k¼ 3 q a a0 : F ¼ Fm am a 0 1:5 a a0 F ¼ Fm : am a0
ð13Þ ð14Þ ð15Þ
In these formulas, k is the modified constant of the Hertz contact theory, am is the maximum indentation during loading, Fm is the maximum contact force at the beginning of unloading, and a0 denotes the permanent indentation in a loading–unloading cycle. 2.1. Sensor equation Piezoelectric sensors monitor the impact-induced strains. The closed circuit charge signal is given by " Z Z # 1 qðtÞ ¼ D3 dA D3 dA þ ; ð16Þ 2 A A z¼z0 z¼z1 where A is the effective electrode surface area of the sensor, z0 and z1 are the z-coordinates of the bottom and top surfaces of the piezoelectric layer, respectively. In addition, D3 denotes the coefficient of the electric displacement in the thickness direction. Assuming that there is no external surface charge on the piezoelectric sensors, the charge is developed from the mechanical strain only. In addition, voltage is assumed to be linearly distributed through the thickness (z-direction) of the piezoelectric layers. We consider the effect of SMA, and the charge signals include the thermal expansion effect. Coefficient D3 in Eq. (16) is determined from Eq. (3) as D3 ¼ fe3 gfeg þ P3 DT ; fe3 g ¼ fe31 ; e32 ; e34 ; e35 ; e36 g:
ð17Þ
Because the piezoelectric effect of PVDF films is directional, it can be expected that by changing the skew angle h (see Fig. 1), the bending, stretching and torsion for motions of the plate are displayed from the sensor charge signal in different combinations due to the relationship between ½e and h: T
½e ¼ ½A½e0 ½T ; where
2
cos h ½A ¼ 4 sin h 0
3 sin h 0 cos h 0 5; 0 1
ð18Þ
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(a) Fig. 2. Piezoelectric stress/charge constants of PVDF film versus skew angle.
½T ¼ 2 cos2 h 6 6 sin2 h 6 6 6 0 6 6 0 4
sin2 h
0
0
cos2 h
0
0
0
cos h
sin h
0
cos h sin h cos h sin h
sin h cos h 0
0
2 cos h sin h
3
7 2 cos h sin h 7 7 7 7; 0 7 7 0 5
cos2 h sin2 h
and ½e0 is the original piezoelectric stress constant. The dependency of the piezoelectric stress constants e on the skew angle h is demonstrated in Fig. 2.
3. Results and discussions To verify the results in this study, numerical data are compared with the data of previous works in [8]. The model is a composite plate with the ply orientation of [0/ )45/45/90]2S . All edges of the plate are clamped, and the plate is subjected to an impact with the velocity of 38.1 m/s. The results in the reference are obtained by 8 8 4 meshes in 3-D element, while 8 8 meshes are chosen in this study. Center deflection of the plate is depicted in Fig. 3 for 0 6 t 6 250 ls. A certain little difference exists between the results of Wu and Chang [8] and the present data. This is due to the different meshes and the simplification of the contact force adopted. However, the numerical results almost coincided. Next, the 16-ply [0°2 =90°2 /+45°2 /)45°2 S laminate with clamped edges is investigated. For this model, the impact is from an aluminum sphere with a diameter of 12.7 mm at a speed of 20 m/s. Impact-induced strain is developed. During the contact of the plate and the sphere, responses of the composite plate are simultaneously monitored by the sensors made from PVDF films which are mounted on the opposite side of the impact. In this example, the materials are graphite/epoxy with the properties as shown in Table 1. The dimensions of the
(b) Fig. 3. Comparison of results: (a) contact force; (b) plate displacement (— present; - - - - Wu and Chang [8]).
plate and piezoelectric sensor are 80 mm 80 mm and 10 mm 10 mm, respectively. Time step Dt is chosen as 5 ls, which is found to be appropriate for sufficient numerical convergence. The displacements of the impactor and the center of the plate during impact are shown in Fig. 4. Fig. 5 depicts the local strains that the sensor is intended to monitor. Components of the strain indicate exx and eyy at the impact point as the function of time. Fig. 6 presents the closed circuit charge q measured from the piezoelectric sensor A as a function of time. It is found that the bonded PVDF patches have very little effect on the dynamic response of the plate. This can be attributed to the flexibility of PVDF films, which prove to be ideal sensors. As can be seen in Figs. 5 and 6, the charge amount of the curve manifests the combinations of the strain components, and the piezoelectric constants may be regarded as weighting factors. For the different skew angles h, these weighting factors are changed, and different combinations of the strain components are achieved, which are clearly reflected by the output charge signal. In addition, Fig. 7 shows that the charge reaches a maximum value at a certain skew angle h. Additionally, the charge may not reach this maximum value for h ¼ 0. As it often occurred in normal cases, other values of h can engender much larger ones that are
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179
Table 1 Properties of materials for the model
Longitudinal Young’s modulus, E1 (GPa) Transverse Young’s modulus, E2 (GPa) Shear modulus in 1-, 2-direction, G12 (GPa) Shear modulus in 2-, 3-direction, G23 (GPa) Poisson’s ratio in 1-, 2-direction, m12 Density, q ðkg=m3 Þ Layer thickness, h (mm) Critical indentation, acr (mm) Piezoelectric strain constant, d31 (pm/v) Piezoelectric strain constant, d32 (pm/v) Piezoelectric strain constant, d33 (pm/v) Pyroelectric coefficient, P ð106 C=m2 KÞ
Plate
PVDF
145.5 10.0 5.687 3.85 0.3 1535.4 0.16 0.08 – – – –
2.0 2.0 0.775 0.775 0.29 1800.0 0.1 – 23.0 3.0 )33.0 30.0
(a)
(b) Fig. 4. History of (a) contact force and (b) displacements. Fig. 5. Strain components: (a) impact-induced strain exx ; (b) impactinduced strain eyy .
more easily detected. This indicates that changing h can improve the sensitivity of the sensor. Fig. 8 illustrates the effectiveness of SMA fibers with the following properties for minimizing transverse deflections of the composite plate-induced impact: E ¼ 70 GPa; m ¼ 0:33;
G ¼ 26:32 GPa; 3
q ¼ 6500:00 kg=m :
In the figure, jsx and jsy indicate the volume fractions of SMA fibers in the global x-direction and y-direction,
respectively. Note that the mechanical properties of the Nitinol correspond to the austenitic phase. SMA fibers are assumed to be subjected to 8% pre-strain with the subsequent temperature increment of DT ¼ 39 °C, and then a complete reverse transformation to austenite and the recovery stress rr ¼ 220 MPa. Generally, SMA fibers have a property to be stiffened or controlled by the addition of heat (generally by applying a current through the fibers). Therefore, the resultant in-plane
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Fig. 6. Charge generated at sensor A with different skew angles: (a) skew angle 0°; (b) skew angle 90°.
Fig. 7. Normalized charge of piezoelectric sensors versus skew angle.
forces adaptively change the structural response of the plate. This approach is referred to as active strain energy tuning (ASET) [5]. The deflection and stresses are reduced by employing pre-strained SMA fibers, which are in the martensitic phase when embedded within the plate. At an elevated temperature, the SMA fibers are transformed into the austenitic phase and tend to contract. However, the contraction is either completely
Fig. 8. Effective of SMA fibers for minimizing deflection-induced impact: (a) SMA fibers orientation in the x-direction; (b) SMA fibers orientation in the y-direction.
prevented or reduced due to a constraint. As a result, a significant tensile recovery stress occurs, which reduces the deformations and stresses. In Fig. 8, the results demonstrate that when the temperature of the plate is elevated, resulting in the reverse transformation, SMA fibers are very effective in controlling the deflection due to the impact. By increasing the SMA fiber volume fraction (js ), dynamic impact-induced deflection is decreased. As a result of recovery stress, maximum deflection for the graphite/epoxy plate is reduced by about 50% with the volume fraction of SMA equal to 0.3. Also, when changing the direction of the SMA fiber, the relative deflection becomes different. We can obtain optimal deflection under the impact by considering appropriate fiber’s volume fraction and direction. Fig. 9 illustrates the effectiveness of SMA fibers for minimizing impact-induced strain by changing both the volume fractions and directions of the SMA fiber. We can reduce the strain by increasing fiber volume fraction depending on its direction. Fig. 10 demonstrates the charge signal generated by each sensor with skew angle 90°, which includes the thermal expansion effect due to the SMA properties. Note also that in all the cases considered above, external temperature is assumed to be equal to that of the
J.-H. Roh, J.-H. Kim / Composite Structures 56 (2002) 175–182
(a)
(b)
181
(a)
(b)
Fig. 9. Effectiveness of SMA fibers for minimizing strain-induced impact: (a) exx with SMA fiber x-direction; (b) eyy with SMA fiber ydirection.
environment. An artificial heating of SMA fibers to produce a desired property may be ineffective and even detrimental to the structure due to the thermal forces generated by the heating. However, this disadvantage can be overcome by changing the composition of SMA. For example, changing the composition of Titanium versus Nickel by just 1% can shift the transformation temperatures to negative [9]. Therefore, designers can produce a material that will be transformed to the austenite phase at room temperature by the adequate tailoring of the shape memory alloys.
(c) Fig. 10. Charge signal including SMA effects: (a) sensor A with skew angle 90°; (b) sensor B with skew angle 90°; (c) sensor C with skew angle 90°.
4. Conclusions The sensing and controlling of hybrid composite plates subjected to low velocity impact has been considered. The results generated in this study illustrate that SMA fibers embedded within the layers of a composite plate can significantly enhance its global resistance to low velocity impact. On the other hand, SMA fibers can become less effective, if it is necessary to increase the temperature to activate them. As a result, the effectiveness of SMA fibers can be further improved by opti-
mizing their distribution throughout the plate. In addition, the feasibility of the piezoelectric sensors to measure strains induced by low velocity impact is investigated numerically. The close circuit charge generated by the piezoelectric sensor is computed, and the results show that it can manifest the impact-induced strain. In addition, the contribution of each strain component can be predicted by changing the PVDF sensor’s skew angle.
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Also, the hybrid smart structure containing the piezoelectric sensors and SMA fiber actuators can detect its damage and adapt to unexpected environmental conditions efficiently without control devices.
Acknowledgements This work was supported by the Brain Korea 21 project.
References [1] Yang D. Shape memory alloy and smart hybrid composites – advanced materials for the 21st century. Mater Des 2000;21: 503–5.
[2] Tan TM, Sun CT. Wave propagation in graphite/epoxy laminates due to impact. J Appl Mech 1985;52:6–12. [3] Sun CT, Chen JK. On the impact of initially stressed composite laminates. J Compos Mater 1985;19:490–504. [4] Birman V. An approach to optimization of shape memory alloy hybrid composite plates subjected to low-velocity impact. Composite B 1996;27:439–46. [5] Rogers CA, Liang C, Jia J. Structural modification of simplysupported laminated plates using embedded shape memory alloy fibers. Comput Struct 1991;38:569–80. [6] Yin L, Wang X-M, Shen Y-P. Damage-monitoring in composite laminates by piezoelectric films. Comput Struct 1996;59:623–30. [7] Yin L, Shen Y. Strain sensing of composite plates subjected to low velocity impact with distributed piezoelectric sensors: a mixed finite element approach. J Sound Vibr 1997;199(1):17–31. [8] Wu H-YT, Chang F-K. Transient dynamic analysis of laminated composite plates subjected to transverse impact. Comput Struct 1991;31:453–66. [9] Funakubo H. In: Shape memory alloys. New York: Gordon and Breach; 1984. p. 77–8.
Hybrid smart composite plate under low velocity impact Jin-Ho Roh, Ji-Hwan Kim
*
School of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul 151-742, Republic of Korea
Abstract This paper presents an approach to the detection of the strains and optimization of shape memory alloys (SMAs) under low velocity impact. It is assumed that the sensor is mounted on the opposite side of the impact process. SMA fibers are embedded within the layers of a composite plate. The first-order shear deformation theory of plates is used, and the finite element method is adopted in the numerical analysis. With different skew angles of the piezoelectric sensor, we can illustrate that the measured charge manifests the strain components induced by the impact. By changing the SMA volume fractions and the temperature increment, we can significantly reduce the deflections. It can be shown that the application of piezoelectric sensors is effective in the monitoring of the transient strain. In addition, the optimization of the distribution of volume fraction of SMA fibers provides further benefits in reducing the deflection of the plate. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Shape memory alloy; Piezoelectric; Impact; Active strain energy tuning; Hybrid smart structure
1. Introduction Intelligent/smart structures have features of integrated sensors and actuators within a host material, and they will have a tremendous effect upon numerous industrial fields. The idea of applying ‘‘smart’’ materials to mechanical and structural systems has been studied by many research workers in various disciplines. Shape memory alloys (SMAs) as the fiber reinforcement or actuator in the composite material can have numerous adaptive capabilities, but their logy behavior is the main reason for the limited applications of SMA. Piezoelectric materials can be used as the sensors and actuators simultaneously due to the piezoelectric effects and fast response of the materials. On the other hand, the displacement is quite small due to the small strain magnitude. Yang [1] discusses smart hybrid composites which incorporate SMA with other structural or functional materials. In addition, he suggests that with the simultaneous use of SMA and a ferroelectric ceramic, the composite material can effectively sense and actuate to reduce the structural vibration without the use of an external control. Owing to their weak impact resistance properties, laminated composite plates are very susceptible to damage when impacted by a solid object. This kind of *
Corresponding author. Tel.: +82-2880-7383; fax: +82-2887-2662. E-mail addresses: [email protected] (J.-H. Roh), jwhkim@ plaza.snu.ac.kr (J.-H. Kim).
damage is internal and it cannot be easily detected from its appearance. Accordingly, numerous experimental and analytical techniques have been developed for composite plates subjected to low velocity impact. Tan and Sun [2] predict the history of a contact force analytically based on the transverse shear deformable plate theory. Subsequently, Sun and Chen [3] investigate the effect of initial stresses on the low velocity dynamic response of the laminates by the finite element method. Smart materials can present an attractive tool for reducing deflections and stresses in the structures subjected to a low velocity impact. In particular, SMAs can generate significant tensile stresses. Reducing the deflection and strain induced by the impact can be achieved as the SMA fibers are embedded within a composite material in a hybrid smart composite. Birman [4] considers an approach to the optimum design of shape memory alloys in hybrid cross-ply composite plates subjected to low velocity impact. The results illustrate that SMA fibers embedded within the layers of a composite plate can significantly enhance its global resistance to low velocity impact. The constitutive relationships and associated micromechanics for a hybrid SMA composite material are considered in a number of studies. Rogers et al. [5] employ a rule of mixtures to develop the micromechanics for SMA composites with graphite/epoxy and nitinol/epoxy plies. Piezoelectric sensors measure the average local strain over the area to which they are bonded, and various investigations are concentrated on the methodology of
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 8 9 - 1
176
J.-H. Roh, J.-H. Kim / Composite Structures 56 (2002) 175–182
using piezoelectric devices as the sensors to monitor structural damage. This concept is exploited by Yin et al. [6] in detecting edge delamination for composite plates subjected to remote uniaxial loading. Yin and Shen [7] investigate numerically the feasibility of using piezoelectric sensors to measure strains under low velocity impact in composites. In the present study, low velocity impact for the hybrid smart plate is considered. SMA fibers are embedded within the layers, and the piezoelectric sensors made from PVDF films are mounted on the opposite side of the impact. For the different skew angles, each sensor measures the charge signal induced by the impact. When the fibers are heated, the SMA fibers try to contract to their normal or memorized length and therefore generate a recovery force. Therefore, the resultant in-plane forces will adaptively change the structural response of the plate. This approach is referred to as active strain energy tuning (ASET), which is used as a control technique. The shape memory effect is employed to reduce transverse deflections of the plates and the global stresses of the material. The first-order shear deformation theory of plates is used, and also the finite element formulation presented to model the dynamic response of the hybrid composite plates. It is shown that SMA fibers can reduce deflection, and that the piezoelectric sensors are effective in monitoring the impact-induced strain.
2. Finite element model Fig. 1 shows the hybrid smart composite plate with the piezoelectric film subjected to impact by the mass ms . The SMA fiber is embedded in a composite matrix. Sensors A, B and C are mounted on the opposite side of impact, and located as shown in Fig. 1(a). The piezoelectric film sensor measures the charge due to the impact for the skew angle h as shown in Fig. 1(b). The Mindlin plate theory is used, and the finite element method is adopted. Each sensor made by a piezoelectric film measures the induced charge signal with respect to skew angles.
Fig. 1. Mechanical model for smart composite plate subjected to impact: (a) hybrid composite plate; (b) piezoelectric film.
By using the nine-node isoparametric shape function, the displacements for the element can be written as n oT ð1Þ fqge ¼ u10 ; v10 ; w10 ; h1x ; h1y ; . . . ; u90 ; v90 ; w90 ; h9x ; h9y ; e
where the subscript e denotes the element, u0 ; v0 ; w0 and hx ; hy stand for the mid-plane displacements and rotational angles, respectively. The recovery tensile stress can be evaluated analytically or experimentally. In this study, experimental stress–temperature curves are employed as in [4]. Thermal effects are incorporated by experimental recovery stress–temperature relationships of the SMA fiber. Thus, the constitutive equation of a piezoelectric includes the effect of thermal expansion. The constitutive equations of the hybrid smart plate with SMA fibers and piezoelectric materials can be written as T
frg ¼ ½Qfeg ½e fEg þ frr gjs ½Qfagjc DT ;
ð2Þ
fDg ¼ ½efeg þ ½nfEg þ fPgDT ;
ð3Þ
where ½Q and ½Q indicate the reduced stiffness matrices for a hybrid composite material and the composite medium, respectively. In addition, ½e is the piezoelectric stress/charge coefficient matrix, ½E is the electric field, frr g is the recovery stress, and fag is the thermal expansion coefficient. The volume fractions of SMA fibers and the composite medium are denoted by js and jc , respectively. While ½D stands for the electric displacement, ½n is the permittivity matrix, and fPg is the thermal stress coefficient. The overall properties of the hybrid composite materials are calculated by using micromechanical equations based on Chamis’ multicell model [4]. The governing equations for SMA fiber-reinforced hybrid composite plates subjected to a combined external, thermal and recovery stress load can be derived by the principle of virtual work dU ¼ dV ;
ð4Þ
where U and V are the strain energy and the work of the external force of the system, respectively. We can express the strain energy U as follows: Z 1 T U¼ feg frg dv: ð5Þ 2 v And work of the external force V is composed of V 0 and V 00 . V 0 is due to in-plane forces and the effect of thermal loads which includes the recovery forces of the Nitinol (Nickel–Titanium-Naval Ordnance Laboratory) fibers, and the heating of the Nitinol fibers. V 00 is due to inertia force. dV 0 can be written as follows: Z Z 2 X 2 X 1 h dw Nab ðoa wÞðob wÞ dA; ð6Þ dV 0 ¼ 2 A a¼1 b¼1
J.-H. Roh, J.-H. Kim / Composite Structures 56 (2002) 175–182 r T r T where Nab ¼ Nab þ Nab . In addition, Nab and Nab denote the recovery tension developed in the Nitinol fibers and compressive in-plane thermal loads in the a- and bdirections, respectively. Thermal loads are generated by temperature change DT , and are developed by both the activation and deactivation processes of the Nitinol fibers. Hence, the thermal load N T are given by the relations 2 T3 2 3 Z hk Nxx axx X 4 NyyT 5 ¼ Qk 4 ayy 5DT jc dz; ð7Þ hkþ1 k axy NxyT
where ½Q is the matrix for the composite material, axx ¼ a11 m2 þ a22 n2 , ayy ¼ a22 m2 þ a11 n2 and axy ¼ 2ða11 a22 Þmn. Symbols axx , ayy and axy denote coefficients of thermal expansion in the x-, and y-directions. Symbol 1 denotes the direction parallel to the fibers while symbol 2 stands for the direction normal to them. The recovery tension, N r , is developed in the Nitinol fibers as 2 2 3 2 r 3 Nxx m 4 Nyyr 5 ¼ rr 4 n2 5js h: ð8Þ Nxyr 2mn The angle of the fiber’ location is denoted by h while m and n denote cos h and sin h, respectively. Now, the virtual of inertia force dV 00 is written as Z Z 00 dV ¼ dwhq€ w dA: ð9Þ A
Finally, the equations of motion for the impactor and hybrid composite plate system are obtained as follows: € s þ F ¼ 0; ms w
ð10Þ
½Mf€ qg þ ð½Ks þ ½Kr DT ½KT Þfqg ¼ fFg;
ð11Þ
where ½M and ½Ks are the system mass and stiffness matrices, and ½Kr and ½KT are the geometric stiffness matrices due to recovery stress and thermal stress, respectively. The force vector, fFg ¼ f0; 0; . . . ; F ; . . . ; 0; T 0g , and F is the contact force between the plate and the impactor. The solution for Eqs. (10) and (11) can be deduced by various methods. In this study, Newmark’s constant-acceleration time integration algorithm will be used. Eq. (11) may be written for each time step as iþ1 ½KfqgtþDt
Dt2 i fFgtþDt þ ½Mfbg; ¼ 4
ð12Þ
where ½K ¼
Dt2 ð½Ks þ ½Kr ½KT DT Þ þ ½M 4
and Dt2 f€ qgr : 4 The same procedure can be used also for solving the equation of motion for the impactor. The evaluation of
fbgr ¼ fqgr þ Dtfq_ gr þ
177
the contact force depends on a contact law which relates the contact force with the indentation. The contact law depends on the permanent indentation after the unloading cycles, and the modified version [7] is to be used as follows: Loading
Unloading Reloading
F ¼ ka1:5 ; 0 < a 6 am ; i1 4 pffiffih 2 r ð1 ms Þ =Es þ 1=E2 : k¼ 3 q a a0 : F ¼ Fm am a 0 1:5 a a0 F ¼ Fm : am a0
ð13Þ ð14Þ ð15Þ
In these formulas, k is the modified constant of the Hertz contact theory, am is the maximum indentation during loading, Fm is the maximum contact force at the beginning of unloading, and a0 denotes the permanent indentation in a loading–unloading cycle. 2.1. Sensor equation Piezoelectric sensors monitor the impact-induced strains. The closed circuit charge signal is given by " Z Z # 1 qðtÞ ¼ D3 dA D3 dA þ ; ð16Þ 2 A A z¼z0 z¼z1 where A is the effective electrode surface area of the sensor, z0 and z1 are the z-coordinates of the bottom and top surfaces of the piezoelectric layer, respectively. In addition, D3 denotes the coefficient of the electric displacement in the thickness direction. Assuming that there is no external surface charge on the piezoelectric sensors, the charge is developed from the mechanical strain only. In addition, voltage is assumed to be linearly distributed through the thickness (z-direction) of the piezoelectric layers. We consider the effect of SMA, and the charge signals include the thermal expansion effect. Coefficient D3 in Eq. (16) is determined from Eq. (3) as D3 ¼ fe3 gfeg þ P3 DT ; fe3 g ¼ fe31 ; e32 ; e34 ; e35 ; e36 g:
ð17Þ
Because the piezoelectric effect of PVDF films is directional, it can be expected that by changing the skew angle h (see Fig. 1), the bending, stretching and torsion for motions of the plate are displayed from the sensor charge signal in different combinations due to the relationship between ½e and h: T
½e ¼ ½A½e0 ½T ; where
2
cos h ½A ¼ 4 sin h 0
3 sin h 0 cos h 0 5; 0 1
ð18Þ
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(a) Fig. 2. Piezoelectric stress/charge constants of PVDF film versus skew angle.
½T ¼ 2 cos2 h 6 6 sin2 h 6 6 6 0 6 6 0 4
sin2 h
0
0
cos2 h
0
0
0
cos h
sin h
0
cos h sin h cos h sin h
sin h cos h 0
0
2 cos h sin h
3
7 2 cos h sin h 7 7 7 7; 0 7 7 0 5
cos2 h sin2 h
and ½e0 is the original piezoelectric stress constant. The dependency of the piezoelectric stress constants e on the skew angle h is demonstrated in Fig. 2.
3. Results and discussions To verify the results in this study, numerical data are compared with the data of previous works in [8]. The model is a composite plate with the ply orientation of [0/ )45/45/90]2S . All edges of the plate are clamped, and the plate is subjected to an impact with the velocity of 38.1 m/s. The results in the reference are obtained by 8 8 4 meshes in 3-D element, while 8 8 meshes are chosen in this study. Center deflection of the plate is depicted in Fig. 3 for 0 6 t 6 250 ls. A certain little difference exists between the results of Wu and Chang [8] and the present data. This is due to the different meshes and the simplification of the contact force adopted. However, the numerical results almost coincided. Next, the 16-ply [0°2 =90°2 /+45°2 /)45°2 S laminate with clamped edges is investigated. For this model, the impact is from an aluminum sphere with a diameter of 12.7 mm at a speed of 20 m/s. Impact-induced strain is developed. During the contact of the plate and the sphere, responses of the composite plate are simultaneously monitored by the sensors made from PVDF films which are mounted on the opposite side of the impact. In this example, the materials are graphite/epoxy with the properties as shown in Table 1. The dimensions of the
(b) Fig. 3. Comparison of results: (a) contact force; (b) plate displacement (— present; - - - - Wu and Chang [8]).
plate and piezoelectric sensor are 80 mm 80 mm and 10 mm 10 mm, respectively. Time step Dt is chosen as 5 ls, which is found to be appropriate for sufficient numerical convergence. The displacements of the impactor and the center of the plate during impact are shown in Fig. 4. Fig. 5 depicts the local strains that the sensor is intended to monitor. Components of the strain indicate exx and eyy at the impact point as the function of time. Fig. 6 presents the closed circuit charge q measured from the piezoelectric sensor A as a function of time. It is found that the bonded PVDF patches have very little effect on the dynamic response of the plate. This can be attributed to the flexibility of PVDF films, which prove to be ideal sensors. As can be seen in Figs. 5 and 6, the charge amount of the curve manifests the combinations of the strain components, and the piezoelectric constants may be regarded as weighting factors. For the different skew angles h, these weighting factors are changed, and different combinations of the strain components are achieved, which are clearly reflected by the output charge signal. In addition, Fig. 7 shows that the charge reaches a maximum value at a certain skew angle h. Additionally, the charge may not reach this maximum value for h ¼ 0. As it often occurred in normal cases, other values of h can engender much larger ones that are
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Table 1 Properties of materials for the model
Longitudinal Young’s modulus, E1 (GPa) Transverse Young’s modulus, E2 (GPa) Shear modulus in 1-, 2-direction, G12 (GPa) Shear modulus in 2-, 3-direction, G23 (GPa) Poisson’s ratio in 1-, 2-direction, m12 Density, q ðkg=m3 Þ Layer thickness, h (mm) Critical indentation, acr (mm) Piezoelectric strain constant, d31 (pm/v) Piezoelectric strain constant, d32 (pm/v) Piezoelectric strain constant, d33 (pm/v) Pyroelectric coefficient, P ð106 C=m2 KÞ
Plate
PVDF
145.5 10.0 5.687 3.85 0.3 1535.4 0.16 0.08 – – – –
2.0 2.0 0.775 0.775 0.29 1800.0 0.1 – 23.0 3.0 )33.0 30.0
(a)
(b) Fig. 4. History of (a) contact force and (b) displacements. Fig. 5. Strain components: (a) impact-induced strain exx ; (b) impactinduced strain eyy .
more easily detected. This indicates that changing h can improve the sensitivity of the sensor. Fig. 8 illustrates the effectiveness of SMA fibers with the following properties for minimizing transverse deflections of the composite plate-induced impact: E ¼ 70 GPa; m ¼ 0:33;
G ¼ 26:32 GPa; 3
q ¼ 6500:00 kg=m :
In the figure, jsx and jsy indicate the volume fractions of SMA fibers in the global x-direction and y-direction,
respectively. Note that the mechanical properties of the Nitinol correspond to the austenitic phase. SMA fibers are assumed to be subjected to 8% pre-strain with the subsequent temperature increment of DT ¼ 39 °C, and then a complete reverse transformation to austenite and the recovery stress rr ¼ 220 MPa. Generally, SMA fibers have a property to be stiffened or controlled by the addition of heat (generally by applying a current through the fibers). Therefore, the resultant in-plane
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Fig. 6. Charge generated at sensor A with different skew angles: (a) skew angle 0°; (b) skew angle 90°.
Fig. 7. Normalized charge of piezoelectric sensors versus skew angle.
forces adaptively change the structural response of the plate. This approach is referred to as active strain energy tuning (ASET) [5]. The deflection and stresses are reduced by employing pre-strained SMA fibers, which are in the martensitic phase when embedded within the plate. At an elevated temperature, the SMA fibers are transformed into the austenitic phase and tend to contract. However, the contraction is either completely
Fig. 8. Effective of SMA fibers for minimizing deflection-induced impact: (a) SMA fibers orientation in the x-direction; (b) SMA fibers orientation in the y-direction.
prevented or reduced due to a constraint. As a result, a significant tensile recovery stress occurs, which reduces the deformations and stresses. In Fig. 8, the results demonstrate that when the temperature of the plate is elevated, resulting in the reverse transformation, SMA fibers are very effective in controlling the deflection due to the impact. By increasing the SMA fiber volume fraction (js ), dynamic impact-induced deflection is decreased. As a result of recovery stress, maximum deflection for the graphite/epoxy plate is reduced by about 50% with the volume fraction of SMA equal to 0.3. Also, when changing the direction of the SMA fiber, the relative deflection becomes different. We can obtain optimal deflection under the impact by considering appropriate fiber’s volume fraction and direction. Fig. 9 illustrates the effectiveness of SMA fibers for minimizing impact-induced strain by changing both the volume fractions and directions of the SMA fiber. We can reduce the strain by increasing fiber volume fraction depending on its direction. Fig. 10 demonstrates the charge signal generated by each sensor with skew angle 90°, which includes the thermal expansion effect due to the SMA properties. Note also that in all the cases considered above, external temperature is assumed to be equal to that of the
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(a)
(b)
181
(a)
(b)
Fig. 9. Effectiveness of SMA fibers for minimizing strain-induced impact: (a) exx with SMA fiber x-direction; (b) eyy with SMA fiber ydirection.
environment. An artificial heating of SMA fibers to produce a desired property may be ineffective and even detrimental to the structure due to the thermal forces generated by the heating. However, this disadvantage can be overcome by changing the composition of SMA. For example, changing the composition of Titanium versus Nickel by just 1% can shift the transformation temperatures to negative [9]. Therefore, designers can produce a material that will be transformed to the austenite phase at room temperature by the adequate tailoring of the shape memory alloys.
(c) Fig. 10. Charge signal including SMA effects: (a) sensor A with skew angle 90°; (b) sensor B with skew angle 90°; (c) sensor C with skew angle 90°.
4. Conclusions The sensing and controlling of hybrid composite plates subjected to low velocity impact has been considered. The results generated in this study illustrate that SMA fibers embedded within the layers of a composite plate can significantly enhance its global resistance to low velocity impact. On the other hand, SMA fibers can become less effective, if it is necessary to increase the temperature to activate them. As a result, the effectiveness of SMA fibers can be further improved by opti-
mizing their distribution throughout the plate. In addition, the feasibility of the piezoelectric sensors to measure strains induced by low velocity impact is investigated numerically. The close circuit charge generated by the piezoelectric sensor is computed, and the results show that it can manifest the impact-induced strain. In addition, the contribution of each strain component can be predicted by changing the PVDF sensor’s skew angle.
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Also, the hybrid smart structure containing the piezoelectric sensors and SMA fiber actuators can detect its damage and adapt to unexpected environmental conditions efficiently without control devices.
Acknowledgements This work was supported by the Brain Korea 21 project.
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