Elastic Waves from a Distributed Surface Source in a Unidirectional Composite Laminate

The response of a unidirectional composite plate of infirritc lateral dimensions to localized dynamic surface sourws is investigated through theoretical modeling and laboratory tests. In the theoretical sitnulations, the material ofthc plate is assumed to be dissipative and transversely isotropic with its symmetry axis parallel to the fibers. The source is assumed to hsrve an arbitrary spatial and time dependence. The associated clastodynamic boundary value problem is solved by means of an integral transform technique followed by numerical evaluation of the inversion intc~als. The laboratory tests are carried out on unidirectional graphite epoxy plates of thicknesses ranging from 1-25 mm and large Iateml dimensions (> 30 cm? excited by means of broadband transducers attached to its surface. The calculated surface response of the plate at different distances and directions from the source is shown to agree very well with the recorded response in the ultrasonic range. INTRODUCTION It is well known that Iaminatcd tibcr reinforced composites often suffer significant internal damage when they arc subjected to Iocalizcd dynamic surface loads. The damage may involve frbcr breakage and debonding as well as delamination bctwccn the individual laminae. Such damage has been observed to occur even at relatively low impact speeds resulting in a severe loss in the load carrying capacity of the larninac. Although the damage is clearly caused by the stresses which develop within the material, the prccisc nature of these stresses and their relationship to the degree and mode ofthc damage are not clearly understood at present. This is particularly true in the dynamic case where the stresses are caused by waves whose propagation characteristics are strongly influenced by the inherent an isotropy and heterogeneity of the composite material. Dynamic response of plates has been studied theoretically by many authors through past decades. The linear elastic solutions of the isotropic or anisotropic plates have been investigated by, c.g, Weaver and Pao (1982), Vasudevan and Mal (1985); Xu and Mal (1987), I,iu et al. (1991a, b), Mrrl and I,ih (1992, 1995). For low frequency response quasi-static and thin plate theories have been used (SCC e.g., Chow(1971 ), Moon(l 973); Sun and l’rm (1984); 1,al (1984)). While nurncrous analytical investigation of wave propagation in plate have been executed, concurrent experimental studies of such wave processes in the laboratory have been far Icss prevalent. Most clXort in this direction were devoted to a cmnpwiwn of pred Icted phase and group velocities of surface waves. lor wrvcfonn analysis, Medick (1960) used a 220 Switl rifle bullet impacting perpendicular to an aluminum plate to generate flexure waves Rwerrtly, German et al. (1989) used a lead break on the surface of a plate to generated wavcy however, there are no available results for comparison between the measured and calculated time history for wave propagation in a cclmposite under dynamic surface loads. In this paper a clas.sieal integral transform technique coupled with the matrix method developed by Mal and Lih (1992) is used for the theoretical simulation of the surface source problem, The integrals involved in the spatial inverse transform arc evaluated by means of a previously developed adaptive integration scheme. The mcrr..urcmcnts are earned out by means of an ultrasonic system called the Fracture Wave 1 letcctor by Digital Wave Corp. Numerical and cxperirncntal rrmr[ts for the response of a unidirectional composite Iarninate due to a quasisine pulsed load are compared. THEORY Material Modeling Composite materials arc known to bc strongly an isotropic as well as dissipative. In tibcr-reinforced composites dissipation is caused by the anisotropy of fiber orientations, and the dissipation of the waves is caused by the viscxrclastic nature of the resin and by scattering from the fibers and other inhomogencities. Both effects can bc modeled in the frequency domain by assuming that the stiffness constants, C,j, are eornplcx and frequcncydependcnt, A possible form of Ci, that can model the essential featu res of the dissipation caused by these factors, has been given in Mal, Bar-Coher~, and Lih (1 992) and will be used here. A brief description of the model is presented for completeness; the details can be found in the cited paper. We recall that the linear constitutive equation for a transversely isotropic elastic solid with its symmetry axis along the xl-axis (Figure 1) can bc expressed in the form II ~11 %2 033 23 031 ~12 c11 C12 C12 0 0 0 %2 C22 C 2 30 0 0 ’312 C23 %2 0 0 0 0 0 0 C4400 0 0 0 0 C550 o 0000C5: Ul,l U2.2 U3,3 ‘2,3 + UI,3 + UI,2 + ‘3,2 U3,1 u?>], (1) where Cli = (ct2 cJ12, oij is the Cauchy’s stress tensor, w is the displacement vector and c1l, CIV czv C23, CSS are the five independent real stiffness constants of the material. We introduce five additional constants al, al, al, ad and as related to Cij and the density of the material, p, through = c221p, a2 = c, / p , a 3 = (c12 + c55)@ al (2) a4 = (C22 c*3y2p z cddip, a5 = c55iP It is WCII known that the quantities da,, <az, J% J% and Jas r~present the velocities of five indcPcndent butk waves that can be transmitted along certain spccitic directions in the transversely isotropic solid. IA Cll, C,2, C21, C+,, C55 denote the urmplex and frequency-dependent stiffness constants of the fiber-reinforced composite and let the complex constants Al,A~, A3, A, be A5 are defmcd through, ,4, = c22/p, A2 = c),/p, ,43 = (c,* + c55yp, A4 = c44@, A5 = ~55ip (3) f(x,,+,lj ~ ._..–— — ——–-3 FIGURE 1. THE SURF’ACE LOAD PROB1.EM FOR A UNIDIREC1”IONAL COMPOSITE Wc assume that the two sets of constants ,4i and a, arc related by the equations