The ultrasonic wave propagation in composite materialand its characteristic evaluationJunjie Chang, Changliang Zheng, Qing-Qing Ni1. IntroductionFRP composite materials were applied to various fields, such as aircraft and space structures, because of the excellent characteristics, e.g., light-weight, high ratio of relative intensity and high ratio of relative rigidity. Despite FRP having such outstanding characteristic, cracks in the matrix and fractures of the fiber make de bonding such kind of damage easy to occur between the fiber and the matrix, or the multi-layers. These damages are difficult to be detected directly by visual inspection from the sample surface, causing trouble to ensure the reliability and safety of the composite material and structures. Meanwhile, health monitoring technologies of materials are indispensable. Among them, the ultrasonic health monitoring technology attracts lots of attentions in recent years. Simulations by finite element method have been performed for the development of apparatus for ultrasonic damage-detection, such as ultrasonic picture inspection and ultrasonic laser, and for the verification of their safety and validity. Researches and calculations on the propagation analysis of the ultrasonic wave in fiber strengthening composite materials have been well conducted and reported [1–8].On the solid interface, two kinds of boundaries can be considered. One is liquid contact in which thin lubricant is placed, and only power and position movement perpendicular to the interface are transmitted. The other one is complete solid combination, which power and position movement both perpendicular to and parallel to the interface are transmitted. Fiber strengthening composite material, the interface between the fiber and the matrix can be considered to be solid contact. In the case of ,debonding existing between the matrix and the fiber, few literatures were found, since the conversions of the transmitted wave mode, reflection wave mode and reflection pulse phase (waveform) make the analysis very complicated. Provided this problem to be solved, the quality of the materials, to some extent, can be estimated from the sound impedance of the reflector and the transmission object, and the optimal damage-detection method can be also assumed in a simulation.In this research, in the simulation of the technique monitoring the health by an ultrasonic wave method, the ultrasonic wave distribution pattern was analyzed with the basic theory for wave propagation by using the model for fiber strengthening composite material. Namely, it aims at obtaining the amplitude of the reflection wave and the amplitude of a transmitted wave, when the longitudinal wave has unit amplitude incidence in model compound material. In the case of an ultrasonic wave propagation inside a model media, the rates of the reflective longitudinal, reflective traverse wave, transmission longitudinal wave and a transmission traverse wave generated at a general incidence angle in the interface (a fiber and exfoliation) were analyzed and reflective coefficient and a transmission coefficient were gotten, respectively. Visualized studies separating into a longitudinal wave and a traverse wave were carried out, and the mechanisms of a longitudinal wave distribution and a traverse-wave distribution were elucidated when the ultrasonic wave propagated inside a composite material.2. Ultrasonic wave equationsConsider a single fiber composite, i.e., a single fiber is embedded in a matrix. Two dimensions analysis is conducted as shown in Fig. 2. In this case, when an ultrasonic wave propagates in this solid media, from Hooke’s law, the stress–strain relationship for two-dimensional plane strain in an isotropic media is written as follows [2]:c σε= （1）202000c λμλλλμμ+⎡⎤⎢⎥=+⎢⎥⎢⎥⎣⎦（2）T xx yy xy σσσσ⎡⎤=⎣⎦ （3）Txx yy xy εεεε⎡⎤=⎣⎦ （4） Where k and l are Lame′ constants, and the T superscript denotes the transposition.The ultrasonic wave equations of motion for two dimensional plane strain in an isotropic media are as follows:2222xy xx x yyxyyu x y tu y x t σσρσσρ∂∂∂+=∂∂∂∂∂∂+=∂∂∂ （5）Where, the first term on the left-hand side of Eq. (5) corresponds to a longitudinal wave, and the second term corresponds to a transverse wave. ρ is density. If the longitudinal wave velocity L c and transverse wave velocity T c are introduced the ultrasonic wave equations of motion for two-dimensional plane strain can be rewritten by()()2222222222222222222222y x x x L T L T y y y x L T L T u u u u c c c c x y x y t u u u u c c c c t y x x y ⎧⎫∂∂∂⎧⎫∂++-⎪⎪⎪⎪∂∂∂∂⎪⎪⎪⎪∂=⎨⎬⎨⎬∂∂∂∂⎪⎪⎪⎪++-⎪⎪⎪⎪∂⎩⎭∂∂∂∂⎩⎭ （6） In the case of a plane advancing wave, the following formula is used to calculate for the oscillating energy generated by the ultrasonic wave per unit time:2p I cρ= （7） 3. Results of analysis and simulation3.1. Transmission energy in different interface shapesWhen an incident vertical wave is obliquely irradiated, four waves as shown in Fig. 3, i.e., reflected longitudinal wave, reflected transverse wave, transmitted longitudinal wave and transmitted transverse wave, would appear on the interface. In other words, the shape of the interface between epoxy and glass may influence the propagation of the ultrasonic wave. For this reason, the model with different interface shapes as shown in Fig. 1 was used to investigate the influence of interface shape on wave propagation behavior. The volume fraction proportion of both materials is 1:1, despite of the different interface shapes of the three models. That is to say, the glass-volume-percentage of all the models is 50%. The properties of each medium used in the analysis are shown in Table 1. As a boundary condition of the model, absorption is considered on the right and left edge, while it is symmetrical (the roller) on the up and down direction. The analytic condition and the input parameters were shown in Table 1.Fig. 2 shows the transmission energy of the ultrasonic wave propagation for these four models shown in Fig. 1.Fig. 1. Four different interface shapes between epoxy and glass.Here the transmission energy was defined by the average energy per unit area, lJ/mm2, at the receiver edge. As seen, in Model 1, the incident ultrasonic wave is perpendicular to the plane interface, and transmitted wave occurs along whole plane, so that the transmission energy is far larger than that in the other models. The full-reflection takes place in part of interface in both Model 2 and Model 3 when the incidence angle is larger than the critical angle because the ultrasonic wave radiates obliquely on a convex or concave interface. About one third of the incident wave experiences full-reflection in Model 2 and Model 3. However, the transmission energy of Model 3 is larger than that of Model 2. A second peak appears in the transmission curve of Model 3. Peak 1 is a reflected wave that propagates as a secondary wave source near the up-down-ward interface (in the glass region), while peak 2 is a transmitted wave in the central part of the glass region. The reason might be that near the interface, a refractive index distribution occurs, resulting in the appearance of the scattered waves, including refraction and reflection waves.The full-reflection takes place in interface of Model 4 (incidence angle is 45_). All primary incident waves were reflected, and the very small transmission energy that shows as figure is because the dispersion wave and the reflected wave penetrated the part as secondary wave source from the vertical neighborhood.3.2. Influence of different fiber conditionsRefractive index distribution occurs near the second phase boundary due to the second phase compounding, resulting in the appearance of the scattered waves, including refraction and reflection in the composite materials strengthened by fibers. In the next, the scattering of the ultrasonic wave shown in Fig. 1 will be taken into consideration. The scatters occur due to fibers embedded in composite materials. The incident wave ()i ψ, propagating in matrix region, is a sinusoidal wave. When the incident wave reaches the fiber, some is transmitted into the fiber, and the other is reflected on the fiber/matrix interface, and becomes a secondary wave source. According to the overlapping principle of wave functions, the whole wave function ()t ψ can be expressed as a sum of the incident wave ()i ψ and the scattered wave ()s ψ.()()()t i s ψψψ=+ （8） Where the scattered wave ()s ψ includes all the waves scattering components generated due to the interface from the known wave ()i ψ.The model figure of the composite materials for the investigation of the scatters was designed as what shown in Fig. 3, where three fibers with different shapes were embedded in the matrix. The size of the model was 1515λλ⨯. The board-shaped glass fiber with thickness 3h λ= was embedded in the center of the matrix of epoxy in Model 1, and was obliquely embedded in Model2. A column shaped glass fiber with a diameter 4.36h λ= was embedded in the center of matrix in Model3. The above three models had a common fiber percentage of 20. The analytic condition and the input parameters were shown in Table 1.For the models in Fig. 3, when the incident wave on the left-hand side of the glass region arrived at the first interface between the epoxy and glass, the transmitted wave and the reflected wave arose. Then the reflected wave propagated to the incidence side, while the transmitted wave propagated to the receiver side and arrived at the second interface of the glass and epoxy through the glass region.The second transmitted wave and the second reflected wave arose at the second interface, and a multiplex reflection occurred in the glass region. For the board-shaped fiber (plane fiber) and the column-shaped fiber (cylindrical fiber), Fig. 4 shows the comparisons of the analytic results in the cases of Model 1 (fiber thickness 3h λ=), Model 2 (fiber thickness 3h λ=, 22θ= _) and Model 3 (fiber diameter 4.36h λ=) in Fig. 3, with an equivalent fiber volume fraction but with a different shape. As seen from the figure, the transmission energy of the Model1 is far larger than that Model2 and Model 3.From Fig. 4, which embedded the board-shaped fiber, two energy peaks were clearly observed by transmission energy curve in Model 1 and Model 3. In Model 1, the strong peaks correspond to the first transmitted wave, and four weak peaks are ascribed to the first reflected wave by the glass fiber. In Model 3, the first energy peak resulted from a transmitted wave through the glass fiber region, while the second energy peak was due to the wave propagating through the upper and lower regions of the epoxy. Consequently, it can be understood why the transmission energy for the board-shaped fiber is larger than that of the column-shaped fiber, when the fiber volume fraction was the same.4. Behavior of wave propagation in composite material4.1. Analysis model and ultrasonic propagation simulationMost of fiber reinforced composites material may be considered as an inhomogeneous body microscopically, and a homogeneous one macroscopically. For the composites with fibers, the fiber array model will be useful to take into account of the reflection and/or transmission of multi interfaces. In order to evaluate the macroscopic characteristic of such a composite material, a two-dimension domain with different fiber arrays was proposed as shown in Fig. 5. In this model, circular glass fibers were embedded with hexagonal in the interior of the epoxy matrix. The size of the model was 2420λλ⨯; the fiber diameter is d. An incident wave of 100 MHz was used. The model for analysis was divided into 14401200⨯ elements (1,72,80,000 total elements). In order to account for the loss of load carrying capacity of the failed elements, the stiffness of such elements are reduced by the use of next method.Fig. 6 shows the series of stress dispersion patterns during the ultrasonic wave propagation for model of fiber reinforced composites in Fig. 5 (fiber diameter 4.36dλ=, without attenuation). When the ultrasonic wave was propagated out reached the fiber, the reflected wave, the transmitted wave, and dispersion wave were appeared clearly (Fig. 6(a)). If a wave motion arrived at the interface between the fiber and the matrix, part of the wave was reflected as asecondary source wave, and at the same time a dispersion wave was generated around the fiber.The other part of the wave was transmitted fiber and propagated to receiver side. The multiplex reflection took place interior of the fiber (Fig. 6(b)). Moreover, the wave which spreads the circumference of the fiber interferes each other among fibers, the propagation situation of the ultrasonic wave become further complicates than that of before (Fig. 6(c)–(e)). From these results, the influence of fiber on propagation and dispersion of an ultrasonic wave in a composite material could be visualized and understood.4.2. Influence of fiber-volume-percentage and with attenuation in matrixWhen diameter of fiber is changed by2,3,4,5=and attenuation with/withoutdλλλλattenuation in matrix, which investigates how the propagation action of the ultrasonic wave in a distributed composite material model. Figs. 7 and 8 have shown the time history curve of reflection energy with/without attenuation in epoxy matrix, that during the ultrasonic wave propagation for model of fiber reinforced composites in Fig. 5, respectively. Fig. 9 has shown the time history curve of transmitted energy with attenuation in epoxy matrix. Fig. 10 has shown that comparison of transmission energy ratio with and/or without attenuation during the ultrasonic wave propagation for model of fiber-reinforced composites in Fig. 5, respectively. A figure in case without attenuation in epoxy matrix is omitted.If the with/without attenuation in epoxy matrix is compared, the peak value of reflected energy curve (in the case of fiber diameter 5=) with attenuation in epoxy matrix (attenuationdλcoefficient 120dB/m/MHz) is smaller about 30% than that without attenuation in epoxy matrix. Moreover, although the reflected energy curve in the figure is displayed only to two peaks, the 2nd peak value is larger than the 1st peak value. The 1st peak value is the energy of the reflected wave from a fiber 3, and the 2nd peak value is the energy of the reflected wave from fibers 1 and 6 (Fig. 5). Disorder arose on the subsequent reflective energy curve, and regularity was lost. Moreover, it follows on the increase in fibers diameter (fiber content) that the energy of a reflected wave increases irrespective of with/without attenuation in epoxy matrix.In the case with attenuation in epoxy matrix, at for the transmitted energy history curve, and the peak value (in the case of fiber diameter d = 2k) in the transmitted energy curve is about half of that without attenuation, and the grade of influence by attenuation in epoxy matrix show up. It becomes clearer from the relation with the existence of energy transmitted and with/without attenuation in epoxy matrix and fiber volume content, which are shown in Fig. 10. As seen this figure, irrespective of with/without attenuation in epoxy matrix, energy transmitted decreases greatly follow increase of fiber volume content. At concurrent, the difference of transmitted energy ratio becomes small by the with/without attenuation in epoxy matrix. It is considered that if the diameter of a fiber increases, since the fiber volume content in a composite material model will also increase, this is because transmitted energy became small.5. ConclusionIn this paper, ultrasonic wave propagation and influence on a received waveform were investigated in model cases of composite materials made of fibers and a matrix with the numerical simulation method. It clarified that propagation mechanism of the complicated ultrasonic wave consisting of reflected wave, transmitted wave and dispersion at the interface from matrix and fiber, and the change interface form of glass fiber and epoxy matrix and the distributed composite material by examining the ultrasonic propagation behaviors. Especially, it was understand that the transmitted energy of an ultrasonic wave was changing largely with change of interface form. Moreover, the influences on the ultrasonic transmitted energy ratio by attenuation of matrix and fiber content, and multiplex reflection of the ultrasonic wave by arrangement of fibers, and the propagation situations of the interference wave generating in thecircumference of fibers and so on were clarified by visualization.。