Material Modeling of Polymer Composites for Numerical Investigations of Continuous Mode Conversion

Abstract

Recent experimental investigations with a scanning laser vibrometer showed irregularities of the wave behavior in assumedly undamaged carbon fiber reinforced plastic (CFRP) plates with transiently excited symmetric (S ₀) and antisymmetric (A ₀) wave groups. This effect is not only visible in complex CFRP structures but also in single UD-layers. Therefore, this particular type of plate is focused. In this chapter at first, the wave propagation in simplified numerical 2D models with varying cross sections as well as with varying material properties is analyzed and therewith reasons for the observed wave propagation phenomenon named “quasi-continuous mode conversion” are discussed. On this basis, an approach for the generation of an enhanced material modeling using a sectored homogenization of the UD-layer is given. Finally, results of the simulation of wave propagation in a UD-layer by means of the introduced material modeling are presented.

The content of the current chapter is primarily based on Dr. Hennings’ research. The results of her investigations are published in a PhD thesis [7] and several conference papers and journal articles (e.g., [8, 9]).

1 Analysis of the Wave Behavior in Simplified Models with Reference to the Continuous Mode Conversion

As mentioned before, experimental investigations of the wave propagation behavior in intact carbon fiber reinforced plastic (CFRP) laminas and laminates exhibit unexpected mode conversion effects, which substantially complicate the detection of defects. In addition to the snapshots of the cross-plies in Sect. 11.2, cf. Figs. 11.2, 12.1 demonstrates the mode conversion effect in an undamaged quasi-isotropic laminate (Fig. 12.1a) as well as in a single unidirectional (UD-) layer (Fig. 12.1b).

Fig. 12.1

Lamb waves in CFRP plates [ f _q = 80 kHz]. (a) Quasi-isotropic laminate [UD255 0^∘/UD500 ±45^∘/UD255 90^∘] _S . (b) UD-layer UD255 [4 ×0^∘]

Identified by their wavelengths, these arising A ₀-waves appear in the whole plate immediately during and after the passing of the S ₀-wave field. This implies a continuous behavior for which reason it is called “continuous” mode conversion. In the investigated structures, these secondary A ₀-waves run almost parallel to the fiber direction (of the near surface layers in case of a laminate). Furthermore, it has been observed that the secondary A ₀-waves either propagate in the same or opposite direction of the primary excited wave groups or are apparently without motion.

Since the conversion effect already appears in the easiest case of a fiber–plastic composite, the wave propagation in these single UD-layers is in the focus of this chapter. In order to get a deeper understanding, the wave behavior is analyzed by means of simplified numerical models, which are presented in the following. Further, it is pointed out that all computations are done using models in the plane strain state for the simulation.

1.1 Aluminum Plates with Changes in Cross Section

At first, the wave propagation in aluminum plates with domains of varied cross sections is scrutinized. Special attention is focused on the influence of the distance between the cross section changes.

1.1.1 Plate with Obstacles

The first numerical model is an aluminum plate (E = 70 GPa, ν = 0. 33, and ρ = 2700 kg/m³) with three uniformly distributed bars bonded to its upper surface, serving as obstacles for the wave propagation. Setup and dimensions of the structure are shown in Fig. 12.2. The plate is excited by a transient two-cycle sinusoidal signal at a frequency of 100 kHz multiplied by a cosine window function. The concentrated load represents the effect of a piezo-actuator and induces the primary S₀- and A₀-wave groups. The numerical computation is done using the spectral finite element method (SEM) in the time domain, cf. Sect. 6.1.2. The numerical model of the aluminum plate is built up by 136 spectral finite elements consisting of 6 × 3 nodes, i.e., the polynomial degree in longitudinal and transverse direction is 5 and 2, respectively. Symmetric boundary conditions are applied to the nodes at the left end of the structure.

Fig. 12.2

Experimental setup (left) and numerical model (right) of an aluminum plate with three obstacles

The deformation of the bottom of the plate resulting from the propagation of S ₀- and A ₀-waves is illustrated in Fig. 12.3. The graphs show the displacements in x ₁-direction (in-plane displacements—dashed line) as well as in x ₃-direction (out-of-plane displacements—solid line) at three different points in time. The first plot (t = 65 µs) displays the interaction of the incident S ₀-wave group with the first obstacle. The symmetric wave field of the S ₀-mode is scattered by the abrupt change of plate thickness within the range of the obstacle, leading to a partial conversion of S ₀- into A ₀-waves. In the following, these arising A ₀-waves are named as secondary A ₀-wave groups. The motion of the primary wave groups (S ₀—white arrow; A ₀—gray arrow) directs from the actuator on the left to the free boundary on the right. The secondary A ₀-wave groups (black arrows) move forward and backward from their point of origin (obstacle). The second plot (t = 78.5 µs) displays the primary S ₀-wave group passing the second obstacle. Naturally, the S ₀-group partly converts into further secondary A ₀-groups, which propagate in both (left and right) directions as well. These forward and backward running secondary A ₀-groups were also identified in experimentally investigated laminates, see Fig. 12.1b. The third plot (t = 90 ms) yet illustrates another important effect. Between the first and second obstacle, standing waves temporarily evolve from a backward and a forward traveling secondary A ₀-wave group. This behavior corresponds to the standing motion pattern of the experimentally analyzed plates in an idealized way.

Fig. 12.3

In- and out-of-plane displacements at the upper side (x ₃ = 0.5 mm) of the plate at various times

1.1.2 Plate with Notches

This computation pursues the investigation on the wave behavior with smaller intervals of the equidistantly distributed discontinuities. The numerical model also consists of a 1-mm-thick aluminum plate (E = 70 GPa, ν = 0. 33, ρ = 2700 kg m^-3), which is furnished with 45 uniformly distributed notches on its upper side. Setup and dimensions of the structure are specified in Fig. 12.4.

Fig. 12.4

Experimental setup (left) and numerical model (right) of the aluminum plate with 45 notches

The plate is excited by a Hann windowed two-cycle sinusoidal signal at a frequency of 25 kHz. Here, the S ₀-wave is initiated by two vertical concentrated forces acting in opposite direction. The numerical model of the structure consists of 1043 spectral finite elements with 3 × 3 nodes each. The allocation of the structure between two notches is depicted in Fig. 12.5.

Fig. 12.5

Distribution of the spectral finite elements and their nodes in the region of the notches

For varied time steps, the in- (gray) and out-of-plane displacements (red or black, respectively) at the bottom side of the plate are shown in Fig. 12.6. In order to highlight the wave behavior in the region of notches (gray area), the in- and out-of-plane displacements are scaled to their maximum. Originally, the amplitudes of the in-plane displacements are five times larger than those of the out-of-plane displacements.

Fig. 12.6

Depiction of the normalized in- and out-of-plane displacements at the bottom of the plate at various times

At first, the red marked out-of-plane displacements are examined. As long as the symmetric waves do not reach the region of the notches, obviously no antisymmetric waves appear in the purely symmetric excited plate. However, once the primary excited S ₀-wave gets to the notches, a mode conversion takes place. Two arising secondary A ₀-wave groups (red-black arrows) propagate in the same direction as the S ₀-wave (gray arrow) and in opposite direction, respectively. Again, a mode conversion is evident when the symmetric wave group leaves the notched region.

Beside the secondary antisymmetric wave groups at the beginning and the end of the notched zone, additional out-of-plane amplitudes appear in the modified area (from t = 80 µs), which initially suggests further mode conversions. Though, these amplitudes are only visible when the symmetric wave passes the notched zone and exhibits a “wavelength” of about 5 mm, which is significantly smaller than those of the A ₀-waves (λ ≈ 20 mm). Hence, at this juncture this is not a result from further mode conversions but from a local raising effect of the out-of-plane amplitudes of the primary excited symmetric wave, which is caused by the asymmetric perturbations (distance of the notches: 5 mm).

The out-of-plane displacements adjusted for the local raising effect are shown as black curves in Fig. 12.6. For this representation, the course of the out-of-plane displacements is transferred by a Fast Fourier Transformation (FFT) from the space domain into the wavenumber domain and next assigned into a diagram of the wavelengths. Afterwards, in this delineation the function values at λ ≈ 5 mm are zeroized. Finally, the resulting course is transformed back into the space domain by an inverse FFT.

1.1.3 Intermediate Results

The numerical computations lead to two important findings. On the one hand, the interaction of the A ₀-waves, arising by a mode conversion at several discontinuities, reflects the wave behavior observed in experiments. On the other hand, in spite of an asymmetric geometry with regular arranged discontinuities no mode conversion from S ₀- to A ₀-wave occurs at the notches. A mode conversion is only observed when the primary excited S ₀-wave enters and leaves the disturbed zone.

In Sect. 12.1.3, investigations on adapted numerical fiber–matrix structures are done to obtain detailed information about this mode conversion phenomenon in CFRP-like structures.

1.2 Conventional Material Modeling of CFRP

As mentioned before, the investigations of the wave propagation in this chapter are performed along the cross section of the structure. Thus, the single layers are modeled separately and no laminate theory (e.g., classical laminate theory, shear deformation theory [2–5, 10–12]) is applied. In doing so, a perfect linking between the individual layers is assumed, so the displacements at the boundary of one layer are directly conveyed to the adjacent layers.

Figure 12.7 shows the approach for the modeling of a CFRP with a layup of four layers in an orientation of [0^∘, 90^∘] _s . The physical structure (left picture) is transferred to a fiber–matrix model on the microscale. Here, both components are described by their individual material properties (center picture). Now, an appropriate homogenization technique is applied in order to obtain the fiber–matrix model on the macroscale (right picture). This procedure leads to a layer-wise homogeneous structure.

Fig. 12.7

Material modeling: from microscopy to fiber–matrix model

Since a variety of homogenization techniques exists, within the scope of this chapter the two applied methods, which are the general rule of mixture and the semiempirical approach of Halpin and Tsai, are briefly displayed. For more information about homogenization techniques, the reader is referred to the literature [2, 6, 10, 13, 14].

1.2.1 General Rule of Mixture

Basis of the general rule of mixture is a highly simplified model of the fiber–matrix composite. The assumptions of this material model are small elastic strains, an isotropic fiber and matrix material, a periodical distribution of the parallel running fibers within the matrix material as well as an ideal bonding between both materials, cf. [2].

Depending on the Young’s (E _f , E _m ) and shear moduli (G _f , G _m ) as well as the Poisson’s (ν _f , ν _m ) and volume ratio (φ _f , φ _m ) of the constituents, the effective material properties of a fiber–matrix composite are computed by

$$\displaystyle\begin{array}{rcl} E_{\vert \vert }& =& E_{f}\varphi _{f} + E_{m}\varphi _{m}\,{}\end{array}$$

(12.1)

$$\displaystyle\begin{array}{rcl} \nu _{\vert \perp }& =& \nu _{f}\varphi _{f} +\nu _{m}\varphi _{m}\,{}\end{array}$$

(12.2)

$$\displaystyle\begin{array}{rcl} E_{\perp \perp }& =& \frac{E_{f}E_{m}} {E_{f}\varphi _{m} +\varphi _{f}E_{m}}\,{}\end{array}$$

(12.3)

$$\displaystyle\begin{array}{rcl} G_{\vert \perp }& =& \frac{G_{f}G_{m}} {G_{f}\varphi _{m} +\varphi _{f}G_{m}}\ .{}\end{array}$$

(12.4)

1.2.2 Semiempirical Homogenization Method of Halpin and Tsai

Semiempirical material models are based upon the equations of the general rule of mixture, which are adapted to experimentally obtain material properties using curve fitting parameters.

Regarding the rectangular fiber geometry by a width(a)-to-height(b) ratio ξ = 2a∕b, the homogenized material parameters are approximately given by

$$\displaystyle\begin{array}{rcl} E_{\vert \vert }& \cong E_{f}\varphi _{f} + E_{m}\varphi _{m}\,&{}\end{array}$$

(12.5)

$$\displaystyle\begin{array}{rcl} \nu _{\vert \perp }& =\nu _{f}\varphi _{f} +\nu _{m}\varphi _{m}&{}\end{array}$$

(12.6)

and by

$$\displaystyle\begin{array}{rcl} \frac{M} {M_{m}} = \frac{1 +\xi \eta \varphi _{f}} {1 -\eta \varphi _{f}}\quad \text{with}\quad \eta = \frac{M_{f}/M_{m} - 1} {M_{f}/M_{m}+\xi } \ .& &{}\end{array}$$

(12.7)

In these equations, M is subsequently replaced by E _{⊥ ⊥}, G _{| ⊥}, respectively, ν _{⊥ ⊥}, M _f by E _{⊥ ⊥} ^f, G _{| ⊥} ^f, respectively, ν _{⊥ ⊥} ^f and M _m is substituted by E _{⊥ ⊥} ^m, G _{| ⊥} ^m, respectively, ν _{⊥ ⊥} ^m in order to get the homogenized material properties. The indices | and ⊥ denote an orientation in and perpendicular to the fiber direction, respectively.

1.3 Fiber–Matrix Models

As shown in diverse experiments (cf. Figs. 11.2, 11.3 and 12.1), the quasi-continuous mode conversion appears when the S ₀-wave group propagates perpendicular to the fiber direction. Therefore, three different 2D plane strain models of a single UD-plate with this fiber orientation are investigated. The first model represents the plate by a homogenized ply (cf. Sect. 12.1.2), the second model consists of periodically distributed fiber–matrix regions, and the third one is built up of randomly distributed fiber–matrix regions. The model dimensions and the excitation signal (Hann windowed two-cycle sinusoidal burst at a frequency of 100 kHz) are illustrated in Fig. 12.8 (left). The material allocation of the analyzed plates is displayed in Fig. 12.8 (right). Every plate consists of 1000 × 4 quadratic spectral finite elements, each with 3 × 3 nodes. Material properties of fiber and matrix as well as of the homogenized layer are shown in Table 12.1. The homogenized material properties are calculated by the general rule of mixture [10] according to the chosen fiber volume ratio of v _f = 0. 5.

Fig. 12.8

Geometry (top left), excitation signal (bottom left), and setup (right) of the investigated fiber–matrix models; length units in [mm]

Table 12.1 Material properties

In- and out-of-plane displacements at the top surface of the plate at t = 0.4 µs are plotted in Fig. 12.9. The first picture shows the propagation of waves in the homogenized ply. As expected, due to the material homogenization only the primary S ₀- (dashed ellipse) and A ₀-wave group (solid ellipse) occur.

Fig. 12.9

In- and out-of-plane displacements at the top surface of a single layer modeled by homogenization (top), periodically distributed (middle), and randomly distributed fiber–matrix regions (bottom) at t = 40 µs

The second numerical model (second model in Fig. 12.8 (right)) is built up by elements ideally allocated with periodically distributed fiber (black quad) and matrix (white quad) material. Since asymmetric inhomogeneities with respect to the center plane lead to a conversion from S ₀- to A ₀-waves, [1], an even number of elements over the height is used. The structural asymmetry and the discrete changes of material properties are deliberately introduced to provoke the continuous mode conversion. The result of this computation is shown in the second plot of Fig. 12.9. Surprisingly, beside the two fundamental wave groups the displacement field offers no additional waves. Reason for this unanticipated behavior seems to be the completely regular pattern of the structure and the size of the S ₀-wave. The wavelength of the S ₀-mode (λ _s ≈ 20 mm) spans a range of about 200 × 4 elements. Thus, there are plenty of element columns within the pressure or tensile range of the S ₀-wave. Figure 12.10a shows a detail of the deformed plate under tensile stress. The cross section stays plane inside the elements, identifiable by the vertically aligned positions of the element center nodes, whereas bending is observable at the edges of fiber and matrix elements. This deformation occurs on a local level and with constant alternation what mutually prevents the conversion of the S ₀- into A ₀-waves on a global level.

Fig. 12.10

Details of the deformation of different fiber matrix structures under constant tensile stress (the same exaggeration factors). (a) Periodical distribution. (b) Random distribution

The third model is motivated by a microscopic inspection of a UD-layer. Figure 12.11 shows two photomicrographs recorded parallel (left) and perpendicular (right) to the fiber direction. As can be seen, fiber orientation and distribution merely approximately follow a regular pattern. Therefore, in the numerical model fiber and matrix material are randomly distributed, see Fig. 12.8 (right). Results of this computation are plotted in the third picture of Fig. 12.9. Beside the primarily excited S ₀- and A ₀-wave groups, additional secondary A ₀-waves arise where the S ₀-wave group is passing the plate. Based on the randomly arranged fiber and matrix elements, a flexural deformation over the cross section (Fig. 12.10b) initiates the conversion of S ₀- into A ₀-waves.

Fig. 12.11

Detail of photomicrographs of a unidirectional layer. (a) Detail in 0^∘. (b) Detail in 90^∘

Concerning the varied phase velocities of the “same” S ₀- and A ₀-waves in the different models (see Fig. 12.9), a brief remark has to be made. As long as the numerical model consists of fiber and matrix elements at a fixed fiber volume ratio, irrelevant of whether fiber and matrix material is periodically or randomly distributed, they are propagating with almost identical phase velocities. A distinct difference occurs when wave propagation is carried out in the homogenized structure. This is due to the application of the general rule of mixture. Numerical tensile tests in propagation direction of the plates yield to the Young’s moduli shown in Table 12.2. The calculated Young’s modulus in the homogenized plate (E ₂ = 5.83 GPa) corresponds to the input value of the computation (see Table 12.1). In comparison, both fiber–matrix models offer higher Young’s moduli. According to this, the approximation of global material properties of a UD-structure using the general rule of mixture is insufficiently accurate. Though, for further simulation the homogenization technique of Halpin and Tsai is used, which provides more accurate results.

Table 12.2 Young’s moduli E ₂ of the investigated fiber–matrix models obtained by a numerical tensile test

1.4 Intermediate Results

The simulation of the wave propagation in a composite with periodically arranged material inhomogeneities and an asymmetric layup sequence confirms the findings of Sect. 12.1.1. If the wavelength of the primary excited S ₀-wave is considerably larger than the distance between the inhomogeneities, no “continuous” mode conversion occurs. Whereas this continuous conversion effect is satisfactorily reproduced if a random distribution of fibers in the matrix material is assumed.

2 Numerical Realization of the Continuous Mode Conversion Effect

Based on the findings of the previous sections, an enhanced material modeling approach, which enables the realistic reproduction of the quasi-CMC effect, is presented in this section.

2.1 Enhanced FE-Material Modeling of UD-Layers

As already mentioned, an assumption of uniformly distributed fibers does not reflect the reality, cf. Fig. 12.11. On closer inspection, obviously, the global fiber volume ratio φ _f of a single layer does not correspond to the local fiber–matrix ratio. This fact is clearly evident in Fig. 12.11b, where the randomly distributed fibers may form regions with a considerably lower fiber volume ratio, e.g., in the upper left corner. Thus, a novel approach with sectored homogenized zones instead of one entirely homogeneous layer is motivated. In Fig. 12.12, the development of the enhanced model by means of a single UD-layer with a thickness of 0.4 mm is outlined.

Fig. 12.12

Approach for the generation of an enhanced material modeling using a sectored homogenization of the UD-layer

In a first step, the fiber–matrix model is viewed on the microscale. The fibers are idealized and depicted as squares (black) with an edge length (8 µm) corresponding to the dimension of a real fiber diameter (5–10 µm). Furthermore, the fibers are randomly distributed (Gaussian distribution) in the matrix material (represented by white squares) in consideration of the global fiber volume ratio. Therefore, a pseudo-random number X(ω) (uniform distribution) with ω ∈ Ω and Ω = [0…1] is generated for every square. Subjected to the fiber volume ratio φ _f , the allocation of fiber or matrix material is given by X(ω) < φ _f or X(ω) > φ _f , respectively. Moreover, it should be noted that each square corresponds to a finite element. The resulting high-resolution numerical model is able to reproduce the QCMC effect. However, due to the vast number of degrees of freedom (DOF) this model is unsuitable for efficient numerical evaluation.

In the next step, the micro-model will be divided into subsets. Both center pictures in Fig. 12.12 show the same detail of the micro-model with their respective subsets, which are composed of an arbitrary square number of fiber–matrix elements and are highlighted by colored frames. Since the primary allocation of fiber and matrix elements is randomly distributed, the fiber volume ratio of every subset can differ from the global ratio.

Subsequently, the material properties (Young’s modulus, Poisson’s ratio, and shear modulus) of each subset are determined by using the semiempirical homogenization method of Halpin and Tsai, cf. [10]. With displacement amplitudes and wave velocity in mind, this method offers the best approximation to the values of the micro-model and therefore is briefly illustrated here.

Due to the fact that now all elements in a subset have the same material parameters, the sectored homogenization allows a coarser discretization as the micro-model, so that every subset in the meso-model can be expressed by only one finite element. The shades of gray in the bottom pictures of Fig. 12.12 correlate to the local fiber volume ratio of the subsets, in which brighter squares reflect a higher matrix concentration than darker ones.

The subsets of the left meso-model show marginal variations in coloring respective fiber volume ratio, whereas the color changes (and with this the different fiber volume ratios) are clearly visible in the right meso-model. The reason for this is the number of fiber and matrix elements per subset. The smaller the number of elements, the stronger is the influence of the random distribution of fiber and matrix, which arises in the deviation from the global fiber volume ratio.

2.2 Wave Propagation in UD-Layers Using the Enhanced FE-Material Modeling

The dimensions of the micro-model of a single UD-layer (fibers in x ₂-direction) are illustrated in Fig. 12.13. The material parameters of fibers and matrix material are listed in Table 12.3. The fiber volume ratio is φ _f = 0. 5. Based on the edge length of 8 µm, the micro-model consists of 12, 500 × 50 squared fiber–matrix elements (FM elements).

Fig. 12.13

Dimensions of the micro-model and a fiber–matrix element (FM element)

Table 12.3 Material properties of fiber and matrix

Originating from the micro-model, different meso-models are created, in which various squared numbers of FM elements (2 × 2/ 5 × 5/ 10 × 10/ 25 × 25/ 50 × 50 elements) are combined to subdomains, cf. Fig. 12.14. As mentioned in the previous section, each subdomain is discretized by one finite element with its homogenized material properties. The computation of wave propagation in these meso-models is supposed to answer the question at which homogenization level of the layer the QCMC effect can be reproduced.

Fig. 12.14

Detail of the micro-model with illustration of the subdomains (red squares)

Figure 12.15 and Table 12.4 show the information of the micro- and meso-models concerning element distribution and local fiber volume ratios. The first rows in Table 12.4 lists the number of subdomains of each model. Every subdomain is represented by a nine-node element (two DOF per node) and thus leads to the total number of DOF for each model shown in the second row.

Fig. 12.15

Frequency distribution of the local fiber volume ratios φ _f ^e in the observed UD-plate (assumed global fiber volume ratio: φ _f = 0. 5)

Table 12.4 Model parameter

Due to the fusion of fiber and matrix elements, there are different fiber volume ratios φ _f ^e in the subdomains. The number of occurring ratios as well as their minimum and maximum are listed in lines 3–5. It is evident that the number of local fiber volume ratios rises with an increasing number of FM elements per subdomain (see Table 12.4 line 3 and Fig. 12.15). Simultaneously, the minimum and maximum values converge to the global fiber volume ratio of the layer (φ _f = 0. 5).

The excitation of the plate takes place at 2.5 mm from the left edge (symmetry axis) at a load of 100 N applied as a two-cycle sine burst signal with a central frequency of 100 kHz. Over two cycles, the sine signal is multiplied by a Hann window. Since the effect of QCMC occurs after the symmetric waves are passing the plate, the structure is excited symmetrically.

Figure 12.16 shows the results of the numerical simulation. The red curves display the out-of-plane displacements (u ₃) of the different meso-models at the top edge of the plate. For comparison also, the displacement amplitudes of the micro-model calculation (black curves) are depicted in the diagrams.

Fig. 12.16

Out-of-plane-displacements [nm] of the UD-plate at the top edge by using variously sized subdomains (t = 49.5 µs)

At the time of t = 49.5 µs, the primary excited S ₀-wave has passed the whole plate and reached the right end of the structure. As expected, due to the random allocation of FM elements the displacement curves of the micro-model show secondary A ₀-wave groups appearing after the symmetric waves have passed the plate. These secondary wave groups are propagating in the same and opposite direction of the primary excited S ₀-wave and moreover they appear locally as standing waves.

As it can be seen in Fig. 12.16, every meso-model is able to reproduce this behavior, except the model with the coarsest discretization (50 × 50 FM elements per subdomain). Even the meso-model with 25 × 25 FM elements per subdomain is able to capture the amplitudes of the secondary A ₀-wave in an excellent manner and shows the peaks of the primary excited S ₀-wave. The offset of the S ₀-displacement curves between the micro- and meso-model is owed to the homogenization method and is not a consequence of the application of subdomains.

The reason for the absence of secondary A ₀-waves in the coarsest meso-model with 50 × 50 FM elements per subdomain cannot be explained by a possibly inadequate discretization. Also these finite elements (length of 0.4 mm) are 10 times smaller than the A ₀-waves (wavelength λ _a ≈ 4 mm). Since this meso-model uses only one finite element across the thickness, the structure gets a symmetric setup with respect to the midplane of the plate and for this reason no conversion from S ₀- to A ₀-mode happens, see [1].

3 Conclusion

The work at hand is concerned with wave propagation in elastic solids, namely in fiber reinforced plastic materials. Special attention is paid to a phenomenon, which is called “continuous” mode conversion and which is accurately investigated by scanning laser vibrometry. For future structurally integrated health monitoring systems, it is mandatory to be aware of the physical processes, in order to be able to interpret the sensor signals from discrete sensors correctly. For the understanding of the phenomenon, it is also important to have appropriate models, which allow for a precise numerical analysis of the wave propagation process. For this reason, various models have been under investigation and the results have been compared. It has been shown that randomly distributed fibers in the matrix material give encouraging results.

Based on those findings, an enhanced material modeling method for single UD-layers is presented. Here, size-varying subsets with homogenized material properties are generated to successfully reproduce the phenomenon of “quasi-continuous mode conversion” in this particular type of CFRP plates. For the homogenization, the semiempirical method of Halpin and Tsai is applied.

Investigations concerning the maximum size of the subregions for simulating the “quasi-continuous mode conversion” yield at least two subregions over the height of the UD-layer, but solely to ensure an asymmetric setup of the numerical model because of the varying material parameters of the subsets. Additionally taking into account common restrictions such as the number of nodes or the polynomial degree per wavelength, the enhanced material model excellently simulates the real propagation behavior in UD-layers and thus motivates the need for a stochastic material model for proper analysis of wave propagation.