A unified formulation for predictive modeling of guided-ultrasonic wave dispersion curves in metallic and composite materials

Transcription

1 Original Article A unified formulation for predictive modeling of guided-ultrasonic wave dispersion curves in metallic and composite materials Journal of Intelligent Material Systems and Structures 1 15 Ó The Author(s) 1 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: 1.11/14589X1559 jim.sagepub.com Darun Barazanchy and Victor Giurgiutiu Abstract To predict dispersion curves it is common to use different solution approaches depending on the material type, isotropic or composite, of the medium in which the wave propagates. The two different solution methods are defined in different domains, frequency wavespeed domain for isotropic materials, and wavenumber wavespeed domain for composites which can lead to difficulties, and unsatisfying results when predicting the dispersion curves for hybrid laminates which contain both isotropic and composite materials. This article, therefore, proposes a unified formulation defined in the wavenumber wavespeed domain for both isotropic and composite materials. The unified formulation, simple, and mathematically straightforward formulation, utilizes Christoffel s equation for a lamina to obtain the eigenvalues and eigenvectors. The eigenvalues and eigenvectors are then used to set up the field matrix from which the dispersion curves could be retrieved. Once the dispersion curves were obtained the waves are grouped using a modeshape analysis. A spline algorithm is applied to obtain a continuous solution from a rough domain which was used to reduce computational time. In addition, this article highlights the challenges faced in the numerical process, and provides a discussions of the methods used to overcome these obstacles. Keywords Dispersion curves, unified formulation, structural health monitoring, isotropic, anisotropic, composite laminates Introduction With the introduction of the Boeing B8 and the Airbus A5 XWB, composite materials have replaced metallic ones as the main components in aircrafts. For these new types of aircraft careful attention needs to be given when traditional non-destructive evaluation (NDE) methods (developed with metals in mind) are applied to composites due to: (a) the anisotropic behavior of composite materials (e.g. for ultrasonic methods one has to take into account that wavespeed is a function of propagation direction); (b) the difference in types of damage that can occur; and (c) the complexity of the damage. Advances are, however, continuously being made to non-destructively test, evaluate, and inspect aerospace composite structures (Georgeson, ; Giurgiutiu and Cuc, 5). Within the ultrasonic NDE research field, methods based on guidedultrasonic waves (GUWs), especially those described by Lamb (191), are preferred due to small amplitude attenuation over long distances (Raghavan and Cesnik, ; Staszewski et al., 4), and are reported on frequently (Barazanchy, 14; Diamanti and Soutis, 1; Giurgiutiu, 14; Kessler, et al., ; Su et al., ). As GUWs are increasingly used, a better understanding of the wave propagation and their dispersive nature, and wavespeed dependence on frequency is desired. A disadvantage in current GUW dispersion curve algorithms is that the solution method used differs based on the material type of the medium that the wave propagates in. Separately, wave propagation in isotropic (Giurgiutiu, 14; Rose, 1999; Viktorov, 19) and in composite (Lowe, 1995; Nayfeh, 1995; Rokhlin et al., 11) materials have been characterized. However, there is no documentation in the literature for a simple and straightforward mathematical formulation which is applicable to both isotropic, and composite materials, while yielding accurate results at the same time. Department of Mechanical Engineering, University of South Carolina, Columbia, USA Corresponding author: Darun Barazanchy, Department of Mechanical Engineering, University of South Carolina, South Main Street, Columbia, SC, 98, USA. dbara@ .sc.edu

2 Journal of Intelligent Material Systems and Structures When exciting GUWs, more specifically Lamb waves, three different types of wave propagate simultaneously in the medium; shear horizontal, shear vertical and pressure waves. For isotropic material the three waves can be decoupled into two parts: (a) shearhorizontal part; and (b) combined shear-vertical and pressure part which form the Lamb waves. The two parts can be solved separately, and combined afterwards to yield all of the propagating waves. For a homogeneous material, like isotropic materials, the wavespeeds will be the same regardless of the direction of the wave propagation. For composite materials, however, two different cases can be distinguished: (a) wave propagation along fiber direction, and (b) off-axis wave propagation, wave propagation at an orientation angle with respect to the fiber direction. In the former case the same decoupling can be applied as for isotropic material due to the same stiffness matrix population for the two materials. For the latter case, no decoupling can be achieved, therefore, the shear-horizontal, shearvertical and pressure waves have to solved simultaneously making the process more difficult. In addition, the wavespeed will differ depending on the orientation angle u, again adding to the complexity when dealing with wave propagation in non-homogeneous materials like composites. Regardless of whether the approach is unified or not, or whether the material is isotropic or composite, when predicting the dispersion curves in N-layered composites, the interface conditions, displacement continuity through the thickness, stress balance for adjacent layers, and the traction free boundary condition on the top and bottom surfaces of the medium need to be satisfied. To satisfy all of the aforementioned conditions, and assemble the N-layered problem, several methods are developed of which the three most common used are: (a) the transfer matrix method (TMM) (Thomson, 195; Haskell, 195); (b) the global matrix method (GMM) (Knopoff, 194); and (c) the stiffness matrix method (SMM) (Rokhlin et al., 11;Wang and Roklin, 1). The TMM utilizes a transfer matrix that describes displacements and stresses at the top of a layer in terms of displacements and stresses at the bottom of the adjacent layer. Repeating the process for all of the layers in the medium, one obtains a compact matrix that relates the boundary condition at the top to the bottom surface of the medium, thereby resulting in a numeric efficient method. However, the TMM can experience numeric instabilities when dealing with high frequencies, large thicknesses, or a combination of the two. Initially formulated for isotropic material the TMM was extended by Nayfeh (1995) to also incorporate anisotropic N-layered media. Knopoff (194), however, proposed the GMM, a method that assembles all of the displacement and stress boundary conditions of each layer into one matrix, the global matrix. The GMM in comparison to the TMM does not encounter numeric instabilities, however, as a drawback of the global matrix is that it does increase in size with an increasing number of layers. The numeric instabilities of the TMM was also studied by Rokhlin and Wang (); Wang and Roklin (1) who introduced the SMM as a result. The SMM relates the stresses at the top and bottom to the displacements at the top and bottom through the layer stiffness matrix. A recursive algorithm is applied to all the layers, and it yields a similar compact matrix as for the TMM, however the SMM is unconditionally stable. Pavlakovic et al. (199) introduced a method based on the GMM to calculate the dispersion curves in any isotropic and composite materials. Later this was developed into a commercially available package, DISPERSE (Pavlakovic and Lowe, ). Bartoli et al. () proposed a semi-analytical finite element (SAFE) approach to obtain the dispersion curves in an arbitrary cross-section that accounted for viscoelastic material damping. Dispersion curves can be obtained in both isotropic and composite materials using the SAFE approach, however, it requires the implementation of a 1D finite element method. WWang and Yuan () reported the formulation of Lamb waves in composites using the D elasticity theory. The reported formulation is mathematically complicated, but able to predict the dispersion in N-layered composites. All the experimentally tested laminates were, however, quasiisotropic. A gap exists in the literature regarding analytic dispersion curves. Recently Pant (14) reported a similar method as Wang and Yuan () based on D elasticity theory, however, Pant (14) included a GLARE panel in their investigation. A numerical issue when implementing the GMM was observed, this lead to subtract 1 from the material orientation when investigating wave propagation in the fiber direction. Another recent investigation reported on multiple dispersion curve algorithms, however, the shear-horizontal, and the combined shear-vertical and pressure waves were decoupled in all of the tested cases (Abdelrahman, 14). To find a stable formulation Abdelrahman (14) proposed the stiffness transfer matrix method (STMM) that combined the merit of both the TMM and SMM. These recent investigations were still based on the methods presented by Nayfeh (1995), Lowe (1995) and Rokhlin et al. (11). However, in recent years, significant attention has not been given to alternative analytic formulation for dispersion curves. This article aims to give such an alternative; the unified formulation. This article proposes the unified formulation: A simple, and straightforward mathematical formulation valid for isotropic, composite, or a combination of the two, with no simplifying assumptions regarding the degree of anisotropy. As highlighted by Moll et al. (1) the probability of detection of damage in anisotropic plate-like structures can be increased when no

3 Barazanchy and Giurgiutiu simplifications regarding the degree of anisotropy are made. In contrast to the work performed by Bartoli et al. () the unified formulation provides a fully analytic approach. The unified formulation utilizes Christoffel s equation for a lamina to obtain the eigenvalues, and eigenvectors. The eigenvalues, and eigenvectors are then used to set up the field matrix from which the dispersion curves could be deduced. Once the dispersion curves were obtained the waves are grouped using a modeshape analysis, and a spline algorithm is applied to obtain a continuous solution over the domain. This article focuses on the analytics behind the proposed formulation and deals with the effects of different material types separately before a composite laminate is considered. The structure of this article is as follows: (a) the theoretical background of the unified formulation is concisely discussed; (b) the effect of different material types (isotropic, orthotropic transversely isotropic, fully orthotropic, and monoclinic) on the lamina eigenvalue are discussed; (c) the dispersion curves of the aforementioned material types on a lamina level are presented; (d) the assembly methods to deal with N-layered composites are discussed and several numerical examples are given; (e) a short comparison is given between the unified formulation and SAFE (Bartoli et al., ), and finally the article ends with the summary and conclusions. Theoretical review In the proposed unified formulation a guided-ultrasonic wave propagating in an arbitrary media is described by its particle displacement, which is a function of the angular frequency v, wavenumber j, and phase velocity v vj (Rokhlin et al., 11). The wave consists of a superposition of waves propagating in multiple directions, see Figure 1 for a schematic representation of the different directions, that is, a wave propagating in the x 1 direction will be the result of a superposition of waves propagating in the x 1 and x direction under x invariant condition, as can be seen by < ^u 1 < ^u 1 u ^u : ; ei(k 1x 1 + k x vt) ^u : ^u ^u where ; eij(x 1 + ax vt) ð1þ k 1 j k aj ðþ Note, k 1 and k in equation (1) are the directional wavenumbers, and a is the ratio between the wavenumbers in the x and x 1 direction, a k k 1. Substituting the derivatives of the displacements, equation (1), with respect to the spatial variables and time variable into the equation of motion, rearranging and dividing out common factors yields c ijlm u m, lj r u i i, j 1,, ðþ (C 11 rv )+C 55 a C 1 + C 45 a (C 1 + C 55 )a C 1 + C 45 a (C rv )+C 44 a 4 (C + C 45 )a 5 (C 1 + C 55 )a (C + C 45 )a (C 55 rv )+C a 8 9 >< ^u 1 > ^u >: >; ^u ð4þ Applying the zero-determinant condition to equation (4), and expanding its determinant resulted in the bi-cubic equation, which after solving yielded the eigenvalues, a a + A 4 A a 4 + A A a + A A ð5þ The bi-cubic coefficients A, A 4, A, A are represented below as function of the material s stiffness matrix, density, and wavespeed A C C 44 C 55 C C45 A 4 (C 44 C 55 C45 )(C 55 rv ) + C C 55 (C rv ) + C C 44 (C 11 rv ) C 1 C 45 C + (C + C 45 )(C 1 + C 55 )C 45 (C 1 + C55 )C 44 (C 45 + C ) C 55 A C (C 11 rv )(C rv ) + C 44 (C 11 rv )(C 55 rv ) + C 55 (C rv )(C 55 rv ) (C 11 rv )(C 45 + C ) (C rv )(C 1 + C55 ) (C 55 rv )(C 55 rv ) ðþ Figure 1. Schematic representation of the wave propagation directions. + C 1 (C 45 + C )(C 1 + C 55 ) C 1 C A ½(C 11 rv )(C rv ) C 1 Š(C 55 rv )

4 4 Journal of Intelligent Material Systems and Structures Subsequently the eigenvalues were used to obtain the eigenvectors, U, after which the displacement field matrix B u was formulated as where B u (x )½b (1) u (x ) b () u (x ) b () u (x )... b (4) u (x ) b (5) u (x ) b () u (x )Š ðþ b (j) u (x )U (j) e ija(j) x j 1,,..., ð8þ To reconstruct the complete wave the eigenvalues and corresponding eigenvectors were substituting into particle displacement equation, equation (1), this yielded u ^u(x )e ij(x 1 vt) X j 1 h j U (j) e ija(j) x! e ij(x 1 vt) ð9þ where h are the partial-wave participation factors. The x -dependent, thickness part was isolated to give >: s 1 s 1 ^u(x )B u (x )h >; ð1þ In a similar fashion the stress field matrix B s was derived; the stress-displacement relationship under x - invariant conditions for a monoclinic lamina was used as given by s 11 u 1, 1 s >< > >< > s u C, ð11þ s u, >: u, 1 + u 1, u, 1 where C is C 11 C 1 C 1 C 1 C C C C C C 44 C 45 sym C >; C ð1þ For guided-wave propagation in an arbitrary media the upper and lower surfaces are traction free. The normal vector on the upper and lower surface is ~e, therefore, the traction on the surface will consist of the normal stress s and the two shear stresses s and s 1. Evaluating equation (11) by taking the stresses which were influenced by the traction on the upper and lower surface yielded s C 1 u 1, 1 + C u, + C u, 1 s C 44 u, + C 45 (u, 1 + u 1, ) s 1 C 45 u, + C 55 (u, 1 + u 1, ) ð1þ Substituting the displacement derivations obtained from equation (9) into equation (1), rearranging and collecting the stress vector yielded 8 9 < ^s ^s(x ) ^s : ; Bs (x )h ð14þ where B s and b (j) s 8 >< s (x )ij >: b (j) ^s 1 are given by B s (x )½b (1) s (x ) b () s (x )b () s (x ) b (4) s (x ) b (5) s (x ) b () s (x )Š C 1 U (j) 1 + C a (j) U (j) + C U (j) C 44 a (j) U (j) + C 45U (j) + C 45a (j) U (j) 1 C 45 a (j) U (j) + C 55U (j) + C 55a (j) U (j) 1 9 > ð15þ ð1þ >; eija(j) x ð1þ Finally, the dispersion curves were retrieved by applying the traction free boundary conditions, stresses at the top and bottom of the medium are zero, and concatenating the two stress field matrices into one to obtain B s () B s h Dh ð18þ (h) The unified formulation, and therefore the dispersion curves were defined in the wavenumber wavespeed solution space thus retrieving the dispersion curves required searching for wavenumber wavespeed pairs which yielded a sign change in the determinant of matrix D. The matrix D, and its corresponding determinant were both complex, therefore, a sign change occurs if, and only if, both the real and imaginary parts of the complex number change sign simultaneously. For a wavespeed-wavenumber combination that result in a singular matrix D, the partial participation factors, h, can be obtained by solving the corresponding eigenvalue problem. Lamina properties Depending on the material type, the behavior of the roots a of equation (4) will differ. Three different cases are identified based on the polynomial discriminant, D, of equation (5): (a) D, all a roots are real and unique; (b) D, all a roots are real and at least two are equal; and (c) D, one a root is real and the other two are complex conjugates (Abramowitz and Stegun, 19). The final solutions to equation (5) are the six a roots that are obtained p by taking the square root for each a root; a ffiffiffiffiffi a. As is conventional, a j a j 1 for j 1,, was used. It is important to note that a are obtained unsorted, to sort the a, the eigenvector corresponding to the positive or negative a value was used in a sorting algorithm.

5 Barazanchy and Giurgiutiu 5 The a behavior will be next discussed separately for each material type: (a) isotropic; (b) orthotropic transversely isotropic; (c) fully orthotropic; and (d) monoclinic. Isotropic Isotropic materials are defined as having an infinite number of material symmetry planes, which results in a material with the same material properties in all directions. The number of independent stiffness constants to characterize the material are therefore only two, C 11 and C 1. In terms of the engineering constants the required constants are the Young s modulus, E, and a Poisson s ratio n. The stiffness matrix population and numeric values for an isotropic material, in this case an aluminum with a Young s modulus of GPa, Poisson s ratio of., and density of kg/m, are given by C 11 C 1 C 1 C 11 C 1 C 11 C C 11 C 1 C sym 11 C 1 4 C 11 C 1 1: 51:1 51:1 1: 51:1 1: : GPa 4 sym : 5 : 5 ð19þ As can be seen from Figure (a) for an isotropic material, in this specific case aluminum, equation (5) produces three real roots of which two are equal to the a, a 1 a 4 ¼ a 5. This behavior was expected after substituting the isotropic stiffness matrix from into equation (4), rearranging, and finding the expressions for a (C 11 rv )+C 55 a (C 1 + C 55 )a (C rv )+C 44 a 4 5 (C 1 + C 55 )a (C 55 rv )+C a 8 9 >< ^u 1 > ^u >: >; ^u The second row of equation () gives a rv C C 44 ðþ ð1þ The first and third row remained connected, and needed to be solved using the zero-determinant condition. Which after expanding and rearranging yields a quadratic equation (C det 11 rv )+C 55 a (C 1 + C 55 )a (C 1 + C 55 )a (C 55 rv )+C a Aa 4 + Ba + C where A, B, and C are given by A C 55 C B (C 11 rv )C +(C 55 rv )C 55 (C 1 + C 55 ) C (C 11 rv )(C 55 rv ) The roots for equation () are given as a B p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 4AC A ðþ ðþ ð4þ ð5þ Expanding, and rearranging equation (5) yields two solution pairs that are given by a rv C 11 C 11 a rv C 55 C 55 ðþ Substituting the isotropic stiffness matrix components into equations (1) and () yields a 1 rv C 11 C 1 C 11 C 1 a 5 rv C 11 C 11 a 4 rv C 11 C 1 C 11 C 1 rv C 11 C 1 1 rv C 11 1 rv C 11 C 1 1 ðaþ ðbþ ðcþ Equation () verifies the a behavior shown in Figure (a), where three roots exist, of which two are equal, equation (a) equation (c) ¼ equation (b). The wavespeeds at which a changes signs were obtained by rewriting equation () and are provided below. More details regarding the aforementioned derivations are provided by Abdelrahman (14) and Nayfeh (1995). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi v 1 C 11 C 1 C 11, v 5, r r sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 4 C 11 C 1 r ð8þ It is important to state that due to the fact that two of the three a values were exactly the same, some numerical challenges had to be overcome. As stated earlier the

6 Journal of Intelligent Material Systems and Structures Figure. a behavior for different types of material, one layer 1 mm in thickness: (a) Isotropic, aluminum; (b) orthotropic transversely isotropic, u 8 CFRP T/914 (Abdelrahman, 14); (c) fully orthotropic, u 8 CFRP (Calomfirescu, 8); and (d) monoclinic, u 58 CFRP T/914 (Abdelrahman, 14). three a values produced six a values, where, in the case of an isotropic material four of the six values were the same in magnitude. Substituting one of those four a values into equation (4) produced an eigenvalue problem which had two eigenvectors satisfying the equation corresponding to two zero eigenvalues. A total of ten, instead of an expected six, eigenvectors were obtained for the six a values. Therefore, the eigenvectors were sorted first based on their type (shear-horizontal or shear-vertical) and pressure, using the vector orthogonality principle. The sorted eigenvectors were matched with their corresponding a values resulting in six distinguished sets of eigenvectors and a values. Applying an uniqueness algorithm to the eigenvectors yielded no more duplicate eigenvectors for each a value. Orthotropic transversely isotropic An orthotropic (transversely isotropic) material is a material that has one plane of isotropy, one plane in which there are an infinite number of planes of symmetry. In comparison to isotropic materials, orthotropic transversely isotropic materials require five independent stiffness constants to be characterized. The five stiffness constants are related to the following engineering constants; the Young s modulus in the 1-direction E 1,the Young s modulus in the -direction E, the Poisson s ratiointhe1-plane,n 1, the shear modulus in the 1- plane, G 1, and the Poisson s ratio in the - plane, or the shear modulus in the - plane, n and G, respectively. An unidirectional carbon fiber reinforced

7 Barazanchy and Giurgiutiu polymer, CFRP T/914 (Abdelrahman, 14) was used to obtain the a behavior shown in Figure (c). The stiffness matrix population and corresponding numeric values, with a density of 15 kg/m are given by C 11 C 1 C 1 C C C C C C sym C 55 4 C 55 5 ð9þ 14:8 : : 1: :5 1: :4 GPa 4 sym 5: 5 5: From Figure (b) it can be seen that for a unidirectional carbon fiber, all the a values are real and unique. This behavior was expected after substituting the stiffness constant components for a transversely isotropic into equations (1) and () which resulted in a 1 rv C C 44 (rv C 55 ) C C a 5 rv C 11 C 11 a 4 rv C 55 C 55 ðaþ ðbþ ðcþ In addition, the wavespeeds at which the a values changed sign were predicted after applying simple arithmetics to equation () which yielded sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi C 55 C 11 C 55 v 1, v 5, v 4 r r r Fully orthotropic ð1þ Fully orthotropic materials have no planes of isotropy, but have two or more planes of symmetry. The number of independent stiffness constants required to characterize a fully orthotropic material, therefore, increases to nine. For wave propagation along the fiber direction the steps shown from equations () to () are still valid. A fully orthotropic material, therefore, has a similar behavior as a orthotropic transversely isotropic material as can be seen from Figure (c). However, there are three unique wavespeeds at which a changes sign due to C 55 ¼ C. The numeric values for the fully orthotropic material, with a density of 15 kg/m used in this case is given by C 11 C 1 C 1 C C C C C 44 sym C 55 4 C 5 ðþ :9 : :8 14: 4: GPa 4 sym 4: 5 1:9 sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi C C 11 C 55 v 1, v 4, v 5 ðþ r r r Monoclinic When a material has only one plane of symmetry it is identified as a monoclinic material. The number of independent stiffness constants required to characterize a monoclinic material is thirteen. The numeric values for the monoclinic case were obtained by rotating the orthotropic stiffness matrix given in equation () by an orientation angle 5, and are given by C 11 C 1 C 1 C 1 C C C C C C C 44 C 45 sym C 55 4 C 5 ð4þ 1: 4:1 : 4 18: :4 1 1: :1 :8 :9 GPa 4 sym 5: 5 : Due to the rotation, the stiffness matrix has a monoclinic form, and the decoupling as shown in equation () was no longer applicable. Therefore, no simple mathematical expressions are available for the a behavior, nor for the wavespeeds at which a changes sign.

8 8 Journal of Intelligent Material Systems and Structures Guided wave solution in a single lamina The guided wave solution in a single lamina consists of finding the non-trivial solutions of the homogeneous system given in equation (18) over a space of wavenumber wavespeed values. This is achieved in two steps: 1. Search the wavenumber wavespeed space for roots of the determinant of equation (18); f (j, v) det(d). This generates a wavenumber wavespeed plot containing the dispersion curves.. For each wavenumber wavespeed pair on a dispersion curve, find the solution of equation (18) and use it to reconstruct the displacement and stress values across the thickness, that is generate the wave modeshapes. These two steps are described next. Dispersion curves in a single lamina Multiple methods exist to search for the roots of the complex determinant of equation (18). One method searches for sign changes in the real and imaginary parts of the complex determinant separately, and then searches for common changes which are the true roots. In this research, however, a different method was developed and implemented, the phase approach. In the phase approach, the idea is that a sign change is associated with an 18 phase change. Hence, the phase of the complex determinant is monitored during the search and a root crossing is identified by an approximate 18 phase change. The search was done for a fixed wavenumber and the difference in phases was determined for two consecutive wavespeeds. where 18 mod(u i, 9) L mod(u i, 9) ð5þ L ju i + 1 u i j ðþ If the difference was within the bound of phase change range, given by the expression in equation (5), the wavenumber wavespeed pair yielded a solution, else the next two consecutive velocities were evaluated until the whole solution space was evaluated. An illustration of the phase change is represented in Figure. Figure (a) represents the phase of the complex determinant at wavespeed i and Figure (b) gives the phase of the complex determinant at wavespeeds i and i + 1. For a sign change to occur the phase at i + 1, u i + 1, must be located at in the green region, for other phase values, for example u i + 1, a sign change does not occur. The mathematical representation of the phase approach is given by equations (5) and (). Figure. Illustration of the phase change approach: (a) initial complex number represented by its phase u i ; (b) consecutive phase u i + 1 when sign change was recorded. Applying the phase approach to the different material types yielded the dispersion curves for each material as is shown in Figure 4. A closer examination of Figure 4, especially Figure 4(a), reveals some missing wavenumber wavespeed pairs which should have yielded a solution. Due to the discretization step size of 1 m/s double roots, wavenumber wavespeed pairs where different waves intersected or different waves neared one another within 1 m/s, were caused resulting in missing pairs in the dispersion curves. The gap of missing pairs was reduced by using a finer discretization size in the wavespeed domain. However, this lead to a higher computational time. A trade-off between computational time and accuracy, therefore, has to be made. Wave modeshapes in a single lamina The displacement, and stress matrices of each solution pair were combined to produce the field matrix B that was required to retrieve the partial-wave participation factors, h B(x ) Bu (x ) B s h (x ) ðþ The partial-wave participation factors were used to calculate the displacement modeshapes through the thickness, z(x ), of the material for each solution pair. The modeshapes for the fundamental wave are shown in Figure 5 and the corresponding partial-wave participation factors are presented in equation (9). For each wave mode the modeshape and partial-wave participation factors have a particular shape, this feature gives one the opportunity to sort the wave modes z(x ) ^u(x ) Bu (x )h ^s(x ) B s B(x (x )h )h ð8þ The accuracy of the modeshapes depends on the accuracy of the wavespeeds. Therefore, a refinement in the wavespeeds domain was required prior to the

9 Barazanchy and Giurgiutiu (a) (b) (c) (d) Figure 4. Dispersion curves obtained using the phase approach in a 1 mm thick plate: (a) Isotropic, aluminum; (b) orthotropic transversely isotropic, u 8 CFRP T/914 (Abdelrahman, 14); (c) fully orthotropic, u 8 CFRP (Calomfirescu, 8); and (d) monoclinic, u 58 CFRP T/914 (Abdelrahman, 14). modeshape calculation. Three methods were analyzed in this investigation: (a) a bisection algorithm; (b) genetic algorithm; (c) a brute force algorithm within a range of 1 m/s for the initially obtained solution. Regardless of the method, the objective of the refinement algorithm was to obtain an accurate wavespeed with the lower and upper wavespeed bound. The upper bound was equal to wavespeed i, the wavespeed at which a sign change occurred, and subsequently the lower bound was equal to wavespeed i 1. The search was therefore confined within a range of 1 m/s. The bisection algorithm was computationally the fastest approach, however, the convergence issue were observed especially for the shear-horizontal S wave. For the genetic algorithm no convergence issue were observed. However, it required significantly more computational time than the bisection algorithm. Due to the fact that the search was confined, evaluating all of the wavespeeds in the domain provided accurate results within acceptable computational time. The brute force algorithm was, therefore, used in this investigation : + i :89 + i :5 + i : + i >< :948i > >< + i > h A h S :5418i : + i + i :4 + :4i >: >; >: >; + i :41 + :4i :4 + i :15 + i :1 + i + i >< :5444 :44i > >< :4 :i > h SHS h SHA : :44i :4 + :i :55 + :99i : + :4i >: >; >: >; :558 + :99i :4 + :4i ð9þ

10 1 Journal of Intelligent Material Systems and Structures 1 ξ 81 1 ξ 81 1 ξ 81 1 ξ 81.9 U 1.9 U 1.9 U 1.9 U 1 Thickness location [mm] U U Thickness location [mm] U U Thickness location [mm] U U Thickness location [mm] U U Amplitude Amplitude Amplitude Amplitude (a) (b) (c) (d) Figure 5. Modeshape for the fundamental waves for an orthotropic material at a j of 81/m: (a) A ; (b) S ; (c) SH A ; and (d) SH S. Once the displacement modeshapes were obtained it was possible to group solution pairs using a modeshape analysis. The modeshape analysis was based on the dotproduct between modeshapes; a high correlation was obtained if, and only if, the modeshapes belonged to the same wave type. At the same time the dot-product between two modeshapes, corresponding to two dissimilar wave types, yielded a low correlation. The modeshape analysis is a more robust method of tracking a dispersion curve than existing methods, such as dispersion curve tracking based on slope and extrapolation of the initial root to find the next root (Pavlakovic et al., 199). It is important to state that the modeshapes were first aligned, before the modeshape analysis algorithm was executed. The alignment was done by rotating the modeshape, a vector with three displacement components fu 1, U, U g T, by the angle of the U 1 component such that all modeshapes had a phase angle of zero for the first component. The modeshape analysis algorithm was executed, the fundamental waves, A, S, SH A, SH S, were grouped, and the spline method was used to obtain a continuous solution over the whole domain as can be seen from Figure. The missing solution pairs in Figure 4 were no longer a problem due to the use of the spline function. Dispersion curves in a laminated composite In an N-layered composite, the layers can consist of different materials, orientation angles, thicknesses, or a combination of the aforementioned. As a consequence of this, the eigenvalues, eigenvectors, and both displacement and stress field matrices are required to be calculated for each layer. The traction free boundary conditions remained applicable at the top and bottom surface of the N-layered composite. The boundary conditions between the layers, referred to as interface conditions hereafter, ensured continuity of displacements and stresses through the thickness of the composite. The interface conditions ensured that displacements and stresses at the top of layer n matched the displacements and stresses at the bottom of layer n 1, that is ^u n (h t n )^u n 1(h b n 1 ) ^s n (h t n )^s n 1(h b n 1 ) n,..., N ð4þ The global matrix method (GMM) and transfer matrix method (TMM), as explained in detail by Giurgiutiu (14), are the most frequent methods used to predict wave propagation in N-layered composites. The unified approach was, therefore, evaluated for these two methods. To verify the GMM and TMM for N-layered laminated composites, the dispersion curves obtained for a single lamina are used as a benchmark. A given lamina was split up into N layers. The overall thickness was kept the same, and thus a direct comparison between the dispersion curves could be made and the numerical effect of the GMM and TMM methodologies could be evaluated. For the isotropic case, the proposed unified approach yielded satisfactory results for a 5-layer laminate using the TMM (Figure (b)) when comparing to the earlier obtained results shown in Figure 4(a). Using the GMM (Figure (a)) revealed more double roots as a result of numeric instabilities when using the GMM method. Furthermore, for the orthotropic laminate, mixed results were obtained as can be seen from Figures (c) and (d). While good results were obtained for the TMM, the opposite was true for the GMM. The GMM suffered from numerical instabilities when calculating the determinant of the matrix in equation (18), especially when the matrix is large in size. To obtain good behavior, the number of layers in GMM had to be reduced to 1. For the next test case, a quasi-isotropic ½ Š s 1 mm thick laminate with eight layers was constructed using the fully orthotropic material discussed in Fully orthotropic. The results for both the GMM and TMM were satisfying as can be seen from Figures 8(a) and (b). It is important to note that the wavenumber domain was limited to 5/m for convenience. Finally, a fiber metal laminate (FML) was build out of the isotropic and orthotropic transversely isotropic material discussed earlier. The results for the FML were satisfactory as shown in Figures 8(c) and (d) for the GMM and TMM, respectively.

11 Barazanchy and Giurgiutiu 11 Figure. Dispersion curves after modeshape analysis and spline algorithm in a 1 mm thick plate: (a) Isotropic, aluminum; (b) orthotropic transversely isotropic, u 8 CFRP T/914 (Abdelrahman, 14); (c) fully orthotropic, u 8 CFRP (Calomfirescu, 8); and (d) monoclinic, u 58 CFRP T/914 (Abdelrahman, 14). Comparing the unified formulation to SAFE The semi-analytic finite element (SAFE) approach developed by Bartoli et al. () is an existing formulation that works for both isotropic and anisotropic material, therefore, this section gives a brief comparison of the two formulations. As with any finite element approach, the accuracy of the solution depends on the size/number of elements used. In the case of SAFE, which required discretization in the thickness direction, the number of elements in thickness will affect the accuracy of the solution. Figure 9 illustrates the effect of number of elements in the thickness direction on the accuracy. Figure 9(a) shows that multiple higher order modes are not retrieved when the number of elements through the thickness is insufficient, increasing the number of elements, as shown in Figure 9(b), provides more accurate results and retrieves all of the waved modes in the domain. The SAFE approach retrieves, when a sufficient number of elements through the thickness is used, all of the wavespeed-wavenumber pairs that form the dispersion curves. In addition to this, the SAFE approach can be extended to account for an arbitrary cross-section (Bartoli et al., ). For the unified formulation the discretization in the wavespeed and wavenumber domain determines the number of double roots. However, the solution will be accurate and all the wave modes in the given solution domain will be retrieved. Summary and conclusions Summary A framework, consisting of: (a) a concise overview of the theory, (b) the numerically implementation of it,

12 1 Journal of Intelligent Material Systems and Structures wavespeed [m/s] (a) (b) MISSING ROOTS (c) (d) Figure. Dispersion curves for 5-layer 1 mm thick laminate using GMM and TMM respectively: (a) GMM, isotropic; (b) TMM, isotropic; (c) GMM, orthotropic; (d) TMM, orthotropic. and (c) initial results, for the proposed unified formulation to predict guided-ultrasonic wave (GUW) dispersion curves was given in this article. The simple, and mathematically straightforward unified formulation was based on Christoffel s equation of a lamina. First, the a values part of the formulated eigenvalue problem were solved using the bi-cubic equation. Second, a values were used to solve the eigenvalue problem that yielded three pair of eigenvalues and eigenvectors. The eigenvectors corresponding to a zero eigenvalue form the solution to the formulated eigenvalue problem. The eigenvectors with their corresponding a values were used to produce the stress and displacement field matrices. The traction free boundary condition at the top and bottom surface of the medium were applied to the stress field matrix, and using the zero-determinant condition the wavenumber wavespeed pairs which formed the dispersion curves were retrieved using a phase approach. A phase approach converted the complex determinants in the whole wavenumber wavespeed domain to a phase angle. If the change in phase angle between two consecutive wavespeeds for a fixed wavenumber were within the phase change range then a dispersion curve pair was found. The phase approach was a quick and simple alternative for finding sign changes between complex numbers compared to searching for a sign change in the real and imaginary parts separately, and finding common sign changes. After the dispersion curves were retrieved a modeshape analysis was used to group the wavenumber wavespeed pairs based on their wave type. The modeshape analysis grouped the wave types based on orthogonality of the modeshapes, and a spline algorithm was used to obtain a continuous solution over the whole domain. Using the aforementioned dispersion curves in a 1 mm thick isotropic, orthotropic transversely isotropic, fully orthotropic and monoclinic materials were successfully obtained. For N-layered media the two

13 Barazanchy and Giurgiutiu (a) (b) (c) (d) Figure 8. Dispersion curves for 1 mm thick laminate using GMM and TMM respectively: (a) and (b) quasi-isotropic ½45 459Š s ; and (c) and (d) fiber metal laminate, orthotropic/isotropic ½Š s. Figure 9. The effect of the number of elements in the thickness direction to the accuracy of SAFE solution when compared to the unified formulation. (a) SAFE approach, 1 element through the thickness. (b) SAFE approach, 15 elements through the thickness.

14 14 Journal of Intelligent Material Systems and Structures most common methods, global matrix method (GMM), and transfer matrix method (TMM) were applied. A short comparison between the SAFE approach and the unified formulation is presented as a test case for the proposed formulation. Finally, numerical examples were given to test the proposed unified formulation. Conclusions The contribution of this article to the literature is as follows: (a) a dispersion curves algorithm based on a unified formulation is proposed and numerical examples are provided to show its working; and (b) a phase approach and modeshape analysis are provided to retrieve the dispersion curves and identify the dispersion curves based on wave type, respectively. The phase approach provided a quick and simple alternative to finding sign changes for complex numbers based on the phase of the complex number. Instead of searching for real and imaginary sign changes separately, and then searching for the common sign changes, the phase approach provided the opportunity to search for sign changes in the real and imaginary part simultaneously. The modeshape analysis on its turn was provided a good method to group wavenumber wavespeed pair based on its corresponding wave type, thereby distinguishing the fundamental wave modes even at locations where different waves intersected. For single lamina the unified formulation was, therefore, able to predict dispersion curves in isotropic, orthotropic transversely isotropic, fully orthotropic, and monoclinic materials. The phase approach, and the modeshape analysis were used to obtain a continuous solution for the whole wavenumber wavespeed domain. For N-layered media four different 1 mm thick laminates were examined: (a) 5-layer isotropic; (b) 5-layer orthotropic; (c) quasi-isotropic, ½45 459Š s ;and(d) a fiber metal laminate (FML) orthotropic/isotropic ½Š s. Satisfying results were obtained for most of the examined laminates and this provided the basis that the unified approach was able to predict dispersion curves in a high number of layered laminates, and in hybrid laminates such as fiber metal laminates. For the 5-layer orthotropic using the GMM, however, unsatisfying results were obtained due to the numerical instabilities that occurred when calculating the determinant required to retrieve the dispersion curves. To obtain satisfying results for the GMM the number of layers had to be reduced to 1. In conclusion the unified formulation yielded promising and satisfying results which can be utilized in future work. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Skolkovo Institute of Science and Technology, Russia and the National Aeronautics and Space Administration (NASA) (grant number NNL15AA1C). References Abdelrahman AK (14) Ultrasonic transduction in metallic and composite structures for structural health monitoring using extensional and shear horizontal piezoelectric wafer active sensors. PhD Thesis, University of South Carolina, USA. Abramowitz M and Stegun I (19) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Vol. 9. New York: Dover. Barazanchy D, Martinez M, Rocha B, et al. 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