Acoustic nonlinearity parameters for transversely isotropic polycrystalline materials

Transcription

1 University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Mechanical & Materials Engineering Faculty Publications Mechanical & Materials Engineering, Deartment of 2015 Acoustic nonlinearity arameters for transversely isotroic olycrystalline materials Christoher M. Kube University of Nebraska-Lincoln, ckube@huskers.unl.edu Joseh A. Turner University of Nebraska-Lincoln, jaturner@unl.edu Follow this and additional works at: htt://digitalcommons.unl.edu/mechengfacub Part of the Mechanical Engineering Commons, and the Physical Sciences and Mathematics Commons Kube, Christoher M. and Turner, Joseh A., "Acoustic nonlinearity arameters for transversely isotroic olycrystalline materials" (2015). Mechanical & Materials Engineering Faculty Publications htt://digitalcommons.unl.edu/mechengfacub/116 This Article is brought to you for free and oen access by the Mechanical & Materials Engineering, Deartment of at DigitalCommons@University of Nebraska - Lincoln. It has been acceted for inclusion in Mechanical & Materials Engineering Faculty Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.

2 Acoustic nonlinearity arameters for transversely isotroic olycrystalline materials Christoher M. Kube a) and Joseh A. Turner Deartment of Mechanical and Materials Engineering, W42 Nebraska Hall, University of Nebraska-Lincoln, Lincoln, Nebraska , USA (Received 1 February 2015; acceted 29 Aril 2015) This article considers olycrystalline materials with macroscoic elastic anisotroy and the effect of the anisotroy on the quadratic nonlinearity arameter used to describe second harmonic generation in solids. The olycrystal is assumed to have transversely isotroic elastic symmetry, which leads to a directional deendence of the nonlinearity arameters. Additionally, the anisotroy leads to second harmonic generation from an inut shear wave. Estimates of the longitudinal and shear wave nonlinearity arameters are given as a function of single-crystal elastic constants, macroscoic anisotroy constants, and roagation direction. An inverse model is resented that relates measured nonlinearity arameters to the macroscoic anisotroy constants. The estimates of the nonlinearity arameters can be used to aroximate the damage-free or baseline nonlinearity arameter of structural comonents, which hels the effort toward absolute measures of material damage. VC 2015 Acoustical Society of America. [htt://dx.doi.org/ / ] [MD] Pages: I. INTRODUCTION A few instances of wave roagation in solids require the constitutive equations of the elastic solid to retain nonlinear variables of strain and stress. Some examles include small amlitude waves in an initially stressed solid, 1 8 finite amlitude interactions between two or more waves, 9 12 and harmonic waves generated from a finite amlitude fundamental wave In an exerimental setting, these examles oen the door for correlating the nonlinear elastic roerties of the solid to wave measurements. Recently, the measurements of the second-harmonic, generated from a high amlitude inut harmonic excitation, have become a oular and imortant tool used to detect and characterize various forms of material damage. 19 A comrehensive review of this toic has recently been given by Matlack et al. 19 The contribution to the second-harmonic amlitude caused by the damage is in addition to the bare nonlinearity accomanying an undamaged material state. The bare nonlinearity is sometimes referred as the virgin nonlinearity and is often assumed to be manifest solely through the lattice anharmonicity. For second-harmonic generation (SHG), the dislacement amlitude of the second-harmonic relative to the fundamental wave dislacement is quantified through the quadratic nonlinearity arameter, b. For the bare nonlinearity arriving from the lattice anharmonicity, the nonlinearity arameter b lattice is found to contain a ratio of second- and third-order elastic constants to second-order elastic constants where the elastic constants follow from the constitutive relation governing lattice deformation. The contribution of a damage arameter b damage is often assumed to be additive where the absolute nonlinearity arameter is b ¼ b lattice þ b damage Thus, the accuracy of a) Electronic mail: ckube@huskers.unl.edu a theoretical model for the absolute b arameter deends on how well b lattice and b damage are reresented. In ractice, absolute measures of b have only been achieved in highly controlled laboratory settings. The extension to in situ absolute measurements of b is extremely difficult because a baseline measure of the bare nonlinearity is tyically unknown. This deficiency amongst others 19 has relegated SHG techniques to relative measures. Thus, many of the theoretical efforts have been aimed toward accurately defining the various contributions to b damage. For materials other than single-crystals, there have been no known attemts to estimate the bare nonlinearity for the undamaged material state beyond an assumtion of isotroic elastic roerties. In rincile, absolute measurements of the nonlinearity arameter are ossible with either an ideal damage-free calibration samle or an accurate theoretical model to redict the baseline. In this article, the bare nonlinearity arameter b 0 is defined to give an estimate of the exected values for a manufactured olycrystalline metal rior to the onset of damage. In other words, b 0 is the nonlinearity arameter exected after a structural member has undergone all manufacturing rocesses and rior to being laced in a functional environment that ossibly leads to damage. The estimate of b 0 allows the olycrystal to be macroscoically anisotroic where the elastic roerties of the individual grains have some degree of referred alignment. The resent article considers the olycrystal s elastic roerties to be transversely isotroic. Most metals have some degree of elastic anisotroy, the case of transversely isotroic texture assumes the grains have a referred orientation about one of their crystallograhic axes while all other axes are randomly oriented. Metals that have been extruded or rolled are often found to be transversely isotroic. Additionally, welded metals often have grains oriented along their [100] crystallograhic axes, 272 J. Acoust. Soc. Am. 17 (6), June /2015/17(6)/272/9/$0.00 VC 2015 Acoustical Society of America Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

3 which is arallel to the directions of thermal gradients during the solidification rocess. 24 By allowing for macroscoic anisotroy, this article attemts to form a starting oint where the bare or virgin nonlinearity can be estimated and alied to in situ measurements of the absolute nonlinearity of structural metal comonents. The assumtion of isotroic material symmetry leads to b 0 being indeendent of roagation direction along with the absence of second-harmonic generation from an inut fundamental shear wave. Allowance for macroscoic anisotroy results in the ossibility of two bulk wave nonlinearity arameters b 0 ql and b0 qsv for quasi-longitudinal and quasi-shear vertical wave modes, resectively. Additionally, both of these wave modes are deendent on the wave roagation direction. II. THEORY The strain energy accomanying a finite deformation of an elastic single-crystal is aroximated by the Taylor exansion of the Lagrangian strain tensor E, 25 q 0 UX ð Þ ¼ 1 2! C ijklðxþe ij E kl þ 1! C ijklmnðxþe ij E kl E mn þ oðe 4 Þ: (1) The elastic tensors C ijkl ðxþ and C ijklmn ðxþ are the fourth- and sixth-rank elastic moduli or stiffness tensors of the elastic solid. The elastic tensors of rank 2n define the elastic constants of order n. For crystallites belonging to oint grou symmetry lower than isotroy, the elastic constants are directionally deendent. The directional deendence of the elastic constants is considered by defining the orientation of the crystallite by X and constructing the tensor transformations, C ijkl ðxþ ¼a i ðxþa jq ðxþa kr ðxþa ls ðxþc qrs ; C ijklmn ðxþ ¼a i ðxþa jq ðxþa kr ðxþa ls ðxþ a mu ðxþa nv ðxþc qrsuv ; (2) where a is a rotation matrix containing elements of the crystallite orientation X. The rotation matrix is constructed by considering three successive rotations of the crystallite axes. a is tyically constructed using a set of Euler angles where each Euler angle describes a rotation about an axis. Equation (2) allows the single-crystal elastic constants to be defined with resect to a frame of reference other than the crystallite axes. The orientation deendence of the elastic constants leads to the orientation deendence of the strain energy in Eq. (1). Ifweconsiderasinglecrystalliteasagrain in a olycrystal, the strain-energy of the olycrystal can be statistically defined as an exectation value or average, U 0 ¼ 1 ð wðxþux ð ÞdX; () 8 2 X where wðxþuðxþ is the robability of finding a crystallite with strain energy UðXÞ on the interval X þ dx: The function wðxþ is the orientation distribution function (ODF), which defines a weighting to the ossible orientations of the crystallites. The ODF is unity when all orientations of the crystallites are equally robable. In this article, we make use of the Voigt assumtion that equates the strain in the olycrystal to the strain in the crystallites, E ¼ E Because E 0 is indeendent of crystallite orientation, the average strain energy can be written as q 0 U 0 ¼ 1 ð wðxþc ijkl ðxþdx E 0 ij 2! E0 kl X þ 1 ð wðxþc ijklmn ðxþdx E ij E kl E mn! X ¼ 1 2! C0 ijkl E0 ij E0 kl þ 1! C0 ijklmn E0 ij E0 kl E0 mn ; (4) where C 0 ijkl ¼ 1 ð 8 2 wðxþc ijkl ðxþdx; X C 0 ijklmn ¼ 1 ð 8 2 wðxþc ijklmn ðxþdx; (5) X are known as the Voigt estimated elastic moduli tensors, which define the second- and third-order elastic constants (SOECs and TOECs) of the olycrystal. For materials belonging to any of the 2 oint grou symmetries, the nonlinear equation of motion can be derived from the strain-energy function. 27 We roceed by defining the equation of motion for the olycrystal of general symmetry 2 u i q 2 ¼ A0 ijkl 2 u k A0 ijklmn ; l where A ijkl and A ijklmn are the fourth- and sixth-rank Huang tensors, A 0 ijkl ¼ C0 ijkl ; A 0 ijklmn ¼ C0 jlmn d ik þ C 0 ijnl d km þ C 0 jknl d im þ C 0 ijklmn ; (7) and C 0 ijkl and C 0 ijklmn are defined in Eq. (5). An aroximate harmonic solution to Eq. (6) was reviously obtained From the harmonic solution, a relationshi between the amlitudes of the first- and second-harmonics was found to deend on the constant b 0 ; which quantifies the second-order nonlinearity on the roagating wave. Following Cantrell, 17 the quadratic nonlinearity arameter b 0 for the olycrystal is given as b 0 ¼ A0 ijklmn^n j^n l^n n^u i^u k^u m A 0 ijkl^n j^n l^u i^u k ; (8) where ^n and ^u are the wave roagation and dislacement directions, resectively. J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters 27 Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

4 At this oint, the elastic symmetry of the olycrystal or the crystallite has not been secified. If we assume that the olycrystal is macroscoically isotroic ½wðXÞ ¼ 1Š and contains crystallites of cubic crystallograhic symmetry, the two indeendent SOECs of the olycrystal are found from Eq. (5) to be 26 C 0 12 ¼ c 12 þ 5 ; C 0 44 ¼ c 44 þ 5 ; (9) where ¼ c 11 c 12 2c 44 is the second-order anisotroy constant for the crystallite. The third, deendent, SOEC of the olycrystal is C 0 11 ¼ C0 12 þ 2C0 44 : The TOECs are obtained from Eq. (5) in a similar manner, C 0 12 ¼ c 12 þ 1 ð 5 d 1 þ 21d 2 Þ; C ¼ c 144 þ 1 ð 5 d 1 þ 7d 2 þ 14d Þ; C ¼ c 456 þ 1 ð 5 d 1 þ 21d Þ; (10) where d 1 ¼ c 111 c 112 þ 2c 12 þ 12c c 155 þ 16c 456 ; d 2 ¼ c 112 c 12 2c 144 ; and d ¼ c 155 c 144 2c 456 are the third-order single-crystal anisotroy constants. The remaining, deendent, TOECs are C ¼ C0 12 þ 6C0 144 þ8c ; C0 112 ¼ C0 12 þ 2C0 144 ; and C0 155 ¼ C0 144 þ 2C0 456 : These TOECs for macroscoically isotroic olycrystals containing cubic crystallites were reviously derived by Chang, 0 Juretschke, 1 and Barsch. 2 The quadratic nonlinearity arameter in Eq. (8) becomes b 0 ¼ C0 11 þ C0 111 C 0 11 ¼ 15c 111 þ 18c 112 þ 2c 12 þ 12c 144 þ 72c 155 þ 16c 456 : (11) 7c ð 11 þ 2c 12 þ 4c 44 Þ The nonlinearity arameter defined using Eq. (11) was found to agree with exeriments erformed on olycrystalline coer and also with exeriments on olycrystalline aluminum. 4 We now consider the case for which the macroscoic elastic roerties of the olycrystal are transversely isotroic. Once again, we restrict the grains to be crystallites of cubic elastic symmetry. For this case, the orientation distribution function wðxþ weights the average in Eq. (5) more heavily for crystallite orientations ðxþ that have some degree of alignment about a single axis while the other two axes are not weighted. For this case, the ODF is 5 wðxþ ¼ wðw; v ¼ cos h; / Þ ¼ 1 þ ffiffiffi 2 16 W 400 0v 2 þ 5ðv 1Þ 2 ðv þ 1Þ 2 cos 4/ þ 5v 4 Š " 26 þ 2 W þ 105v 2 15v # v2 1 11v 2 1 cos 4/ ðv 1Þ 2 ðv þ 1Þ 2 þ 21v 6 ; (12) where W 400 and W 600 are macroscoic anisotroy constants and w; h; and / are Euler angles. Equation (12) is an exansion of sherical harmonics derived by Roe and is simlified and truncated from a more general case. 5 Details of the simlification can be found in Roe 5 and Johnson. 6 Carrying out the integrations in Eq. (5) leads to the SOECs of a transversely isotroic olycrystal, C 0 11 ¼ c 12 þ 2c 44 þ 5 þ ffiffi ; C 0 12 ¼ c 12 þ 5 þ ffiffi 42 5 ; C 0 1 ¼ c 12 þ ffiffiffi ; C 0 ¼ c 12 þ 2c 44 þ 5 þ ffiffi 22 5 ; C 0 44 ¼ c 44 þ ffiffiffi ; C 0 66 ¼ c 44 þ 5 þ ffiffi 42 5 : (1) Only five of the SOECs are indeendent where C 0 66 ¼ðC 0 11 C0 12Þ=2: The TOECs are found to be 274 J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

5 C ¼ c 12 þ 6c 144 þ 8c 456 þ 5 5d ffiffiffi ð 1 þ 21d 2 þ 84d Þþ 6 2 d 1 77 þ d 2 5 þ 4d 20 2 d 1 5 C ¼ c 12 þ 2c 144 þ 1 5 d ffiffiffi ð 1 þ 5d 2 þ 28d Þþ 4 2 9d 1 85 þ d 2 7 þ 4d 4 2 d 1 5 C 0 11 ¼ c 12 þ 2c 144 þ 1 5 d ffiffiffi ð 1 þ 5d 2 þ 28d Þ 4 2 d 1 55 þ d 2 7 þ 16d þ 24 2 d 1 5 C 0 12 ¼ c 12 þ 1 5 d ffiffi ð 1 þ 21d 2 Þ 4 2 d 1 55 þ d 2 þ 8 2 d 1 5 C 0 1 ¼ c 12 þ 2c 144 þ 1 5 d ffiffiffi ð 1 þ 5d 2 þ 28d Þ 64 2 d 1 85 þ d 2 2 d 1 5 C ¼ c 144 þ 1 5 d ffiffi ð 1 þ 7d 2 þ 14d Þ 4 2 d 1 55 þ 4d 2 5 þ d þ 8 2 d 1 5 C ¼ c 144 þ 2c 456 þ 1 5 d ffiffi ð 1 þ 7d 2 þ 56d Þ 4 2 d 1 55 þ 4d 2 5 þ 17d þ 24 2 d 1 5 C ¼ c 144 þ 2c 456 þ 1 5 d ffiffi ð 1 þ 7d 2 þ 56d Þþ 4 2 9d 1 85 þ d 2 5 þ 8d 4 2 d 1 5 C 0 ¼ c 12 þ 6c 144 þ 8c 456 þ 5 5d ffiffiffi ð 1 þ 21d 2 þ 84d Þþ 96 2 d 1 77 þ d 2 5 þ 4d þ 64 2 d 1 5 C 0 55 ¼ c 144 þ 2c 456 þ 1 5 d ffiffiffi ð 1 þ 7d 2 þ 56d Þ d 1 85 þ d 2 5 þ d 2 2 d 1 5 C 0 66 ¼ c 144 þ 1 5 d ffiffi ð 1 þ 7d 2 þ 14d Þ 4 2 d 1 55 d 2 5 þ 8d þ 8 2 d 1 5 C ¼ c 456 þ 1 5 d ffiffi ð 1 þ 21d Þ 4 2 d 1 55 þ d þ 8 2 d : (14) Of the 12 constants given, nine are indeendent where C ¼ðC0 111 C0 112 Þ=4; C0 66 ¼ðC0 11 C0 12 Þ=4; and C0 456 ¼ð C0 144 þc 0 155Þ=2: Thus, the transversely isotroic olycrystal exhibits Curie symmetry ð1mþ: The nonlinearity arameters for the transversely isotroic olycrystal follow from Eq. (8) where the inner roducts on the Huang coefficients in exanded form are A 0 ijkl^n j^n l^u i^u k ¼ C 0 11 ð^n2 1^u2 1 þ ^n2 2^u2 2 Þþ2C012^n 1^n 2^u 1^u 2 þ 2C 0 1^n 1^n ^u 1^u þ C 0 ^n2 ^u2 þ C 0 44 ½ð^n 1^u þ ^n ^u 1 Þ 2 þð^n 2^u þ ^n ^u 2 Þ 2 Šþ2C 0 66^n 1^n 2^u 1^u 2 ; (15) and A 0 ijklmn^n j^n l^n n^u i^u k ^u m ¼ C 0 11 ð^n 1^u 1 þ ^n 2^u 2 þ ^n 1^u 1^u 2 2 þ ^n 1^u 1^u 2 þ ^n 2^u2 1^u 2 þ ^n 2^u 2^u 2 Þ þ C 0 12 ð^n 1^n 2 2^u 1 þ ^n2 1^n 2^u 2 þ ^n 1^n 2 2^u 1^u 2 2 þ ^n2 1^n 2^u 2 1^u 2 þ ^n 1^n 2 2^u 1^u 2 þ ^n2 1^n 2^u 2^u 2 Þ þ C 0 1 ð^n 1^n 2 ^u 1 þ ^n 2^n 2 ^u 2 þ ^n2 1^n ^u þ ^n2 2^n ^u þ ^n 1^n 2 ^u 1^u 2 2 þ ^n 1^n 2 ^u 1^u 2 þ ^n 2^n 2 ^u2 1^u 2 þ ^n 2 1^n ^u 2 1^u þ ^n 2 1^n ^u 2 2^u þ ^n 2 2^n ^u 2 1^u þ ^n 2^n 2 ^u 2^u 2 þ ^n2 2^n ^u 2 2^u Þ þ C 0 ð^n ^u þ ^n ^u2 1^u þ ^n ^u2 2^u Þþ6C 0 44 ð^n 1^n 2 ^u 1 þ ^n 2^n 2 ^u 2 þ ^n2 1^n ^u þ ^n 2 2^n ^u þ ^n 1^n 2 ^u 1^u 2 2 þ ^n 1^n 2 ^u 1^u 2 þ ^n 2^n 2 ^u2 1^u 2 þ ^n 2 1^n ^u 2 1^u þ ^n 2 1^n ^u 2 2^u þ ^n 2 2^n ^u 2 1^u þ ^n 2^n 2 ^u 2^u 2 þ ^n2 2^n ^u 2 2^u Þþ6C 0 66 ð^n 1^n 2 2^u 1 þ ^n2 1^n 2^u 2 þ ^n 1^n 2 2^u 1^u 2 2 þ ^n 2 1^n 2^u 2 1^u 2 þ ^n 1^n 2 2^u 1^u 2 þ ^n2 1^n 2^u 2^u 2 ÞþC0 111 ð^n 1^u 1 þ ^n 2^u 2 ÞþC0 112 ð^n 1^n 2 2^u 1^u 2 2 þ ^n2 1^n 2^u 2 1^u 2Þ þ C 0 11 ð^n2 1^n ^u 2 1^u þ ^n 2 2^n ^u 2 2^u Þþ6C 0 12^n 1^n 2^n ^u 1^u 2^u þ C 0 1 ð^n 1^n 2 ^u 1^u 2 þ ^n 2^n 2 ^u 2^u 2 Þ þ C ð^n 1^n 2 2^u 1^u 2 þ ^n 1^n 2 ^u 1^u 2 2 þ ^n 2^n 2 ^u2 1^u 2 þ ^n 2 1^n 2^u 2^u 2 þ ^n 1^n 2^n ^u 1^u 2^u Þ þ C ð^n 1^n 2 ^u 1 þ ^n 2^n 2 ^u 2 þ ^n 1^u 1^u 2 þ ^n 2^u 2^u 2 þ 2^n2 1^n ^u 2 1^u þ 2^n 2 2^n ^u 2 2^u Þ þ C ð^n 1^n 2 2^u 1 þ ^n2 1^n 2^u 2 þ ^n 1^u 1^u 2 2 þ ^n 2^u2 1^u 2 þ 2^n 1^n 2 2^u 1^u 2 2 þ 2^n2 1^n 2^u 2 1^u 2ÞþC 0 ^n ^u þ C 0 55 ð^n2 1^n ^u þ ^n2 2^n ^u þ ^n ^u2 1^u þ ^n ^u2 2^u þ 2^n 1^n 2 ^u 1^u 2 þ 2^n 2^n 2 ^u 2^u 2 Þ þ C 0 66 ð^n2 1^n ^u 2 2^u þ ^n 2 2^n ^u 2 1^u þ 2^n 1^n 2^n ^u 1^u 2^u Þ þ 6C ð^n 1^n 2 2^u 1^u 2 þ ^n 1^n 2 ^u 1^u 2 2 þ ^n 2^n 2 ^u2 1^u 2 þ ^n 2 1^n 2^u 2^u 2 þ ^n2 1^n ^u 2 2^u þ ^n 2 2^n ^u 2 1^u þ 2^n 1^n 2^n ^u 1^u 2^u Þ: (16) J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters 275 Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

6 ðc 0 ijkl^n j^n l qv 2 d ik Þu k ¼ 0; (17) where v is the hase velocity of the wave. In order to solve for the dislacement directions, the roagation direction is defined in sherical coordinates as seen in Fig. 1, which yields ^n ¼ ^x cos / 0 sin h 0 þ ^y sin / 0 sin h 0 þ ^z cos h 0 ; (18) FIG. 1. The sherical coordinate system used to define the roagation direction ^n and rotated dislacement directions. The macroscale anisotroy of the olycrystal causes the wave dislacement directions ^u to be deendent on the roagation direction ^n: The dislacement directions must satisfy the Christoffel equation, 7 where the ^y-direction is into the age and / 0 is the angle formed between ^x and ^y: The second-order elastic constants needed to define C 0 ijkl are defined in Eq. (1) where ^z is the axis of alignment. The three dislacement solutions of ^u are 8 40 ^u ql ¼ ^x cos / 0 sin c 0 þ ^y sin / 0 sin c 0 þ ^z cos c 0 ; ^u qsv ¼ ^x cos / 0 cos c 0 þ ^y sin / 0 cos c 0 ^z sin c 0 ; and ^u SH ¼ ^x sin / 0 ^y cos / 0 ; (19) where # c 0 ¼ h 0 þ w 0 ðh 0 Þ ¼ h 0 "sin 1 2 atan H cos 2 h 0 C h 0 þ C0 1 þ 2C0 44 H cos 2h 0 cos 2 h 0 þ C 0 11 þ 2C0 1 þ ; (20) C0 44 and H ¼ C C0 1 þ C0 4C0 44 : Thus, the effect of the transverse isotroy leads to a rotation w 0 of the quasilongitudinal (ql) and shear-vertical (qsv) dislacement directions away from the roagation direction ^n as seen in Fig. 1. Additionally, the anisotroy leads to non-zero values of b 0 qsv for the shear-vertical wave when the roagation direction is not arallel or erendicular to the transverse isotroy axis. Exressions of the longitudinal ½b 0 ql ðh 0ÞŠ or shearvertical ½b 0 qsv ðh 0ÞŠ nonlinearity arameters are generally too comlex to be given exlicitly. However, the longitudinal nonlinearity arameter b 0 ql for roagation direction arallel ðh 0 ¼ 0Þ and erendicular ðh 0 ¼ =2Þ simlify considerably due to symmetry and can be exressed as b 0 L b0 qlð h 0 ¼ 0Þ ¼ C0 þ C0 ; C 0 b 0 L1 b0 qlð h 0 ¼ =2Þ ¼ C0 11 þ C0 111 : (21) C 0 11 The exressions in Eq. (21) may even be inverted to give the anisotroy coefficients in terms of the nonlinearity arameters b 0 L and b0 L1 ; W 400 ¼ 7 ffiffiffi 2 h i d ð 1 þ11d 2 þ44d Þþðþ6b 0 L1 þ5b0 L Þ ½ 105 ð c12 þ6c 144 þ8c 456 Þ þ9ð5d 1 þ21d 2 þ84d Þþð5c 12 þ10c 44 þþð6þ16b 0 L1 þ5b0 L ÞŠ; 26 h i 1 n W 600 ¼ d ð 1 þ11d 2 þ44d Þþðþ6b 0 L1 d þ5b0 L Þ 95d ð 1 þ11d 2 þ44d Þð5d 1 þ21d 2 þ84d Þ 1 h i þ21ð5d 1 þ11d 2 þ44d Þ 5ðc 12 þ6c 144 þ8c 456 Þþðc 12 þ2c 44 Þð15þ8b 0 L1 b0 L Þ h þ 1155ðc 12 þ6c 44 þc 12 þ6c 144 þ9c 456 Þþ85ðc 12 þ2c 44 Þðb 0 L1 þb0 L þb0 L1 b0 L Þ þ 77ðc 12 þ6c 144 þ8c 456 Þð8b 0 L b0 L1 Þþ15d 1ð96þ27b 0 L1 þ5b0 L i Þ þ69ðd 2 þ4d Þð6þb 0 L1 þb0 L Þ þ21 2 ð9þb 0 L1 þb0 L1 b0 L o: Þ (22) Thus, quantitative measurements of the olycrystal anisotroy can be found from exerimentally measuring b 0 L and b0 L1 values and use of the single-crystal elastic constants, which are tabulated for many materials J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

7 TABLE I. Texture coefficients for random and aligned orientations of the crystallograhic axes. random [100] [110] [111] W ffiffi 2 /(2 2 ) 7 ffiffi 2 /(128 2 ) 6 ffiffi 2 /(512 2 ) ffiffiffiffiffi W /(64 2 ) 1 ffiffiffiffiffi 26 /(512 2 ) 85 ffiffiffiffiffi 26 /( ) III. RESULTS AND DISCUSSION To illustrate the theory more clearly, the elastic constants and nonlinearity arameters of a olycrystal with transversely isotroic textures are calculated for olycrystalline iron and aluminum. We consider transversely isotroic textures where one secific crystallograhic axis is erfectly aligned for every grain in the olycrystal. The air of Euler angles ðw; hþ allows the samle axes to be aligned with the crystallite axes. In order to calculate the elastic constant C 0 ðw; hþ; the transverse symmetry axis along the ^z-direction is rotated using the Euler angles w and h to coincide with the crystallograhic axis that is chosen to be the erfectly aligned axis. Then, the angular average of Eq. (2) is erformed about the aligned axis, C 0 ðw; hþ ¼ 1 2 ð 2 0 a a q a r a s C qrs d/: (2) For alignment about any crystallograhic axis, Eq. (2) can be equated with the constant C 0 defined in Eq. (1) in order to calculate the coefficient W 400 : Using the value for W 400 ; this rocess is reeated using the average C 0 ðw; hþ ¼ 1 2 ð 2 0 a a q a r a s a u a v C qrsuv d/ (24) and Eq. (14) to obtain the value for W 600 : Once W 400 and W 600 are known, all of the SOECs and TOECs in Eqs. (1) and (14) can be determined. For this article, the three crystallograhic axes considered are chosen to be along the [100], [110], and [111] directions. For these cases the following Euler angle airs are used: ðw ¼ =2; h ¼ =2Þ aligns ^z with [100], ðw ¼ =4; h ¼ =2Þ aligns ^z with [110], and ðw ¼ =4; h ¼ =4Þ aligns ^z with [111]. This rocedure leads to the three values of W 400 and W 600 given in Table I for the individual cases of erfect alignment about the crystallograhic axes [100], [110], and [111]. The examle texture of erfect alignment about [100] is considered a common hysical texture for cast or extruded metal arts and its effects on attenuation coefficients were reviously considered. 24,9 In what follows, we consider olycrystalline iron and aluminum with [100], [110], and [111] textures. For these materials, the second- and third-order single-crystal elastic constants and anisotroy constants are given in Table II. The wave dislacements of the ql and qsv modes are calculated using Eqs. (19) and (20). Figure 2 gives the rotation angle w 0 as a function of the angle h 0 ; which we use to define the wave roagation direction ^n in Eq. (18). 42 For iron, small deviations away from h 0 ¼ 0orh 0 ¼ =2 cause the ql and qsv dislacement directions to rotate by an angle close to w 0 ¼ =4 for the [100] and [110] textures considered. The high degree of anisotroy ¼ c 11 c 12 2c 44 for iron contributes to the large rotations. The rotations for aluminum are not as extreme due to the smaller anisotroy. For both materials, w 0 ½100Š > w 0 ½110Š > w 0 ½111Š; which follows from the values of the stiffnesses along these axes c 11 > c 12 > c 44 : Using these rotations, the roagation and dislacement directions in Eqs. (18) and (19) are well defined, which allows the longitudinal b 0 ql ðh 0Þ and shear-vertical b 0 qsv ðh 0Þ to be evaluated from Eq. (8). Figure gives the relation between the ql and qsv nonlinearity arameters as a function of roagation angle h 0. Polycrystalline iron and aluminum are considered in Figs. (a) and (b) and Figs. (c) and (d), resectively. The nonlinearity arameter b 0 ql has three unique values for the textures considered when the wave roagates either arallel ðh 0 ¼ 0Þ or erendicular ðh 0 ¼ =2Þ to the transversely isotroic axis. These values corresond to the directions used to define b 0 L1 and b0 L in Eq. (21). The quantitative values for these cases are given in Table III. Longitudinal roagation arallel to the aligned [111] axis generates the largest second-harmonic comonent for aluminum while roducing the weakest comonent for iron. Conversely, the largest second-harmonic for roagation erendicular to the aligned axis in iron is observed when the grains are aligned about the [111] axis. For this roagation direction in aluminum, the [111] axis serves as the weakest for SHG. As the roagation direction increases from h 0 ¼ 0toh 0 ¼ =2; the value of jb 0 qlj follows an oscillating trend with a distinct local maxima is observed for iron at only h ¼ =2 for alignment about [111]. For iron, most values of jb 0 qlj for the transversely isotroic case fall beneath the value of jb 0 qlj for a macroscoically isotroic olycrystal with randomly orientated grains. However, for aluminum, the values of jb 0 qlj for the transversely isotroic case reside near the isotroic estimate. The larger deviation for iron is caused by the larger values of crystallite anisotroy ð; d 1 ; d 2 ; and d Þ when comared with aluminum as shown in Table II. Metals with a stronger degree of crystallite anisotroy than iron are exected to have a larger deviation from the isotroic case. The measures of single-crystal anisotroy ð; d 1 ; d 2 ; and d Þ could be used to determine the range of validity of the assumtion of macroscoic isotroy. For the cases of h 0 ¼ 0 and h 0 ¼ =2; no secondharmonic is generated from the inut fundamental shear wave. This exectation follows from Norris 7 who showed TABLE II. Single-crystal elastic constants for iron (Ref. 41 ) and aluminum (Ref. 41) (GPa). c 11 c 12 c 44 c 111 c 112 c 12 c 144 c 155 c 456 d 1 d 2 d Fe Al J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters 277 Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. 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8 FIG. 2. Rotation angle w 0 versus incident angle h 0 : for iron (a), and aluminum (b). The textures considered were: random ( ), [110] ( ), [111] ( ), and [100] ( ). that shear wave roagation within elastic symmetry lanes is not caable of roducing harmonics. However, secondharmonic shear waves are observed for roagation directions that deviate only slightly from h 0 ¼ 0toh 0 ¼ =2: Again, non-constant behavior is observed for increasing angle h 0 : For many angles, the amlitude of the secondharmonic is redicted to be near the same magnitude as the second-harmonic roduced from a fundamental longitudinal wave. Iron is shown to generate larger shear wave secondharmonics than aluminum, likely because of the larger single-crystal anisotroy. For aluminum with the aligned axes [111], certain angles h 0 cause jb 0 qsvj to vanish. This outcome is not because ^n or ^u qsv vanish, rather, the combinations of the calculated elastic constants C 0 ijkl and C0 ijklmn for aluminum bring Eq. (16) to zero for these angles of h 0 : The three textures considered here are for erfect alignment of the crystallograhic axes. It is imortant to note that the derivation allows for different degrees of texture from random alignment, when W 400 ¼ W 600 ¼ 0; to erfect alignment. Alignment about other crystallograhic directions will likely cause the angular deendence of b 0 to look considerably different than the results in Fig.. Paroni 4 has derived the bounds on the texture coefficients 7 ffiffiffi 2 =ð48 2 ÞW ffiffi 2 =ð2 2 Þ and 1 ffiffiffiffiffi 26 =ð512 2 ÞW 600 ffiffiffiffiffi 26 =ð6 2 Þ: Similarly, jb 0 j is also bounded where the three cases considered here. It was observed that the term A 0 ijklmn^n j^n l^n n^u i^u k^u m has a much stronger deendence on h 0 than A 0 ijkl^n j^n l^u i^u k : In many instances, A 0 ijkl^n j^n l^u i^u k may be allowed to be aroximated by its value for the case of randomly oriented crystallites or determined exerimentally from hase velocity measurements. Using such aroximations may hel the inversion of measurement data to solve for W 400 and W 600 : The resented model is based on a Voigt-tye assumtion of equivalent Lagrangian strains. Other more comlicated homogenization schemes 44,45 could rove to be more FIG.. Nonlinearity arameters versus angle of roagation: jb 0 qlj for iron (a), jb 0 qsv j for iron (b), jb0 qlj for aluminum (c), and jb 0 qsvj for aluminum (d). The textures considered were: random ( ), [110] ( ), [111] ( ), and [100] ( ). 278 J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

9 TABLE III. Values of the longitudinal nonlinearity arameter for h 0 ¼ 0 and h 0 ¼ /2. b 0 ql (h 0 ¼ 0) accurate but are also much more comlex and ossibly intractable. The alicability of the resent Voigt-tye scheme is suorted by revious exerimental results on macroscoically isotroic olycrystals.,4 IV. CONCLUSION b 0 ql (h 0 ¼ /2) Random [100] [110] [111] Random [100] [110] [111] Fe Al In this article, values of the exected quadratic nonlinearity arameter b 0 for a damage-free olycrystal that is macroscoically anisotroic have been derived. We consider the anisotroy to be of transversely isotroic symmetry. Both longitudinal and shear wave modes are redicted to generate second-harmonics. Because of the macroscoic anisotroy, both nonlinearity arameters b 0 ql and b0 qsv deend on the roagation direction with resect to the aligned symmetry axis. In order to arrive at estimates of b 0 ; the elastic roerties were homogenized using a Voigt-tye rocedure. The estimates for b 0 ql for the cases of roagation arallel and erendicular to the aligned symmetry axis were shown in Eq. (22) to be invertible and could be used to exerimentally measure the macroscoic anisotroy coefficients W 400 and W 600 : Having a damage-free estimate of b 0 allows measurement baselines to be established, which circumvents the need for calibration samles. Thus, this work could hel imrove techniques for in situ measures of the absolute nonlinearity arameter b for alications in materials characterization and nondestructive evaluation. 1 D. S. Hughes and J. L. Kelly, Second-order elastic deformation of solids, Phys. Rev. 92, (195). 2 R. H. Bergman and R. A. Shahbender, Effect of statically alied stresses on the velocity of roagation of ultrasonic waves, J. Al. Phys. 29, (1958). R. W. Bensen and V. J. Raelson, Acoustoelasticity A new technique, Prod. Eng. 0(29), (1959). 4 R. N. Thurston and K. Brugger, Third-order elastic constants and the velocity of small amlitude waves in homogeneously stressed media, Phys. Rev. 1, (1964). 5 Y.-H. Pao, W. Sachse, and H. Fukuoka, Acoustoelasticity and ultrasonic measurements of residual stress, in Physical Acoustics, edited by W. P. Mason and R. N. Thurston (Academic, New York, 1984), Vol. 17, C.-S. Man and W. Y. Lu, Towards an acoustoelastic theory for measurement of residual stress, J. Elast. 17, (1987). 7 R. Paroni and C.-S. Man, Two micromechanical models in acoustoelasticity: A comarative study, J. Elast. 59, (2000). 8 M. Huang, H. Zhan, X. Lin, and H. Tang, Constitutive relation of weakly anisotroic olycrystal with microstructure and initial stress, Acta. Mech. Sin. 2, (2007). 9 G. L. Jones and D. R. Kobett, Interaction of elastic waves in an isotroic solid, J. Acoust. Soc. Am. 5, 5 10 (196). 10 F. R. Rollins, Interaction of ultrasonic waves in solid media, Al. Phys. Lett. 2, (196). 11 A. J. Croxford, P. D. Wilson, B. W. Drinkwater, and P. B. Nagy, The use of non-collinear mixing for nonlinear ultrasonic detection of lasticity and fatigue, J. Acoust. Soc. Am. 126, EL117 EL122 (2009). 12 M. Liu, G. Tang, L. J. Jacobs, and J. Qu, Measuring acoustic nonlinearity arameter using collinear wave mixing, J. Al. Phys. 112, (2012). 1 M. A. Breazeale and D. O. Thomson, Finite-amlitude ultrasonic waves in aluminum, Al. Phys. Lett., (196). 14 M. A. Breazeale and M. A. Ford, Ultrasonic studies of the nonlinear behavior of solids, J. Al. Phys. 6, (1965). 15 A. C. Holt and J. Ford, Theory of ultrasonic ulse measurements of thirdorder elastic constants for cubic crystals, J. Al. Phys. 8, (1967). 16 D. C. Wallace, Thermoelastic theory of stressed crystals and higher-order elastic constants, in Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1970), Vol. 25, J. H. Cantrell, Crystalline structure and symmetry deendence of acoustic nonlinearity arameters, J. Al. Phys. 76, (1994). 18 J.-Y. Kim, J. Qu, L. J. Jacobs, J. W. Littles, and M. F. Savage, Acoustic nonlinearity arameter due to microlasticity, J. Nondestruct. Eval. 25, 28 6 (2006). 19 K. H. Matlack, J.-Y. Kim, L. J. Jacobs, and J. Qu, Review of second harmonic generation measurement techniques for material state determination in metals, J. Nondestruct. Eval. 4, 1 2 (2014). 20 J. H. Cantrell and W. T. Yost, Nonlinear ultrasonic characterization of fatigue microstructures, Int. J. Fatigue 2, S487 S490 (2001). 21 J. H. Cantrell, Ultrasonic harmonic generation from fatigue-induced dislocation substructures in lanar sli metals and assessment of remaining fatigue life, J. Al. Phys. 106, (2009). 22 Z. Chen and J. Qu, Dislocation-induced acoustic nonlinearity arameter in crystalline solids, J. Al. Phys. 114, (201). 2 J. H. Cantrell and W. T. Yost, Acoustic nonlinearity and cumulative lastic shear strain in cyclically loaded metals, J. Al. Phys. 11, (201). 24 S. Ahmed and R. B. Thomson, Proagation of elastic waves in equiaxed stainless steel olycrystals with aligned [001] axes, J. Acoust. Soc. Am. 99, (1996). 25 Indices take on the values 1, 2, and. Summation convention for each reeated index is assumed. 26 W. Voigt, Theoretische Studien uber die Elasticit atsverh altnisse der Krystalle ( Theoretical studies of the elastic behavior of crystals ), Abh. Kgl. Ges. Wiss. G ottingen. 4, 51 (1887). 27 A. N. Norris, Finite amlitude waves in solids, in Nonlinear Acoustics, edited by M. F. Hamilton and D. T. Blackstock (Acoustical Society of America, New York, 1997), L. K. Zarembo and V. A. Krasil nikov, Nonlinear henomenon in the roagation of elastic waves, Sov. Phys. Us. 1, (1971). 29 R. N. Thurston, Waves in solids, in Mechanics of Solids, edited by C. Truesdell (Sringer-Verlag, Berlin, 1974), Vol. 4, R. Chang, Relationshis between the nonlinear elastic constants of monocrystalline and olycrystalline solids of cubic symmetry, Al. Phys. Lett. 11, (1967). 1 H. J. Juretschke, Third-order elastic constants of olycrystalline media, Al. Phys. Lett. 12, (1968). 2 G. R. Barsch, Relation between third-order elastic constants of single crystals and olycrystals, J. Al. Phys. 9, (1968). D. J. Barnard, Variation of nonlinearity arameter at low fundamental amlitudes, Al. Phys. Lett. 74, (1999). 4 W. T. Yost and J. H. Cantrell, Anomalous nonlinearity arameters of solids at low acoustic drive amlitudes, Al. Phys. Lett. 94, (2009). 5 R.-J. Roe, Descrition of crystallite orientation in olycrystalline materials III. General solution to ole figure inversion, J. Al. Phys. 6, (1965). 6 G. C. Johnson, Acoustoelastic resonse of olycrystalline aggregate with orthotroic texture, J. Al. Mech. 52, (1985). 7 A. N. Norris, Symmetry conditions for third order elastic moduli and imlications in nonlinear wave theory, J. Elast. 25, (1991). 8 F. I. Fedorov, Theory of Elastic Waves in Crystals (Plenum, New York, 1968). 9 J. A. Turner, Elastic wave roagation and scattering in heterogeneous, anisotroic media: Textured olycrystalline materials, J. Acoust. Soc. Am. 106, (1999). 40 L. Thomsen, Weak elastic anisotroy, Geohysics 51, (1986). J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters 279 Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40

10 41 A. G. Every and A. K. McCurdy, Second and higher-order elastic constants, in Landolt-B ornstein Numerical Data and Functional Relationshis in Science and Technology New Series Grou III: Crystal and Solid State Physics, edited by O. Madelung and D. F. Nelson (Sringer-Verlag, Berlin, 1992), Vol. 29, The axial symmetry about ^z removes the / 0 deendence and allows / 0 to be arbitrarily chosen to generate the results. 4 R. Paroni, Otimal bounds on texture coefficients, J. Elast. 60, 19 4 (2000). 44 T. K. Ballabh, M. Paul, T. R. Middya, and A. N. Basu, Theoretical multile-scattering calculation of nonlinear elastic constants of disordered solids, Phys. Rev. B 45, (1992). 45 V. A. Lubarda, New estimates of the third-order elastic constants for isotroic aggregates of cubic crystals, J. Mech. Phys. Solids 45, (1997). 280 J. Acoust. Soc. Am., Vol. 17, No. 6, June 2015 C. M. Kube and J. A. Turner: Acoustic nonlinearity arameters Redistribution subject to ASA license or coyright; see htt://acousticalsociety.org/content/terms. Download to IP: On: Tue, 2 Jun :55:40