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Second-order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media
International Journal of Engineering Science 47 (2009) 499–511
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Second-order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media Addolorata Marasco Department of Mathematics and Applications, R. Caccippoli University of Naples Federico II, via Cintia, 80126 Naples, Italy
a r t i c l e
i n f o
Article history: Received 17 April 2008 Received in revised form 15 July 2008 Accepted 22 August 2008 Available online 11 October 2008
Keywords: Ordinary waves Nonlinear elastic incompressible media Perturbation method Second-order effects
a b s t r a c t In this paper we propose a perturbation method to investigate the propagation of the ordinary waves in second-order elastic, isotropic, incompressible, and homogeneous materials. This method allows us to determine the first-order terms of the speeds and the amplitudes both of the principal waves and the waves in any propagation direction, when the undisturbed region is subjected to an arbitrary isochoric deformation. Another application of this method is presented when the undisturbed region is subjected to a simple shear. Finally, some numerical results are presented in Mooney–Rivlin materials. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Owing to the important applications in geophysics, engineering, nondestructive testing of materials, electronic signal processing devices, etc., wave propagation in nonlinear elastic, isotropic, and homogeneous media has been investigated in many papers. However, general results in finite elasticity have been obtained only for principal waves, i.e., for the ordinary waves which propagate along the principal axes of strain (see [1,2] which contain detailed references to other contributions). These waves, when they exist, can be only longitudinal or transverse, and if the form of the stress relation is known, then the Ericksen’s formulae supply the unique value for their speeds. Conversely, if these speeds are known functions of the three principal stretches, for example by experiments, then the response coefficients are uniquely determined. All that supplies the first simple and definite connection between the existence of principal waves and physically reasonable static response of an elastic isotropic material in finite deformations (see [1, p. 278–294]).1 In spite of the deep relation existing between the wave propagation and the constitutive equations in nonlinear elasticity, it is not an easy task to deduce by experiments the constitutive equations of an isotropic elastic material. In fact, we have to determine, e.g. using particular static deformations or the wave propagation, functions depending on the principal invariants of the left Cauchy–Green tensor B. For these reasons, many authors studied the wave propagation in special classes of materials (hyperelastic Zahorski, Mooney–Rivlin, Blatz–Ko, neo-Hookean, St. Venant–Kirchhoff materials, etc.), which are described by simple constitutive relations, and for particular deformations (see [3–13]).2 Alternatively, for arbitrary sufficiently small deformations, it is possible to analyze the above problem in second-order elasticity in which a second-order
E-mail address: [email protected] In detail, Ericksen proved that some constitutive inequalities upon the response functions of isotropic elastic materials, are necessary and sufficient conditions so that the speeds of all principal waves be real. 2 In many other papers, for instance in [14–17], the elastic waves are studied by solving analytically or numerically the elastic equations of motion in particular situations. 1
0020-7225/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2008.08.009
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expansion of a constitutive relation is considered and all terms of third or higher order in the displacement gradient are systematically neglected in all the involved equations (as well as in Signorini’s and Green and Spatt’s perturbation methods). In this last case, the costitutive equations are determined by only a few material constants (five for hyperelastic compressible materials, two for incompressible bodies). On the other hand, the second-order constitutive relations, usually proposed in the context of isotropic materials (for example, see [18–24]) provided always a good description of the mechanical response of an elastic body for sufficiently small deformations (see [25]). Moreover, many experimental phenomena as Kelvin and Poynting effects, representing, respectively, density changes under imposed shear or longitudinal extension in an elastic solid, which are characteristic of the finite elasticity, occur already in second-order elasticity (see [26–29]). Likewise, also in the second-order elasticity, the speeds of the waves depend both on the direction of the propagations and on the deformations (see [30,31] and the results in these papers). As a consequence of the above considerations, in the papers [30,31] the propagation of the acceleration waves in isotropic, compressible, and homogeneous materials is carried out in the contest of second-order elasticity. In particular, in [30], we propose a Signorini’s perturbation method which allows us to determine the first-order terms of the speeds as well as the amplitude of the waves in any arbitrary direction of propagation.3 This procedure reduces the eigenvalue problem for the acoustic tensor to two simpler problems: the first coincides with the eigenvalue problem for the acoustic tensor in linear elasticity, whereas the second supplies the corresponding corrective first-order terms of the velocities and of the waves amplitudes. Moreover, the analysis of the wave propagation along any direction, is carried out when the undisturbed region is subjected to a simple extension or to a simple shear. More generally, in [31], we determine the first-order terms of the speeds and the amplitudes of the waves in any propagation direction, when the undisturbed region is subjected to an arbitrary deformation. The method proposed in [30] is devoted only to compressible engineering materials, e.g., steel or aluminum. Nevertheless, since the incompressible materials are used extensively in cars, aircrafts, buildings, bridges, railroads, medical equipments, and biomedical engineering, the wave propagation analysis have also been carried out on incompressible bodies (see [3–9]). For incompressible materials the presence of an arbitrary hydrostatic pressure in the stress relations push us to modify the method presented in [30]. In particular, in this paper, we propose a generalized Green and Spratt’s perturbation method in order to investigate the propagation of the (necessarily transverse) waves in second-order elastic, isotropic, incompressible, and homogeneous materials (see [33]). Similarly to [30], the proposed method allows us again to reduce the eigenvalue problem for the acoustic tensor, equipped with the orthogonality conditions between the amplitudes and the propagation directions, to two simpler problems: the first coincides with the eigenvalue problem for the acoustic tensor in linear elasticity, whereas the second supplies the first-order terms to add to the speeds and the amplitudes of the linear elasticity. The paper is organized into seven sections including the introduction. In Section 2 we recall some general results about the theory of waves in isotropic, homogeneous, elastic, and incompressible media. Moreover, to formulate the eigenvalue problem for the acoustic tensor in the reference configuration, we recall the relations among the speeds, the amplitudes, and the directions of propagation of the waves in the actual configuration and in the reference configuration. The Section 3 is devoted to the description of the perturbation method which allows us to determine the explicit form of the first two dynamic compatibility equations for the transverse waves, and the first-order relations among some quantities in the reference configuration and the corresponding ones in the actual configuration. In Section 4 we calculate the first-order terms of the speeds and the amplitudes of the principal transverse waves in an eigenvector basis of the tensor B. In particular, we verify that the speeds coincide with those evaluated by Truesdell in [2]. In Section 5 we determine the speeds and the amplitudes of the waves in any direction of propagation, when the undisturbed region is subjected to an arbitrary isochoric deformation. The analysis of the principal transverse waves and of their propagation speeds, when the undisturbed region is subjected to a simple shear, is carried out in Section 6. Finally, in Section 7 some numerical experiments lead us to evaluate the second-order effects on the speeds of the waves in some conventional Mooney–Rivlin incompressible materials. 2. Ordinary waves in incompressible elastic media In this section we expose briefly the theory of the ordinary waves in incompressible elastic materials and we resume the main results on the principal transverse wave propagation. Let S be an elastic, homogeneous, incompressible, and isotropic continuous system in the reference configuration C . In the sequel, we neglect all the thermal phenomena associated with the evolution of S. If X is any point of C and x is the corresponding position in the actual configuration CðtÞ, then the finite deformation of S in going from C to CðtÞ can be equivalently expressed by the finite deformation x ¼ xðX; tÞ or by the displacement u ¼ uðX; tÞ. It is well known that for incompressible materials, in which only isochoric deformations are possible, the constitutive equation of Cauchy stress tensor T has the form
e T ¼ pI þ f 1 B þ f 1 B1 pI þ T;
ð1Þ
e depends on the deformation gradient F ¼ r x, B ¼ FF is the left Cauchy–Green tensor, f 1 ; f 1 are functions of the two where T principal invariants IB and IIB of B, and p is an indeterminate pressure (see [2]). T
3 In [32] we proved that the five constitutive constants of a second-order isotropic, compressible, and homogeneous material, can be determined by simple experiments based on equilibrium problems in second-order elasticity.
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The local balance of momentum is given by the equation
q x€ ¼ rp þ r Te þ q b in CðtÞ;
ð2Þ
where q q is the mass density of S, since J ¼ det F ¼ 1, and b is the body force density. Let RðtÞ be a moving surface in the actual configuration CðtÞ of equation gðx; tÞ ¼ GðXðx; tÞ; tÞ ¼ 0. If the solution xðX; tÞ of the second-order equation (2) exhibits a discontinuity in some or all of its second-order derivatives across the surface RðtÞ, xðX; tÞ is said to represent an ordinary wave and RðtÞ is the wavefront which divides the region CðtÞ into the perturbed and undisturbed regions C ðtÞ and C þ ðtÞ, respectively. Moreover, the vector a, characterizing the strength of the discontinuity the normal speed of the second derivatives of xðX; tÞ across RðtÞ, is called amplitude of the singularity. We denote by cN ¼ og=ot jrgj g of propagation of RðtÞ, and by U N ¼ cN x_ N the local speed of the wavefront. Finally, if a is parallel to the unit normal N ¼ jr rgj to the surface RðtÞ, then the wave is said to be longitudinal; if it is normal, transverse. We recall that the constraint of incompressibility renders longitudinal waves impossible. Then, in an incompressible material, all the singular surfaces are necessarily transversal: a N ¼ 0. If Eq. (2) is written in the regions C ðtÞ and C þ ðtÞ, its limits for x towards a point r 2 RðtÞ are considered and the obtained results are subtracted, then we obtain the jump system associated with (2)
oT~ ij oHlM op € q sxi tr ¼ þ ; oxi r oHlM oxj r
i ¼ 1; 2; 3:
ð3Þ
r
By recalling the kinematic relations of second-order singular surfaces (see [34, p. 119])
s€xi tr ¼ U 2N ai ;
oHlM ¼ F hM Nj Nh al oxj r
ð4Þ
and that across such a surface the jump of the first derivatives of the pressure is
op ¼ AN i ; oxi r the jump system (3) becomes
e il al ; i ¼ 1; 2; 3; q U 2N ai ¼ ANi þ Q
ð5Þ
where
e e il ¼ o T ij F hM Nj Nh : Q oHlM Taking the scalar product of (5) by N and recalling that a N ¼ 0, we have
e jl al : A ¼ Nj Q Finally, we obtain the following equation for the amplitudes and speeds
ðQil q U 2N dil Þal ¼ 0;
ð6Þ
e jl e il Ni Nj Q Qil ¼ Q
ð7Þ
where
is the acoustic tensor. The algebraic conditions (6) express the well-known Hadamard’s result: Given an undisturbed state xþ ðX; tÞ towards which the ordinary wave RðtÞ propagates, then due to the continuity of xðX; tÞ across RðtÞ, the matrix Q is a known function of t and r 2 RðtÞ. Furthermore, given a propagation direction N, the speeds of propagation U N are such that q U 2N are the eigenvalues of the acoustic tensor and the discontinuity vectors a are its eigenvectors. For the isotropic incompressible materials, the acoustical axes for principal waves are principal axes of strain. Moreover, the principal waves are necessarily transverse, and the classical compatibility conditions of this waves hold only for stretches for which det B ¼ 1.4 Ericksen’s formulae on the speeds of propagation of the transverse principal waves are expressed in terms of the functions f 1 ; f 1 and their derivatives (see [2, p. 293]). However, since it is practically impossible to determine experimentally these functions, in [2] Truesdell finds the first terms in the expansions of the exact formulae of the local speeds only of transverse principal waves in a basis of eigenvectors of B (see [2, p. 293]). Therefore, in the above results and in the first-order approximation, there is no information about the speeds of propagation along directions which are not principal axes of strain. In the next section we propose a perturbation method, similar to the method exposed in [30], to overcome these difficulties, at least in second-order elasticity. This approach allows us to obtain the first-order speeds along any direction and for all 4 In [2], the authors derive a sequence of universal relations connecting the principal wave speeds in a basis of eigenvectors of B for any isotropic elastic material (compatibility conditions).
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
‘‘sufficiently small” deformations, leading us to the same results which Truesdell obtained for the first-order speeds of propagations of the transverse principal waves. To this end, it is useful to recall the following relations:
jrgj jr Gj 1 U N CU N ; U N ¼ U N U N ; jr Gj jrgj C 1 ai ¼ 2 ai ; ai ¼ C2 ai ; U N ¼
C
ð8Þ ð9Þ
where the vectors U N and ai are the local speed and the amplitude of the singularity of the surface R ðtÞ in the reference G configuration C . We also recall that the relations between the two unit normal vectors N and N ¼ jr r Gj are expressed by the formulae
Ni ¼
1
C
F 1 Li N L ;
NL ¼ CF iL Ni ;
ð10Þ
where, since N i N i ¼ 1, it is
C¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 F 1 Li F Mi N L N M :
Owing to (8)–(10), instead of analyzing (6), it is more convenient to study the following equation in the reference configuration C
ðQ il q U 2N dil Þal ¼ 0; where Q il
i ¼ 1; 2; 3;
ð11Þ
e il Q ^ il , and ¼Q
e e il ¼ o T ij F 1 NL NM ; Q oHlM Lj
b il ¼ 1 F 1 F 1 NL NM Q e jl : Q Mj 2 Li
C
3. Second-order effects in incompressible isotropic materials The Cauchy stress tensor (1) of an elastic, isotropic, incompressible, and homogeneous body, up to second-order terms, has the form (see [2, p. 241])
TðHÞ ¼ pI þ 2lE þ lHHT þ b1 E2 ;
ð12Þ
T
where E ¼ þ H Þ is the infinitesimal strain tensor, l is a Lame’s coefficient, and b1 is a second-order constitutive constant. Taking in the account the following derivation formulae (see [30]): 1 ðH 2
oEiL 1 ¼ ðdij dLM þ dLj diM Þ; oHjM 2 oðHLa Hai Þ ¼ HMi dLj þ HLj diM ; oHjM oðEia EaL Þ 1 ¼ ðEML dij þ EjL diM þ Eij dLM þ EiM dLj Þ oHjM 2 and that up to first-order terms we have
F1 ¼ I H þ Oð1Þ;
1
C2
1 ¼ F 1 hi F ki N h N k ¼ 1 2Hhi N h N i þ Oð1Þ
by simple but tedious calculations we obtain the first-order form of the tensor Q
1 1 1 Q ij ðH; N Þ ¼ lðNi Nj dij Þ þ b1 Eij þ ½ð4l þ b1 ÞEih þ 2lHhi Nj Nh ðb1 Ejh 2lHhj ÞN i Nh 2 2 2 1 Hhk ½2Ni Nj ð4l þ b1 Þ b1 dij Nh Nk : 2
ð13Þ
In order to reduce the starting problem (11), in which we have introduced (13), to a family of more simple problems, we suppose that a suitable nondimensional analysis of (2) leads us to introduce a small parameter depending on the acting forces, the material characteristic, and geometry of C (see [35]). Moreover, according to the Green and Spratt’s method [33], we suppose that the displacement u and the pressure p can be written in the following form (see [32,30])
u ¼ uð1Þ þ Oð1Þ; so that we have also
p ¼ pð1Þ þ Oð1Þ;
ð14Þ
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
H ¼ Hð1Þ þ Oð1Þ;
ð15Þ
ð1Þ
ð1Þ
where H is the nondimensional displacement gradient relative to u . To apply a perturbation method to the dynamic compatibility conditions (11), we assume that the quantities Q ij ðHðÞ; N ðÞÞ; aj ðÞ; kðÞ, and N ðÞ are analytic functions of the small parameter , so that, up to first-order terms, we obtain
k ¼ kð0Þ þ kð1Þ ;
ð1Þ a ¼ að0Þ þ a ;
ð1Þ N ðÞ ¼ Nð0Þ þ N :
ð16Þ
Moreover, in the same approximation, we have ð0Þ ð0Þ C ¼ 1 Eð1Þ ij N i N j
and the relations (8) and ð10Þ2 become ð1Þ
ð0Þ
ð0Þ
ð1Þ
U N ¼ ð1 Eij Ni Nj ÞU N ; NL ¼
ð0Þ NL
þ
ð1Þ ðNL
þ
ð0Þ
ð0Þ
U N ¼ ð1 þ Eij Ni Nj ÞU N ;
ð1Þ ð0Þ HiL Ni
ð17Þ
ð1Þ ð0Þ ð0Þ ð0Þ Ehk Nh Nk NL Þ:
ð18Þ
We recall that the vectors NðÞ and N ðÞ are unitary, this means that in our approximation result N ð0Þ ð0Þ ð1Þ and similarly Nð0Þ N ¼ 1; N N ¼ 0. Owing to these results, (13) assumes the following form: ð0Þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð0Þ
Q ij ðN ðÞ; 0Þ ¼ lðNi Nj dij Þ lðNi Nj þ Ni N j Þ þ
ð0Þ
N
ð0Þ
ð0Þ
¼ 1; N
N
ð1Þ
¼ 0,
1 1 ð1Þ ð0Þ ð0Þ b Eð1Þ þ ½ð4l þ b1 ÞEð1Þ ih þ 2lH hi N j N h 2 1 ij 2
1 1 ð1Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ðb1 Eð1Þ jh 2lH hj ÞN i N h H hk ½2N i N j ð4l þ b1 Þ b1 dij N h N k 2 2 ð0Þ ð1Þ Q ij þ Q ij :
ð19Þ
Since the waves are necessarily transverse, in view of (9) and ð10Þ1 , the condition a N ¼ 0 becomes
ai F 1 Li N L ¼ 0; so that in the first-order approximation we have
(
ð0Þ að0Þ N ¼ 0;
ð20Þ
ð0Þ ð1Þ ð0Þ ð1Þ ð0Þ að1Þ N þ a ðN N H Þ ¼ 0:
Finally, when we impose the conditions (20) and we neglect terms of higher order than one, from the dynamic compatibility Eq. (11), we derive the following systems: ð0Þ
ðl kð0Þ Þai ¼ 0; ð0Þ
ðl k
ð1Þ Þai
ð0Þ kð1Þ ai
ð21Þ 1 1 ð0Þ ð0Þ ð1Þ ð0Þ ð0Þ ð1Þ ð1Þ ð0Þ ð0Þ þ b1 ai Hhk Nh Nk þ aj b1 ðEij þ Ejh Ni Nh Þ ¼ 0: 2 2
ð22Þ
We note that when we associate the condition ð20Þ1 to the system (21), we obtain the usual eigenvalue equations of the linear elasticity. Consequently, from (21), we have that l=q is the square of the propagation speed, and condition ð20Þ1 supð0Þ with l and we consider only the first-order plies the amplitude of the discontinuity að0Þ . Similarly, when we identify k ð1Þ isochoric deformations, i.e. IHð1Þ ¼ 0, the Eqs. (22) and ð20Þ2 in the four unknowns aj , j ¼ 1; 2; 3, and kð1Þ , determine, at least ð1Þ ð1Þ in principle, the direction of a and the value of k . We recall that in nonlinear elasticity the tensor Q could be not symmetric for any direction and for any deformation. In particular, in our approximation it results ð1Þ
ð1Þ
ð0Þ
ð0Þ
ð1Þ
ð0Þ2
ð0Þ2
ð1Þ
ð0Þ
ð0Þ
ð1Þ
ð0Þ
ð0Þ
Q 12 Q 21 ¼ ð2l þ b1 Þ½ðE11 E22 ÞN1 N2 E12 ðN1 N2 Þ þ ð2l þ b1 Þ½E13 N2 N3 E23 N1 N3 ; Q 13 Q 31 ¼ ð2l þ Q 23 Q 32 ¼ ð2l þ
ð1Þ b1 Þ½ðE11 ð1Þ b1 Þ½ðE22
ð1Þ ð0Þ ð0Þ E33 ÞN1 N3 ð1Þ ð0Þ ð0Þ E33 ÞN2 N3
þ þ
ð1Þ ð0Þ ð0Þ E12 N2 N3 ð1Þ ð0Þ ð0Þ E12 N1 N3
ð2l þ ð2l þ
ð0Þ2 ð0Þ2 ð1Þ ð1Þ ð0Þ ð0Þ b1 Þ½ðN1 N 3 ÞE13 þ E23 N1 N2 ; ð1Þ ð0Þ ð0Þ ð0Þ2 ð0Þ2 ð1Þ b1 Þ½E13 N1 N2 þ ðN2 N 3 ÞE23 ;
ð23Þ ð24Þ ð25Þ
where the components of the infinitesimal strain tensor must satisfy the condition IEð1Þ ¼ 0. Moreover, as in finite elasticity, þ að1;hÞ the eigenvector of the tensor Q the acoustic tensor (19) is singular. When Q is symmetric and we denote with að0;hÞ ð0;hÞ ð1;hÞ belonging to the eigenvalue k þ k , then we have to satisfy the following orthogonality conditions
ðað0;hÞ þ að1;hÞ Þ ðað0;kÞ þ að1;kÞ Þ ¼ 0;
h–k;
which in turn are equivalent to the following others
(
að0;hÞ að0;kÞ ¼ 0; að1;kÞ þ að1;hÞ að0;kÞ ¼ 0: að0;hÞ
h–k:
ð26Þ
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
4. First-order local speeds of principal waves In this section, using (21) and (22), we analyze the transverse principal waves and we prove that the velocities of these waves coincide with the speeds obtained by Truesdell in [2] with a different approach.5 Let K1 ; K2 ; K3 be the eigenvalues of the left Cauchy–Green tensor B: Noting that for isochoric deformations det B ¼ 1, we have K1 K2 K3 ¼ 1. In the corresponding basis of eigenvectors B ¼ he1 ; e2 ; e3 i, the tensor B is diagonal and the directions of the principal axes of strain are
N1 ð1; 0; 0Þ;
N2 ð0; 1; 0Þ;
N3 ð0; 0; 1Þ:
ð27Þ
In this new basis B, up to first-order terms, Ki ¼ 1 þ K
0 B Eð1Þ ¼ B @
K
ð1Þ 1 =2
0
0
0
K2ð1Þ =2
0
0
0
Kð1Þ 3 =2
1
0
C C; A
B Hð1Þ ¼ B @
ð1Þ i ;i
ð1Þ 1 =2 ð1Þ H12 ð1Þ H13
ð1Þ
¼ 1; 2; 3, and, since B ¼ I þ 2E , we have ð1Þ
H12
K
K2ð1Þ =2 ð1Þ
H23
ð1Þ
H13
1
C ð1Þ H23 C A; ð1Þ K3 =2
where, since IHð1Þ ¼ 0, result K1 þ K2 þ K3 ¼ 0. Owing to (18), the corresponding directions of (27) are ð1Þ
ð1Þ
N1 ð1; 0; 0Þ þ ðH12 ; H13 ; 0Þ; N2 ð0; 1; 0Þ þ N3 ð0; 0; 1Þ þ
ð28Þ
ð1Þ ð1Þ ðH12 ; 0; H23 Þ; ð1Þ ð1Þ ðH13 ; H23 ; 0Þ:
In the sequel we analyze the wave propagation along each of the above directions. ð1Þ
ð1Þ
4.1. Waves propagation along N1 ðÞ ð1 þ H12 ; H13 ; 0Þ Along this vector the tensor Q writes
0
0 B0 Q ¼ @
0 ð1Þ ð1Þ l þ 12 b1 ðE11 þ E22 Þ
0 ð1Þ E11
ð1Þ E22
1
0 0 ð1Þ b1 ðE11
lþ 1 2
0
þ
ð1Þ E33 Þ
C A;
ð1Þ E33
where þ þ ¼ 0. When we impose the conditions (20) and (26), the Eqs. (21) and (22) lead to the following results6:
b1 ð1Þ b ð1Þ ð1Þ ðE þ E33 Þ ¼ l 1 E22 ; 2 11 2 b1 ð1Þ b1 ð1Þ ð1Þ k2 ¼ l þ ðE11 þ E22 Þ ¼ l E33 ; 2 2
ð1Þ
k1 ¼ l þ
a1 ð0; 0; 1Þ þ ð0; a2 ; 0Þ; ð1Þ
a2 ð0; 1; 0Þ þ ð0; 0; a2 Þ:
Therefore, the transverse waves propagate with the speeds
q U 21 ¼ l
b1 ð1Þ E ; 2 22
q U 22 ¼ l
b1 ð1Þ E ; 2 33
or equivalently, by using ð17Þ2 , we obtain
q U 21 ¼ l þ 2lEð1Þ 11
b1 ð1Þ E22 ; 2
q U 22 ¼ l þ 2lEð1Þ 11
b1 ð1Þ E33 : 2
ð29Þ
In particular, relation ð29Þ2 coincides with the formula (78.15) p. 293 in [2] provided that we identify the quantities ri with ð1Þ Eii and we recall that IEð1Þ ¼ 0. ð1Þ
ð1Þ
4.2. Waves propagation along N2 ðÞ ðH12 ; 1; H23 Þ Along this direction the tensor Q has the form
0 B Q ¼ @
ð1Þ l þ 12 b1 ðEð1Þ 0 11 þ E22 Þ
0 0
ð1Þ
ð1Þ
1
0
0
0
0
ð1Þ b1 ðE22
l þ 12
þ
ð1Þ E33 Þ
C A;
ð1Þ
where E11 þ E22 þ E33 ¼ 0. 5 6
This can be seen provided that we consider the relation b1 ¼ 4lðb 12Þ between the constitutive constant b1 to the constant b used in [2]. ð1Þ ð1Þ ð1Þ When E22 ¼ E33 , we obtain k ¼ l b21 E22 ; a ð0; a2 ; a3 Þ.
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
505
When we impose the conditions (20) and (26), Eqs. (21) and (22) supply the following results: ð1Þ
ð1Þ
If E11 –E33 , then
b1 ð1Þ b ð1Þ ð1Þ ðE þ E33 Þ ¼ l 1 E11 ; 2 22 2 b1 ð1Þ b1 ð1Þ ð1Þ k2 ¼ l þ ðE11 þ E22 Þ ¼ l E33 ; 2 2
ð1Þ
k1 ¼ l þ
a1 ð0; 0; 1Þ þ ða1 ; 0; 0Þ; ð1Þ
a2 ð1; 0; 0Þ þ ð0; 0; a1 Þ:
Owing to ð17Þ2 , the waves propagate in CðtÞ with the following speeds:
q U 21 ¼ l þ 2lEð1Þ 22 ð1Þ
b1 ð1Þ E11 ; 2
q U 22 ¼ l þ 2lEð1Þ 22
b1 ð1Þ E33 : 2
ð30Þ
ð1Þ
If E11 ¼ E33 , then we obtain
k3 ¼ l
b1 ð1Þ E ; 2 33
ð0Þ
ð0Þ
ð1Þ
ð1Þ
a3 ða1 ; 0; a3 Þ þ ða1 ; 0; a3 Þ
and, finally
q U 23 ¼ l þ 2lEð1Þ 22
b1 ð1Þ E11 : 2
ð31Þ
ð1Þ
ð1Þ
4.3. Waves propagation along N3 ðÞ ðH13 ; H23 ; 1Þ Along this direction the tensor Q writes
0 B Q ¼ @
ð1Þ l þ 12 b1 ðEð1Þ 11 þ E33 Þ
0
ð1Þ E22
1 C
ð1Þ l þ 12 b1 ðEð1Þ 0 A; 22 þ E33 Þ
0 0
ð1Þ E11
0
0
0
ð1Þ E33
where þ þ ¼ 0. When we impose the conditions (20) and (26), Eqs. (21) and (22) lead to the following results7: ð1Þ
ð1Þ
If E11 –E22 , then
b1 ð1Þ b ð1Þ ð1Þ ðE þ E33 Þ ¼ l 1 E11 ; 2 22 2 b1 ð1Þ b1 ð1Þ ð1Þ k2 ¼ l þ ðE11 þ E33 Þ ¼ l E22 ; 2 2
ð1Þ
k1 ¼ l þ
by using ð17Þ2 , result
q U 21 ¼ l þ 2lEð1Þ 33 ð1Þ
b1 ð1Þ E11 ; 2
a1 ð0; 1; 0Þ þ ða1 ; 0; 0Þ; ð1Þ
a2 ð1; 0; 0Þ þ ð0; a1 ; 0Þ
q U 22 ¼ l þ 2lEð1Þ 33
b1 ð1Þ E22 : 2
ð32Þ
ð1Þ
If E11 ¼ E22 , then we obtain
k3 ¼ l
b1 ð1Þ E ; 2 22
q U 23 ¼ l þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
a3 ða1 ; a2 ; 0Þ þ ða1 ; a2 ; 0Þ; b ð1Þ ð1Þ 2lE33 1 E11 : 2
ð33Þ
5. First-order transverse waves along an arbitrary direction N In this section, we calculate the first-order speeds of propagation in any direction, which is not a principal direction, and for sufficiently small deformations. Without loss of generality, we choose in C the unit vector N ð1; 0; 0Þ. To this vector, according to ð18Þ2 , corresponds the ð1Þ ð1Þ unit vector N ð1; H12 ; H13 Þ in CðtÞ. We note that N is a principal direction of strain if and only if ð1Þ
E12 ¼ 0; 7
ð1Þ
ð1Þ
ð1Þ
E13 ¼ 0:
ð34Þ ð1Þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð0Þ
ð1Þ
ð1Þ
When E11 ¼ E22 , we obtain k ¼ l b21 E22 ; a ða1 ; a2 ; 0Þ þ ða1 ; a2 ; a1 ðH13 þ E33 ÞÞ.
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
Then, in order not to fall in the same situation we have already considered in the above section, we suppose that conditions (34) are not verified. Along the direction N , the relations (23)–(25) become ð1Þ
Q 12 Q 21 ¼ ð2l þ b1 ÞE12 ; ð1Þ
Q 13 Q 31 ¼ ð2l þ b1 ÞE13 ; Q 23 Q 32 ¼ 0; so that the tensor Q is symmetric for any deformation if and only if 2l þ b1 ¼ 0. For any other second-order elastic material, Q is symmetric only if the deformation verifies (34) that, as we have already remarked, are equivalent to the request that N is an eigenvector of B. Imposing the conditions (20) and IEð1Þ ¼ 0, the Eqs. (21) and (22) supply the following results: ð1Þ
ð1Þ
ð1Þ
If E23 ¼ 0; E22 ¼ E33 , then
q U 2 ¼ l
b1 ð1Þ E ; 2 22
ð0Þ
ð0Þ
ð0Þ
ð1Þ
ð0Þ
ð1Þ
ð1Þ
ð1Þ
a ð0; a2 ; a3 Þ þ ða2 H12 þ a3 H13 ; a2 ; a3 Þ;
and, by using ð17Þ2 , we obtain
q U 2 ¼ l þ 2lEð1Þ 11 ð1Þ
ð1Þ
b1 ð1Þ E22 : 2
ð35Þ
ð1Þ
If E23 ¼ 0; E22 –E33 , then
b1 ð1Þ E ; 2 22 b ð1Þ ¼ l 1 E33 ; 2
ð1Þ
ð1Þ
q U 21 ¼ l
a1 ð0; 0; 1Þ þ ðH13 ; a2 ; 0Þ;
q U 22
a2 ð0; 1; 0Þ þ ðH12 ; 0; a3 Þ
ð1Þ
ð1Þ
and the waves propagate in CðtÞ with the speeds
ð1Þ
ð1Þ
b1 ð1Þ E Þ; 2 33
ð1Þ q U 21 ¼ l þ ð2lE11
q U 22 ¼ l þ 2lEð1Þ 11
b1 ð1Þ E22 : 2
ð36Þ
ð1Þ
If E23 –0; E22 –E33 , then
q U 2 ¼ l þ ð0Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 ð1Þ ð1Þ ð1Þ ð1Þ E11 4ðE23 Þ2 þ ðE22 E33 Þ2 ; 4
ð0Þ
ð0Þ
ð1Þ
ð0Þ
ð1Þ
ð1Þ
ð1Þ
a ð0; a2 ; a3 Þ þ ða2 H12 þ a3 H13 ; a2 ; a3 Þ; ð0Þ
ð0Þ
where a3 –0, and a2 ¼
ð0Þ a3 ð1Þ 2E23
q U 2 ¼ l þ 2lEð1Þ 11 þ ð1Þ
ð1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ½E22 E33 4ðE23 Þ2 þ ðE22 E33 Þ2 . Then, owing to ð17Þ2 , it results
b1 ð1Þ ðE 2 33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ 4ðE23 Þ2 þ ðE22 E33 Þ2 Þ :
ð37Þ
ð1Þ
If E23 –0; E22 ¼ E33 , then
b1 ð1Þ ð1Þ ðE E23 Þ; 2 22 ð0Þ ð0Þ ð0Þ ð1Þ ð1Þ ð1Þ ð1Þ a ð0; a2 ; a2 Þ þ ða2 ðH12 H13 Þ; a2 ; a2 Þ
q U 2 ¼ l
and, finally
q U 2 ¼ l þ ½2lEð1Þ 11
b1 ð1Þ ð1Þ ðE E23 Þ: 2 22
ð38Þ
The above results, show that, for any fixed direction of propagation, the speeds of the waves depend stongly on the deformations. 6. Simple shear In this section we analyze the finite isochoric deformation
x1 ¼ X 1 þ X 2 tan h;
x2 ¼ X 2 ;
x3 ¼ X 3 ;
ð39Þ
where h is the constant angle of shear. The particles move only in the X 1 -direction and their displacement is proportional to X 2 -coordinate. Planes, which are parallel to X 1 ¼ 0, are rotated about an axis parallel to X 3 -axis through the angle h. The deformation gradient and the left Cauchy–Green tensor are
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
0
1 1 k 0 B C F ¼ @ 0 1 0 A; 0 0 1
0
2
1þk B B¼@ k 0
507
1 k 0 C 1 0 A; 0 1
where k ¼ tan h, and the displacement gradient is
0
0 k
0
1
B C H ¼ @ 0 0 0 A: 0 0 0 Then, when we identify h with the small parameter , we have that
0
H
ð1Þ
1 0 1 0 B C ¼ @ 0 0 0 A; 0 0 0
0
1 0 1 0 B C B ¼ I þ @ 1 0 0 A þ Oð1Þ: 0 0 0
The unit directions of the principal axes of strain are
N1 ðÞ ð0; 0; 1Þ; 1 1 1 1 N2 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 ; 2 2 2 2 1 1 1 1 N3 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 2 2 2 2
ð40Þ
and owing to (18), the directions which correspond in C to these axes are
N1 ðÞ ð0; 0; 1Þ; 1 1 1 1 N2 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 ; 2 2 2 2 1 1 1 1 N3 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 : 2 2 2 2
ð41Þ
Along the direction N1 ðÞ ð0; 0; 1Þ, the tensor Q is symmetric. Then, Eqs. (21) and (22), when we impose the conditions (20) and (26), lead to the following results8:
b1 ; 4 b q U 2 q U 2 ¼ l 1 ; 4
q U 2 q U 2 ¼ l þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
a1 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; 0Þ; ð0Þ
ð0Þ
ð1Þ
ð42Þ
ð1Þ
a2 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; 0Þ:
pffiffiffi pffiffiffi pffiffiffi pffiffiffi Along the vector N2 ðÞ ð1= 2; 1= 2; 0Þ þ ð1= 2; 1= 2; 0Þ, the tensor Q is symmetric. Solving the Eqs. (21) and (22), in view of (20) and (26), we obtain the following results:
b1 ð0Þ ð1Þ ð1Þ ; a1 ð0; 0; a3 Þ þ ða1 ; a1 ; 0Þ; 4 ð0Þ ð0Þ ð0Þ ð0Þ ð1Þ a2 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; a3 Þ
k1 ¼ l k2 ¼ l;
and, owing to ð17Þ2 , in the actual configuration CðtÞ the waves propagate with the speeds
1 4
q U 21 ¼ l l; q U 22 ¼ l ð4l þ b1 Þ:
ð43Þ
Along the last direction N3 ðÞ ðp1ffiffi2 ; p1ffiffi2 ; 0Þ þ ð p1ffiffi2 ; p1ffiffi2 ; 0Þ, the tensor Q is symmetric. Solving (21) and (22), in view of (20), (26), and ð17Þ2 , we obtain the following results:
b1 ð0Þ ð1Þ ð1Þ ; a1 ð0; 0; a3 Þ þ ða1 ; a1 ; 0Þ; 4 ð0Þ ð0Þ ð0Þ ð0Þ ð1Þ k2 ¼ l; a2 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; a3 Þ; 1 q U 22 ¼ l ð4l b1 Þ; q U 21 ¼ l l: 4 k1 ¼ l þ
8
For the sake of simplicity, from now on we take unit amplitudes.
ð44Þ
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
7. Some numerical results In this section, we evaluate the first-order speeds of the waves determined in Sections 2–4 for arbitrary small deformapffiffiffiffiffiffiffiffiffiffiffi tions, and compare them with the constant value l=q of the speeds in incompressible linear elasticity. In particular, we consider some incompressible materials for which the constitutive equations are described by the traditional Mooney–Rivlin strain–energy function
W ¼ c1 ðIB 3Þ þ c2 ðIIB 3Þ;
ð45Þ
where, in our notations,
c1 ¼
1 1 ð4l þ b1 Þ; c2 ¼ b1 : 8 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
If we denote by
U 2 =q the first-order speeds of the waves determined in the previous sections, the first-order terms
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
ð46Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi of the quantity ½ð U 2 =q l=q Þ= l=q 100% measure the nonlinear effects on the wave propagation respect to the linear theory of the elasticity. In the sequel, we evaluate the percentage effects (46) for some incompressible materials for which the constants c1 and c2 in (45) are been experimentally calculated in the papers [36–38] (by acoustoelastic analysis, indentation tests, shrinkage test, and hot tensile test). In order to consider only arbitrary small deformations, owing to nondimensional analysis of the Section 2, it is sufficient ð1Þ to choose deformations for which Ei;j ’ 1, for all i and j, whereas the small parameter 1. Table 1 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [36, p. 847]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ c1 c2 E11 E22 E33 ðU 2 lÞ=2l 100% ð1Þ
ð1Þ
Principal waves along the direction N 1 ðÞ ð1 þ H12 ; H13 ; 0Þ 0.86 0.155 0.1 1 0.86 0.155 0.04 1 ð1Þ
0.55 0.55
0.45 0.45
Eq. (29) 11.21, 10.99 4.48, 4.39 Eq. (30)
Eq. (31)
7.7, 4.51 3.08, 1.8 – –
– – 11.1 4.44
Eq. (32)
Eq. (33)
6.7, 3.3 2.68, 1.32 – –
– – 11.1 4.44
ð1Þ
Principal waves along the direction N2 ðÞ ðH12 ; 1; H23 Þ 0.86 0.155 0.1 1 0.86 0.155 0.04 1 0.86 0.155 0.1 0.5 0.86 0.155 0.04 0.5 ð1Þ
0.55 0.55 1 1
0.45 0.45 0.5 0.5
ð1Þ
Principal waves along the direction N3 ðÞ ðH13 ; H23 ; 1Þ 0.86 0.155 0.1 1 0.86 0.155 0.04 1 0.86 0.155 0.1 0.5 0.86 0.155 0.04 0.5
0.55 0.55 0.5 0.5
0.45 0.45 1 1
Table 2 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [38, p. 1204]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ c1 c2 E11 E22 E33 ðU 2 lÞ=2l 100% ð1Þ
ð1Þ
Principal waves along the direction N 1 ðÞ ð1 þ H12 ; H13 ; 0Þ 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1 ð1Þ
0.45 0.45
Eq. (29) 9.86, 9.89 3.94, 3.95 Eq. (30)
Eq. (31)
14.25, 14.39 5.7, 5.75 – –
– – 9.87 3.95
Eq. (32)
Eq. (33)
3.75, 4.15 1.5, 1.66 – –
– – 19.75 7.9
ð1Þ
Principal waves along the direction N2 ðÞ ðH12 ; 1; H23 Þ 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1 1.34 0.0342 0.1 0.5 1.34 0.0342 0.04 0.5 ð1Þ
0.55 0.55
1.45 1.45 1 1
0.45 0.45 0.5 0.5
ð1Þ
Principal waves along the direction N3 ðÞ ðH13 ; H23 ; 1Þ 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1
0.6 0.6 1 1
0.4 0.4 2 2
509
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
Table 3 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
Principal waves along the direction N 1 ðÞ ð1 þ H12 ; H13 ; 0Þ 0.0456 0.2331 0.1 1 0.0456 0.2331 0.04 1
0.5 0.5
1.5 1.5
Eq. (29) 16.22, 8.65 6.49, 3.46
c1
ð1Þ
c2
ð1Þ
ð1Þ
ð1Þ
ð1Þ
Eq. (30)
Eq. (31)
6.43, 10.97 2.57,4.39 – –
– – 3.78 1.51
Eq. (32)
Eq. (33)
7.93, 11.34 3.17, 4.53 – –
– – 3.78 1.51
ð1Þ
Principal waves along the direction N2 ðÞ ðH12 ; 1; H23 Þ 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.2331 0.1 0.0456 0.2331 0.04 ð1Þ
1 1 0.5 0.5
0.6 0.6 1 1
0.4 0.4 0.5 0.5
ð1Þ
Principal waves along the direction N3 ðÞ ðH13 ; H23 ; 1Þ 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.2331 0.1 0.0456 0.2331 0.04
1 1 0.5 0.5
0.55 0.55 0.5 0.5
0.45 0.45 1 1
Table 4 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) c1
c2
Waves along the direction N ðÞ ð1; 0; 0Þ 0.0014 0.292 0.1 0.0014 0.292 0.04 0.0014 0.292 0.1 0.0014 0.292 0.04 0.0014 0.0014 0.0014 0.0014
0.292 0.292 0.292 0.292
0.1 0.04 0.1 0.04
E23
E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
0 0 0 0
1 1 0.3 0.3
0.5 0.5 0.8 0.8
0.5 0.5 0.5 0.5
Eq. (35) 4.98 1.99 – –
Eq. (36) – – 8.02, 5.03 3.21, 2.01
1 1 1 1
0.6 0.6 1 1
1 1 0.5 0.5
0.4 0.4 0.5 0.5
Eq. (37) 24.56, 24.5 9.82, 9.8 – –
Eq. (38) – – 25.07, 5.07 10.03, 2.03
ð1Þ
ð1Þ
ð1Þ
Table 5 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) c1
c2
Waves along the direction N ðÞ ð1; 0; 0Þ 0.0303 0.4503 0.1 0.0303 0.4503 0.04 0.0303 0.4503 0.1 0.0303 0.4503 0.04 0.0303 0.0303 0.0303 0.0303
0.4503 0.4503 0.4503 0.4503
0.1 0.04 0.1 0.04
E23
E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
0 0 0 0
1 1 0.3 0.3
0.5 0.5 0.8 0.8
0.5 0.5 0.5 0.5
Eq. (35) 4.64 1.85 – –
Eq. (36) – – 8.36, 5.57 3.34, 2.23
1 1 1 1
0.6 0.6 1 1
1 1 0.5 0.5
0.4 0.4 0.5 0.5
Eq. (37) 26.6, 25.74 10.64, 10.3 – –
Eq. (38) – – 26.08, 6.08 10.43, 2.43
ð1Þ
ð1Þ
ð1Þ
In Tables 1–3 we evaluate the values of the quantities (46) relative to the first-order speeds (29)–(33) of principal waves, for different values of constants c1 and c2 . The numerical results show that the absolute percentage nonlinear effects on the wave speeds vary in the range 3.3–11.21 if c1 ¼ 0:86; c2 ¼ 0:155, and ¼ 0:1, or in the range 1.32–4.48 if ¼ 0:04 (see Table 1). Moreover, if c1 ¼ 1:34; c2 ¼ 0:0342, the absolute percentage of nonlinear effects on the wave speeds vary either in the range 3.75–19.75 if ¼ 0:1, or in the range 1.5–7.9, if ¼ 0:04 (see Table 2). Finally, if c1 ¼ 0:0456; c2 ¼ 0:2331, the absolute percentage of nonlinear effects on the wave speeds vary either in the range 3.78–16.22 if ¼ 0:1 or in the range 1.51–6.49 if ¼ 0:04 (see Table 3). Likewise, the numerical values of (46), which refer to the first-order speeds (35)–(38), are listed in Tables 4–6. The numerical results show that the absolute percentage of nonlinear effects on the wave speeds vary either in the range 4.98–25.07 if c1 ¼ 0:0014; c2 ¼ 0:292, and ¼ 0:1, or in the range 1.99–10.03 if ¼ 0:04 (see Table 4). For c1 ¼ 0:0303; c2 ¼ 0:4503, the percentage of nonlinear effects on the wave speeds vary either in the range 4.64–26.6 if ¼ 0:1, or in the range 1.85–10.43 if
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
Table 6 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) c1
c2
Waves along the direction N ðÞ ð1; 0; 0Þ 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.0456 0.0456 0.0456
0.2331 0.2331 0.2331 0.2331
0.1 0.04 0.1 0.04
E23
E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
0 0 0 0
1 1 0.3 0.3
0.5 0.5 0.8 0.8
0.5 0.5 0.5 0.5
Eq. (35) 3.78 1.51 – –
Eq. (36) – – 9.22, 6.94 3.69, 2.78
1 1 1 1
0.6 0.6 1 1
1 1 0.5 0.5
0.4 0.4 0.5 0.5
Eq. (37) 31.81, 28.89 12.72, 11.56 – –
Eq. (38) – – 28.65, 8.65 11.46, 3.46
ð1Þ
ð1Þ
ð1Þ
Table 7 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [38, p. 1204], and [37, p. 404]) c1
c2
Simple shear 0.6871 0.6871 0.0303 0.0303 0.0456 0.0456
0.6871 0.6871 0.4503 0.4503 0.2331 0.2331
0.1 0.04 0.1 0.04 0.1 0.04
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100% Eq. (42) 2.5, 2.5 1, 1 5.36, 5.36 2.14, 2.14 6.21, 6.21 2.49, 2.49
Eq. (43) 5, 2.5 2, 1 5, 0.36 2, 0.14 5, 1.21 2, 0.49
Eq. (44) 7.5, 5 3, 2 10.36, 5 4.14, 2 11.21, 5 4.49, 2
¼ 0:04 (see Table 5). For the values c1 ¼ 0:0456; c2 ¼ 0:2331, the absolute percentage of nonlinear effects relative to the wave speeds vary either in the range 3.78–31.81 if ¼ 0:1, or in the range 1.51–12.72 if ¼ 0:04 (see Table 6). Finally, in Table 7 we calculate the quantities (46) relative to the speeds (42)–(44), for different values of constants c1 and c2 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
C. Wang, C. Truesdell, Introduction to Rational Elasticity, Noordhoff, Groningen, 1973. C. Truesdell, W. Noll, The Nonlinear Field Theories of Mechanics. Handbuch der Physik, Band III/3 Springer-Verlag, Berlin, 1965. J.L. Ericksen, On the propagation of waves in isotropic incompressible perfectly elastic materials, J. Rational Mech. Anal. 2 (1953) 329–337. R. Hill, Acceleration waves in solids, J. Mech. Phys. Solids 10 (1962) 1–16. N.H. Scott, Acceleration waves in constrained elastic materials, Arch. Rational Mech. Anal. 58 (1) (1975) 57–75. N.H. Scott, Acceleration waves in incompressible solids viscoelastic waves, Quart. J. Mech. Appl. Math. 29 (1976) 295–310. N.H. Scott, M. Hayes, Constant amplitude acceleration waves in a prestrained incompressible isotropic elastic solid, Math. Mech. Solids 2 (3) (1997) 291–295. T. Gültop, On the propagation of acceleration waves in incompressible hyperelastic solids, J. Sound Vib. 264 (2) (2003) 377–389. M. Major, Velocity of acceleration wave propagating in hyperelastic Zahorski and Mooney–Rivlin materials, J. Theoret. Appl. Mech. 43 (4) (2005) 777– 787. H. Cohen, C.C. Wang, Principal waves in monotropic laminated bodies, Arch. Rational Mech. Anal. 137 (1) (1997) 27–47. T. Gültop, Existence of shear bands in hyperelastic solids, Mech. Res. Commun. 29/5 (2002) 431–436. E.R. Ferreira, Ph. Boulanger, Superposition of transverse and longitudinal finite-amplitude waves in a deformed Blatz–Ko material, Math. Mech. Solids 12 (5) (2007) 543–558. M. Ciarletta, B. Straughan, V. Zampoli, Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation, Int. J. Eng. Sci. 45 (9) (2007) 736–743. M. Destrade, G. Saccomandi, Finite-amplitude inhomogeneous waves in Mooney–Rivlin viscoelastic solids, Wave Motion 40 (3) (2004) 251–262. M. Destrade, G. Saccomandi, Finite amplitude elastic waves propagating in compressible solids, Phys. Rev. E (3) 72 (2005) 12. no. 1 016620. H. Kobayashi, R. Vanderby, New strain energy function for acoustoelastic analysis of dilatational waves in nearly incompressible hyper-elastic materials, Trans. ASME J. Appl. Mech. 72 (6) (2005) 843–851. J.J. Rushchitsky, C. Cattani, Similarities and differences in the description of the evolution of quadratically nonlinear hyperelastic plane waves by Murnaghan and Signorini potentials, Prikl. Mekh. 42 (9) (2006) 41–58 (in Russian, translation in Int. Appl. Mech. 42 (9) (2006) 997–1010). F.D. Murnaghan, Finite deformations of an elastic solid, Am. J. Math. 59 (2) (1937) 235–260. R.S. Rivlin, The solution of problems in second-order elasticity theory, J. Rational Mech. Anal. 2 (1953) 53–81. R.A. Toupin, B. Bernstein, Sound waves in deformed perfectly elastic materials, acoustoelastic effect, J. Acoust. Soc. Am. 33 (1961) 216–225. R.W. Ogden, Non-linear elastic deformations. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press, John Wiley and Sons, Inc., New York, 1984, xv + 532 pp., ISBN:0-85312-273-3. K.A. Lindsay, The second order deformation of an incompressible isotropic slab under torsion, Quart. J. Mech. Appl. Math. 45 (4) (1992) 529–544. D.M. Haughton, K.A. Lindsay, The second-order deformation of a finite compressible isotropic elastic annulus subjected to circular shearing, Proceedings of the Royal Society of London Series A 442 (1993) 621–639. D.M. Haughton, K.A. Lindsay, The second order deformation of a finite incompressible isotropic elastic annulus subjected to circular shearing, Acta Mech. 104 (3–4) (1994) 125–141. T.J. Van Dyke, A. Hoger, A comparison of second-order constitutive theories for hyperelastic materials, Int. J. Solids Struct. 37 (41) (2000) 5873–5917.
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511
[26] H. Moon, C. Truesdell, Interpretation of adscititious inequalities through the effects pure shear stress produces upon an isotropic elastic solid, Arch. Rational Mech. Anal. 55 (1974) 1–17. [27] I. Choi, R.T. Shield, Second-order effects in problems for a class of elastic materials, Z. Angew. Math. Phys. 32 (4) (1981) 361–381. [28] F. dell’Isola, G.C. Ruta, R.C. Batra, A second-order solution of Saint–Venant’s problem for an elastic pretwisted bar using Signorini’s perturbation method, J. Elast. 49 (2) (1998) 113–127. [29] R.C. Batra, F. dell’Isola, G.C. Ruta, Second-order solution of Saint–Venant’s problem for an elastic bar predeformed in flexure, Int. J. Non-Linear Mech. 40 (2–3) (2005) 411–422. [30] A. Marasco, A. Romano, On the acceleration waves in second-order elastic, isotropic, compressible, and homogeneous materials, Math. Comp. Modell. (2008), doi:10.1016/j.mcm.2008.06.005. [31] A. Marasco, On the first-order speeds in any directions of ordinary waves in second-order isotropic, compressible, and homogeneous materials, Math. Comp. Modell., in press. [32] G.L. Iaccarino, A. Marasco, A. Romano, Signorini’s method for live loads and 2nd order effects, Int. J. Eng. Sci. 44 (5–6) (2006) 312–324. [33] A.E. Green, E.B. Spratt, Second-order effects in the deformation of elastic bodies, Proc. Roy. Soc. Lond. A 224 (1954) 347–361. [34] A. Romano, R. Lancellotta, A. Marasco, Continuum Mechanics Using Mathematica: Fundamentals, Methods, and Scientific Computing, Birchauser Boston, 2006. [35] A. Romano, A. Marasco, A Methodological Approach to Continuum Mechanics: Advanced Topics and Research Trends. Birchauser Boston-Basel-Berlin, in press. [36] H. Kobayashi, R. Vanderby, New strain energy function for acoustoelastic analysis of dilatational waves in nearly incompressible, hyper-elastic materials, ASME J. Appl. Mech. 72 (2005) 843–851. [37] Y. Dong, R.J.T. Lin, D. Bhattacharyya, Determination of critical material parameters for numerical simulation of acrylic sheet forming, J. Mater. Sci. 40 (2005) 399–410. [38] A.E. Giannakopoulos, A. Triantafyllou, Spherical indentation of incompressible rubber-like materials, J. Mech. Phys. Solids 55 (2007) 1196–1211.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Second-order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media Addolorata Marasco Department of Mathematics and Applications, R. Caccippoli University of Naples Federico II, via Cintia, 80126 Naples, Italy
a r t i c l e
i n f o
Article history: Received 17 April 2008 Received in revised form 15 July 2008 Accepted 22 August 2008 Available online 11 October 2008
Keywords: Ordinary waves Nonlinear elastic incompressible media Perturbation method Second-order effects
a b s t r a c t In this paper we propose a perturbation method to investigate the propagation of the ordinary waves in second-order elastic, isotropic, incompressible, and homogeneous materials. This method allows us to determine the first-order terms of the speeds and the amplitudes both of the principal waves and the waves in any propagation direction, when the undisturbed region is subjected to an arbitrary isochoric deformation. Another application of this method is presented when the undisturbed region is subjected to a simple shear. Finally, some numerical results are presented in Mooney–Rivlin materials. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Owing to the important applications in geophysics, engineering, nondestructive testing of materials, electronic signal processing devices, etc., wave propagation in nonlinear elastic, isotropic, and homogeneous media has been investigated in many papers. However, general results in finite elasticity have been obtained only for principal waves, i.e., for the ordinary waves which propagate along the principal axes of strain (see [1,2] which contain detailed references to other contributions). These waves, when they exist, can be only longitudinal or transverse, and if the form of the stress relation is known, then the Ericksen’s formulae supply the unique value for their speeds. Conversely, if these speeds are known functions of the three principal stretches, for example by experiments, then the response coefficients are uniquely determined. All that supplies the first simple and definite connection between the existence of principal waves and physically reasonable static response of an elastic isotropic material in finite deformations (see [1, p. 278–294]).1 In spite of the deep relation existing between the wave propagation and the constitutive equations in nonlinear elasticity, it is not an easy task to deduce by experiments the constitutive equations of an isotropic elastic material. In fact, we have to determine, e.g. using particular static deformations or the wave propagation, functions depending on the principal invariants of the left Cauchy–Green tensor B. For these reasons, many authors studied the wave propagation in special classes of materials (hyperelastic Zahorski, Mooney–Rivlin, Blatz–Ko, neo-Hookean, St. Venant–Kirchhoff materials, etc.), which are described by simple constitutive relations, and for particular deformations (see [3–13]).2 Alternatively, for arbitrary sufficiently small deformations, it is possible to analyze the above problem in second-order elasticity in which a second-order
E-mail address: [email protected] In detail, Ericksen proved that some constitutive inequalities upon the response functions of isotropic elastic materials, are necessary and sufficient conditions so that the speeds of all principal waves be real. 2 In many other papers, for instance in [14–17], the elastic waves are studied by solving analytically or numerically the elastic equations of motion in particular situations. 1
0020-7225/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2008.08.009
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expansion of a constitutive relation is considered and all terms of third or higher order in the displacement gradient are systematically neglected in all the involved equations (as well as in Signorini’s and Green and Spatt’s perturbation methods). In this last case, the costitutive equations are determined by only a few material constants (five for hyperelastic compressible materials, two for incompressible bodies). On the other hand, the second-order constitutive relations, usually proposed in the context of isotropic materials (for example, see [18–24]) provided always a good description of the mechanical response of an elastic body for sufficiently small deformations (see [25]). Moreover, many experimental phenomena as Kelvin and Poynting effects, representing, respectively, density changes under imposed shear or longitudinal extension in an elastic solid, which are characteristic of the finite elasticity, occur already in second-order elasticity (see [26–29]). Likewise, also in the second-order elasticity, the speeds of the waves depend both on the direction of the propagations and on the deformations (see [30,31] and the results in these papers). As a consequence of the above considerations, in the papers [30,31] the propagation of the acceleration waves in isotropic, compressible, and homogeneous materials is carried out in the contest of second-order elasticity. In particular, in [30], we propose a Signorini’s perturbation method which allows us to determine the first-order terms of the speeds as well as the amplitude of the waves in any arbitrary direction of propagation.3 This procedure reduces the eigenvalue problem for the acoustic tensor to two simpler problems: the first coincides with the eigenvalue problem for the acoustic tensor in linear elasticity, whereas the second supplies the corresponding corrective first-order terms of the velocities and of the waves amplitudes. Moreover, the analysis of the wave propagation along any direction, is carried out when the undisturbed region is subjected to a simple extension or to a simple shear. More generally, in [31], we determine the first-order terms of the speeds and the amplitudes of the waves in any propagation direction, when the undisturbed region is subjected to an arbitrary deformation. The method proposed in [30] is devoted only to compressible engineering materials, e.g., steel or aluminum. Nevertheless, since the incompressible materials are used extensively in cars, aircrafts, buildings, bridges, railroads, medical equipments, and biomedical engineering, the wave propagation analysis have also been carried out on incompressible bodies (see [3–9]). For incompressible materials the presence of an arbitrary hydrostatic pressure in the stress relations push us to modify the method presented in [30]. In particular, in this paper, we propose a generalized Green and Spratt’s perturbation method in order to investigate the propagation of the (necessarily transverse) waves in second-order elastic, isotropic, incompressible, and homogeneous materials (see [33]). Similarly to [30], the proposed method allows us again to reduce the eigenvalue problem for the acoustic tensor, equipped with the orthogonality conditions between the amplitudes and the propagation directions, to two simpler problems: the first coincides with the eigenvalue problem for the acoustic tensor in linear elasticity, whereas the second supplies the first-order terms to add to the speeds and the amplitudes of the linear elasticity. The paper is organized into seven sections including the introduction. In Section 2 we recall some general results about the theory of waves in isotropic, homogeneous, elastic, and incompressible media. Moreover, to formulate the eigenvalue problem for the acoustic tensor in the reference configuration, we recall the relations among the speeds, the amplitudes, and the directions of propagation of the waves in the actual configuration and in the reference configuration. The Section 3 is devoted to the description of the perturbation method which allows us to determine the explicit form of the first two dynamic compatibility equations for the transverse waves, and the first-order relations among some quantities in the reference configuration and the corresponding ones in the actual configuration. In Section 4 we calculate the first-order terms of the speeds and the amplitudes of the principal transverse waves in an eigenvector basis of the tensor B. In particular, we verify that the speeds coincide with those evaluated by Truesdell in [2]. In Section 5 we determine the speeds and the amplitudes of the waves in any direction of propagation, when the undisturbed region is subjected to an arbitrary isochoric deformation. The analysis of the principal transverse waves and of their propagation speeds, when the undisturbed region is subjected to a simple shear, is carried out in Section 6. Finally, in Section 7 some numerical experiments lead us to evaluate the second-order effects on the speeds of the waves in some conventional Mooney–Rivlin incompressible materials. 2. Ordinary waves in incompressible elastic media In this section we expose briefly the theory of the ordinary waves in incompressible elastic materials and we resume the main results on the principal transverse wave propagation. Let S be an elastic, homogeneous, incompressible, and isotropic continuous system in the reference configuration C . In the sequel, we neglect all the thermal phenomena associated with the evolution of S. If X is any point of C and x is the corresponding position in the actual configuration CðtÞ, then the finite deformation of S in going from C to CðtÞ can be equivalently expressed by the finite deformation x ¼ xðX; tÞ or by the displacement u ¼ uðX; tÞ. It is well known that for incompressible materials, in which only isochoric deformations are possible, the constitutive equation of Cauchy stress tensor T has the form
e T ¼ pI þ f 1 B þ f 1 B1 pI þ T;
ð1Þ
e depends on the deformation gradient F ¼ r x, B ¼ FF is the left Cauchy–Green tensor, f 1 ; f 1 are functions of the two where T principal invariants IB and IIB of B, and p is an indeterminate pressure (see [2]). T
3 In [32] we proved that the five constitutive constants of a second-order isotropic, compressible, and homogeneous material, can be determined by simple experiments based on equilibrium problems in second-order elasticity.
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The local balance of momentum is given by the equation
q x€ ¼ rp þ r Te þ q b in CðtÞ;
ð2Þ
where q q is the mass density of S, since J ¼ det F ¼ 1, and b is the body force density. Let RðtÞ be a moving surface in the actual configuration CðtÞ of equation gðx; tÞ ¼ GðXðx; tÞ; tÞ ¼ 0. If the solution xðX; tÞ of the second-order equation (2) exhibits a discontinuity in some or all of its second-order derivatives across the surface RðtÞ, xðX; tÞ is said to represent an ordinary wave and RðtÞ is the wavefront which divides the region CðtÞ into the perturbed and undisturbed regions C ðtÞ and C þ ðtÞ, respectively. Moreover, the vector a, characterizing the strength of the discontinuity the normal speed of the second derivatives of xðX; tÞ across RðtÞ, is called amplitude of the singularity. We denote by cN ¼ og=ot jrgj g of propagation of RðtÞ, and by U N ¼ cN x_ N the local speed of the wavefront. Finally, if a is parallel to the unit normal N ¼ jr rgj to the surface RðtÞ, then the wave is said to be longitudinal; if it is normal, transverse. We recall that the constraint of incompressibility renders longitudinal waves impossible. Then, in an incompressible material, all the singular surfaces are necessarily transversal: a N ¼ 0. If Eq. (2) is written in the regions C ðtÞ and C þ ðtÞ, its limits for x towards a point r 2 RðtÞ are considered and the obtained results are subtracted, then we obtain the jump system associated with (2)
oT~ ij oHlM op € q sxi tr ¼ þ ; oxi r oHlM oxj r
i ¼ 1; 2; 3:
ð3Þ
r
By recalling the kinematic relations of second-order singular surfaces (see [34, p. 119])
s€xi tr ¼ U 2N ai ;
oHlM ¼ F hM Nj Nh al oxj r
ð4Þ
and that across such a surface the jump of the first derivatives of the pressure is
op ¼ AN i ; oxi r the jump system (3) becomes
e il al ; i ¼ 1; 2; 3; q U 2N ai ¼ ANi þ Q
ð5Þ
where
e e il ¼ o T ij F hM Nj Nh : Q oHlM Taking the scalar product of (5) by N and recalling that a N ¼ 0, we have
e jl al : A ¼ Nj Q Finally, we obtain the following equation for the amplitudes and speeds
ðQil q U 2N dil Þal ¼ 0;
ð6Þ
e jl e il Ni Nj Q Qil ¼ Q
ð7Þ
where
is the acoustic tensor. The algebraic conditions (6) express the well-known Hadamard’s result: Given an undisturbed state xþ ðX; tÞ towards which the ordinary wave RðtÞ propagates, then due to the continuity of xðX; tÞ across RðtÞ, the matrix Q is a known function of t and r 2 RðtÞ. Furthermore, given a propagation direction N, the speeds of propagation U N are such that q U 2N are the eigenvalues of the acoustic tensor and the discontinuity vectors a are its eigenvectors. For the isotropic incompressible materials, the acoustical axes for principal waves are principal axes of strain. Moreover, the principal waves are necessarily transverse, and the classical compatibility conditions of this waves hold only for stretches for which det B ¼ 1.4 Ericksen’s formulae on the speeds of propagation of the transverse principal waves are expressed in terms of the functions f 1 ; f 1 and their derivatives (see [2, p. 293]). However, since it is practically impossible to determine experimentally these functions, in [2] Truesdell finds the first terms in the expansions of the exact formulae of the local speeds only of transverse principal waves in a basis of eigenvectors of B (see [2, p. 293]). Therefore, in the above results and in the first-order approximation, there is no information about the speeds of propagation along directions which are not principal axes of strain. In the next section we propose a perturbation method, similar to the method exposed in [30], to overcome these difficulties, at least in second-order elasticity. This approach allows us to obtain the first-order speeds along any direction and for all 4 In [2], the authors derive a sequence of universal relations connecting the principal wave speeds in a basis of eigenvectors of B for any isotropic elastic material (compatibility conditions).
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
‘‘sufficiently small” deformations, leading us to the same results which Truesdell obtained for the first-order speeds of propagations of the transverse principal waves. To this end, it is useful to recall the following relations:
jrgj jr Gj 1 U N CU N ; U N ¼ U N U N ; jr Gj jrgj C 1 ai ¼ 2 ai ; ai ¼ C2 ai ; U N ¼
C
ð8Þ ð9Þ
where the vectors U N and ai are the local speed and the amplitude of the singularity of the surface R ðtÞ in the reference G configuration C . We also recall that the relations between the two unit normal vectors N and N ¼ jr r Gj are expressed by the formulae
Ni ¼
1
C
F 1 Li N L ;
NL ¼ CF iL Ni ;
ð10Þ
where, since N i N i ¼ 1, it is
C¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 F 1 Li F Mi N L N M :
Owing to (8)–(10), instead of analyzing (6), it is more convenient to study the following equation in the reference configuration C
ðQ il q U 2N dil Þal ¼ 0; where Q il
i ¼ 1; 2; 3;
ð11Þ
e il Q ^ il , and ¼Q
e e il ¼ o T ij F 1 NL NM ; Q oHlM Lj
b il ¼ 1 F 1 F 1 NL NM Q e jl : Q Mj 2 Li
C
3. Second-order effects in incompressible isotropic materials The Cauchy stress tensor (1) of an elastic, isotropic, incompressible, and homogeneous body, up to second-order terms, has the form (see [2, p. 241])
TðHÞ ¼ pI þ 2lE þ lHHT þ b1 E2 ;
ð12Þ
T
where E ¼ þ H Þ is the infinitesimal strain tensor, l is a Lame’s coefficient, and b1 is a second-order constitutive constant. Taking in the account the following derivation formulae (see [30]): 1 ðH 2
oEiL 1 ¼ ðdij dLM þ dLj diM Þ; oHjM 2 oðHLa Hai Þ ¼ HMi dLj þ HLj diM ; oHjM oðEia EaL Þ 1 ¼ ðEML dij þ EjL diM þ Eij dLM þ EiM dLj Þ oHjM 2 and that up to first-order terms we have
F1 ¼ I H þ Oð1Þ;
1
C2
1 ¼ F 1 hi F ki N h N k ¼ 1 2Hhi N h N i þ Oð1Þ
by simple but tedious calculations we obtain the first-order form of the tensor Q
1 1 1 Q ij ðH; N Þ ¼ lðNi Nj dij Þ þ b1 Eij þ ½ð4l þ b1 ÞEih þ 2lHhi Nj Nh ðb1 Ejh 2lHhj ÞN i Nh 2 2 2 1 Hhk ½2Ni Nj ð4l þ b1 Þ b1 dij Nh Nk : 2
ð13Þ
In order to reduce the starting problem (11), in which we have introduced (13), to a family of more simple problems, we suppose that a suitable nondimensional analysis of (2) leads us to introduce a small parameter depending on the acting forces, the material characteristic, and geometry of C (see [35]). Moreover, according to the Green and Spratt’s method [33], we suppose that the displacement u and the pressure p can be written in the following form (see [32,30])
u ¼ uð1Þ þ Oð1Þ; so that we have also
p ¼ pð1Þ þ Oð1Þ;
ð14Þ
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
H ¼ Hð1Þ þ Oð1Þ;
ð15Þ
ð1Þ
ð1Þ
where H is the nondimensional displacement gradient relative to u . To apply a perturbation method to the dynamic compatibility conditions (11), we assume that the quantities Q ij ðHðÞ; N ðÞÞ; aj ðÞ; kðÞ, and N ðÞ are analytic functions of the small parameter , so that, up to first-order terms, we obtain
k ¼ kð0Þ þ kð1Þ ;
ð1Þ a ¼ að0Þ þ a ;
ð1Þ N ðÞ ¼ Nð0Þ þ N :
ð16Þ
Moreover, in the same approximation, we have ð0Þ ð0Þ C ¼ 1 Eð1Þ ij N i N j
and the relations (8) and ð10Þ2 become ð1Þ
ð0Þ
ð0Þ
ð1Þ
U N ¼ ð1 Eij Ni Nj ÞU N ; NL ¼
ð0Þ NL
þ
ð1Þ ðNL
þ
ð0Þ
ð0Þ
U N ¼ ð1 þ Eij Ni Nj ÞU N ;
ð1Þ ð0Þ HiL Ni
ð17Þ
ð1Þ ð0Þ ð0Þ ð0Þ Ehk Nh Nk NL Þ:
ð18Þ
We recall that the vectors NðÞ and N ðÞ are unitary, this means that in our approximation result N ð0Þ ð0Þ ð1Þ and similarly Nð0Þ N ¼ 1; N N ¼ 0. Owing to these results, (13) assumes the following form: ð0Þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð0Þ
Q ij ðN ðÞ; 0Þ ¼ lðNi Nj dij Þ lðNi Nj þ Ni N j Þ þ
ð0Þ
N
ð0Þ
ð0Þ
¼ 1; N
N
ð1Þ
¼ 0,
1 1 ð1Þ ð0Þ ð0Þ b Eð1Þ þ ½ð4l þ b1 ÞEð1Þ ih þ 2lH hi N j N h 2 1 ij 2
1 1 ð1Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ðb1 Eð1Þ jh 2lH hj ÞN i N h H hk ½2N i N j ð4l þ b1 Þ b1 dij N h N k 2 2 ð0Þ ð1Þ Q ij þ Q ij :
ð19Þ
Since the waves are necessarily transverse, in view of (9) and ð10Þ1 , the condition a N ¼ 0 becomes
ai F 1 Li N L ¼ 0; so that in the first-order approximation we have
(
ð0Þ að0Þ N ¼ 0;
ð20Þ
ð0Þ ð1Þ ð0Þ ð1Þ ð0Þ að1Þ N þ a ðN N H Þ ¼ 0:
Finally, when we impose the conditions (20) and we neglect terms of higher order than one, from the dynamic compatibility Eq. (11), we derive the following systems: ð0Þ
ðl kð0Þ Þai ¼ 0; ð0Þ
ðl k
ð1Þ Þai
ð0Þ kð1Þ ai
ð21Þ 1 1 ð0Þ ð0Þ ð1Þ ð0Þ ð0Þ ð1Þ ð1Þ ð0Þ ð0Þ þ b1 ai Hhk Nh Nk þ aj b1 ðEij þ Ejh Ni Nh Þ ¼ 0: 2 2
ð22Þ
We note that when we associate the condition ð20Þ1 to the system (21), we obtain the usual eigenvalue equations of the linear elasticity. Consequently, from (21), we have that l=q is the square of the propagation speed, and condition ð20Þ1 supð0Þ with l and we consider only the first-order plies the amplitude of the discontinuity að0Þ . Similarly, when we identify k ð1Þ isochoric deformations, i.e. IHð1Þ ¼ 0, the Eqs. (22) and ð20Þ2 in the four unknowns aj , j ¼ 1; 2; 3, and kð1Þ , determine, at least ð1Þ ð1Þ in principle, the direction of a and the value of k . We recall that in nonlinear elasticity the tensor Q could be not symmetric for any direction and for any deformation. In particular, in our approximation it results ð1Þ
ð1Þ
ð0Þ
ð0Þ
ð1Þ
ð0Þ2
ð0Þ2
ð1Þ
ð0Þ
ð0Þ
ð1Þ
ð0Þ
ð0Þ
Q 12 Q 21 ¼ ð2l þ b1 Þ½ðE11 E22 ÞN1 N2 E12 ðN1 N2 Þ þ ð2l þ b1 Þ½E13 N2 N3 E23 N1 N3 ; Q 13 Q 31 ¼ ð2l þ Q 23 Q 32 ¼ ð2l þ
ð1Þ b1 Þ½ðE11 ð1Þ b1 Þ½ðE22
ð1Þ ð0Þ ð0Þ E33 ÞN1 N3 ð1Þ ð0Þ ð0Þ E33 ÞN2 N3
þ þ
ð1Þ ð0Þ ð0Þ E12 N2 N3 ð1Þ ð0Þ ð0Þ E12 N1 N3
ð2l þ ð2l þ
ð0Þ2 ð0Þ2 ð1Þ ð1Þ ð0Þ ð0Þ b1 Þ½ðN1 N 3 ÞE13 þ E23 N1 N2 ; ð1Þ ð0Þ ð0Þ ð0Þ2 ð0Þ2 ð1Þ b1 Þ½E13 N1 N2 þ ðN2 N 3 ÞE23 ;
ð23Þ ð24Þ ð25Þ
where the components of the infinitesimal strain tensor must satisfy the condition IEð1Þ ¼ 0. Moreover, as in finite elasticity, þ að1;hÞ the eigenvector of the tensor Q the acoustic tensor (19) is singular. When Q is symmetric and we denote with að0;hÞ ð0;hÞ ð1;hÞ belonging to the eigenvalue k þ k , then we have to satisfy the following orthogonality conditions
ðað0;hÞ þ að1;hÞ Þ ðað0;kÞ þ að1;kÞ Þ ¼ 0;
h–k;
which in turn are equivalent to the following others
(
að0;hÞ að0;kÞ ¼ 0; að1;kÞ þ að1;hÞ að0;kÞ ¼ 0: að0;hÞ
h–k:
ð26Þ
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A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
4. First-order local speeds of principal waves In this section, using (21) and (22), we analyze the transverse principal waves and we prove that the velocities of these waves coincide with the speeds obtained by Truesdell in [2] with a different approach.5 Let K1 ; K2 ; K3 be the eigenvalues of the left Cauchy–Green tensor B: Noting that for isochoric deformations det B ¼ 1, we have K1 K2 K3 ¼ 1. In the corresponding basis of eigenvectors B ¼ he1 ; e2 ; e3 i, the tensor B is diagonal and the directions of the principal axes of strain are
N1 ð1; 0; 0Þ;
N2 ð0; 1; 0Þ;
N3 ð0; 0; 1Þ:
ð27Þ
In this new basis B, up to first-order terms, Ki ¼ 1 þ K
0 B Eð1Þ ¼ B @
K
ð1Þ 1 =2
0
0
0
K2ð1Þ =2
0
0
0
Kð1Þ 3 =2
1
0
C C; A
B Hð1Þ ¼ B @
ð1Þ i ;i
ð1Þ 1 =2 ð1Þ H12 ð1Þ H13
ð1Þ
¼ 1; 2; 3, and, since B ¼ I þ 2E , we have ð1Þ
H12
K
K2ð1Þ =2 ð1Þ
H23
ð1Þ
H13
1
C ð1Þ H23 C A; ð1Þ K3 =2
where, since IHð1Þ ¼ 0, result K1 þ K2 þ K3 ¼ 0. Owing to (18), the corresponding directions of (27) are ð1Þ
ð1Þ
N1 ð1; 0; 0Þ þ ðH12 ; H13 ; 0Þ; N2 ð0; 1; 0Þ þ N3 ð0; 0; 1Þ þ
ð28Þ
ð1Þ ð1Þ ðH12 ; 0; H23 Þ; ð1Þ ð1Þ ðH13 ; H23 ; 0Þ:
In the sequel we analyze the wave propagation along each of the above directions. ð1Þ
ð1Þ
4.1. Waves propagation along N1 ðÞ ð1 þ H12 ; H13 ; 0Þ Along this vector the tensor Q writes
0
0 B0 Q ¼ @
0 ð1Þ ð1Þ l þ 12 b1 ðE11 þ E22 Þ
0 ð1Þ E11
ð1Þ E22
1
0 0 ð1Þ b1 ðE11
lþ 1 2
0
þ
ð1Þ E33 Þ
C A;
ð1Þ E33
where þ þ ¼ 0. When we impose the conditions (20) and (26), the Eqs. (21) and (22) lead to the following results6:
b1 ð1Þ b ð1Þ ð1Þ ðE þ E33 Þ ¼ l 1 E22 ; 2 11 2 b1 ð1Þ b1 ð1Þ ð1Þ k2 ¼ l þ ðE11 þ E22 Þ ¼ l E33 ; 2 2
ð1Þ
k1 ¼ l þ
a1 ð0; 0; 1Þ þ ð0; a2 ; 0Þ; ð1Þ
a2 ð0; 1; 0Þ þ ð0; 0; a2 Þ:
Therefore, the transverse waves propagate with the speeds
q U 21 ¼ l
b1 ð1Þ E ; 2 22
q U 22 ¼ l
b1 ð1Þ E ; 2 33
or equivalently, by using ð17Þ2 , we obtain
q U 21 ¼ l þ 2lEð1Þ 11
b1 ð1Þ E22 ; 2
q U 22 ¼ l þ 2lEð1Þ 11
b1 ð1Þ E33 : 2
ð29Þ
In particular, relation ð29Þ2 coincides with the formula (78.15) p. 293 in [2] provided that we identify the quantities ri with ð1Þ Eii and we recall that IEð1Þ ¼ 0. ð1Þ
ð1Þ
4.2. Waves propagation along N2 ðÞ ðH12 ; 1; H23 Þ Along this direction the tensor Q has the form
0 B Q ¼ @
ð1Þ l þ 12 b1 ðEð1Þ 0 11 þ E22 Þ
0 0
ð1Þ
ð1Þ
1
0
0
0
0
ð1Þ b1 ðE22
l þ 12
þ
ð1Þ E33 Þ
C A;
ð1Þ
where E11 þ E22 þ E33 ¼ 0. 5 6
This can be seen provided that we consider the relation b1 ¼ 4lðb 12Þ between the constitutive constant b1 to the constant b used in [2]. ð1Þ ð1Þ ð1Þ When E22 ¼ E33 , we obtain k ¼ l b21 E22 ; a ð0; a2 ; a3 Þ.
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
505
When we impose the conditions (20) and (26), Eqs. (21) and (22) supply the following results: ð1Þ
ð1Þ
If E11 –E33 , then
b1 ð1Þ b ð1Þ ð1Þ ðE þ E33 Þ ¼ l 1 E11 ; 2 22 2 b1 ð1Þ b1 ð1Þ ð1Þ k2 ¼ l þ ðE11 þ E22 Þ ¼ l E33 ; 2 2
ð1Þ
k1 ¼ l þ
a1 ð0; 0; 1Þ þ ða1 ; 0; 0Þ; ð1Þ
a2 ð1; 0; 0Þ þ ð0; 0; a1 Þ:
Owing to ð17Þ2 , the waves propagate in CðtÞ with the following speeds:
q U 21 ¼ l þ 2lEð1Þ 22 ð1Þ
b1 ð1Þ E11 ; 2
q U 22 ¼ l þ 2lEð1Þ 22
b1 ð1Þ E33 : 2
ð30Þ
ð1Þ
If E11 ¼ E33 , then we obtain
k3 ¼ l
b1 ð1Þ E ; 2 33
ð0Þ
ð0Þ
ð1Þ
ð1Þ
a3 ða1 ; 0; a3 Þ þ ða1 ; 0; a3 Þ
and, finally
q U 23 ¼ l þ 2lEð1Þ 22
b1 ð1Þ E11 : 2
ð31Þ
ð1Þ
ð1Þ
4.3. Waves propagation along N3 ðÞ ðH13 ; H23 ; 1Þ Along this direction the tensor Q writes
0 B Q ¼ @
ð1Þ l þ 12 b1 ðEð1Þ 11 þ E33 Þ
0
ð1Þ E22
1 C
ð1Þ l þ 12 b1 ðEð1Þ 0 A; 22 þ E33 Þ
0 0
ð1Þ E11
0
0
0
ð1Þ E33
where þ þ ¼ 0. When we impose the conditions (20) and (26), Eqs. (21) and (22) lead to the following results7: ð1Þ
ð1Þ
If E11 –E22 , then
b1 ð1Þ b ð1Þ ð1Þ ðE þ E33 Þ ¼ l 1 E11 ; 2 22 2 b1 ð1Þ b1 ð1Þ ð1Þ k2 ¼ l þ ðE11 þ E33 Þ ¼ l E22 ; 2 2
ð1Þ
k1 ¼ l þ
by using ð17Þ2 , result
q U 21 ¼ l þ 2lEð1Þ 33 ð1Þ
b1 ð1Þ E11 ; 2
a1 ð0; 1; 0Þ þ ða1 ; 0; 0Þ; ð1Þ
a2 ð1; 0; 0Þ þ ð0; a1 ; 0Þ
q U 22 ¼ l þ 2lEð1Þ 33
b1 ð1Þ E22 : 2
ð32Þ
ð1Þ
If E11 ¼ E22 , then we obtain
k3 ¼ l
b1 ð1Þ E ; 2 22
q U 23 ¼ l þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
a3 ða1 ; a2 ; 0Þ þ ða1 ; a2 ; 0Þ; b ð1Þ ð1Þ 2lE33 1 E11 : 2
ð33Þ
5. First-order transverse waves along an arbitrary direction N In this section, we calculate the first-order speeds of propagation in any direction, which is not a principal direction, and for sufficiently small deformations. Without loss of generality, we choose in C the unit vector N ð1; 0; 0Þ. To this vector, according to ð18Þ2 , corresponds the ð1Þ ð1Þ unit vector N ð1; H12 ; H13 Þ in CðtÞ. We note that N is a principal direction of strain if and only if ð1Þ
E12 ¼ 0; 7
ð1Þ
ð1Þ
ð1Þ
E13 ¼ 0:
ð34Þ ð1Þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
ð0Þ
ð1Þ
ð1Þ
When E11 ¼ E22 , we obtain k ¼ l b21 E22 ; a ða1 ; a2 ; 0Þ þ ða1 ; a2 ; a1 ðH13 þ E33 ÞÞ.
506
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
Then, in order not to fall in the same situation we have already considered in the above section, we suppose that conditions (34) are not verified. Along the direction N , the relations (23)–(25) become ð1Þ
Q 12 Q 21 ¼ ð2l þ b1 ÞE12 ; ð1Þ
Q 13 Q 31 ¼ ð2l þ b1 ÞE13 ; Q 23 Q 32 ¼ 0; so that the tensor Q is symmetric for any deformation if and only if 2l þ b1 ¼ 0. For any other second-order elastic material, Q is symmetric only if the deformation verifies (34) that, as we have already remarked, are equivalent to the request that N is an eigenvector of B. Imposing the conditions (20) and IEð1Þ ¼ 0, the Eqs. (21) and (22) supply the following results: ð1Þ
ð1Þ
ð1Þ
If E23 ¼ 0; E22 ¼ E33 , then
q U 2 ¼ l
b1 ð1Þ E ; 2 22
ð0Þ
ð0Þ
ð0Þ
ð1Þ
ð0Þ
ð1Þ
ð1Þ
ð1Þ
a ð0; a2 ; a3 Þ þ ða2 H12 þ a3 H13 ; a2 ; a3 Þ;
and, by using ð17Þ2 , we obtain
q U 2 ¼ l þ 2lEð1Þ 11 ð1Þ
ð1Þ
b1 ð1Þ E22 : 2
ð35Þ
ð1Þ
If E23 ¼ 0; E22 –E33 , then
b1 ð1Þ E ; 2 22 b ð1Þ ¼ l 1 E33 ; 2
ð1Þ
ð1Þ
q U 21 ¼ l
a1 ð0; 0; 1Þ þ ðH13 ; a2 ; 0Þ;
q U 22
a2 ð0; 1; 0Þ þ ðH12 ; 0; a3 Þ
ð1Þ
ð1Þ
and the waves propagate in CðtÞ with the speeds
ð1Þ
ð1Þ
b1 ð1Þ E Þ; 2 33
ð1Þ q U 21 ¼ l þ ð2lE11
q U 22 ¼ l þ 2lEð1Þ 11
b1 ð1Þ E22 : 2
ð36Þ
ð1Þ
If E23 –0; E22 –E33 , then
q U 2 ¼ l þ ð0Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 ð1Þ ð1Þ ð1Þ ð1Þ E11 4ðE23 Þ2 þ ðE22 E33 Þ2 ; 4
ð0Þ
ð0Þ
ð1Þ
ð0Þ
ð1Þ
ð1Þ
ð1Þ
a ð0; a2 ; a3 Þ þ ða2 H12 þ a3 H13 ; a2 ; a3 Þ; ð0Þ
ð0Þ
where a3 –0, and a2 ¼
ð0Þ a3 ð1Þ 2E23
q U 2 ¼ l þ 2lEð1Þ 11 þ ð1Þ
ð1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ½E22 E33 4ðE23 Þ2 þ ðE22 E33 Þ2 . Then, owing to ð17Þ2 , it results
b1 ð1Þ ðE 2 33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ 4ðE23 Þ2 þ ðE22 E33 Þ2 Þ :
ð37Þ
ð1Þ
If E23 –0; E22 ¼ E33 , then
b1 ð1Þ ð1Þ ðE E23 Þ; 2 22 ð0Þ ð0Þ ð0Þ ð1Þ ð1Þ ð1Þ ð1Þ a ð0; a2 ; a2 Þ þ ða2 ðH12 H13 Þ; a2 ; a2 Þ
q U 2 ¼ l
and, finally
q U 2 ¼ l þ ½2lEð1Þ 11
b1 ð1Þ ð1Þ ðE E23 Þ: 2 22
ð38Þ
The above results, show that, for any fixed direction of propagation, the speeds of the waves depend stongly on the deformations. 6. Simple shear In this section we analyze the finite isochoric deformation
x1 ¼ X 1 þ X 2 tan h;
x2 ¼ X 2 ;
x3 ¼ X 3 ;
ð39Þ
where h is the constant angle of shear. The particles move only in the X 1 -direction and their displacement is proportional to X 2 -coordinate. Planes, which are parallel to X 1 ¼ 0, are rotated about an axis parallel to X 3 -axis through the angle h. The deformation gradient and the left Cauchy–Green tensor are
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
0
1 1 k 0 B C F ¼ @ 0 1 0 A; 0 0 1
0
2
1þk B B¼@ k 0
507
1 k 0 C 1 0 A; 0 1
where k ¼ tan h, and the displacement gradient is
0
0 k
0
1
B C H ¼ @ 0 0 0 A: 0 0 0 Then, when we identify h with the small parameter , we have that
0
H
ð1Þ
1 0 1 0 B C ¼ @ 0 0 0 A; 0 0 0
0
1 0 1 0 B C B ¼ I þ @ 1 0 0 A þ Oð1Þ: 0 0 0
The unit directions of the principal axes of strain are
N1 ðÞ ð0; 0; 1Þ; 1 1 1 1 N2 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 ; 2 2 2 2 1 1 1 1 N3 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 2 2 2 2
ð40Þ
and owing to (18), the directions which correspond in C to these axes are
N1 ðÞ ð0; 0; 1Þ; 1 1 1 1 N2 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 ; 2 2 2 2 1 1 1 1 N3 ðÞ pffiffiffi ; pffiffiffi ; 0 þ pffiffiffi ; pffiffiffi ; 0 : 2 2 2 2
ð41Þ
Along the direction N1 ðÞ ð0; 0; 1Þ, the tensor Q is symmetric. Then, Eqs. (21) and (22), when we impose the conditions (20) and (26), lead to the following results8:
b1 ; 4 b q U 2 q U 2 ¼ l 1 ; 4
q U 2 q U 2 ¼ l þ
ð0Þ
ð0Þ
ð1Þ
ð1Þ
a1 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; 0Þ; ð0Þ
ð0Þ
ð1Þ
ð42Þ
ð1Þ
a2 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; 0Þ:
pffiffiffi pffiffiffi pffiffiffi pffiffiffi Along the vector N2 ðÞ ð1= 2; 1= 2; 0Þ þ ð1= 2; 1= 2; 0Þ, the tensor Q is symmetric. Solving the Eqs. (21) and (22), in view of (20) and (26), we obtain the following results:
b1 ð0Þ ð1Þ ð1Þ ; a1 ð0; 0; a3 Þ þ ða1 ; a1 ; 0Þ; 4 ð0Þ ð0Þ ð0Þ ð0Þ ð1Þ a2 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; a3 Þ
k1 ¼ l k2 ¼ l;
and, owing to ð17Þ2 , in the actual configuration CðtÞ the waves propagate with the speeds
1 4
q U 21 ¼ l l; q U 22 ¼ l ð4l þ b1 Þ:
ð43Þ
Along the last direction N3 ðÞ ðp1ffiffi2 ; p1ffiffi2 ; 0Þ þ ð p1ffiffi2 ; p1ffiffi2 ; 0Þ, the tensor Q is symmetric. Solving (21) and (22), in view of (20), (26), and ð17Þ2 , we obtain the following results:
b1 ð0Þ ð1Þ ð1Þ ; a1 ð0; 0; a3 Þ þ ða1 ; a1 ; 0Þ; 4 ð0Þ ð0Þ ð0Þ ð0Þ ð1Þ k2 ¼ l; a2 ða1 ; a1 ; 0Þ þ ða1 ; a1 ; a3 Þ; 1 q U 22 ¼ l ð4l b1 Þ; q U 21 ¼ l l: 4 k1 ¼ l þ
8
For the sake of simplicity, from now on we take unit amplitudes.
ð44Þ
508
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
7. Some numerical results In this section, we evaluate the first-order speeds of the waves determined in Sections 2–4 for arbitrary small deformapffiffiffiffiffiffiffiffiffiffiffi tions, and compare them with the constant value l=q of the speeds in incompressible linear elasticity. In particular, we consider some incompressible materials for which the constitutive equations are described by the traditional Mooney–Rivlin strain–energy function
W ¼ c1 ðIB 3Þ þ c2 ðIIB 3Þ;
ð45Þ
where, in our notations,
c1 ¼
1 1 ð4l þ b1 Þ; c2 ¼ b1 : 8 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
If we denote by
U 2 =q the first-order speeds of the waves determined in the previous sections, the first-order terms
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
ð46Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi of the quantity ½ð U 2 =q l=q Þ= l=q 100% measure the nonlinear effects on the wave propagation respect to the linear theory of the elasticity. In the sequel, we evaluate the percentage effects (46) for some incompressible materials for which the constants c1 and c2 in (45) are been experimentally calculated in the papers [36–38] (by acoustoelastic analysis, indentation tests, shrinkage test, and hot tensile test). In order to consider only arbitrary small deformations, owing to nondimensional analysis of the Section 2, it is sufficient ð1Þ to choose deformations for which Ei;j ’ 1, for all i and j, whereas the small parameter 1. Table 1 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [36, p. 847]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ c1 c2 E11 E22 E33 ðU 2 lÞ=2l 100% ð1Þ
ð1Þ
Principal waves along the direction N 1 ðÞ ð1 þ H12 ; H13 ; 0Þ 0.86 0.155 0.1 1 0.86 0.155 0.04 1 ð1Þ
0.55 0.55
0.45 0.45
Eq. (29) 11.21, 10.99 4.48, 4.39 Eq. (30)
Eq. (31)
7.7, 4.51 3.08, 1.8 – –
– – 11.1 4.44
Eq. (32)
Eq. (33)
6.7, 3.3 2.68, 1.32 – –
– – 11.1 4.44
ð1Þ
Principal waves along the direction N2 ðÞ ðH12 ; 1; H23 Þ 0.86 0.155 0.1 1 0.86 0.155 0.04 1 0.86 0.155 0.1 0.5 0.86 0.155 0.04 0.5 ð1Þ
0.55 0.55 1 1
0.45 0.45 0.5 0.5
ð1Þ
Principal waves along the direction N3 ðÞ ðH13 ; H23 ; 1Þ 0.86 0.155 0.1 1 0.86 0.155 0.04 1 0.86 0.155 0.1 0.5 0.86 0.155 0.04 0.5
0.55 0.55 0.5 0.5
0.45 0.45 1 1
Table 2 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [38, p. 1204]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð1Þ ð1Þ c1 c2 E11 E22 E33 ðU 2 lÞ=2l 100% ð1Þ
ð1Þ
Principal waves along the direction N 1 ðÞ ð1 þ H12 ; H13 ; 0Þ 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1 ð1Þ
0.45 0.45
Eq. (29) 9.86, 9.89 3.94, 3.95 Eq. (30)
Eq. (31)
14.25, 14.39 5.7, 5.75 – –
– – 9.87 3.95
Eq. (32)
Eq. (33)
3.75, 4.15 1.5, 1.66 – –
– – 19.75 7.9
ð1Þ
Principal waves along the direction N2 ðÞ ðH12 ; 1; H23 Þ 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1 1.34 0.0342 0.1 0.5 1.34 0.0342 0.04 0.5 ð1Þ
0.55 0.55
1.45 1.45 1 1
0.45 0.45 0.5 0.5
ð1Þ
Principal waves along the direction N3 ðÞ ðH13 ; H23 ; 1Þ 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1 1.34 0.0342 0.1 1 1.34 0.0342 0.04 1
0.6 0.6 1 1
0.4 0.4 2 2
509
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
Table 3 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
Principal waves along the direction N 1 ðÞ ð1 þ H12 ; H13 ; 0Þ 0.0456 0.2331 0.1 1 0.0456 0.2331 0.04 1
0.5 0.5
1.5 1.5
Eq. (29) 16.22, 8.65 6.49, 3.46
c1
ð1Þ
c2
ð1Þ
ð1Þ
ð1Þ
ð1Þ
Eq. (30)
Eq. (31)
6.43, 10.97 2.57,4.39 – –
– – 3.78 1.51
Eq. (32)
Eq. (33)
7.93, 11.34 3.17, 4.53 – –
– – 3.78 1.51
ð1Þ
Principal waves along the direction N2 ðÞ ðH12 ; 1; H23 Þ 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.2331 0.1 0.0456 0.2331 0.04 ð1Þ
1 1 0.5 0.5
0.6 0.6 1 1
0.4 0.4 0.5 0.5
ð1Þ
Principal waves along the direction N3 ðÞ ðH13 ; H23 ; 1Þ 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.2331 0.1 0.0456 0.2331 0.04
1 1 0.5 0.5
0.55 0.55 0.5 0.5
0.45 0.45 1 1
Table 4 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) c1
c2
Waves along the direction N ðÞ ð1; 0; 0Þ 0.0014 0.292 0.1 0.0014 0.292 0.04 0.0014 0.292 0.1 0.0014 0.292 0.04 0.0014 0.0014 0.0014 0.0014
0.292 0.292 0.292 0.292
0.1 0.04 0.1 0.04
E23
E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
0 0 0 0
1 1 0.3 0.3
0.5 0.5 0.8 0.8
0.5 0.5 0.5 0.5
Eq. (35) 4.98 1.99 – –
Eq. (36) – – 8.02, 5.03 3.21, 2.01
1 1 1 1
0.6 0.6 1 1
1 1 0.5 0.5
0.4 0.4 0.5 0.5
Eq. (37) 24.56, 24.5 9.82, 9.8 – –
Eq. (38) – – 25.07, 5.07 10.03, 2.03
ð1Þ
ð1Þ
ð1Þ
Table 5 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) c1
c2
Waves along the direction N ðÞ ð1; 0; 0Þ 0.0303 0.4503 0.1 0.0303 0.4503 0.04 0.0303 0.4503 0.1 0.0303 0.4503 0.04 0.0303 0.0303 0.0303 0.0303
0.4503 0.4503 0.4503 0.4503
0.1 0.04 0.1 0.04
E23
E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
0 0 0 0
1 1 0.3 0.3
0.5 0.5 0.8 0.8
0.5 0.5 0.5 0.5
Eq. (35) 4.64 1.85 – –
Eq. (36) – – 8.36, 5.57 3.34, 2.23
1 1 1 1
0.6 0.6 1 1
1 1 0.5 0.5
0.4 0.4 0.5 0.5
Eq. (37) 26.6, 25.74 10.64, 10.3 – –
Eq. (38) – – 26.08, 6.08 10.43, 2.43
ð1Þ
ð1Þ
ð1Þ
In Tables 1–3 we evaluate the values of the quantities (46) relative to the first-order speeds (29)–(33) of principal waves, for different values of constants c1 and c2 . The numerical results show that the absolute percentage nonlinear effects on the wave speeds vary in the range 3.3–11.21 if c1 ¼ 0:86; c2 ¼ 0:155, and ¼ 0:1, or in the range 1.32–4.48 if ¼ 0:04 (see Table 1). Moreover, if c1 ¼ 1:34; c2 ¼ 0:0342, the absolute percentage of nonlinear effects on the wave speeds vary either in the range 3.75–19.75 if ¼ 0:1, or in the range 1.5–7.9, if ¼ 0:04 (see Table 2). Finally, if c1 ¼ 0:0456; c2 ¼ 0:2331, the absolute percentage of nonlinear effects on the wave speeds vary either in the range 3.78–16.22 if ¼ 0:1 or in the range 1.51–6.49 if ¼ 0:04 (see Table 3). Likewise, the numerical values of (46), which refer to the first-order speeds (35)–(38), are listed in Tables 4–6. The numerical results show that the absolute percentage of nonlinear effects on the wave speeds vary either in the range 4.98–25.07 if c1 ¼ 0:0014; c2 ¼ 0:292, and ¼ 0:1, or in the range 1.99–10.03 if ¼ 0:04 (see Table 4). For c1 ¼ 0:0303; c2 ¼ 0:4503, the percentage of nonlinear effects on the wave speeds vary either in the range 4.64–26.6 if ¼ 0:1, or in the range 1.85–10.43 if
510
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
Table 6 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [37, p. 404]) c1
c2
Waves along the direction N ðÞ ð1; 0; 0Þ 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.2331 0.1 0.0456 0.2331 0.04 0.0456 0.0456 0.0456 0.0456
0.2331 0.2331 0.2331 0.2331
0.1 0.04 0.1 0.04
E23
E11
E22
ð1Þ
E33
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100%
0 0 0 0
1 1 0.3 0.3
0.5 0.5 0.8 0.8
0.5 0.5 0.5 0.5
Eq. (35) 3.78 1.51 – –
Eq. (36) – – 9.22, 6.94 3.69, 2.78
1 1 1 1
0.6 0.6 1 1
1 1 0.5 0.5
0.4 0.4 0.5 0.5
Eq. (37) 31.81, 28.89 12.72, 11.56 – –
Eq. (38) – – 28.65, 8.65 11.46, 3.46
ð1Þ
ð1Þ
ð1Þ
Table 7 Absolute percentage of nonlinear effects on the wave speeds in a Mooney–Rivlin material characterized by the constants c1 and c2 (MPa) (see [38, p. 1204], and [37, p. 404]) c1
c2
Simple shear 0.6871 0.6871 0.0303 0.0303 0.0456 0.0456
0.6871 0.6871 0.4503 0.4503 0.2331 0.2331
0.1 0.04 0.1 0.04 0.1 0.04
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU 2 lÞ=2l 100% Eq. (42) 2.5, 2.5 1, 1 5.36, 5.36 2.14, 2.14 6.21, 6.21 2.49, 2.49
Eq. (43) 5, 2.5 2, 1 5, 0.36 2, 0.14 5, 1.21 2, 0.49
Eq. (44) 7.5, 5 3, 2 10.36, 5 4.14, 2 11.21, 5 4.49, 2
¼ 0:04 (see Table 5). For the values c1 ¼ 0:0456; c2 ¼ 0:2331, the absolute percentage of nonlinear effects relative to the wave speeds vary either in the range 3.78–31.81 if ¼ 0:1, or in the range 1.51–12.72 if ¼ 0:04 (see Table 6). Finally, in Table 7 we calculate the quantities (46) relative to the speeds (42)–(44), for different values of constants c1 and c2 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
C. Wang, C. Truesdell, Introduction to Rational Elasticity, Noordhoff, Groningen, 1973. C. Truesdell, W. Noll, The Nonlinear Field Theories of Mechanics. Handbuch der Physik, Band III/3 Springer-Verlag, Berlin, 1965. J.L. Ericksen, On the propagation of waves in isotropic incompressible perfectly elastic materials, J. Rational Mech. Anal. 2 (1953) 329–337. R. Hill, Acceleration waves in solids, J. Mech. Phys. Solids 10 (1962) 1–16. N.H. Scott, Acceleration waves in constrained elastic materials, Arch. Rational Mech. Anal. 58 (1) (1975) 57–75. N.H. Scott, Acceleration waves in incompressible solids viscoelastic waves, Quart. J. Mech. Appl. Math. 29 (1976) 295–310. N.H. Scott, M. Hayes, Constant amplitude acceleration waves in a prestrained incompressible isotropic elastic solid, Math. Mech. Solids 2 (3) (1997) 291–295. T. Gültop, On the propagation of acceleration waves in incompressible hyperelastic solids, J. Sound Vib. 264 (2) (2003) 377–389. M. Major, Velocity of acceleration wave propagating in hyperelastic Zahorski and Mooney–Rivlin materials, J. Theoret. Appl. Mech. 43 (4) (2005) 777– 787. H. Cohen, C.C. Wang, Principal waves in monotropic laminated bodies, Arch. Rational Mech. Anal. 137 (1) (1997) 27–47. T. Gültop, Existence of shear bands in hyperelastic solids, Mech. Res. Commun. 29/5 (2002) 431–436. E.R. Ferreira, Ph. Boulanger, Superposition of transverse and longitudinal finite-amplitude waves in a deformed Blatz–Ko material, Math. Mech. Solids 12 (5) (2007) 543–558. M. Ciarletta, B. Straughan, V. Zampoli, Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation, Int. J. Eng. Sci. 45 (9) (2007) 736–743. M. Destrade, G. Saccomandi, Finite-amplitude inhomogeneous waves in Mooney–Rivlin viscoelastic solids, Wave Motion 40 (3) (2004) 251–262. M. Destrade, G. Saccomandi, Finite amplitude elastic waves propagating in compressible solids, Phys. Rev. E (3) 72 (2005) 12. no. 1 016620. H. Kobayashi, R. Vanderby, New strain energy function for acoustoelastic analysis of dilatational waves in nearly incompressible hyper-elastic materials, Trans. ASME J. Appl. Mech. 72 (6) (2005) 843–851. J.J. Rushchitsky, C. Cattani, Similarities and differences in the description of the evolution of quadratically nonlinear hyperelastic plane waves by Murnaghan and Signorini potentials, Prikl. Mekh. 42 (9) (2006) 41–58 (in Russian, translation in Int. Appl. Mech. 42 (9) (2006) 997–1010). F.D. Murnaghan, Finite deformations of an elastic solid, Am. J. Math. 59 (2) (1937) 235–260. R.S. Rivlin, The solution of problems in second-order elasticity theory, J. Rational Mech. Anal. 2 (1953) 53–81. R.A. Toupin, B. Bernstein, Sound waves in deformed perfectly elastic materials, acoustoelastic effect, J. Acoust. Soc. Am. 33 (1961) 216–225. R.W. Ogden, Non-linear elastic deformations. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press, John Wiley and Sons, Inc., New York, 1984, xv + 532 pp., ISBN:0-85312-273-3. K.A. Lindsay, The second order deformation of an incompressible isotropic slab under torsion, Quart. J. Mech. Appl. Math. 45 (4) (1992) 529–544. D.M. Haughton, K.A. Lindsay, The second-order deformation of a finite compressible isotropic elastic annulus subjected to circular shearing, Proceedings of the Royal Society of London Series A 442 (1993) 621–639. D.M. Haughton, K.A. Lindsay, The second order deformation of a finite incompressible isotropic elastic annulus subjected to circular shearing, Acta Mech. 104 (3–4) (1994) 125–141. T.J. Van Dyke, A. Hoger, A comparison of second-order constitutive theories for hyperelastic materials, Int. J. Solids Struct. 37 (41) (2000) 5873–5917.
A. Marasco / International Journal of Engineering Science 47 (2009) 499–511
511
[26] H. Moon, C. Truesdell, Interpretation of adscititious inequalities through the effects pure shear stress produces upon an isotropic elastic solid, Arch. Rational Mech. Anal. 55 (1974) 1–17. [27] I. Choi, R.T. Shield, Second-order effects in problems for a class of elastic materials, Z. Angew. Math. Phys. 32 (4) (1981) 361–381. [28] F. dell’Isola, G.C. Ruta, R.C. Batra, A second-order solution of Saint–Venant’s problem for an elastic pretwisted bar using Signorini’s perturbation method, J. Elast. 49 (2) (1998) 113–127. [29] R.C. Batra, F. dell’Isola, G.C. Ruta, Second-order solution of Saint–Venant’s problem for an elastic bar predeformed in flexure, Int. J. Non-Linear Mech. 40 (2–3) (2005) 411–422. [30] A. Marasco, A. Romano, On the acceleration waves in second-order elastic, isotropic, compressible, and homogeneous materials, Math. Comp. Modell. (2008), doi:10.1016/j.mcm.2008.06.005. [31] A. Marasco, On the first-order speeds in any directions of ordinary waves in second-order isotropic, compressible, and homogeneous materials, Math. Comp. Modell., in press. [32] G.L. Iaccarino, A. Marasco, A. Romano, Signorini’s method for live loads and 2nd order effects, Int. J. Eng. Sci. 44 (5–6) (2006) 312–324. [33] A.E. Green, E.B. Spratt, Second-order effects in the deformation of elastic bodies, Proc. Roy. Soc. Lond. A 224 (1954) 347–361. [34] A. Romano, R. Lancellotta, A. Marasco, Continuum Mechanics Using Mathematica: Fundamentals, Methods, and Scientific Computing, Birchauser Boston, 2006. [35] A. Romano, A. Marasco, A Methodological Approach to Continuum Mechanics: Advanced Topics and Research Trends. Birchauser Boston-Basel-Berlin, in press. [36] H. Kobayashi, R. Vanderby, New strain energy function for acoustoelastic analysis of dilatational waves in nearly incompressible, hyper-elastic materials, ASME J. Appl. Mech. 72 (2005) 843–851. [37] Y. Dong, R.J.T. Lin, D. Bhattacharyya, Determination of critical material parameters for numerical simulation of acrylic sheet forming, J. Mater. Sci. 40 (2005) 399–410. [38] A.E. Giannakopoulos, A. Triantafyllou, Spherical indentation of incompressible rubber-like materials, J. Mech. Phys. Solids 55 (2007) 1196–1211.