- Email: [email protected]
Dispersion phenomena in symmetric pre-stressed layered elastic structures
International Journal of Solids and Structures 58 (2015) 220–232
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Dispersion phenomena in symmetric pre-stressed layered elastic structures M.I. Lashhab a, G.A. Rogerson b,⇑, K.J. Sandiford c a
University of Almergeb, Faculty of Science, Zliten, Libya School of Computing and Mathematics, University of Keele, Keele, United Kingdom c School of Computing, Science and Engineering, University of Salford, Salford, United Kingdom b
a r t i c l e
i n f o
Article history: Received 14 November 2014 Available online 14 January 2015 Keywords: Pre-stress Layered media Waves Dispersion Short waves
a b s t r a c t The dispersion relation associated with a symmetric three layer structure, composed of compressible, pre-stressed elastic layers, is derived. This mathematically elaborate transcendental equation gives phase speed as an implicit function of wave number. Numerical solutions are established to show a wide range of dispersion behaviour which is delicately dependent on the material parameters and pre-stress in each layer. Particularly interesting behaviour is observed within the short wave (high wave number) regime, with six possible cases of short wave liming behaviour shown possible. Within each of these, a short wave asymptotic analysis is carried out, resulting in a set of approximations which provide explicit relationships between phase speed and wave number. It is envisaged that these approximations may prove helpful to approximate numerical truncation errors associated with impact response, as well as providing excellent first approximations for particularly (numerically) challenging sets of material parameters. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Many modern technological applications exist in which layered elastic material components are employed. Amongst the most common, we cite the use of layered composites within the aerospace industry and rubber-like vibration control devices used for earthquake protection in bridges and tall buildings. There are also examples of naturally occurring layered structures in geo and biomechanics. With these and other applications in mind, a significant research effort has in recent times been focussed on elucidation of the dynamic properties of layered elastic media. As the complexity of application has increased, considerable effort has been put into a complete understanding of dispersion in layered media. Seemingly the first attempt to consider non-fundamental modes was made in Lamb (1917) for a plane section of an isotropic plate. A more complete study of higher modes of an isotropic plate was later presented in Mindlin (1960). Because of the very complicated structure of the underlying dispersion relations, most of the results in these early papers were obtained through use of numerical computations.
⇑ Corresponding author. E-mail addresses: [email protected] (M.I. Lashhab), [email protected] (G.A. Rogerson), [email protected] (K.J. Sandiford). http://dx.doi.org/10.1016/j.ijsolstr.2015.01.006 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
In this present contribution, we aim to add to current understanding by investigating wave propagation in a symmetric 3-layer laminate, with each layer composed of compressible, pre-stressed elastic material. The paper will specifically focus on symmetric (extensional-type) waves and as such generalise the constitutive framework of a study done some years ago for incompressible pre-stressed elastic layers, see Rogerson and Sandiford (2000). Within this present paper a thorough investigation of the dispersion relation associated with the aforementioned structure is carried out. An initial numerical investigation is used to enable short wave asymptotic approximations, giving phase speed explicitly in terms of wave number, to be established. We remark that a detailed understanding of the behaviour of the dispersion relation is a critical pre-requisite in determining dynamic response. Although traditionally much effort has been afforded to fundamental modes, there is very good motivation to not neglect the harmonics. For example, the solution of any impact problem will generally involve contributions from each mode over the entire wave number regime. Moreover, important features, such as surface or interfacial waves, may arise from the cumulative affects of the harmonics, rather than result from the short wave limit of a single mode, see Rogerson (1992). We also remark that Kaplunov et al. (1998) and Kaplunov and Markushevich (1993) show that for certain types of vibration the contributions of higher modes is significant.
221
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
The problem of dispersion in layered pre-stressed media has seen a number of publications over the last twenty years. Plane incremental waves in incompressible and compressible elastic single-layer plates (with traction free boundaries) were discussed in Ogden and Roxburgh (1993) and Roxburgh and Ogden (1994), respectively. An asymptotic short wave analysis for a single layer plate was carried out in Rogerson (1997), Sandiford and Rogerson (2000) and Nolde et al. (2004) for incompressible, nearly incompressible and compressible materials, respectively. An extension to three dimensional motion was carried in Pichugin and Rogerson (2002) in respect of incompressible elastic single layer plates. Many of the aforementioned papers are a good source of references to earlier work. The paper is organised as follows. In Section 2 a very brief review of the underlying constitutive theory, and equations of motion, is presented. The dispersion relation is derived in Section 3. A numerical analysis of the dispersion relation is presented in Section 4. From this numerical analysis it is shown that the short wave limiting behaviour may be classified into six distinct cases. Each of these cases are analysed in Section 5, within which short wave asymptotic approximations are established in each case. These approximations, give phase speed as an explicit function of wave number and material parameters for each mode and are shown to provide excellent agreement with the numerical solution over a surprisingly large wave number regime. It is envisaged that these might provide some help in the numerical inversion of the highly oscillatory wave number integrals associated with impact problems. 2. Governing equations Only a brief summary of the relevant underlying theory is presented; for further details the reader is referred to [8]. We begin by considering a homogeneous elastic body B, possessing a natural isotropic unstressed state B0 . A purely homogeneous static deformation is imposed, resulting in the pre-stressed equilibrium state Be . Upon Be , we superimpose a small time dependent motion ui ðX A ; tÞ, resulting in the current material state Bt . Position vectors xi ðX A ; tÞ of a representative particle are denoted by X A ; xi ðX A Þ and ~ xi expressible in the form in B0 ; Be and Bt respectively, with ~
x~i ðX A ; tÞ ¼ xi ðX A Þ þ ui ðX A ; tÞ:
ð1Þ
The deformation gradients associated with the deformations and take the compoB0 ! Bt and B0 ! Be are denoted by F and F nent forms
F iA ¼
@ x~i ; @X A
@xi : FiA ¼ @X A
ð2Þ
may be related through On making use of Eqs. (1) and (2), F and F
F iA ¼ ðdij þ ui;j ÞFjA :
ð3Þ
The most general isotropic strain-energy function is of the form
WðFÞ ¼ WðI1 ; I2 ; I3 Þ
ð4Þ
with I1 ; I2 and I3 principal invariants of the left Cauchy–Green strain tensor. The equations of infinitesimal incremental motion may be written as
piA;A ¼ q0 u€i ; piA ¼
@W @F iA
ð5Þ
with q0 the density per unit volume of B0 ; piA components of the first Piola Kirchhoff stress tensor and a superimposed dot indicating differentiation with respect to time. A linearised form (5) is obtainable in the form
€i Apiba ua;bp ¼ qe u
ð6Þ
with qe the material density per unit volume of Be and Apiba the fourth order elasticity tensor, defined in the component form
Apiba ¼ J 1 FbC FpA
@ 2 W @F iA @F aC
ð7Þ
:
F¼F
The components of A allow especially simple representation relative to the principal axes of the primary static deformation. The only non-zero components have the form Aiijj ; Aijij or Aijji with i; j 2 f1; 2; 3g. These are given in terms of the principal stretches km , with m 2 f1; 2; 3g, as
Aiijj ¼ J 1 ki kj
Aijij ¼
@2W ; @ki @kj
ð8Þ
8 1 > ki @W kj @W > @ki @kj > :1 A 2
iiii
k2i
k2i k2j
Aiijj þ J
Aijji ¼ Ajiij ¼ Aijij J 1 ki
1
ki @W @ki
@W 0 @ki
i – j ki – kj ; ð9Þ i – j ki – kj ;
i – j:
ð10Þ
We also need a linearised measure of incremental traction, obtainable as
si ¼ Ajilk uk;l nj
ð11Þ
with n the outward unit normal to a material surface in Be . Our aim is to consider wave propagation in layered media, layers having finite thickness in one spatial direction and infinite lateral extent in the other two. Each layer is composed of elastic material characterised by the strain energy function (4). A Cartesian coordinate system Ox1 x2 x3 is chosen, coincident with the principal axes of deformation in Be , with Ox2 normal to the upper traction free surface. A state of plane strain is assumed, with both u1 and u2 independent of x3 and u3 0. On making use of (8)–(10) in (6), the two non-trivial equations of motion are expressed as
€1 ; A1111 u1;11 þ ðA1122 þ A2112 Þu2;21 þ A2121 u1;22 ¼ qe u
ð12Þ
€2 : ðA1221 þ A2211 Þu1;12 þ A1212 u2;11 þ A2222 u2;22 ¼ qe u
ð13Þ
We now seek solutions of (12) and (13) in the form
ðu1 ; u2 Þ ¼ ðU; VÞekqx2 eikðx1 v tÞ
ð14Þ
yielding two linear homogeneous equations with non-trivial solutions if
a22 c2 q4 þ fa22 ðv 2 a11 Þ þ c2 ðv 2 c1 Þ þ b2 gq2 þ ðv 2 a11 Þðv 2 c1 Þ ¼ 0 with
v
2
ð15Þ
¼ qe v and aij ; ci and b defined by 2
aij ¼ Aiijj ; i; j 2 f1; 2g; c1 ¼ A1212 ; c2 ¼ A2121 ; b ¼ a12 þ c2 r2
ð16Þ
and with r2 ¼ c2 A1221 the principal Cauchy stress along Ox2 . If the two roots of equation (15) are denoted by q21 and q22 , we note these may be either real, purely imaginary or complex conjugates, U and V may be presented as linear combinations of four generally independent solutions. On making use of equations (12) and (13), U and V may be expressed as
U¼
2 X
U ð2m1Þ ekqm x2 þ U ð2mÞ ekqm x2 ;
ð17Þ
m¼1
V¼
2 X gðqm Þ ð2m1Þ kqm x2 qm U e U ð2mÞ ekqm x2 d ; b m¼1
ð18Þ
222
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
where gðqm Þ ¼ a11 v 2 c2 q2m and U ðiÞ ; i 2 f1; 2; 3; 4g, are disposable constants. Solutions (17) and (18), with (11), are now employed to cast the linearised incremental traction components as
s1 k
s2 ik
¼
2 X f ðqm Þ ð2m1Þ kqm x2 qm U e U ð2mÞ ekqm x2 ; b m¼1
ð19Þ
¼
2 X hðqm Þ ð2m1Þ kqm x2 U e þ U ð2mÞ ekqm x2 ; b m¼1
ð20Þ
1
1
1
2
1
1
2
2
ð25Þ
An explicit representation of the dispersion relation is obtained as
where
f ðqm Þ ¼ bc2 q2m þ ðc2 r2 Þgðqm Þ;
Equations (24) will yield non-trivial solutions provided q1 hðq1 ÞC 1 q1 hðq1 ÞS1 q2 hðq2 ÞC 2 q2 hðq2 ÞS2 0 0 f ðq ÞS f ðq1 ÞC 1 f ðq2 ÞS2 f ðq2 ÞC 2 0 0 1 1 ~ ~ ~ ~ q1 b 0 q2 b 0 p1 bC 1 p2 bC 2 ¼ 0: 0 gðq1 Þ 0 gðq2 Þ g~ðp1 Þ~S1 g~ðp2 Þ~S2 ~f ðp Þ~S1 ~f ðp Þ~S2 0 f ðq1 Þ 0 f ðq2 Þ 1 2 q hðq Þ ~ ÞC ~ ÞC ~ 1 p hðp ~2 0 q hðq Þ 0 p hðp
hðqm Þ ¼ a22 gðqm Þ a12 b:
ð21Þ
We note that q ¼ 0 corresponds to propagation in unbounded media, Eq. (15) then yielding v 21 ¼ a11 or c1 . These relate to two body waves propagating along Ox1 , the first longitudinal, the second shear.
We consider a 4-ply (3 layer) symmetric laminated structure, formed of two identical outer layers of thickness h, perfectly bonded to an inner core of thickness 2d, all formed of isotropic pre-stressed elastic materials of the same density. The axes of the left Cauchy-Green strain tensor are assumed coincident for all layers, with one normal to the upper surface. The material parameters of the (identical) outer layers are as denoted in the preceding section, with the inner core’s parameters distinguished by imposing an over tilde. Accordingly, solutions for the inner core are sought in the form
ð22Þ
If (22) is inserted into (12) and (13), displacement and traction components may be inferred from (17)–(20). Continuity requires that ri ¼ r~ i ; i ¼ 1; 2, with the (infinitesimal) boundary and continuity conditions given by
s1 ¼ s2 ¼ 0 at x2 ¼ ðd þ hÞ; ~ 1 ; U2 ¼ U ~ 2 ; s1 ¼ s ~ 1 ; s2 ¼ s ~2 at x2 ¼ d: U1 ¼ U
ð23Þ
the six equations 2 X f ðqm Þ qm ðU ð2m1Þ ekqm ðdþhÞ U ð2mÞ ekqm ðdþhÞ Þ ¼ 0; b m¼1 2 X hðqm Þ ð2m1Þ kqm ðdþhÞ ðU e þ U ð2mÞ ekqm ðdþhÞ Þ ¼ 0; b m¼1 2 X ~ m ¼ 0; ~ ð2m1Þ C U ð2m1Þ ekqm d þ U ð2mÞ ekqm d U 2 X gðqm Þ ð2m1Þ kqm d g~ðpm Þ ~ ð2m1Þ ~ U ðU e U ð2mÞ ekqm d Þ Sm ¼ 0; ~ bqm bp m m¼1 2 ~f ðp Þ X f ðqm Þ ð2m1Þ kqm d m ~ ð2m1Þ ~ Sm ¼ 0; ðU e U ð2mÞ ekqm d Þ U ~ bqm bp m m¼1 2 ~ X hðp hðqm Þ ð2m1Þ kqm d m Þ ~ ð2m1Þ ~ U ðU e þ U ð2mÞ ekqm d Þ C m ¼ 0: ~ b b m¼1
q2 f ðq1 Þhðq2 ÞS1 S2 gD3 þ fq1 f ðq2 Þhðq1 ÞS1 S2 q2 f ðq1 Þhðq2 ÞC 1 C 2 gD4 fq1 f ðq2 Þhðq1 ÞC 2 S1 q2 f ðq1 Þhðq2 ÞS2 C 1 gD5 ¼ 0;
ð26Þ
~ Þ bhðq ~ D1 ¼ ~S2 C~ 1 p1 fð f ðq2 Þg~ðp2 Þ ~f ðp2 Þgðq2 ÞÞðbhðp 1 2 ÞÞg ~ ~ ~ 2 p fð f ðq Þg~ðp Þ f ðp Þgðq ÞÞðbhðp Þ bhðq ~ ~S1 C 2 2 1 1 2 2 2 ÞÞg; ~ ~ ~ ~ ~ D2 ¼ C 1 C 2 p1 p2 bfð f ðq1 Þgðq2 Þ f ðq2 Þgðq1 ÞÞðhðp1 Þ hðp2 ÞÞg; ~ Þ bhðq ~ D3 ¼ ~S2 C~ 1 p1 q2 fð f ðq1 Þg~ðp2 Þ ~f ðp2 Þgðq1 ÞÞðbhðp 1 2 ÞÞg ~ Þ bhðq ~ 2 p q fð f ðq Þg~ðp Þ ~f ðp Þgðq ÞÞðbhðp ~ ÞÞg; þ ~S1 C 2 2
1
1
1
1
2
2
~ Þ bhðq ~ D4 ¼ ~S2 C~ 1 p1 q1 fð f ðq2 Þg~ðp2 Þ ~f ðp2 Þgðq2 ÞÞðbhðp 1 1 ÞÞg ~ ~ ~ ~ ~ S1 C 2 p q fð f ðq Þg~ðp Þ f ðp Þgðq ÞÞðbhðp Þ bhðq ÞÞg; 2 1
2
1
1
2
2
1
D5 ¼ ~S1 ~S2 q1 q2 bfð~f ðp1 Þg~ðp2 Þ ~f ðp2 Þg~ðp1 ÞÞðhðq1 Þ hðq2 ÞÞg; ~ Þ bhðq ~ D6 ¼ S~2 C~ 1 p1 fð f ðq1 Þg~ðp2 Þ ~f ðp2 Þgðq1 ÞÞðbhðp 1 1 ÞÞg ~ ~ ~ ~ ~ S1 C 2 p fð f ðq Þg~ðp Þ f ðp Þgðq ÞÞðbhðp Þ bhðq ÞÞg: 2
1
1
1
1
2
1
ð27Þ 4. Numerical solutions of the dispersion relations
Inserting appropriate forms of (17)–(20) into (23) yields a system of twelve equations in twelve unknowns. In view of symmetry, this may be reduced to two systems of six equations in six unknowns, one analogous to the classical extensional wave problem, one with the flexural problem. It is the former wave type that we shall focus attention upon in the remainder of this paper. For symmetric ~ 2 and s ~1 vanish at x2 ¼ 0, implying (extensional-like) solutions, U ð1Þ ð2Þ ~ ~ ~ ð3Þ ¼ U ~ ð4Þ , thus yielding from (18) and (19) that U ¼ U and U
m¼1
q2 f ðq1 Þhðq2 ÞS1 C 2 gD2 þ fq1 f ðq2 Þhðq1 ÞC 1 C 2
within which
3. Derivation of the dispersion relation
~ VÞe ~ kpx2 eikðx1 v tÞ : ðu1 ; u2 Þ ¼ ðU;
q1 q2 ff ðq1 Þhðq1 ÞD1 þ f ðq2 Þhðq2 ÞD6 g fq1 f ðq2 Þhðq1 ÞC 1 S2
ð24Þ
Throughout this section, various numerical solutions are presented. The parameters are chosen to demonstrate the range of material response, rather than model any specific material. In examining the numerical behaviour of the dispersion relation, we first use the modified Varga strain energy function
W ¼ lðk1 þ k2 þ k3 3Þ þ
1 jðJ 1Þ2 ; 2
J ¼ det F ¼ k1 k2 k3
ð28Þ
with l and j shear and bulk moduli, respectively. Fig. 1 presents a against scaled wave number kh for the plot of scaled phase speed v first 28 branches of the dispersion relation (26). First note that in the low wave number limit ðkh; kd ! 0Þ only the fundamental mode retains finite wave speed, with all harmonics having an associated infinite limit. In the high wave number limit ðkh; kd ! 1Þ the fundamental mode tends to a distinct limit, which is in fact the R 0:67, of the material of the outer Rayleigh surface wave speed, v layers. In this high wave number limit, all harmonics asymptote, from above, to the longitudinal wave speed associated with outer layers. It is observed numerically that in this case one of q1 or q2 is imaginary, the other real, with p1 and p2 both real. As kh; kd ! 1, the magnitude of the imaginary q tends to zero, with 2 ¼ v 21 ¼ a11 . Similar the limiting wave speed, from (15), given by v behaviour is also observed when p1 and p2 are complex conjugates and we therefore define, without loss of generality, case 1 as:
^1 ; q2 real; p1 and p2 real or complex conjugates; case ð1Þ : q1 ¼ iq ^1 ! 0 and v 2 ! v 21 ¼ a11 : as kh; kd ! 1 q
223
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
A further prominent feature in Fig. 1 is the flattening of the disper¼ sion curves to form so-called ghost lines around v ¼ v~ 1 1:17; v v 2 1:543. These are longitudinal and shear wave fronts associated with the material of the inner core. A second case, analogous to case 1, occurs when all harmonics asymptote, from above, to the scaled longitudinal wave speed ~ 1 . This case is classified through associated with the inner core, v
^1 ; p2 real; q1 and q2 real or complex conjugates; case ð4Þ : p1 ¼ ip ~1 : ^1 ! 0 and v 2 ! v~ 22 ¼ c as kh; kd ! 1 p The final strain energy function employed is the Blatz–Ko strain energy
W¼
^1 ; p2 real; q1 and q2 real or complex conjugates; case ð2Þ : p1 ¼ ip ~ 11 : ^1 ! 0 and v 2 ! v~ 21 ¼ a as kh; kd ! 1 p The second form of strain energy employed is the modified Neo-Hookean
W ¼ c1 ðI1 3Þ þ
1 jðJ 1Þ2 2
ð29Þ
with c1 a material parameter and j is the bulk modulus. In Fig. 2, we again note a single finite long wave limit and distinct short wave limit of the fundamental mode. All harmonics are noted to tend to the shear wave speed associated with the two outer layers. In this case, it is observed numerically that one of q1 or q2 is imaginary, the other real, whilst p1 and p2 are both real. Moreover, as kh; kd ! 1, the magnitude of the imaginary q tends to zero. The limiting wave speed of the harmonics is therefore obtainable from equation (15) 2 ¼ v 22 ¼ c1 0:685. This case may therefore classified as as v follows
^1 ; q2 real; p1 and p2 real or complex conjugates; case ð3Þ : q1 ¼ iq
2 J2
! þ 2J 5 :
ð30Þ
Fig. 3 illustrates a plot for the first twenty-eight branches of the dispersion relation (26) for layers with the Blatz–Ko strain energy (30). The fundamental mode and harmonics exhibit clear oscillatory behaviour in the high wave number region. This behaviour is a possible feature of pre-stressed materials which is never present within the classic linear isotropic framework, see Rogerson (1998). The short wave limiting speed of the harmonics gives rise to a wave front which is in general neither purely longitudinal nor purely shear in nature. In the limit kh; kd ! 1 both p1 and p2 are imaginary and jp1 j ! jp2 j, whilst q1 and q2 are complex conjugates and retain order 1 magnitude. This limiting behaviour is also observed when q1 and q2 are both real. The limiting wave speed of the harmonics, denoted by v~ 3 , is obtained by allowing the discriminant to vanish in the appropriate form of equation (15). We thus describe this case through the requirements
^1 ; p2 ¼ ip ^2 ; q1 and q2 real or complex conjugates; case ð5Þ : p1 ¼ ip as kh; kd ! 1 jp1 j ! jp2 j and v ! v~ 3 ;
^1 ! 0 and v 2 ! v 22 ¼ c1 : as kh; kd ! 1 q Shear and longitudinal wave fronts are again starting to form, around v 2:31; v 2:44, with another front also sharply defined 0:98. around v An analogous case, to that just presented as case 3, may also be classified within which the short wave limit of all harmonics is the shear wave speed associated with the inner core; this case is classified through
l I2
v
~ 23
¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 ðc ~ a ~2 a ~ 22 Þðc ~1 c ~2 a ~ 11 a ~ 22 Þ b ~2 þ a ~ 22 Þ þ 2b ~2 T~ 0 ~ 22 c ðc ~2 a ~ 22 Þ ðc
2
; ð31Þ
where
~2 ðc ~1 a ~ 11 Þðc ~2 a ~ 22 Þ: T~ 0 ¼ b
ð32Þ
A final case, analogous to case 5, is characterised by
2
1.8
1.6
1.4
v¯
1.2
1
0.8
0.6
0.4
0
5
10
15
20
25
30
kh against kh for a laminate formed of layers of materials with strain energy (28): outer layers Fig. 1. v l~ ¼ 2:5; j~ ¼ 1:5; ~k1 ¼ 1:97; ~k2 ¼ 0:99; ~k3 ¼ 0:47: v 1 ¼ 0:946; v 2 ¼ 1:543; v~ 1 ¼ 1:169; v~ 2 ¼ 1:546; v R ¼ 0:67.
l ¼ 2:0; j ¼ 1:0; k1 ¼ 1:6; k2 ¼ 0:8; k3 ¼ 0:7; inner core
224
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
3
2.5
2
v¯ 1.5
1
0.5
0
5
10
15
20
25
30
35
40
kh against kh for laminate formed layers of material with strain energy (29); outer layers c1 ¼ 0:2; Fig. 2. v ~3 ¼ 2:06; v ~ ¼ 0:4; ~ 1 ¼ 0:98; v 2 ¼ 0:685; v ~ 1 ¼ 2:44; v ~ 2 ¼ 2:31; v R ¼ 0:59. ~c1 ¼ 1:4; j k1 ¼ 1:7; ~ k2 ¼ 0:43, k
^1 ; q2 ¼ iq ^2 ; p1 and p2 real or complex conjugates; case ð6Þ : q1 ¼ iq as kh; kd ! 1 jq1 j ! jq2 j and v ! v 3 ;
v 23 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc2 a22 Þðc1 c2 a11 a22 Þ b2 ðc2 þ a22 Þ þ 2b a22 c2 T 0 ðc2 a22 Þ2
: ð33Þ
j ¼ 0:3; k1 ¼ 1:4; k2 ¼ 1:2; k3 ¼ 1:0; inner core
4.1. Some general remarks We note that there are six possible short wave phase speed limits for the harmonics. These are three wave speeds associated with the material of the inner core and three with the outer layers. The actual limiting wave speed is dependent on the material parameters and pre-stress. We define the limiting wave speed of the outer L ) to be the wave speed that the harmonics would tend layers (v
3.5
3
2.5
2
v 1.5
1
0.5
0
5
10
15
20
25
30
kh Fig. 3. Phase speed against scaled wave number for laminate formed of layers of material with strain energy (30); with v 3 ¼ 1:20 and v~ 3 ¼ 0:22.
v 1 ¼ 1:53; v 2 ¼ 1:24; v~ 1 ¼ 3:04 and v~ 2 ¼ 0:48 and
225
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
to in a single layer plate of the same material. Its value is one of three, dependent on aij ; ci and b, and given by
8 > < v 1 v L ¼ > v 2 : v 3
if c2 ðc1 a11 Þ > b2 and a11 < c1 ; if a22 ða11 c1 Þ > b2 and a11 > c1 ;
ð34Þ
otherwise
with the corresponding limit for the inner core identified by
8 > < v~ 1 v~ L ¼ > v~ 2 : v~ 3
~2 and a ~2 ðc ~1 a ~ 11 Þ > b ~ 11 < c ~1 ; if c ~2 and a ~ 22 ða ~ 11 c ~1 Þ > b ~ 11 > c ~1 ; if a
ð35Þ
otherwise:
1 ; v 2 ; v 3 Þ. However, it The limiting wave speed v L is in general minðv 1 ; v 2 Þ but v 3 < minðv 1 ; v 2 Þ. is exceptionally possible that v L ¼ minðv 3 be the limit is It will be remembered that the requirement for v that q1 and q2 are both imaginary and jq1 j ! jq2 j as kh ! 1. The 3 is less than both v 1 and v 2 but not case just described, when v the limiting wave speed, arises when jq1 j ! jq2 j but q1 and q2 are both real. The inequalities in (34) then give the range of pre-stress within which the appropriate requirements are satisfied for the given limit to occur. The actual limit of the harmonics in our symmetric 4-ply laminated plate is the lower of v L and v~ L , and, in the vast majority of cases, the lowest of the six possible values.
materials of the inner core and outer layers, respectively, and defined in (34) and (35). It is clear that in general there are six possible wave speed limits for the harmonics as kh; kd ! 1 and that the actual limit is dependent on material parameters. Each of these cases will be analysed in turn. !v 1 with a11 < c1 and c2 ðc1 a11 Þ b2 > 0 5.2.1. Case 1: v The first case we consider is when the limiting wave speed of the harmonics is the longitudinal wave speed associated with the outer layers. Numerical calculations indicate that for all harmonics v 2 approaches a11 from above and therefore, from (15), it is clear that the requirements outlined in case 1 on q1 ; q2 and p1 ; p2 are 2 < c1 are applicable. Accordingly, valid within the region a11 < v we seek to expand the dispersion relation (26) around the small ^1 and then use (15) to establish that quantity q
v 2 ¼ a11 þ c2 þ
2 þ Oðq ^21 Þ; q2 ¼ q
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðc1 a11 Þ b2
The numerical indications of short wave behaviour, discussed in Section 4, are now investigated.
where
n o ð1Þ ð1Þ ð1Þ ð1Þ fq1 f ðq2 Þhðq1 Þ q2 f ðq1 Þhðq2 Þg D2 þ D3 þ D4 D5 ¼ 0;
ð36Þ
in which a superscript ð1Þ indicates that Eq. (26) has been divided ~1C ~ 2 and the subsequent hyperbolic tangents throughout by C 1 C 2 C replaced with unity. After a little algebraic manipulation it can readily be established that the first factor of (36) is in fact the Rayleigh surface wave equation for the material of the outer layer, see Dowaikh and Ogden (1991a). We also note that the second factor is the Stoneley interfacial wave equation, see Dowaikh and Ogden (1991b), associated with the interface between the materials of the outer layers and the inner core. If both waves exist, in general the fundamental mode will tend to the least of these two wave speeds, with the first harmonic tending to the second. An important exception arises when the surface wave speed is greater than the shear and/or longitudinal wave speed of the core. In this case, a ghost line occurs in the dispersion relation, with associated flat maxima of the local group velocity modes in adjacent wave number regions, see Rogerson (1998). Moreover, in any numerical algorithm to determine impact response, based on integral transforms, great care must be taken so as not to miss such an important feature, when it arises through the combination of many harmonics over a large wave number regime, rather than the more commonly reported short wave contribution of a single mode.
^21 Þ; 1 þ Oðq p1 ¼ p
^21 Þ; 2 þ Oðq p2 ¼ p
ð38Þ
2 ; p 1 and p 2 are order 1 quantities defined by within which q
2 ¼ q
We first consider the short wave limit when q1 and q2 and p1 and p2 , are either real or complex conjugate pairs. In this case as kh; kd ! 1, the dispersion relation (26) tends to
ð37Þ
Expansions for q2 ; p1 and p2 are obtained using Eq. (37) with the appropriate form of Eq. (15), yielding
5. Asymptotic analysis
5.1. Surface and interfacial waves
! b2 ^2 þ Oðq ^41 Þ: q a11 c1 1
a22 c2
;
22 ¼ Fða11 Þ; p1 2 ; p
ð39Þ
~2 ~ 22 c ~2 Fðv 2 Þ ¼ a ~ 22 ða ~ 11 v 2 Þ þ c ~2 ðc ~1 v 2 Þ b 2a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a~ 22 ðv 2 a~ 11 Þ þ c~2 ðv 2 c~1 Þ þ b~2 4a~ 22 c~2 ðv 2 a~ 11 Þðv 2 c~1 Þ: To avoid proliferation of symbols an over-bar will be used to indicate the order 1 terms within expansions of q21 ; q22 ; p21 and p22 in a number of the limiting cases, the exact definition of each will change as necessary. The appropriate form of the dispersion relation for large wave number is now obtained by making use of (37) and (38) in (26), to provide
^1 hÞ ðq 2 hðq 2 Þðc2 r2 Þða11 c1 Þf1 Þ q ^21 ðða22 ða11 c1 Þ tan ðkq ^ a12 bÞf ðq2 Þg1 Þg ¼ q1 fq2 hðq2 Þðc2 r2 Þða11 c1 Þg1 2 Þf1 g þ Oðq ^21 Þ; ðða22 ða11 c1 Þ a12 bÞf ðq
ð40Þ
within which f1 and g1 are the order 1 quantities
~ p ~ q 2 p 2 ÞÞg 2 fððc2 r2 Þða11 c1 Þg~ðp 1 Þ a11 c1 ~f ðp 1 ÞÞðbhð 2 Þ bhð f1 ¼ q ~ p ~ q 2 p 2 ÞÞg 1 fððc r2 Þða11 c Þg~ðp 2 Þ a11 c ~f ðp 2 ÞÞðbhð 1 Þ bhð q 2
1
1
~ p ~ p ~ c r2 Þða11 c Þgðq 2 Þ f ðq 2 Þa11 c1 Þðhð 1 p 2 bfðð 1 Þ hð 2 ÞÞg; p 2 1 ~ p ~ a22 ða11 c Þ 1 Þ bðð g1 ¼ p1 fð f ðq2 Þg~ðp2 Þ gðq2 Þ~f ðp2 ÞÞðbhð 1 ~ p ~ a22 ða11 2 Þg~ðp 2 Þ~f ðp 2 fð f ðq 1 Þ gðq 1 ÞÞðbhð 2 Þ bðð a12 bÞÞg p 2 bfð~f ðp 1 Þg~ðp 2 Þ ~f ðp 2 Þg~ðp 1 ÞÞððða22 ða11 c1 Þ a12 bÞÞg q 2 ÞÞg: c1 Þ a12 bÞ hðq ^1 hÞ Oðq ^1 Þ, implyFrom equation (40), it is deduced that tanðkq ^1 hÞ ! 0 as q ^1 ! 0, and thus ing that in the limit kh ! 1; tanðkq
^1 ¼ q
np 2 þ OðkhÞ : kh
ð41Þ
5.2. Short wavelength limit ðkh ! 1Þ
Inserting Eq. (41) into (37) yields the second order approximation for the phase speed of the nth harmonic
The numerical results indicate that the short wavelength limiting wave speed for all harmonics is in general the lower of two wave speeds, termed limiting wave speed, associated with the
v 2n ¼ a11 þ c2 þ
! np2 b2 þ ; a11 c1 kh
ð42Þ
226
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
where here and hereafter n ¼ 1; 2; 3 . . .. A higher order expansion for the phase speed is obtained by setting
np / 3 þ 1 þ OðkhÞ ; kh ðkhÞ2
^1 ¼ q
^1 hÞ ¼ tanðkq
/1 3 þ OðkhÞ ; kh
ð43Þ
where /1 is to be determined. If these two expansions are inserted into Eq. (40), and like powers of kh equated, it is found that
/1 ¼
2 Þ f1 ða22 ða11 c1 Þ a12 bÞf ðq np: þ 2 hðq 2 Þðc2 r2 Þða11 c1 Þ g1 q
ð44Þ
On inserting Eq. (44) into Eq. (43)1, and on making use of equation (37), it may be shown that
v
2n
! np2 b2 ¼ a11 þ c2 þ a11 c1 kh
2 Þ f1 2 ða22 ða11 c1 Þ a12 bÞf ðq þ 1þ þ 2 Þðc2 r2 Þða11 c1 Þ g1 2 hðq kh q
ð45Þ
We remark that the asymptotic orders in Eq. (40) will change in, and close to, the special case c2 ¼ r2 . Appropriately specialised asymptotic expansions could readily be obtained for this case. A comparison of the third order expansions with numerical solutions are shown in Fig. 4. Good agreement with the numerical solutions is observed over a surprisingly large wave number regime.
~2 > 0 ~ 11 > c ~1 and a ~ 22 ða ~ 11 c ~1 Þ b ~ 2 with a 5.2.4. Case 4: v ! v Using the same analysis we obtain
v
2n
!
np2 ~2 b 2g ¼ c~1 þ a~22 1 þ 4 þ . . . ; n ¼ 1; 2; 3; . . . a~ 11 c~1 kd f4 kd ð49Þ
with f4 and g4 inferred from definitions of f1 and g1 . Good agreement is obtained even in the low wave number ranges, with expansions accurate over a much greater range of kh; kd than one might expect, see Fig. 5. ~2 < 0 ~ 11 > c ~1 and a ~ 22 ða ~ 11 c ~1 Þ b ~ 3 with a 5.2.5. Case 5: v ! v In this case we know numerically that as kh; kd ! 1 both p1 and p2 are imaginary and that jp1 j ! jp2 j, whilst q1 and q2 are either real or complex conjugates. The limit kh; kd ! 1 may therefore be examined by setting
~ ~ þ b; p21 ¼ a
v
!
np2 ~2 b 2g2 ~ 11 þ c ~2 þ þ ; ¼a 1þ a~ 11 c~1 kd f2 kd
~¼ a
a~ 22 ðv 2 a~ 11 Þ þ c~2 ðv 2 c~1 Þ þ b~2 ~ ; b ~ 22 c ~2 2a
¼
By making use of Eq. (51), in conjunction with (15), it is possible to ~ obtain the following quadratic for a
a~ 22 c~2 ða~ 22 c~2 Þ2 a~2 þ 2a~ 22 c~2 ð2b~2 ða~ 11 c~1 Þða~ 22 c~2 ÞÞa~ ~ þ ðc ~2 Þða ~2 Þ ¼ 0; ~ 22 c ~ 2 ða ~ 22 þ c ~2 Þ2 b ~2 ða ~ 11 c ~1 Þ þ b ~ 22 ða ~ 11 c ~1 Þ b a
~ p 2 p 1 Þða 1 Þððc ~ 11 c ~1 Þ gðq ~2 r2 Þða ~ 11 c ~1 ÞÞÞðbhð 2 fð f ðq 2 Þ f2 ¼ q ~ q 2 ÞÞg q1 p2 fð f ðq2 Þða ~ 11 c ~1 Þ gðq2 Þððc ~2 r2 Þða ~ 11 bhð ~ p ~ q ~1 ÞÞÞðbhð 1 ÞÞg q 1 q 2 bfðððc ~2 r2 Þða ~ 11 c ~1 ÞÞg~ðp 2 Þ bhð 2 Þ c ~ 11 c ~1 ÞÞðhðq 1 Þ hðq 2 ÞÞg; 2 Þða ~f ðp
ð52Þ ~ implies that which for small b
~2 þ Oðb ~4 Þ; ~¼a ~0 þ a ~1 b a
~ q ~ f ðq 1 Þgðq 2 Þ gðq 1 Þf ðq 2 ÞÞðða 1 ÞÞg ~ 22 ða ~ 11 c ~1 Þ a ~ 12 bÞ hð g2 ¼ p2 bfð ~ q 2 ÞÞg q 1 fð f ðq 2 Þg~ðp 2 ÞÞðbðða ~ 22 ða ~ 11 c ~1 Þ 2 Þ ~f ðp 2 Þgðq bhð ~ q ~ 12 bÞ bhð 1 ÞÞg: a 1 and q 2 may be inferred from Eq. (39) by inter2 ; q The form of p changing the material parameters of the outer layers and the inner core. 2
5.2.3. Case 3: v ! v 2 with a11 > c1 and a22 ða11 c1 Þ b > 0 By the same method previously used, we obtain
b
2 ¼ q
a22 c2
;
2
~ 22 c ~2 Þ ða
;
sffiffiffiffiffiffiffiffiffiffiffiffi 1 a~ 22 c~2 ~1 ¼ ða ~ 22 þ c ~2 Þ a ~2 T 0 2 b
ð54Þ
with T 0 defined in Eq. (32). On inserting Eqs. (50) and (53) into (26), ~ it is deduced that for high wave number and expanding for small b, ~ that either and small b
~2 Þ ¼ 0; 1 hðq 1 Þf ðq 2 Þ q 2 hðq 2 Þf ðq 1 Þ þ Oðb q
ð55Þ
~ a ~ a 3 þ q 3 ÞbSð 5 bCð 1 C 2 C ~0 Þ þ C ~0 Þ ðq
np2 kh
21 ; p 22 ¼ Fðc1 Þ: p
~0 ¼ a
qffiffiffiffiffiffiffiffiffi ~2 ~2 ~ 22 þ c ~2 Þ a~b c~1~ T 0 b ða 22 2
or
~ bÞ ~ ðq ~ þ Oðb ~2 Þ; 6 bCð 2 þ q 4 ÞSðbÞ 2 C 1 C ¼C ð47Þ
within which f3 and g3 may be readily inferred by replacing the def2 ; p 1 and p 2 , as given below, into the expressions for f1 initions of q and g1 presented just after Eq. (40)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a22 ða11 c1 Þ b2
ð53Þ
where
2 fð f ðq 1 Þg~ðp 1 Þ~f ðp ~ 22 ða ~ 11 c ~1 Þ a ~ 12 bÞ 2 Þ gðq 2 ÞÞðbðða þq
a11 c1
2 Þ f3 2 ða22 ða11 c1 Þ a12 bÞÞf ðq þ þ ... 1þ 2 hðq 2 Þðc2 r2 Þða11 c1 Þ g3 kh q
:
~ 22 c ~2 2a
ð51Þ
where f2 and g2 are given by
v 2n ¼ c1 þ a22
ð50Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 a~ 22 ðv 2 a~ 11 Þ þ c~2 ðq~ v 2 c~1 Þ þ b~2 4a~ 22 c~2 ðv 2 a~ 11 Þðv 2 c~1 Þ
ð46Þ
!
~ P 0; b
thus
n
¼ 1; 2; 3; . . . ;
2
~ > 0; a
~ are real and b ~ ! 0 as kh; kd ! 1. The values of a ~ and b ~ and where a ~ b may be obtained explicitly from the appropriate form of Eq. (15),
~2 > 0 ~ 11 < c ~1 and c ~ 2 ðc ~1 a ~ 11 Þ b ~!v ~ 1 with a 5.2.2. Case 2: v Using the same analysis we obtain
2n
~ ~ b; p22 ¼ a
ð48Þ
ð56Þ
m (m ¼ 1; . . . ; 6) are order one terms given in within which C ~ and SðbÞ ~ are defined by ~ ~0 Þ; CðbÞ Appendix A, Cða0 Þ; Sða
pffiffiffiffiffi ~0 kd; ~0 Þ ¼ cos 2 a Cða ! ~ bkd ~ ¼ cos p ffiffiffiffiffi ; CðbÞ ~0 a
pffiffiffiffiffi ~0 kd; ~0 Þ ¼ sin 2 a Sða ! ~ bkd ~ ¼ sin p ffiffiffiffiffi SðbÞ ~0 a
1 and q 2 are order 1 terms found by setting v ¼ v~ 3 in Eq. (15). and q Eq. (55), at leading order, is the appropriately specialised form of
227
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
1.15
Numerical solutions Asymptotic expansions
1.1
1.05
v 1
0.95
0.9
0
5
10
15
20
25
30
35
40
45
50
kh Fig. 4. Comparison of numerical solutions with third order asymptotic expansions (45) for case 1. The same material parameters as in Fig. 1 have been used.
0.9
Numerical solutions Asymptotic expansions
0.85
0.8
0.75
v
0.7
0.65
0.6
0.55 0
10
20
30
40
50
60
kh Fig. 5. Comparison of numerical solutions with third order asymptotic expansions (49) for case 4. The same material parameters as in Fig. 2 have been used.
the Rayleigh surface wave equation. From equation (56), it is readily ~ b, ~ implying that SðbÞ ~ ! 0 as b ~ ! 0, and thus deduced that SðbÞ
pffiffiffiffiffi ~0 2 ~ ¼ np a b þ OðkdÞ : kd
np2 1 ~ 2 þ 2a ~0 1 þ a ~1 ~ 11 a ~ 22 þ c ~1 c ~2 b ~ 22 c ~2 a a a22 þ c~2 kd
v 2n ¼ ~
þ ð58Þ
ð57Þ
A second order expansion for the phase speed for the nth harmonic may now be obtained by inserting Eq. (57) into (51)1 and (52), to obtain
A higher order expansion is obtainable by now setting
~ ¼ np b
pffiffiffiffiffi ~0 a / 3 þ 5 2 þ OðkdÞ kd ðkdÞ
ð59Þ
228
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
from which we infer that
~ bkd sin pffiffiffiffiffi ~0 a
!
/ 3 ¼ ð1Þn pffiffiffiffiffi5 þ OðkdÞ ; ~0 kd a n
¼ ð1Þ þ OðkdÞ
~ bkd cos pffiffiffiffiffi ~0 a
We now make use of Eq. (67), in conjunction with the appropriate form of Eq. (15), to obtain expansions for p1 and p2 , given by
!
2
2
ð60Þ
with /5 to be determined. On inserting Eqs. (59) and (60) into Eq. (58) and comparing like powers of kd, it is obtained that
( /5 ¼ np
) 2 þ ðSða 5 þ q 6 Þ þ Cða 1Þ ~0 Þðq 2 C 1 C ~0 ÞC ð1Þn C : 3 þ q 4 2 C 1 C ð1Þn q
v
4
n 1 ¼ a~ a~ þ c~ c~ b~2 þ 2a~ 22 c~2 a~ 22 þ c~2 11 22 1 2 !!) np2 ^5 2/ ~1 a ~0 ~0 þ a a 1 þ pffiffiffiffiffi þ ; kd ~0 kd a
p2 ¼ p20 þ p22 þ Oðb Þ;
ð68Þ
where
p10 ¼
1 k10 þ l0 2 ; ~2 a ~ 22 2c
p12 ¼
p20 ¼
1 k10 l0 2 ; ~2 a ~ 22 2c
p22 ¼
ð61Þ
Eq. (61) can now be used in conjunction with Eqs. (51)1, (53) and (59), to obtain
2n
4
p1 ¼ p10 þ p12 b þ Oðb Þ;
k12 þ l2 1
1
~2 a ~ 22 Þ2 ðk10 þ l0 Þ2 ð2c
;
k12 l2 1
1
~2 a ~ 22 Þ2 ðk10 l0 Þ2 ð2c
and within which
~2 ; ~ 22 ða ~ 11 a1 Þ c ~2 ða1 c ~1 Þ b k10 ¼ a ð62Þ
^ 5 ¼ /5 . The appearance of sinusoidal terms in the third order where / np term of Eq. (62) indicates that the asymptotic expansions will give a crude approximation to second order. Fig. 6 shows that once the asymptotic expansions are taken to third order, and the trigonometric terms included, good agreement with the numerical solutions is obtained.
~ 22 þ c ~2 Þ; k12 ¼ a2 ða
a1 ¼ ða11 a22 þ c1 c2 b2 þ 2a22 c2 a0 Þ=ða22 þ c2 Þ; a2 ¼
2a22 c2 a1 ; a22 þ c2
l20 ¼ b~4 þ 2b~2 ða~ 22 ða1 a~ 11 Þ þ c~2 ða1 c~1 ÞÞ 2
5.2.6. Case 6: v ! v 3 with a11 > c1 and a22 ða11 c1 Þ b2 < 0 The final case to consider arises when both q1 and q2 are imaginary and jq1 j ! jq2 j as kh; kd ! 1, whilst p1 and p2 are either both real or complex conjugates. The limit kh; kd ! 1 is therefore examined by setting
q21 ¼ a þ b;
q22 ¼ a b;
a > 0;
b P 0;
ð63Þ
where a and b are real and b ! 0 as kh; kd ! 1. Following the previous analysis we see that 2
4
a ¼ a0 þ a1 b þ Oðb Þ
ð64Þ
for small b, with the forms of a0 and a1 inferred from Eq. (54). In this case it is deduced that for high wave number equation (26) takes the form
^1 q ^2 fhðq1 Þf ðq1 ÞD1 þ hðq2 Þf ðq2 ÞD6 g fq ^1 hðq1 Þf ðq2 Þ q ^2 hðq2 Þf ðq1 ÞðCðbÞSða0 ÞD2 ðCða0 ÞSðbÞD2 þ Sða0 ÞCðbÞD5 Þ q
~ 22 ða1 a ~ 11 Þ þ c ~ 2 ðc ~1 a1 ÞÞ ; þ ða ~2 ða ~ 22 þ c ~2 Þ 2a2 ða ~ 22 c ~2 Þðc ~2 ða1 c ~1 Þ a ~ 22 ða1 2l0 l2 ¼ 2a2 b ~ 11 ÞÞ: a On making use of Eqs. (66)–(68), in conjunction with (65), the dispersion relation in the high wave number region may be cast in the relatively simple form 2
A1 þ ðA2 þ Cða0 ÞA3 Sða0 ÞA5 Þb 2 3 ¼ CðbÞ A1 þ A4 b Sða0 ÞA6 b þ Oðb Þ;
where Cða0 Þ; Sða0 Þ; CðbÞ and SðbÞ are trigonometric terms which may be inferred from the definitions given after equation (56), and Am are order 1 quantities defined in Appendix B. It is readily deduced from equation (69) that at leading order
bkh bkh 1 cos pffiffiffiffiffi OðbÞ sin pffiffiffiffiffi ; a0 a0
^1 hðq2 Þf ðq2 Þðq ^2 ÞCða0 ÞCðbÞD3 SðbÞCða0 ÞD5 Þg þ fq ^1 Sða0 ÞSðbÞD4 q ^2 hðq2 Þf ðq1 Þðq ^2 Sða0 ÞSðbÞD3 þq ^1 Cða0 ÞCðbÞD4 Þg þq
thus implying that cos
¼0
ð65Þ
^1 and q ^2 are approximated by within which q
bkh p ffiffiffi ffi a0
ð66Þ
3
with n ¼ ð4a0 a1 1Þ=8a20 . It is found that the leading order term of
! 1 as kh; kd ! 1, and therefore
pffiffiffiffiffi np 2 b ¼ 2 a0 þ OðkhÞ : kh
v 2n ¼
Eq. (65) vanishes in the limit, necessitating inclusion of Oðb Þ terms in all expansions. By making use of the appropriate form of Eqs. (51)1 and (64) we obtain
n o 1 a11 a22 þ c1 c2 b2 þ 2a22 c2 ða0 þ a1 b2 Þ a22 þ c2 4
ð71Þ
np2 1 a11 a22 þ c1 c2 b2 þ 2a22 c2 a0 1 þ 4a1 a22 þ c2 kd þ ð72Þ
2
þ Oðb Þ:
ð70Þ
A second order approximation to the phase speed is obtainable by inserting Eq. (71) into Eq. (67)1, to deduce that
pffiffiffiffiffi b 2 3 a0 pffiffiffiffiffi þ nb þ Oðb Þ; 2 a0 pffiffiffiffiffi b ^2 ¼ a0 þ pffiffiffiffiffi þ nb2 þ Oðb3 Þ q 2 a0
^1 ¼ q
v 2n ¼
ð69Þ
ð67Þ
We now seek a higher order expansion by setting
pffiffiffiffiffi np / 3 b ¼ 2 a0 þ 6 þ OðkhÞ kh ðkhÞ2 from which it is deduced that
ð73Þ
229
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
0.7
Numerical solutions Asymptotic expansions
0.65
0.6
v
0.55
0.5
0.45
0.4
0
10
20
30
40
50
60
kh Fig. 6. Comparison of numerical solutions with third order asymptotic expansions (62) for case 5. The same material parameters as in Fig. 3 have been used.
bkh / 3 sin pffiffiffiffiffi ¼ pffiffiffiffiffi6 þ OðkhÞ ; a0 a0 kh ¼1
/26 2a0 ðkhÞ
bkh cos pffiffiffiffiffi a0
of kh, reveals that the equation is identically zero at leading order. At the next order the following quadratic equation for /6 is revealed
4
2
þ OðkhÞ ;
ð74Þ
where /6 is to be determined and it is noted that it is now necessary 2
to include an OðkhÞ term in the expansion (74)2 due to the vanishing of the leading order term prior to the derivation of Eq. (69). Inserting Eqs. (73) and (74) into Eq. (69), and comparing like powers
A1 2 / þ 2npA6 /6 þ 4a0 ðnpÞ2 fA2 A4 þ A3 Cða0 Þ A5 Sða0 Þg ¼ 0: 2a0 6
ð75Þ
We note that quadratic equation for /6 is obtained in this case, whilst in the previous five cases a single value for /1 ; . . . ; /5 was obtained. Interestingly, it has been verified numerically that for particular values of n the two solutions indicated in Eq. (75)
0.7
Numerical solutions Asymptotic expansions
0.65
0.6
0.55
v
0.5
0.45
0.4
0.35 0
10
20
30
40
50
60
kh Fig. 7. Comparison of numerical solutions with second order asymptotic expansions (72) for case 6 for Blatz–Ko material (30), with ~ 1 ¼ 2:24; v 2 ¼ 1:33; v ~ 2 ¼ 1:29; v 3 ¼ 0:17. k3 ¼ 1:0, ~ k1 ¼ 0:8; ~ k2 ¼ 0:8; ~ k3 ¼ 0:43 and v 1 ¼ 0:64; v
l ¼ 0:4; l~ ¼ 0:3; k1 ¼ 1:8; k2 ¼ 0:5;
230
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
corresponds to two distinct branches of the dispersion relation. This is not too surprising, as will be seen in Figs. 7 and 8 that the harmonics group together to form distinct pairs in the high wave number region. Eq. (75) may now be used, in conjunction with Eqs. (72) and (67)1, to obtain
8 2 p 2 n 2 /^ o > 6ffi < v 23 þ nþ1 a0 1 þ 4a1 nkdp pffiffiffi þ . . . n odd; 2 kd a0 v 2n ¼ > n o þ
^ 6ffi : v 2 þ np 2 a0 1 þ 4a1 np 2 p/ffiffiffi þ . . . n ev en; 3 2kd kd a0
ð76Þ
^ 6 ¼ / =np; / ^ þ and / ^ representing solutions of Eq. (75) where / 6 associated with the positive and the negative square roots, respectively. The asymptotic expansions derived for the phase speed are superimposed upon the numerical solutions in Figs. 7 and 8. Fig. 7 shows the second order expansions. Each value on n used in (72) generates an approximate solutions that passes between the 2nth and ð2n 1Þth branches of the numerical dispersion curves, clearly indicating that the second order expansion is not sufficient to describe the behaviour of the numerical solutions accurately. When the expansions are taken to higher order, as in Fig. 8, the oscillations are fully accounted for by the trigonometric functions within the third order term of Eq. (76) and the two roots of the quadratic equation in /6 now provide an asymptotic approximation for each harmonic. These provide a highly reasonable approximation to the phase speed both in the high and moderate kh region. 6. Concluding remarks
Appendix A The constants Ci ; i ¼ 1; . . . ; 6, are defined by
1 q 2 b g~0 ~f 1 þ g~11~f 0 ðhðq 2 Þ hðq 1 ÞÞ; þq
~ 11 g~0 ¼ a
~f 0 ¼ ðc ~c ~2 r2 Þg~0 b ~2 a0 ; ~0 ¼ a ~ ~ 22 g~0 a ~ 12 b; h
~f 1 ¼ b ~c ~ 2 ðc ~2 r2 Þg~1 ;
~1 ¼ a ~ 22 g~1 : h
Appendix B The constants Ai ; i ¼ 1; . . . ; 6, are defined by
A1 ¼ h0 f 0 a0 D10 ; A2 ¼
3 1 f 0 ð4a20 h2 h0 þ 8a20 nh0 Þ þ 4a0 ðf 1 h1 þ f 2 h0 Þ D10 4a0
a0 ðf 0 h1 þ f 1 h0 ÞD11 h0 f 0 a0 D12 ;
A4 ¼
1 ðf h0 2a0 ðf 1 h0 f 0 h11 ÞÞD10 4a0 0 1 D31 ð2a0 ðf 1 h0 f 0 h1 Þ f 0 h0 Þ; 2
pffiffiffiffiffi 1 D10 2a0 ðf 0 h2 f 1 h1 þ f 2 h0 Þ þ f 0 h1 f 1 h0 þ 4 a0 nf 0 h0 2 1 h0 f 0 D31 þ h0 f 0 a0 D32 ; 2
1 A5 ¼ pffiffiffiffiffi ð2a0 ðf 1 h0 f 0 h1 Þ f 0 h0 ÞD21 ; 2 a0
pffiffiffiffiffi A6 ¼ h10 f 10 a0 D22 : m
Within the above definitions, Dim represents the coefficients of b within Di , which may be obtained by inserting the expansions given ~1C ~2, in Eqs. (63) and (68) into Eq. (27), dividing throughout by C 1 C 2 C and replacing the resulting hyperbolic tangents with unity. These are given explicitly by:
~ ~ ~ ~ ~ D10 ¼ 2ðbðp 20 h10 ðf 0 g 10 g 0 f 10 Þ þ p10 h10 ðg 0 f 20 f 0 g 20 ÞÞ ~ ~10 ðf g~20 h ~20 g ~f 20 Þ þ p h ~ ~ ~ þ bðp10 h 0 0 20 20 ðg 0 f 10 f 0 g 10 ÞÞÞ;
0
1
0
11
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D12 ¼ 2ðbðp12 h 0 0 22 20 ðg 0 f 10 f 0 g 10 Þ ~12 þ g~22 h ~10 Þ þ h ~10 ðf g~20 g ~f 22 Þ ~f 20 ðg h ~ þ p10 ðf 0 ðg~20 h 2 0 0 12 ~10 ÞÞ þ p ðh ~20 ðg ~f 12 f g~12 Þ g~10 ðf h ~20 f h ~22 Þ g h
pffiffiffiffiffi ~0 b ; ~h 1 Þ~f 0 f ðq 1 Þg~0 hðq 2 Þb a0 gðq
12
20
0
0
2
0
~22 þ g h ~ ~ ~ ~ þ ~f 10 ðg 0 h 12 20 ÞÞÞ þ bðp22 h10 ðf 0 g 10 g 0 f 10 Þ
~0 b ; ~h 1 Þb hðq
1 n 2 a0 o 1 n 2 Þ gðq 1 Þð~f 0 þ 2a0 ~f 1 Þ þ f ðq 1 Þð2a0 g~1 g~0 Þ ; pffiffiffiffiffi b2 hðq 2 a0
a11 a22 þ c1 c2 b2 þ 2a22 c2 ða0 þ a2 Þ ~ ~2 ; þ c2 a0 ; g~1 ¼ c a22 þ c2
þ p10 ð~f 20 ðg 0 h11 þ g 11 h10 Þ g~20 ðf 1 h10 f 0 h11 ÞÞÞÞ;
1 q 2 b g~0 ~f 1 þ g~11~f 10 ðhðq 2 Þ hðq 1 ÞÞ; q
~0 ~f 0 þ 2a0 ðh ~0 ~f 1 þ h ~1 ~f 0 Þ þ f ðq ~0 g~0 h ~ 1 Þ g~0 h ~0 5 ¼ pffiffiffiffiffi b gðq 1 Þ h 1 Þ 2a0 ðg~1 h C 1
where
20
~1 b 2 ¼ a0 h ~ðf ðq 2 Þgðq 1 Þ f ðq 1 Þgðq 2 ÞÞ C
pffiffiffiffiffi
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D11 ¼ 2ðbðp10 h 1 11 20 20 ðg 11 f 10 f 1 g 10 ÞÞ ~ ~ þ bðp ðg~10 ðf h11 þ f h10 Þ f 10 ðg h11 g h10 ÞÞ
~1 b 1 ¼ a0 h ~ðf ðq 2 Þgðq 1 Þ f ðq 1 Þgðq 2 ÞÞ C
4 ¼ a0 gðq 2 Þ~f 0 f ðq 2 Þg~0 C
2 a0 o o 1 n~ ~ 1 Þ gðq 2 Þð~f 0 þ 2a0 ~f 1 Þ þ f ðq 2 Þð2a0 g~1 g~0 Þ ; q g~0 h þ pffiffiffiffiffi bhð 0 2 a0
A3 ¼
In this paper, the dispersion of symmetric (extensional-like) waves in a symmetric three-layer structure has been investigated. The investigation has been carried out within the framework of compressible, pre-stressed elastic media, with each layered assumed composed of such material. The dispersion relation is a mathematically elaborate transcendental equation, derived from a six by six determinant, which gives phase speed (or frequency) as an implicit function of wave number. A thorough numerical analysis reveals the nature of the dispersion relation and provides a sign post for the derivation of some short wave approximation of the harmonics in each of the six different cases which are shown to exist. These approximations provide can explicit relationship between phase speed and wave number and are shown to provide highly accurate approximations over a wide wave number range. Having these explicit approximations may prove helpful to approximate numerical truncation errors, as well as provide an excellent first approximation in numerical dispersion schemes for particularly challenging sets of material parameters.
3 ¼ C
n
1 ~ ~f þ 2a ðh ~ ~f þ h ~ ~f Þ þ f ðq ~ g~ h ~ 6 ¼ p 2 Þ h 2 Þ 2a0 ðg~1 h ffiffiffiffiffi b gðq C 0 0 0 0 1 1 0 0 0 1Þ
þ p12 h10 ðg 0 ~f 20 f 0 g~20 Þ þ p20 ðf 0 ðh10 g~12 þ h12 g~10 Þ þ g~10 ðf 1 h11 o
þ f 2 h10 Þ g 0 ðh10 ~f 12 h12~f 10 Þ ~f 10 ðg 11 h11 h10 g 12 ÞÞ þ p10 ðh10 ðg 0 ~f 22 f 2 g~20 Þ g~20 ðf 0 h12 f 1 h11 Þ h10 ðf 0 g~22 þ g 12~f 20 Þ þ ~f 20 ðg 0 h12 þ g 11 h11 ÞÞÞÞ;
231
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
0.7
Numerical solutions Asymptotic expansions
0.65
0.6
0.55
v
0.5
0.45
0.4
0.35 0
10
20
30
40
50
60
kh Fig. 8. Comparison of numerical solutions with third order asymptotic expansions (76) for case 6. The same material parameters from Fig. 7 have been used.
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D31 ¼ 2ðbðp10 h 1 11 20 20 ðg 11 f 10 f 1 g 10 ÞÞ ~ ~ ~ ~ ~ þ bðp 20 ðh10 ðf 1 g 10 g 11 f 10 Þ h11 ðf 0 g 10 þ g 0 f 10 ÞÞ ~ þ p ðh11 ðf g~20 g f 20 Þ h10 ðf g~20 þ g ~f 20 ÞÞÞÞ; 10
0
0
1
11
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D32 ¼ 2ðbðp12 h 0 0 22 20 ðg 0 f 10 f 0 g 10 Þ ~ ~ ~ ~ ~ 12 þ p10 ðf 0 ðg~20 h12 þ g~22 h10 Þ þ h10 ðf 2 g~20 g 0 f 22 Þ ~f 20 ðg 0 h ~10 ÞÞ þ p ðh ~20 ðg ~f 12 f g~12 Þ g~10 ðf h ~20 f h ~22 Þ g h 12
20
0
0
2
0
~22 þ g h ~ ~ ~ ~ þ ~f 10 ðg 0 h 12 20 ÞÞÞ þ bðp22 h10 ðf 0 g 10 g 0 f 10 Þ þ p12 h10 ðg 0 ~f 20 f 0 g~20 Þ þ p20 ðf 0 ðh10 g~12 þ h12 g~10 Þ g~10 ðf 1 h11 þ f 2 h10 Þ g 0 ðh10 ~f 12 h12 ~f 10 Þ þ ~f 10 ðg 11 h11 h10 g 12 ÞÞ þ p10 ðg~20 ðf 1 h11 f 0 h12 Þ h10 ðf 0 g~22 f 2 g~20 Þ þ g 0 ðh10 ~f 22 þ h12~f 20 Þ ~f 20 ðg 11 h11 þ h10 g 12 ÞÞÞÞ; ~10 h ~20 Þðf g f g Þ þ 2a0 h11 bðg~20 ~f 10 g~10 ~f 20 Þ; ~ h D21 ¼ 2p10 p20 bð 1 0 0 11 ~10 h ~ 20 Þðf g f g Þ: ~ h D22 ¼ 2a0 h11 bðg~20 ~f 10 g~10 ~f 20 Þ þ 2p10 p20 bð 1 0 0 11 m Similarly, g m ; f m ; hm ; g~im ; ~f im and g~im are the Oðb Þ term within ~ ~ expansions of gðqÞ; f ðqÞ; hðqÞ; g~ðpi Þ; f ðpi Þ and hðp1 Þ, respectively, thus
2
3
f ðq1 Þ; f ðq2 Þ ¼ f 0 f 1 b þ f 2 b þ Oðb Þ; 2
3
gðq1 Þ; gðq2 Þ ¼ g 0 g 11 b þ g 12 b þ Oðb Þ; 2
3
hðq1 Þ; hðq2 Þ ¼ h10 h11 b þ h12 b þ Oðb Þ; ~f ðp Þ ¼ ~f 10 þ ~f 12 b2 þ Oðb4 Þ; 1 2 4 g~ðp1 Þ ¼ g~10 þ g~12 b þ Oðb Þ;
~f ðp Þ ¼ ~f 20 þ ~f 22 b2 þ Oðb4 Þ; 2 2 4 g~ðp2 Þ ¼ g~20 þ g~22 b þ Oðb Þ;
~ Þ¼h ~10 þ h ~12 b2 þ Oðb4 Þ; hðp 1
~ Þ¼h ~20 þ h ~22 b2 þ Oðb4 Þ; hðp 2
where
f 0 ¼ g 0 ðc2 r2 Þ a0 c2 b;
f 1 ¼ c2 b þ ðc2 r2 Þg 11 ;
f2
¼ g 12 ðc2 r2 Þ c2 a1 b; g 0 ¼ a11 a1 þ c2 a0 ; h10 ¼ g 0 a22 a12 b;
g 11 ¼ c2 ;
g 12 ¼ a2 þ c2 a1 ;
h11 ¼ a22 g 11 ;
h12 ¼ a22 g 12 ;
~ 11 a1 c ~2 p210 ; g~10 ¼ a
~ 11 a1 c ~2 p220 ; g~20 ¼ a
~2 p10 p12 ; g~12 ¼ 2a2 c
~2 p20 p22 ; g~22 ¼ 2a2 c
~f 10 ¼ b ~c ~2 p210 þ ðc ~2 r2 Þg~10 ;
~f 12 ¼ 2b ~c ~2 p10 p12 þ ðc ~2 r2 Þg~12 ;
~f 20 ¼ b ~c ~2 p220 þ ðc ~2 r2 Þg~20 ;
~f 22 ¼ 2b ~c ~2 p20 p22 þ ðc ~2 r2 Þg~22 ;
~10 ¼ a ~ ~ 22 g~10 a ~ 12 b; h ~12 ¼ a ~ 22 g~12 ; h
~20 ¼ a ~ ~ 22 g~20 a ~ 12 b; h
~22 ¼ a ~ 22 g~22 : h
References Dowaikh, M., Ogden, R., 1991a. On surface waves and deformations in a compressible elastic half space. Stab. Appl. Anal. Cont. Media 1, 27–45 (vol. 31, no. 12, pp. 1611–1639). Dowaikh, M., Ogden, R., 1991b. Interfacial waves and deformations in pre-stressed elastic media. Proc. R. Soc. A 433, 313–328 (vol. 31, no. 12, pp. 1611–1639). Kaplunov, J., Markushevich, D., 1993. Plane vibrations and radiation of an elastic layer lying on a liquid half-space. Wave Motion 17 (3), 199–211. Kaplunov, J., Kossovitch, L., Nolde, E., 1998. Dynamics of Thin Walled Elastic Bodies. Academic Press. Lamb, H., 1917. On waves in an elastic plate. Proc. R. Soc. A 93 (648), 114–128 (containing papers of a mathematical and physical character). Mindlin, R., 1960. Waves and vibrations in isotropic, elastic plates. Structural Mechanics. Pergamon Press Inc., Oxford, New York, pp. 199–232. Nolde, E., Prikazchikova, L., Rogerson, G., 2004. Dispersion of small amplitude waves in a pre-stressed, compressible elastic plate. J. Elast. 75 (1), 1–29.
232
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
Ogden, R., Roxburgh, D., 1993. The effect of pre-stress on the vibration and stability of elastic plates. Int. J. Eng. Sci. 31 (12), 1611–1639. Pichugin, A., Rogerson, G., 2002. An asymptotic membrane-like theory for longwave motion in a pre-stressed elastic plate. Proc. R. Soc. A 458 (2022), 1447– 1468. Rogerson, G., 1992. Penetration of impact waves in a six-ply fibre composite laminate. J. Sound Vib. 158 (1), 105–120. Rogerson, G., 1997. Some asymptotic expansions of the dispersion relation for an incompressible elastic plate. Int. J. Solids Struct. 34 (22), 2785–2802.
Rogerson, G., 1998. On the existence of surface waves and propagation of plate waves in pre-stressed fibre-reinforced composites. J. Mech. Phys. Solids 46 (1998), 1581–1612. Rogerson, G.A., Sandiford, K.J., 2000. The effect of finite primary deformations on harmonic waves in layered elastic media. Int. J. Solids Struct. 37, 2059–2087. Roxburgh, D., Ogden, R., 1994. Stability and vibration of pre-stressed compressible elastic plates. Int. J. Sci. 32 (3), 427–454. Sandiford, K., Rogerson, G., 2000. Some dynamic properties of a pre-stressed, nearly incompressible (rubber-like) elastic layer. Int. J. Non-linear Mech. 35 (5), 849–868.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Dispersion phenomena in symmetric pre-stressed layered elastic structures M.I. Lashhab a, G.A. Rogerson b,⇑, K.J. Sandiford c a
University of Almergeb, Faculty of Science, Zliten, Libya School of Computing and Mathematics, University of Keele, Keele, United Kingdom c School of Computing, Science and Engineering, University of Salford, Salford, United Kingdom b
a r t i c l e
i n f o
Article history: Received 14 November 2014 Available online 14 January 2015 Keywords: Pre-stress Layered media Waves Dispersion Short waves
a b s t r a c t The dispersion relation associated with a symmetric three layer structure, composed of compressible, pre-stressed elastic layers, is derived. This mathematically elaborate transcendental equation gives phase speed as an implicit function of wave number. Numerical solutions are established to show a wide range of dispersion behaviour which is delicately dependent on the material parameters and pre-stress in each layer. Particularly interesting behaviour is observed within the short wave (high wave number) regime, with six possible cases of short wave liming behaviour shown possible. Within each of these, a short wave asymptotic analysis is carried out, resulting in a set of approximations which provide explicit relationships between phase speed and wave number. It is envisaged that these approximations may prove helpful to approximate numerical truncation errors associated with impact response, as well as providing excellent first approximations for particularly (numerically) challenging sets of material parameters. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Many modern technological applications exist in which layered elastic material components are employed. Amongst the most common, we cite the use of layered composites within the aerospace industry and rubber-like vibration control devices used for earthquake protection in bridges and tall buildings. There are also examples of naturally occurring layered structures in geo and biomechanics. With these and other applications in mind, a significant research effort has in recent times been focussed on elucidation of the dynamic properties of layered elastic media. As the complexity of application has increased, considerable effort has been put into a complete understanding of dispersion in layered media. Seemingly the first attempt to consider non-fundamental modes was made in Lamb (1917) for a plane section of an isotropic plate. A more complete study of higher modes of an isotropic plate was later presented in Mindlin (1960). Because of the very complicated structure of the underlying dispersion relations, most of the results in these early papers were obtained through use of numerical computations.
⇑ Corresponding author. E-mail addresses: [email protected] (M.I. Lashhab), [email protected] (G.A. Rogerson), [email protected] (K.J. Sandiford). http://dx.doi.org/10.1016/j.ijsolstr.2015.01.006 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.
In this present contribution, we aim to add to current understanding by investigating wave propagation in a symmetric 3-layer laminate, with each layer composed of compressible, pre-stressed elastic material. The paper will specifically focus on symmetric (extensional-type) waves and as such generalise the constitutive framework of a study done some years ago for incompressible pre-stressed elastic layers, see Rogerson and Sandiford (2000). Within this present paper a thorough investigation of the dispersion relation associated with the aforementioned structure is carried out. An initial numerical investigation is used to enable short wave asymptotic approximations, giving phase speed explicitly in terms of wave number, to be established. We remark that a detailed understanding of the behaviour of the dispersion relation is a critical pre-requisite in determining dynamic response. Although traditionally much effort has been afforded to fundamental modes, there is very good motivation to not neglect the harmonics. For example, the solution of any impact problem will generally involve contributions from each mode over the entire wave number regime. Moreover, important features, such as surface or interfacial waves, may arise from the cumulative affects of the harmonics, rather than result from the short wave limit of a single mode, see Rogerson (1992). We also remark that Kaplunov et al. (1998) and Kaplunov and Markushevich (1993) show that for certain types of vibration the contributions of higher modes is significant.
221
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
The problem of dispersion in layered pre-stressed media has seen a number of publications over the last twenty years. Plane incremental waves in incompressible and compressible elastic single-layer plates (with traction free boundaries) were discussed in Ogden and Roxburgh (1993) and Roxburgh and Ogden (1994), respectively. An asymptotic short wave analysis for a single layer plate was carried out in Rogerson (1997), Sandiford and Rogerson (2000) and Nolde et al. (2004) for incompressible, nearly incompressible and compressible materials, respectively. An extension to three dimensional motion was carried in Pichugin and Rogerson (2002) in respect of incompressible elastic single layer plates. Many of the aforementioned papers are a good source of references to earlier work. The paper is organised as follows. In Section 2 a very brief review of the underlying constitutive theory, and equations of motion, is presented. The dispersion relation is derived in Section 3. A numerical analysis of the dispersion relation is presented in Section 4. From this numerical analysis it is shown that the short wave limiting behaviour may be classified into six distinct cases. Each of these cases are analysed in Section 5, within which short wave asymptotic approximations are established in each case. These approximations, give phase speed as an explicit function of wave number and material parameters for each mode and are shown to provide excellent agreement with the numerical solution over a surprisingly large wave number regime. It is envisaged that these might provide some help in the numerical inversion of the highly oscillatory wave number integrals associated with impact problems. 2. Governing equations Only a brief summary of the relevant underlying theory is presented; for further details the reader is referred to [8]. We begin by considering a homogeneous elastic body B, possessing a natural isotropic unstressed state B0 . A purely homogeneous static deformation is imposed, resulting in the pre-stressed equilibrium state Be . Upon Be , we superimpose a small time dependent motion ui ðX A ; tÞ, resulting in the current material state Bt . Position vectors xi ðX A ; tÞ of a representative particle are denoted by X A ; xi ðX A Þ and ~ xi expressible in the form in B0 ; Be and Bt respectively, with ~
x~i ðX A ; tÞ ¼ xi ðX A Þ þ ui ðX A ; tÞ:
ð1Þ
The deformation gradients associated with the deformations and take the compoB0 ! Bt and B0 ! Be are denoted by F and F nent forms
F iA ¼
@ x~i ; @X A
@xi : FiA ¼ @X A
ð2Þ
may be related through On making use of Eqs. (1) and (2), F and F
F iA ¼ ðdij þ ui;j ÞFjA :
ð3Þ
The most general isotropic strain-energy function is of the form
WðFÞ ¼ WðI1 ; I2 ; I3 Þ
ð4Þ
with I1 ; I2 and I3 principal invariants of the left Cauchy–Green strain tensor. The equations of infinitesimal incremental motion may be written as
piA;A ¼ q0 u€i ; piA ¼
@W @F iA
ð5Þ
with q0 the density per unit volume of B0 ; piA components of the first Piola Kirchhoff stress tensor and a superimposed dot indicating differentiation with respect to time. A linearised form (5) is obtainable in the form
€i Apiba ua;bp ¼ qe u
ð6Þ
with qe the material density per unit volume of Be and Apiba the fourth order elasticity tensor, defined in the component form
Apiba ¼ J 1 FbC FpA
@ 2 W @F iA @F aC
ð7Þ
:
F¼F
The components of A allow especially simple representation relative to the principal axes of the primary static deformation. The only non-zero components have the form Aiijj ; Aijij or Aijji with i; j 2 f1; 2; 3g. These are given in terms of the principal stretches km , with m 2 f1; 2; 3g, as
Aiijj ¼ J 1 ki kj
Aijij ¼
@2W ; @ki @kj
ð8Þ
8 1 > ki @W kj @W > @ki @kj
iiii
k2i
k2i k2j
Aiijj þ J
Aijji ¼ Ajiij ¼ Aijij J 1 ki
1
ki @W @ki
@W 0 @ki
i – j ki – kj ; ð9Þ i – j ki – kj ;
i – j:
ð10Þ
We also need a linearised measure of incremental traction, obtainable as
si ¼ Ajilk uk;l nj
ð11Þ
with n the outward unit normal to a material surface in Be . Our aim is to consider wave propagation in layered media, layers having finite thickness in one spatial direction and infinite lateral extent in the other two. Each layer is composed of elastic material characterised by the strain energy function (4). A Cartesian coordinate system Ox1 x2 x3 is chosen, coincident with the principal axes of deformation in Be , with Ox2 normal to the upper traction free surface. A state of plane strain is assumed, with both u1 and u2 independent of x3 and u3 0. On making use of (8)–(10) in (6), the two non-trivial equations of motion are expressed as
€1 ; A1111 u1;11 þ ðA1122 þ A2112 Þu2;21 þ A2121 u1;22 ¼ qe u
ð12Þ
€2 : ðA1221 þ A2211 Þu1;12 þ A1212 u2;11 þ A2222 u2;22 ¼ qe u
ð13Þ
We now seek solutions of (12) and (13) in the form
ðu1 ; u2 Þ ¼ ðU; VÞekqx2 eikðx1 v tÞ
ð14Þ
yielding two linear homogeneous equations with non-trivial solutions if
a22 c2 q4 þ fa22 ðv 2 a11 Þ þ c2 ðv 2 c1 Þ þ b2 gq2 þ ðv 2 a11 Þðv 2 c1 Þ ¼ 0 with
v
2
ð15Þ
¼ qe v and aij ; ci and b defined by 2
aij ¼ Aiijj ; i; j 2 f1; 2g; c1 ¼ A1212 ; c2 ¼ A2121 ; b ¼ a12 þ c2 r2
ð16Þ
and with r2 ¼ c2 A1221 the principal Cauchy stress along Ox2 . If the two roots of equation (15) are denoted by q21 and q22 , we note these may be either real, purely imaginary or complex conjugates, U and V may be presented as linear combinations of four generally independent solutions. On making use of equations (12) and (13), U and V may be expressed as
U¼
2 X
U ð2m1Þ ekqm x2 þ U ð2mÞ ekqm x2 ;
ð17Þ
m¼1
V¼
2 X gðqm Þ ð2m1Þ kqm x2 qm U e U ð2mÞ ekqm x2 d ; b m¼1
ð18Þ
222
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
where gðqm Þ ¼ a11 v 2 c2 q2m and U ðiÞ ; i 2 f1; 2; 3; 4g, are disposable constants. Solutions (17) and (18), with (11), are now employed to cast the linearised incremental traction components as
s1 k
s2 ik
¼
2 X f ðqm Þ ð2m1Þ kqm x2 qm U e U ð2mÞ ekqm x2 ; b m¼1
ð19Þ
¼
2 X hðqm Þ ð2m1Þ kqm x2 U e þ U ð2mÞ ekqm x2 ; b m¼1
ð20Þ
1
1
1
2
1
1
2
2
ð25Þ
An explicit representation of the dispersion relation is obtained as
where
f ðqm Þ ¼ bc2 q2m þ ðc2 r2 Þgðqm Þ;
Equations (24) will yield non-trivial solutions provided q1 hðq1 ÞC 1 q1 hðq1 ÞS1 q2 hðq2 ÞC 2 q2 hðq2 ÞS2 0 0 f ðq ÞS f ðq1 ÞC 1 f ðq2 ÞS2 f ðq2 ÞC 2 0 0 1 1 ~ ~ ~ ~ q1 b 0 q2 b 0 p1 bC 1 p2 bC 2 ¼ 0: 0 gðq1 Þ 0 gðq2 Þ g~ðp1 Þ~S1 g~ðp2 Þ~S2 ~f ðp Þ~S1 ~f ðp Þ~S2 0 f ðq1 Þ 0 f ðq2 Þ 1 2 q hðq Þ ~ ÞC ~ ÞC ~ 1 p hðp ~2 0 q hðq Þ 0 p hðp
hðqm Þ ¼ a22 gðqm Þ a12 b:
ð21Þ
We note that q ¼ 0 corresponds to propagation in unbounded media, Eq. (15) then yielding v 21 ¼ a11 or c1 . These relate to two body waves propagating along Ox1 , the first longitudinal, the second shear.
We consider a 4-ply (3 layer) symmetric laminated structure, formed of two identical outer layers of thickness h, perfectly bonded to an inner core of thickness 2d, all formed of isotropic pre-stressed elastic materials of the same density. The axes of the left Cauchy-Green strain tensor are assumed coincident for all layers, with one normal to the upper surface. The material parameters of the (identical) outer layers are as denoted in the preceding section, with the inner core’s parameters distinguished by imposing an over tilde. Accordingly, solutions for the inner core are sought in the form
ð22Þ
If (22) is inserted into (12) and (13), displacement and traction components may be inferred from (17)–(20). Continuity requires that ri ¼ r~ i ; i ¼ 1; 2, with the (infinitesimal) boundary and continuity conditions given by
s1 ¼ s2 ¼ 0 at x2 ¼ ðd þ hÞ; ~ 1 ; U2 ¼ U ~ 2 ; s1 ¼ s ~ 1 ; s2 ¼ s ~2 at x2 ¼ d: U1 ¼ U
ð23Þ
the six equations 2 X f ðqm Þ qm ðU ð2m1Þ ekqm ðdþhÞ U ð2mÞ ekqm ðdþhÞ Þ ¼ 0; b m¼1 2 X hðqm Þ ð2m1Þ kqm ðdþhÞ ðU e þ U ð2mÞ ekqm ðdþhÞ Þ ¼ 0; b m¼1 2 X ~ m ¼ 0; ~ ð2m1Þ C U ð2m1Þ ekqm d þ U ð2mÞ ekqm d U 2 X gðqm Þ ð2m1Þ kqm d g~ðpm Þ ~ ð2m1Þ ~ U ðU e U ð2mÞ ekqm d Þ Sm ¼ 0; ~ bqm bp m m¼1 2 ~f ðp Þ X f ðqm Þ ð2m1Þ kqm d m ~ ð2m1Þ ~ Sm ¼ 0; ðU e U ð2mÞ ekqm d Þ U ~ bqm bp m m¼1 2 ~ X hðp hðqm Þ ð2m1Þ kqm d m Þ ~ ð2m1Þ ~ U ðU e þ U ð2mÞ ekqm d Þ C m ¼ 0: ~ b b m¼1
q2 f ðq1 Þhðq2 ÞS1 S2 gD3 þ fq1 f ðq2 Þhðq1 ÞS1 S2 q2 f ðq1 Þhðq2 ÞC 1 C 2 gD4 fq1 f ðq2 Þhðq1 ÞC 2 S1 q2 f ðq1 Þhðq2 ÞS2 C 1 gD5 ¼ 0;
ð26Þ
~ Þ bhðq ~ D1 ¼ ~S2 C~ 1 p1 fð f ðq2 Þg~ðp2 Þ ~f ðp2 Þgðq2 ÞÞðbhðp 1 2 ÞÞg ~ ~ ~ 2 p fð f ðq Þg~ðp Þ f ðp Þgðq ÞÞðbhðp Þ bhðq ~ ~S1 C 2 2 1 1 2 2 2 ÞÞg; ~ ~ ~ ~ ~ D2 ¼ C 1 C 2 p1 p2 bfð f ðq1 Þgðq2 Þ f ðq2 Þgðq1 ÞÞðhðp1 Þ hðp2 ÞÞg; ~ Þ bhðq ~ D3 ¼ ~S2 C~ 1 p1 q2 fð f ðq1 Þg~ðp2 Þ ~f ðp2 Þgðq1 ÞÞðbhðp 1 2 ÞÞg ~ Þ bhðq ~ 2 p q fð f ðq Þg~ðp Þ ~f ðp Þgðq ÞÞðbhðp ~ ÞÞg; þ ~S1 C 2 2
1
1
1
1
2
2
~ Þ bhðq ~ D4 ¼ ~S2 C~ 1 p1 q1 fð f ðq2 Þg~ðp2 Þ ~f ðp2 Þgðq2 ÞÞðbhðp 1 1 ÞÞg ~ ~ ~ ~ ~ S1 C 2 p q fð f ðq Þg~ðp Þ f ðp Þgðq ÞÞðbhðp Þ bhðq ÞÞg; 2 1
2
1
1
2
2
1
D5 ¼ ~S1 ~S2 q1 q2 bfð~f ðp1 Þg~ðp2 Þ ~f ðp2 Þg~ðp1 ÞÞðhðq1 Þ hðq2 ÞÞg; ~ Þ bhðq ~ D6 ¼ S~2 C~ 1 p1 fð f ðq1 Þg~ðp2 Þ ~f ðp2 Þgðq1 ÞÞðbhðp 1 1 ÞÞg ~ ~ ~ ~ ~ S1 C 2 p fð f ðq Þg~ðp Þ f ðp Þgðq ÞÞðbhðp Þ bhðq ÞÞg: 2
1
1
1
1
2
1
ð27Þ 4. Numerical solutions of the dispersion relations
Inserting appropriate forms of (17)–(20) into (23) yields a system of twelve equations in twelve unknowns. In view of symmetry, this may be reduced to two systems of six equations in six unknowns, one analogous to the classical extensional wave problem, one with the flexural problem. It is the former wave type that we shall focus attention upon in the remainder of this paper. For symmetric ~ 2 and s ~1 vanish at x2 ¼ 0, implying (extensional-like) solutions, U ð1Þ ð2Þ ~ ~ ~ ð3Þ ¼ U ~ ð4Þ , thus yielding from (18) and (19) that U ¼ U and U
m¼1
q2 f ðq1 Þhðq2 ÞS1 C 2 gD2 þ fq1 f ðq2 Þhðq1 ÞC 1 C 2
within which
3. Derivation of the dispersion relation
~ VÞe ~ kpx2 eikðx1 v tÞ : ðu1 ; u2 Þ ¼ ðU;
q1 q2 ff ðq1 Þhðq1 ÞD1 þ f ðq2 Þhðq2 ÞD6 g fq1 f ðq2 Þhðq1 ÞC 1 S2
ð24Þ
Throughout this section, various numerical solutions are presented. The parameters are chosen to demonstrate the range of material response, rather than model any specific material. In examining the numerical behaviour of the dispersion relation, we first use the modified Varga strain energy function
W ¼ lðk1 þ k2 þ k3 3Þ þ
1 jðJ 1Þ2 ; 2
J ¼ det F ¼ k1 k2 k3
ð28Þ
with l and j shear and bulk moduli, respectively. Fig. 1 presents a against scaled wave number kh for the plot of scaled phase speed v first 28 branches of the dispersion relation (26). First note that in the low wave number limit ðkh; kd ! 0Þ only the fundamental mode retains finite wave speed, with all harmonics having an associated infinite limit. In the high wave number limit ðkh; kd ! 1Þ the fundamental mode tends to a distinct limit, which is in fact the R 0:67, of the material of the outer Rayleigh surface wave speed, v layers. In this high wave number limit, all harmonics asymptote, from above, to the longitudinal wave speed associated with outer layers. It is observed numerically that in this case one of q1 or q2 is imaginary, the other real, with p1 and p2 both real. As kh; kd ! 1, the magnitude of the imaginary q tends to zero, with 2 ¼ v 21 ¼ a11 . Similar the limiting wave speed, from (15), given by v behaviour is also observed when p1 and p2 are complex conjugates and we therefore define, without loss of generality, case 1 as:
^1 ; q2 real; p1 and p2 real or complex conjugates; case ð1Þ : q1 ¼ iq ^1 ! 0 and v 2 ! v 21 ¼ a11 : as kh; kd ! 1 q
223
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
A further prominent feature in Fig. 1 is the flattening of the disper¼ sion curves to form so-called ghost lines around v ¼ v~ 1 1:17; v v 2 1:543. These are longitudinal and shear wave fronts associated with the material of the inner core. A second case, analogous to case 1, occurs when all harmonics asymptote, from above, to the scaled longitudinal wave speed ~ 1 . This case is classified through associated with the inner core, v
^1 ; p2 real; q1 and q2 real or complex conjugates; case ð4Þ : p1 ¼ ip ~1 : ^1 ! 0 and v 2 ! v~ 22 ¼ c as kh; kd ! 1 p The final strain energy function employed is the Blatz–Ko strain energy
W¼
^1 ; p2 real; q1 and q2 real or complex conjugates; case ð2Þ : p1 ¼ ip ~ 11 : ^1 ! 0 and v 2 ! v~ 21 ¼ a as kh; kd ! 1 p The second form of strain energy employed is the modified Neo-Hookean
W ¼ c1 ðI1 3Þ þ
1 jðJ 1Þ2 2
ð29Þ
with c1 a material parameter and j is the bulk modulus. In Fig. 2, we again note a single finite long wave limit and distinct short wave limit of the fundamental mode. All harmonics are noted to tend to the shear wave speed associated with the two outer layers. In this case, it is observed numerically that one of q1 or q2 is imaginary, the other real, whilst p1 and p2 are both real. Moreover, as kh; kd ! 1, the magnitude of the imaginary q tends to zero. The limiting wave speed of the harmonics is therefore obtainable from equation (15) 2 ¼ v 22 ¼ c1 0:685. This case may therefore classified as as v follows
^1 ; q2 real; p1 and p2 real or complex conjugates; case ð3Þ : q1 ¼ iq
2 J2
! þ 2J 5 :
ð30Þ
Fig. 3 illustrates a plot for the first twenty-eight branches of the dispersion relation (26) for layers with the Blatz–Ko strain energy (30). The fundamental mode and harmonics exhibit clear oscillatory behaviour in the high wave number region. This behaviour is a possible feature of pre-stressed materials which is never present within the classic linear isotropic framework, see Rogerson (1998). The short wave limiting speed of the harmonics gives rise to a wave front which is in general neither purely longitudinal nor purely shear in nature. In the limit kh; kd ! 1 both p1 and p2 are imaginary and jp1 j ! jp2 j, whilst q1 and q2 are complex conjugates and retain order 1 magnitude. This limiting behaviour is also observed when q1 and q2 are both real. The limiting wave speed of the harmonics, denoted by v~ 3 , is obtained by allowing the discriminant to vanish in the appropriate form of equation (15). We thus describe this case through the requirements
^1 ; p2 ¼ ip ^2 ; q1 and q2 real or complex conjugates; case ð5Þ : p1 ¼ ip as kh; kd ! 1 jp1 j ! jp2 j and v ! v~ 3 ;
^1 ! 0 and v 2 ! v 22 ¼ c1 : as kh; kd ! 1 q Shear and longitudinal wave fronts are again starting to form, around v 2:31; v 2:44, with another front also sharply defined 0:98. around v An analogous case, to that just presented as case 3, may also be classified within which the short wave limit of all harmonics is the shear wave speed associated with the inner core; this case is classified through
l I2
v
~ 23
¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 ðc ~ a ~2 a ~ 22 Þðc ~1 c ~2 a ~ 11 a ~ 22 Þ b ~2 þ a ~ 22 Þ þ 2b ~2 T~ 0 ~ 22 c ðc ~2 a ~ 22 Þ ðc
2
; ð31Þ
where
~2 ðc ~1 a ~ 11 Þðc ~2 a ~ 22 Þ: T~ 0 ¼ b
ð32Þ
A final case, analogous to case 5, is characterised by
2
1.8
1.6
1.4
v¯
1.2
1
0.8
0.6
0.4
0
5
10
15
20
25
30
kh against kh for a laminate formed of layers of materials with strain energy (28): outer layers Fig. 1. v l~ ¼ 2:5; j~ ¼ 1:5; ~k1 ¼ 1:97; ~k2 ¼ 0:99; ~k3 ¼ 0:47: v 1 ¼ 0:946; v 2 ¼ 1:543; v~ 1 ¼ 1:169; v~ 2 ¼ 1:546; v R ¼ 0:67.
l ¼ 2:0; j ¼ 1:0; k1 ¼ 1:6; k2 ¼ 0:8; k3 ¼ 0:7; inner core
224
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
3
2.5
2
v¯ 1.5
1
0.5
0
5
10
15
20
25
30
35
40
kh against kh for laminate formed layers of material with strain energy (29); outer layers c1 ¼ 0:2; Fig. 2. v ~3 ¼ 2:06; v ~ ¼ 0:4; ~ 1 ¼ 0:98; v 2 ¼ 0:685; v ~ 1 ¼ 2:44; v ~ 2 ¼ 2:31; v R ¼ 0:59. ~c1 ¼ 1:4; j k1 ¼ 1:7; ~ k2 ¼ 0:43, k
^1 ; q2 ¼ iq ^2 ; p1 and p2 real or complex conjugates; case ð6Þ : q1 ¼ iq as kh; kd ! 1 jq1 j ! jq2 j and v ! v 3 ;
v 23 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc2 a22 Þðc1 c2 a11 a22 Þ b2 ðc2 þ a22 Þ þ 2b a22 c2 T 0 ðc2 a22 Þ2
: ð33Þ
j ¼ 0:3; k1 ¼ 1:4; k2 ¼ 1:2; k3 ¼ 1:0; inner core
4.1. Some general remarks We note that there are six possible short wave phase speed limits for the harmonics. These are three wave speeds associated with the material of the inner core and three with the outer layers. The actual limiting wave speed is dependent on the material parameters and pre-stress. We define the limiting wave speed of the outer L ) to be the wave speed that the harmonics would tend layers (v
3.5
3
2.5
2
v 1.5
1
0.5
0
5
10
15
20
25
30
kh Fig. 3. Phase speed against scaled wave number for laminate formed of layers of material with strain energy (30); with v 3 ¼ 1:20 and v~ 3 ¼ 0:22.
v 1 ¼ 1:53; v 2 ¼ 1:24; v~ 1 ¼ 3:04 and v~ 2 ¼ 0:48 and
225
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
to in a single layer plate of the same material. Its value is one of three, dependent on aij ; ci and b, and given by
8 > < v 1 v L ¼ > v 2 : v 3
if c2 ðc1 a11 Þ > b2 and a11 < c1 ; if a22 ða11 c1 Þ > b2 and a11 > c1 ;
ð34Þ
otherwise
with the corresponding limit for the inner core identified by
8 > < v~ 1 v~ L ¼ > v~ 2 : v~ 3
~2 and a ~2 ðc ~1 a ~ 11 Þ > b ~ 11 < c ~1 ; if c ~2 and a ~ 22 ða ~ 11 c ~1 Þ > b ~ 11 > c ~1 ; if a
ð35Þ
otherwise:
1 ; v 2 ; v 3 Þ. However, it The limiting wave speed v L is in general minðv 1 ; v 2 Þ but v 3 < minðv 1 ; v 2 Þ. is exceptionally possible that v L ¼ minðv 3 be the limit is It will be remembered that the requirement for v that q1 and q2 are both imaginary and jq1 j ! jq2 j as kh ! 1. The 3 is less than both v 1 and v 2 but not case just described, when v the limiting wave speed, arises when jq1 j ! jq2 j but q1 and q2 are both real. The inequalities in (34) then give the range of pre-stress within which the appropriate requirements are satisfied for the given limit to occur. The actual limit of the harmonics in our symmetric 4-ply laminated plate is the lower of v L and v~ L , and, in the vast majority of cases, the lowest of the six possible values.
materials of the inner core and outer layers, respectively, and defined in (34) and (35). It is clear that in general there are six possible wave speed limits for the harmonics as kh; kd ! 1 and that the actual limit is dependent on material parameters. Each of these cases will be analysed in turn. !v 1 with a11 < c1 and c2 ðc1 a11 Þ b2 > 0 5.2.1. Case 1: v The first case we consider is when the limiting wave speed of the harmonics is the longitudinal wave speed associated with the outer layers. Numerical calculations indicate that for all harmonics v 2 approaches a11 from above and therefore, from (15), it is clear that the requirements outlined in case 1 on q1 ; q2 and p1 ; p2 are 2 < c1 are applicable. Accordingly, valid within the region a11 < v we seek to expand the dispersion relation (26) around the small ^1 and then use (15) to establish that quantity q
v 2 ¼ a11 þ c2 þ
2 þ Oðq ^21 Þ; q2 ¼ q
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðc1 a11 Þ b2
The numerical indications of short wave behaviour, discussed in Section 4, are now investigated.
where
n o ð1Þ ð1Þ ð1Þ ð1Þ fq1 f ðq2 Þhðq1 Þ q2 f ðq1 Þhðq2 Þg D2 þ D3 þ D4 D5 ¼ 0;
ð36Þ
in which a superscript ð1Þ indicates that Eq. (26) has been divided ~1C ~ 2 and the subsequent hyperbolic tangents throughout by C 1 C 2 C replaced with unity. After a little algebraic manipulation it can readily be established that the first factor of (36) is in fact the Rayleigh surface wave equation for the material of the outer layer, see Dowaikh and Ogden (1991a). We also note that the second factor is the Stoneley interfacial wave equation, see Dowaikh and Ogden (1991b), associated with the interface between the materials of the outer layers and the inner core. If both waves exist, in general the fundamental mode will tend to the least of these two wave speeds, with the first harmonic tending to the second. An important exception arises when the surface wave speed is greater than the shear and/or longitudinal wave speed of the core. In this case, a ghost line occurs in the dispersion relation, with associated flat maxima of the local group velocity modes in adjacent wave number regions, see Rogerson (1998). Moreover, in any numerical algorithm to determine impact response, based on integral transforms, great care must be taken so as not to miss such an important feature, when it arises through the combination of many harmonics over a large wave number regime, rather than the more commonly reported short wave contribution of a single mode.
^21 Þ; 1 þ Oðq p1 ¼ p
^21 Þ; 2 þ Oðq p2 ¼ p
ð38Þ
2 ; p 1 and p 2 are order 1 quantities defined by within which q
2 ¼ q
We first consider the short wave limit when q1 and q2 and p1 and p2 , are either real or complex conjugate pairs. In this case as kh; kd ! 1, the dispersion relation (26) tends to
ð37Þ
Expansions for q2 ; p1 and p2 are obtained using Eq. (37) with the appropriate form of Eq. (15), yielding
5. Asymptotic analysis
5.1. Surface and interfacial waves
! b2 ^2 þ Oðq ^41 Þ: q a11 c1 1
a22 c2
;
22 ¼ Fða11 Þ; p1 2 ; p
ð39Þ
~2 ~ 22 c ~2 Fðv 2 Þ ¼ a ~ 22 ða ~ 11 v 2 Þ þ c ~2 ðc ~1 v 2 Þ b 2a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a~ 22 ðv 2 a~ 11 Þ þ c~2 ðv 2 c~1 Þ þ b~2 4a~ 22 c~2 ðv 2 a~ 11 Þðv 2 c~1 Þ: To avoid proliferation of symbols an over-bar will be used to indicate the order 1 terms within expansions of q21 ; q22 ; p21 and p22 in a number of the limiting cases, the exact definition of each will change as necessary. The appropriate form of the dispersion relation for large wave number is now obtained by making use of (37) and (38) in (26), to provide
^1 hÞ ðq 2 hðq 2 Þðc2 r2 Þða11 c1 Þf1 Þ q ^21 ðða22 ða11 c1 Þ tan ðkq ^ a12 bÞf ðq2 Þg1 Þg ¼ q1 fq2 hðq2 Þðc2 r2 Þða11 c1 Þg1 2 Þf1 g þ Oðq ^21 Þ; ðða22 ða11 c1 Þ a12 bÞf ðq
ð40Þ
within which f1 and g1 are the order 1 quantities
~ p ~ q 2 p 2 ÞÞg 2 fððc2 r2 Þða11 c1 Þg~ðp 1 Þ a11 c1 ~f ðp 1 ÞÞðbhð 2 Þ bhð f1 ¼ q ~ p ~ q 2 p 2 ÞÞg 1 fððc r2 Þða11 c Þg~ðp 2 Þ a11 c ~f ðp 2 ÞÞðbhð 1 Þ bhð q 2
1
1
~ p ~ p ~ c r2 Þða11 c Þgðq 2 Þ f ðq 2 Þa11 c1 Þðhð 1 p 2 bfðð 1 Þ hð 2 ÞÞg; p 2 1 ~ p ~ a22 ða11 c Þ 1 Þ bðð g1 ¼ p1 fð f ðq2 Þg~ðp2 Þ gðq2 Þ~f ðp2 ÞÞðbhð 1 ~ p ~ a22 ða11 2 Þg~ðp 2 Þ~f ðp 2 fð f ðq 1 Þ gðq 1 ÞÞðbhð 2 Þ bðð a12 bÞÞg p 2 bfð~f ðp 1 Þg~ðp 2 Þ ~f ðp 2 Þg~ðp 1 ÞÞððða22 ða11 c1 Þ a12 bÞÞg q 2 ÞÞg: c1 Þ a12 bÞ hðq ^1 hÞ Oðq ^1 Þ, implyFrom equation (40), it is deduced that tanðkq ^1 hÞ ! 0 as q ^1 ! 0, and thus ing that in the limit kh ! 1; tanðkq
^1 ¼ q
np 2 þ OðkhÞ : kh
ð41Þ
5.2. Short wavelength limit ðkh ! 1Þ
Inserting Eq. (41) into (37) yields the second order approximation for the phase speed of the nth harmonic
The numerical results indicate that the short wavelength limiting wave speed for all harmonics is in general the lower of two wave speeds, termed limiting wave speed, associated with the
v 2n ¼ a11 þ c2 þ
! np2 b2 þ ; a11 c1 kh
ð42Þ
226
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
where here and hereafter n ¼ 1; 2; 3 . . .. A higher order expansion for the phase speed is obtained by setting
np / 3 þ 1 þ OðkhÞ ; kh ðkhÞ2
^1 ¼ q
^1 hÞ ¼ tanðkq
/1 3 þ OðkhÞ ; kh
ð43Þ
where /1 is to be determined. If these two expansions are inserted into Eq. (40), and like powers of kh equated, it is found that
/1 ¼
2 Þ f1 ða22 ða11 c1 Þ a12 bÞf ðq np: þ 2 hðq 2 Þðc2 r2 Þða11 c1 Þ g1 q
ð44Þ
On inserting Eq. (44) into Eq. (43)1, and on making use of equation (37), it may be shown that
v
2n
! np2 b2 ¼ a11 þ c2 þ a11 c1 kh
2 Þ f1 2 ða22 ða11 c1 Þ a12 bÞf ðq þ 1þ þ 2 Þðc2 r2 Þða11 c1 Þ g1 2 hðq kh q
ð45Þ
We remark that the asymptotic orders in Eq. (40) will change in, and close to, the special case c2 ¼ r2 . Appropriately specialised asymptotic expansions could readily be obtained for this case. A comparison of the third order expansions with numerical solutions are shown in Fig. 4. Good agreement with the numerical solutions is observed over a surprisingly large wave number regime.
~2 > 0 ~ 11 > c ~1 and a ~ 22 ða ~ 11 c ~1 Þ b ~ 2 with a 5.2.4. Case 4: v ! v Using the same analysis we obtain
v
2n
!
np2 ~2 b 2g ¼ c~1 þ a~22 1 þ 4 þ . . . ; n ¼ 1; 2; 3; . . . a~ 11 c~1 kd f4 kd ð49Þ
with f4 and g4 inferred from definitions of f1 and g1 . Good agreement is obtained even in the low wave number ranges, with expansions accurate over a much greater range of kh; kd than one might expect, see Fig. 5. ~2 < 0 ~ 11 > c ~1 and a ~ 22 ða ~ 11 c ~1 Þ b ~ 3 with a 5.2.5. Case 5: v ! v In this case we know numerically that as kh; kd ! 1 both p1 and p2 are imaginary and that jp1 j ! jp2 j, whilst q1 and q2 are either real or complex conjugates. The limit kh; kd ! 1 may therefore be examined by setting
~ ~ þ b; p21 ¼ a
v
!
np2 ~2 b 2g2 ~ 11 þ c ~2 þ þ ; ¼a 1þ a~ 11 c~1 kd f2 kd
~¼ a
a~ 22 ðv 2 a~ 11 Þ þ c~2 ðv 2 c~1 Þ þ b~2 ~ ; b ~ 22 c ~2 2a
¼
By making use of Eq. (51), in conjunction with (15), it is possible to ~ obtain the following quadratic for a
a~ 22 c~2 ða~ 22 c~2 Þ2 a~2 þ 2a~ 22 c~2 ð2b~2 ða~ 11 c~1 Þða~ 22 c~2 ÞÞa~ ~ þ ðc ~2 Þða ~2 Þ ¼ 0; ~ 22 c ~ 2 ða ~ 22 þ c ~2 Þ2 b ~2 ða ~ 11 c ~1 Þ þ b ~ 22 ða ~ 11 c ~1 Þ b a
~ p 2 p 1 Þða 1 Þððc ~ 11 c ~1 Þ gðq ~2 r2 Þða ~ 11 c ~1 ÞÞÞðbhð 2 fð f ðq 2 Þ f2 ¼ q ~ q 2 ÞÞg q1 p2 fð f ðq2 Þða ~ 11 c ~1 Þ gðq2 Þððc ~2 r2 Þða ~ 11 bhð ~ p ~ q ~1 ÞÞÞðbhð 1 ÞÞg q 1 q 2 bfðððc ~2 r2 Þða ~ 11 c ~1 ÞÞg~ðp 2 Þ bhð 2 Þ c ~ 11 c ~1 ÞÞðhðq 1 Þ hðq 2 ÞÞg; 2 Þða ~f ðp
ð52Þ ~ implies that which for small b
~2 þ Oðb ~4 Þ; ~¼a ~0 þ a ~1 b a
~ q ~ f ðq 1 Þgðq 2 Þ gðq 1 Þf ðq 2 ÞÞðða 1 ÞÞg ~ 22 ða ~ 11 c ~1 Þ a ~ 12 bÞ hð g2 ¼ p2 bfð ~ q 2 ÞÞg q 1 fð f ðq 2 Þg~ðp 2 ÞÞðbðða ~ 22 ða ~ 11 c ~1 Þ 2 Þ ~f ðp 2 Þgðq bhð ~ q ~ 12 bÞ bhð 1 ÞÞg: a 1 and q 2 may be inferred from Eq. (39) by inter2 ; q The form of p changing the material parameters of the outer layers and the inner core. 2
5.2.3. Case 3: v ! v 2 with a11 > c1 and a22 ða11 c1 Þ b > 0 By the same method previously used, we obtain
b
2 ¼ q
a22 c2
;
2
~ 22 c ~2 Þ ða
;
sffiffiffiffiffiffiffiffiffiffiffiffi 1 a~ 22 c~2 ~1 ¼ ða ~ 22 þ c ~2 Þ a ~2 T 0 2 b
ð54Þ
with T 0 defined in Eq. (32). On inserting Eqs. (50) and (53) into (26), ~ it is deduced that for high wave number and expanding for small b, ~ that either and small b
~2 Þ ¼ 0; 1 hðq 1 Þf ðq 2 Þ q 2 hðq 2 Þf ðq 1 Þ þ Oðb q
ð55Þ
~ a ~ a 3 þ q 3 ÞbSð 5 bCð 1 C 2 C ~0 Þ þ C ~0 Þ ðq
np2 kh
21 ; p 22 ¼ Fðc1 Þ: p
~0 ¼ a
qffiffiffiffiffiffiffiffiffi ~2 ~2 ~ 22 þ c ~2 Þ a~b c~1~ T 0 b ða 22 2
or
~ bÞ ~ ðq ~ þ Oðb ~2 Þ; 6 bCð 2 þ q 4 ÞSðbÞ 2 C 1 C ¼C ð47Þ
within which f3 and g3 may be readily inferred by replacing the def2 ; p 1 and p 2 , as given below, into the expressions for f1 initions of q and g1 presented just after Eq. (40)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a22 ða11 c1 Þ b2
ð53Þ
where
2 fð f ðq 1 Þg~ðp 1 Þ~f ðp ~ 22 ða ~ 11 c ~1 Þ a ~ 12 bÞ 2 Þ gðq 2 ÞÞðbðða þq
a11 c1
2 Þ f3 2 ða22 ða11 c1 Þ a12 bÞÞf ðq þ þ ... 1þ 2 hðq 2 Þðc2 r2 Þða11 c1 Þ g3 kh q
:
~ 22 c ~2 2a
ð51Þ
where f2 and g2 are given by
v 2n ¼ c1 þ a22
ð50Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 a~ 22 ðv 2 a~ 11 Þ þ c~2 ðq~ v 2 c~1 Þ þ b~2 4a~ 22 c~2 ðv 2 a~ 11 Þðv 2 c~1 Þ
ð46Þ
!
~ P 0; b
thus
n
¼ 1; 2; 3; . . . ;
2
~ > 0; a
~ are real and b ~ ! 0 as kh; kd ! 1. The values of a ~ and b ~ and where a ~ b may be obtained explicitly from the appropriate form of Eq. (15),
~2 > 0 ~ 11 < c ~1 and c ~ 2 ðc ~1 a ~ 11 Þ b ~!v ~ 1 with a 5.2.2. Case 2: v Using the same analysis we obtain
2n
~ ~ b; p22 ¼ a
ð48Þ
ð56Þ
m (m ¼ 1; . . . ; 6) are order one terms given in within which C ~ and SðbÞ ~ are defined by ~ ~0 Þ; CðbÞ Appendix A, Cða0 Þ; Sða
pffiffiffiffiffi ~0 kd; ~0 Þ ¼ cos 2 a Cða ! ~ bkd ~ ¼ cos p ffiffiffiffiffi ; CðbÞ ~0 a
pffiffiffiffiffi ~0 kd; ~0 Þ ¼ sin 2 a Sða ! ~ bkd ~ ¼ sin p ffiffiffiffiffi SðbÞ ~0 a
1 and q 2 are order 1 terms found by setting v ¼ v~ 3 in Eq. (15). and q Eq. (55), at leading order, is the appropriately specialised form of
227
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
1.15
Numerical solutions Asymptotic expansions
1.1
1.05
v 1
0.95
0.9
0
5
10
15
20
25
30
35
40
45
50
kh Fig. 4. Comparison of numerical solutions with third order asymptotic expansions (45) for case 1. The same material parameters as in Fig. 1 have been used.
0.9
Numerical solutions Asymptotic expansions
0.85
0.8
0.75
v
0.7
0.65
0.6
0.55 0
10
20
30
40
50
60
kh Fig. 5. Comparison of numerical solutions with third order asymptotic expansions (49) for case 4. The same material parameters as in Fig. 2 have been used.
the Rayleigh surface wave equation. From equation (56), it is readily ~ b, ~ implying that SðbÞ ~ ! 0 as b ~ ! 0, and thus deduced that SðbÞ
pffiffiffiffiffi ~0 2 ~ ¼ np a b þ OðkdÞ : kd
np2 1 ~ 2 þ 2a ~0 1 þ a ~1 ~ 11 a ~ 22 þ c ~1 c ~2 b ~ 22 c ~2 a a a22 þ c~2 kd
v 2n ¼ ~
þ ð58Þ
ð57Þ
A second order expansion for the phase speed for the nth harmonic may now be obtained by inserting Eq. (57) into (51)1 and (52), to obtain
A higher order expansion is obtainable by now setting
~ ¼ np b
pffiffiffiffiffi ~0 a / 3 þ 5 2 þ OðkdÞ kd ðkdÞ
ð59Þ
228
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
from which we infer that
~ bkd sin pffiffiffiffiffi ~0 a
!
/ 3 ¼ ð1Þn pffiffiffiffiffi5 þ OðkdÞ ; ~0 kd a n
¼ ð1Þ þ OðkdÞ
~ bkd cos pffiffiffiffiffi ~0 a
We now make use of Eq. (67), in conjunction with the appropriate form of Eq. (15), to obtain expansions for p1 and p2 , given by
!
2
2
ð60Þ
with /5 to be determined. On inserting Eqs. (59) and (60) into Eq. (58) and comparing like powers of kd, it is obtained that
( /5 ¼ np
) 2 þ ðSða 5 þ q 6 Þ þ Cða 1Þ ~0 Þðq 2 C 1 C ~0 ÞC ð1Þn C : 3 þ q 4 2 C 1 C ð1Þn q
v
4
n 1 ¼ a~ a~ þ c~ c~ b~2 þ 2a~ 22 c~2 a~ 22 þ c~2 11 22 1 2 !!) np2 ^5 2/ ~1 a ~0 ~0 þ a a 1 þ pffiffiffiffiffi þ ; kd ~0 kd a
p2 ¼ p20 þ p22 þ Oðb Þ;
ð68Þ
where
p10 ¼
1 k10 þ l0 2 ; ~2 a ~ 22 2c
p12 ¼
p20 ¼
1 k10 l0 2 ; ~2 a ~ 22 2c
p22 ¼
ð61Þ
Eq. (61) can now be used in conjunction with Eqs. (51)1, (53) and (59), to obtain
2n
4
p1 ¼ p10 þ p12 b þ Oðb Þ;
k12 þ l2 1
1
~2 a ~ 22 Þ2 ðk10 þ l0 Þ2 ð2c
;
k12 l2 1
1
~2 a ~ 22 Þ2 ðk10 l0 Þ2 ð2c
and within which
~2 ; ~ 22 ða ~ 11 a1 Þ c ~2 ða1 c ~1 Þ b k10 ¼ a ð62Þ
^ 5 ¼ /5 . The appearance of sinusoidal terms in the third order where / np term of Eq. (62) indicates that the asymptotic expansions will give a crude approximation to second order. Fig. 6 shows that once the asymptotic expansions are taken to third order, and the trigonometric terms included, good agreement with the numerical solutions is obtained.
~ 22 þ c ~2 Þ; k12 ¼ a2 ða
a1 ¼ ða11 a22 þ c1 c2 b2 þ 2a22 c2 a0 Þ=ða22 þ c2 Þ; a2 ¼
2a22 c2 a1 ; a22 þ c2
l20 ¼ b~4 þ 2b~2 ða~ 22 ða1 a~ 11 Þ þ c~2 ða1 c~1 ÞÞ 2
5.2.6. Case 6: v ! v 3 with a11 > c1 and a22 ða11 c1 Þ b2 < 0 The final case to consider arises when both q1 and q2 are imaginary and jq1 j ! jq2 j as kh; kd ! 1, whilst p1 and p2 are either both real or complex conjugates. The limit kh; kd ! 1 is therefore examined by setting
q21 ¼ a þ b;
q22 ¼ a b;
a > 0;
b P 0;
ð63Þ
where a and b are real and b ! 0 as kh; kd ! 1. Following the previous analysis we see that 2
4
a ¼ a0 þ a1 b þ Oðb Þ
ð64Þ
for small b, with the forms of a0 and a1 inferred from Eq. (54). In this case it is deduced that for high wave number equation (26) takes the form
^1 q ^2 fhðq1 Þf ðq1 ÞD1 þ hðq2 Þf ðq2 ÞD6 g fq ^1 hðq1 Þf ðq2 Þ q ^2 hðq2 Þf ðq1 ÞðCðbÞSða0 ÞD2 ðCða0 ÞSðbÞD2 þ Sða0 ÞCðbÞD5 Þ q
~ 22 ða1 a ~ 11 Þ þ c ~ 2 ðc ~1 a1 ÞÞ ; þ ða ~2 ða ~ 22 þ c ~2 Þ 2a2 ða ~ 22 c ~2 Þðc ~2 ða1 c ~1 Þ a ~ 22 ða1 2l0 l2 ¼ 2a2 b ~ 11 ÞÞ: a On making use of Eqs. (66)–(68), in conjunction with (65), the dispersion relation in the high wave number region may be cast in the relatively simple form 2
A1 þ ðA2 þ Cða0 ÞA3 Sða0 ÞA5 Þb 2 3 ¼ CðbÞ A1 þ A4 b Sða0 ÞA6 b þ Oðb Þ;
where Cða0 Þ; Sða0 Þ; CðbÞ and SðbÞ are trigonometric terms which may be inferred from the definitions given after equation (56), and Am are order 1 quantities defined in Appendix B. It is readily deduced from equation (69) that at leading order
bkh bkh 1 cos pffiffiffiffiffi OðbÞ sin pffiffiffiffiffi ; a0 a0
^1 hðq2 Þf ðq2 Þðq ^2 ÞCða0 ÞCðbÞD3 SðbÞCða0 ÞD5 Þg þ fq ^1 Sða0 ÞSðbÞD4 q ^2 hðq2 Þf ðq1 Þðq ^2 Sða0 ÞSðbÞD3 þq ^1 Cða0 ÞCðbÞD4 Þg þq
thus implying that cos
¼0
ð65Þ
^1 and q ^2 are approximated by within which q
bkh p ffiffiffi ffi a0
ð66Þ
3
with n ¼ ð4a0 a1 1Þ=8a20 . It is found that the leading order term of
! 1 as kh; kd ! 1, and therefore
pffiffiffiffiffi np 2 b ¼ 2 a0 þ OðkhÞ : kh
v 2n ¼
Eq. (65) vanishes in the limit, necessitating inclusion of Oðb Þ terms in all expansions. By making use of the appropriate form of Eqs. (51)1 and (64) we obtain
n o 1 a11 a22 þ c1 c2 b2 þ 2a22 c2 ða0 þ a1 b2 Þ a22 þ c2 4
ð71Þ
np2 1 a11 a22 þ c1 c2 b2 þ 2a22 c2 a0 1 þ 4a1 a22 þ c2 kd þ ð72Þ
2
þ Oðb Þ:
ð70Þ
A second order approximation to the phase speed is obtainable by inserting Eq. (71) into Eq. (67)1, to deduce that
pffiffiffiffiffi b 2 3 a0 pffiffiffiffiffi þ nb þ Oðb Þ; 2 a0 pffiffiffiffiffi b ^2 ¼ a0 þ pffiffiffiffiffi þ nb2 þ Oðb3 Þ q 2 a0
^1 ¼ q
v 2n ¼
ð69Þ
ð67Þ
We now seek a higher order expansion by setting
pffiffiffiffiffi np / 3 b ¼ 2 a0 þ 6 þ OðkhÞ kh ðkhÞ2 from which it is deduced that
ð73Þ
229
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
0.7
Numerical solutions Asymptotic expansions
0.65
0.6
v
0.55
0.5
0.45
0.4
0
10
20
30
40
50
60
kh Fig. 6. Comparison of numerical solutions with third order asymptotic expansions (62) for case 5. The same material parameters as in Fig. 3 have been used.
bkh / 3 sin pffiffiffiffiffi ¼ pffiffiffiffiffi6 þ OðkhÞ ; a0 a0 kh ¼1
/26 2a0 ðkhÞ
bkh cos pffiffiffiffiffi a0
of kh, reveals that the equation is identically zero at leading order. At the next order the following quadratic equation for /6 is revealed
4
2
þ OðkhÞ ;
ð74Þ
where /6 is to be determined and it is noted that it is now necessary 2
to include an OðkhÞ term in the expansion (74)2 due to the vanishing of the leading order term prior to the derivation of Eq. (69). Inserting Eqs. (73) and (74) into Eq. (69), and comparing like powers
A1 2 / þ 2npA6 /6 þ 4a0 ðnpÞ2 fA2 A4 þ A3 Cða0 Þ A5 Sða0 Þg ¼ 0: 2a0 6
ð75Þ
We note that quadratic equation for /6 is obtained in this case, whilst in the previous five cases a single value for /1 ; . . . ; /5 was obtained. Interestingly, it has been verified numerically that for particular values of n the two solutions indicated in Eq. (75)
0.7
Numerical solutions Asymptotic expansions
0.65
0.6
0.55
v
0.5
0.45
0.4
0.35 0
10
20
30
40
50
60
kh Fig. 7. Comparison of numerical solutions with second order asymptotic expansions (72) for case 6 for Blatz–Ko material (30), with ~ 1 ¼ 2:24; v 2 ¼ 1:33; v ~ 2 ¼ 1:29; v 3 ¼ 0:17. k3 ¼ 1:0, ~ k1 ¼ 0:8; ~ k2 ¼ 0:8; ~ k3 ¼ 0:43 and v 1 ¼ 0:64; v
l ¼ 0:4; l~ ¼ 0:3; k1 ¼ 1:8; k2 ¼ 0:5;
230
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
corresponds to two distinct branches of the dispersion relation. This is not too surprising, as will be seen in Figs. 7 and 8 that the harmonics group together to form distinct pairs in the high wave number region. Eq. (75) may now be used, in conjunction with Eqs. (72) and (67)1, to obtain
8 2 p 2 n 2 /^ o > 6ffi < v 23 þ nþ1 a0 1 þ 4a1 nkdp pffiffiffi þ . . . n odd; 2 kd a0 v 2n ¼ > n o þ
^ 6ffi : v 2 þ np 2 a0 1 þ 4a1 np 2 p/ffiffiffi þ . . . n ev en; 3 2kd kd a0
ð76Þ
^ 6 ¼ / =np; / ^ þ and / ^ representing solutions of Eq. (75) where / 6 associated with the positive and the negative square roots, respectively. The asymptotic expansions derived for the phase speed are superimposed upon the numerical solutions in Figs. 7 and 8. Fig. 7 shows the second order expansions. Each value on n used in (72) generates an approximate solutions that passes between the 2nth and ð2n 1Þth branches of the numerical dispersion curves, clearly indicating that the second order expansion is not sufficient to describe the behaviour of the numerical solutions accurately. When the expansions are taken to higher order, as in Fig. 8, the oscillations are fully accounted for by the trigonometric functions within the third order term of Eq. (76) and the two roots of the quadratic equation in /6 now provide an asymptotic approximation for each harmonic. These provide a highly reasonable approximation to the phase speed both in the high and moderate kh region. 6. Concluding remarks
Appendix A The constants Ci ; i ¼ 1; . . . ; 6, are defined by
1 q 2 b g~0 ~f 1 þ g~11~f 0 ðhðq 2 Þ hðq 1 ÞÞ; þq
~ 11 g~0 ¼ a
~f 0 ¼ ðc ~c ~2 r2 Þg~0 b ~2 a0 ; ~0 ¼ a ~ ~ 22 g~0 a ~ 12 b; h
~f 1 ¼ b ~c ~ 2 ðc ~2 r2 Þg~1 ;
~1 ¼ a ~ 22 g~1 : h
Appendix B The constants Ai ; i ¼ 1; . . . ; 6, are defined by
A1 ¼ h0 f 0 a0 D10 ; A2 ¼
3 1 f 0 ð4a20 h2 h0 þ 8a20 nh0 Þ þ 4a0 ðf 1 h1 þ f 2 h0 Þ D10 4a0
a0 ðf 0 h1 þ f 1 h0 ÞD11 h0 f 0 a0 D12 ;
A4 ¼
1 ðf h0 2a0 ðf 1 h0 f 0 h11 ÞÞD10 4a0 0 1 D31 ð2a0 ðf 1 h0 f 0 h1 Þ f 0 h0 Þ; 2
pffiffiffiffiffi 1 D10 2a0 ðf 0 h2 f 1 h1 þ f 2 h0 Þ þ f 0 h1 f 1 h0 þ 4 a0 nf 0 h0 2 1 h0 f 0 D31 þ h0 f 0 a0 D32 ; 2
1 A5 ¼ pffiffiffiffiffi ð2a0 ðf 1 h0 f 0 h1 Þ f 0 h0 ÞD21 ; 2 a0
pffiffiffiffiffi A6 ¼ h10 f 10 a0 D22 : m
Within the above definitions, Dim represents the coefficients of b within Di , which may be obtained by inserting the expansions given ~1C ~2, in Eqs. (63) and (68) into Eq. (27), dividing throughout by C 1 C 2 C and replacing the resulting hyperbolic tangents with unity. These are given explicitly by:
~ ~ ~ ~ ~ D10 ¼ 2ðbðp 20 h10 ðf 0 g 10 g 0 f 10 Þ þ p10 h10 ðg 0 f 20 f 0 g 20 ÞÞ ~ ~10 ðf g~20 h ~20 g ~f 20 Þ þ p h ~ ~ ~ þ bðp10 h 0 0 20 20 ðg 0 f 10 f 0 g 10 ÞÞÞ;
0
1
0
11
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D12 ¼ 2ðbðp12 h 0 0 22 20 ðg 0 f 10 f 0 g 10 Þ ~12 þ g~22 h ~10 Þ þ h ~10 ðf g~20 g ~f 22 Þ ~f 20 ðg h ~ þ p10 ðf 0 ðg~20 h 2 0 0 12 ~10 ÞÞ þ p ðh ~20 ðg ~f 12 f g~12 Þ g~10 ðf h ~20 f h ~22 Þ g h
pffiffiffiffiffi ~0 b ; ~h 1 Þ~f 0 f ðq 1 Þg~0 hðq 2 Þb a0 gðq
12
20
0
0
2
0
~22 þ g h ~ ~ ~ ~ þ ~f 10 ðg 0 h 12 20 ÞÞÞ þ bðp22 h10 ðf 0 g 10 g 0 f 10 Þ
~0 b ; ~h 1 Þb hðq
1 n 2 a0 o 1 n 2 Þ gðq 1 Þð~f 0 þ 2a0 ~f 1 Þ þ f ðq 1 Þð2a0 g~1 g~0 Þ ; pffiffiffiffiffi b2 hðq 2 a0
a11 a22 þ c1 c2 b2 þ 2a22 c2 ða0 þ a2 Þ ~ ~2 ; þ c2 a0 ; g~1 ¼ c a22 þ c2
þ p10 ð~f 20 ðg 0 h11 þ g 11 h10 Þ g~20 ðf 1 h10 f 0 h11 ÞÞÞÞ;
1 q 2 b g~0 ~f 1 þ g~11~f 10 ðhðq 2 Þ hðq 1 ÞÞ; q
~0 ~f 0 þ 2a0 ðh ~0 ~f 1 þ h ~1 ~f 0 Þ þ f ðq ~0 g~0 h ~ 1 Þ g~0 h ~0 5 ¼ pffiffiffiffiffi b gðq 1 Þ h 1 Þ 2a0 ðg~1 h C 1
where
20
~1 b 2 ¼ a0 h ~ðf ðq 2 Þgðq 1 Þ f ðq 1 Þgðq 2 ÞÞ C
pffiffiffiffiffi
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D11 ¼ 2ðbðp10 h 1 11 20 20 ðg 11 f 10 f 1 g 10 ÞÞ ~ ~ þ bðp ðg~10 ðf h11 þ f h10 Þ f 10 ðg h11 g h10 ÞÞ
~1 b 1 ¼ a0 h ~ðf ðq 2 Þgðq 1 Þ f ðq 1 Þgðq 2 ÞÞ C
4 ¼ a0 gðq 2 Þ~f 0 f ðq 2 Þg~0 C
2 a0 o o 1 n~ ~ 1 Þ gðq 2 Þð~f 0 þ 2a0 ~f 1 Þ þ f ðq 2 Þð2a0 g~1 g~0 Þ ; q g~0 h þ pffiffiffiffiffi bhð 0 2 a0
A3 ¼
In this paper, the dispersion of symmetric (extensional-like) waves in a symmetric three-layer structure has been investigated. The investigation has been carried out within the framework of compressible, pre-stressed elastic media, with each layered assumed composed of such material. The dispersion relation is a mathematically elaborate transcendental equation, derived from a six by six determinant, which gives phase speed (or frequency) as an implicit function of wave number. A thorough numerical analysis reveals the nature of the dispersion relation and provides a sign post for the derivation of some short wave approximation of the harmonics in each of the six different cases which are shown to exist. These approximations provide can explicit relationship between phase speed and wave number and are shown to provide highly accurate approximations over a wide wave number range. Having these explicit approximations may prove helpful to approximate numerical truncation errors, as well as provide an excellent first approximation in numerical dispersion schemes for particularly challenging sets of material parameters.
3 ¼ C
n
1 ~ ~f þ 2a ðh ~ ~f þ h ~ ~f Þ þ f ðq ~ g~ h ~ 6 ¼ p 2 Þ h 2 Þ 2a0 ðg~1 h ffiffiffiffiffi b gðq C 0 0 0 0 1 1 0 0 0 1Þ
þ p12 h10 ðg 0 ~f 20 f 0 g~20 Þ þ p20 ðf 0 ðh10 g~12 þ h12 g~10 Þ þ g~10 ðf 1 h11 o
þ f 2 h10 Þ g 0 ðh10 ~f 12 h12~f 10 Þ ~f 10 ðg 11 h11 h10 g 12 ÞÞ þ p10 ðh10 ðg 0 ~f 22 f 2 g~20 Þ g~20 ðf 0 h12 f 1 h11 Þ h10 ðf 0 g~22 þ g 12~f 20 Þ þ ~f 20 ðg 0 h12 þ g 11 h11 ÞÞÞÞ;
231
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
0.7
Numerical solutions Asymptotic expansions
0.65
0.6
0.55
v
0.5
0.45
0.4
0.35 0
10
20
30
40
50
60
kh Fig. 8. Comparison of numerical solutions with third order asymptotic expansions (76) for case 6. The same material parameters from Fig. 7 have been used.
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D31 ¼ 2ðbðp10 h 1 11 20 20 ðg 11 f 10 f 1 g 10 ÞÞ ~ ~ ~ ~ ~ þ bðp 20 ðh10 ðf 1 g 10 g 11 f 10 Þ h11 ðf 0 g 10 þ g 0 f 10 ÞÞ ~ þ p ðh11 ðf g~20 g f 20 Þ h10 ðf g~20 þ g ~f 20 ÞÞÞÞ; 10
0
0
1
11
~10 ðf g~20 g ~f 20 Þ þ p h ~ ~ ~ D32 ¼ 2ðbðp12 h 0 0 22 20 ðg 0 f 10 f 0 g 10 Þ ~ ~ ~ ~ ~ 12 þ p10 ðf 0 ðg~20 h12 þ g~22 h10 Þ þ h10 ðf 2 g~20 g 0 f 22 Þ ~f 20 ðg 0 h ~10 ÞÞ þ p ðh ~20 ðg ~f 12 f g~12 Þ g~10 ðf h ~20 f h ~22 Þ g h 12
20
0
0
2
0
~22 þ g h ~ ~ ~ ~ þ ~f 10 ðg 0 h 12 20 ÞÞÞ þ bðp22 h10 ðf 0 g 10 g 0 f 10 Þ þ p12 h10 ðg 0 ~f 20 f 0 g~20 Þ þ p20 ðf 0 ðh10 g~12 þ h12 g~10 Þ g~10 ðf 1 h11 þ f 2 h10 Þ g 0 ðh10 ~f 12 h12 ~f 10 Þ þ ~f 10 ðg 11 h11 h10 g 12 ÞÞ þ p10 ðg~20 ðf 1 h11 f 0 h12 Þ h10 ðf 0 g~22 f 2 g~20 Þ þ g 0 ðh10 ~f 22 þ h12~f 20 Þ ~f 20 ðg 11 h11 þ h10 g 12 ÞÞÞÞ; ~10 h ~20 Þðf g f g Þ þ 2a0 h11 bðg~20 ~f 10 g~10 ~f 20 Þ; ~ h D21 ¼ 2p10 p20 bð 1 0 0 11 ~10 h ~ 20 Þðf g f g Þ: ~ h D22 ¼ 2a0 h11 bðg~20 ~f 10 g~10 ~f 20 Þ þ 2p10 p20 bð 1 0 0 11 m Similarly, g m ; f m ; hm ; g~im ; ~f im and g~im are the Oðb Þ term within ~ ~ expansions of gðqÞ; f ðqÞ; hðqÞ; g~ðpi Þ; f ðpi Þ and hðp1 Þ, respectively, thus
2
3
f ðq1 Þ; f ðq2 Þ ¼ f 0 f 1 b þ f 2 b þ Oðb Þ; 2
3
gðq1 Þ; gðq2 Þ ¼ g 0 g 11 b þ g 12 b þ Oðb Þ; 2
3
hðq1 Þ; hðq2 Þ ¼ h10 h11 b þ h12 b þ Oðb Þ; ~f ðp Þ ¼ ~f 10 þ ~f 12 b2 þ Oðb4 Þ; 1 2 4 g~ðp1 Þ ¼ g~10 þ g~12 b þ Oðb Þ;
~f ðp Þ ¼ ~f 20 þ ~f 22 b2 þ Oðb4 Þ; 2 2 4 g~ðp2 Þ ¼ g~20 þ g~22 b þ Oðb Þ;
~ Þ¼h ~10 þ h ~12 b2 þ Oðb4 Þ; hðp 1
~ Þ¼h ~20 þ h ~22 b2 þ Oðb4 Þ; hðp 2
where
f 0 ¼ g 0 ðc2 r2 Þ a0 c2 b;
f 1 ¼ c2 b þ ðc2 r2 Þg 11 ;
f2
¼ g 12 ðc2 r2 Þ c2 a1 b; g 0 ¼ a11 a1 þ c2 a0 ; h10 ¼ g 0 a22 a12 b;
g 11 ¼ c2 ;
g 12 ¼ a2 þ c2 a1 ;
h11 ¼ a22 g 11 ;
h12 ¼ a22 g 12 ;
~ 11 a1 c ~2 p210 ; g~10 ¼ a
~ 11 a1 c ~2 p220 ; g~20 ¼ a
~2 p10 p12 ; g~12 ¼ 2a2 c
~2 p20 p22 ; g~22 ¼ 2a2 c
~f 10 ¼ b ~c ~2 p210 þ ðc ~2 r2 Þg~10 ;
~f 12 ¼ 2b ~c ~2 p10 p12 þ ðc ~2 r2 Þg~12 ;
~f 20 ¼ b ~c ~2 p220 þ ðc ~2 r2 Þg~20 ;
~f 22 ¼ 2b ~c ~2 p20 p22 þ ðc ~2 r2 Þg~22 ;
~10 ¼ a ~ ~ 22 g~10 a ~ 12 b; h ~12 ¼ a ~ 22 g~12 ; h
~20 ¼ a ~ ~ 22 g~20 a ~ 12 b; h
~22 ¼ a ~ 22 g~22 : h
References Dowaikh, M., Ogden, R., 1991a. On surface waves and deformations in a compressible elastic half space. Stab. Appl. Anal. Cont. Media 1, 27–45 (vol. 31, no. 12, pp. 1611–1639). Dowaikh, M., Ogden, R., 1991b. Interfacial waves and deformations in pre-stressed elastic media. Proc. R. Soc. A 433, 313–328 (vol. 31, no. 12, pp. 1611–1639). Kaplunov, J., Markushevich, D., 1993. Plane vibrations and radiation of an elastic layer lying on a liquid half-space. Wave Motion 17 (3), 199–211. Kaplunov, J., Kossovitch, L., Nolde, E., 1998. Dynamics of Thin Walled Elastic Bodies. Academic Press. Lamb, H., 1917. On waves in an elastic plate. Proc. R. Soc. A 93 (648), 114–128 (containing papers of a mathematical and physical character). Mindlin, R., 1960. Waves and vibrations in isotropic, elastic plates. Structural Mechanics. Pergamon Press Inc., Oxford, New York, pp. 199–232. Nolde, E., Prikazchikova, L., Rogerson, G., 2004. Dispersion of small amplitude waves in a pre-stressed, compressible elastic plate. J. Elast. 75 (1), 1–29.
232
M.I. Lashhab et al. / International Journal of Solids and Structures 58 (2015) 220–232
Ogden, R., Roxburgh, D., 1993. The effect of pre-stress on the vibration and stability of elastic plates. Int. J. Eng. Sci. 31 (12), 1611–1639. Pichugin, A., Rogerson, G., 2002. An asymptotic membrane-like theory for longwave motion in a pre-stressed elastic plate. Proc. R. Soc. A 458 (2022), 1447– 1468. Rogerson, G., 1992. Penetration of impact waves in a six-ply fibre composite laminate. J. Sound Vib. 158 (1), 105–120. Rogerson, G., 1997. Some asymptotic expansions of the dispersion relation for an incompressible elastic plate. Int. J. Solids Struct. 34 (22), 2785–2802.
Rogerson, G., 1998. On the existence of surface waves and propagation of plate waves in pre-stressed fibre-reinforced composites. J. Mech. Phys. Solids 46 (1998), 1581–1612. Rogerson, G.A., Sandiford, K.J., 2000. The effect of finite primary deformations on harmonic waves in layered elastic media. Int. J. Solids Struct. 37, 2059–2087. Roxburgh, D., Ogden, R., 1994. Stability and vibration of pre-stressed compressible elastic plates. Int. J. Sci. 32 (3), 427–454. Sandiford, K., Rogerson, G., 2000. Some dynamic properties of a pre-stressed, nearly incompressible (rubber-like) elastic layer. Int. J. Non-linear Mech. 35 (5), 849–868.