Propagation of elastic waves in anisotropic solids is solved through a pure stress formalism. The stress solutions lead to the three wave solutions expected from the displacement formulation plus three non-propagating stresses with zero wave speed which do not satisfy the strain compatibility conditions. This work provides a different perspective to modeling elastic waves and is expected to be better suited for certain types of problems.

The displacement form of the differential equation of motion governing elastic wave propagation traces back to Navier's exposition in 1821 and later memoir.1 Hadamard's treatise2 is the foundation of the modern theory of elastic waves in ansiotropic solids, which formulated the Fresnel-Hadamard propagation condition, now more commonly known as the Christoffel equation. The vast majority of research activity in elastic waves in anisotropic solids has since been approached through the displacement formulation.3–7 Stress-based solutions to elastostatic problems are well-known, e.g., solutions that satisfy the Beltrami-Michell compatibility relations.8 A dual elastodynamic treatment using pure stress language is not common. Some work in this direction did begin to appear in the late 1950s.9 Recently, Ostoja-Starzewski offered an extensive review of the topic covering the time period since then and gave several new extensions, which elucidated several benefits of considering stress formulations.10 To our knowledge, a dual pure stress formulation analogous to the displacement-based Fresnel-Hadamard propagation condition or the Christoffel equation has not been developed. The aforementioned is developed here in addition to its solutions, which give the phase velocities and stress components of the stress wave. The phase velocities and stress components are functions of newly formed invariants containing elastic compliance constants and the propagation direction.

This work offers an alternative way to consider waves in elastic solids that could lead to new and unexpected results. As an example, the stress-based analysis indicates that in addition to the expected propagating stress wave solutions there are non-propagating stresses, i.e., with zero velocity, which correspond identically to all possible incompatible strains. The present formulation provides a simultaneous solution to the propagating (compatible) and non-propagating (incompatible) wave solutions for waves in anisotropic solids.

Traditionally, elastic wave motion is described in terms of the displacement of a traveling wave,2u(x,t)=ug(n·xvt), where u is a constant vector and g is twice differentiable with respect to space and time. In an infinite medium, Cauchy's law of motion governs the wave displacement,

divσ=ρü.
(1)

The linear elastic constitutive relation is Hooke's law, σij=cijkluk,l, with cijkl=cjikl=cklij. Upon substitution of Hooke's law and executing the derivatives in Eq. (1) leads to the Christoffel equation

(Qλδ)ů=0,
(2)

where Q=Qik=cijklnjnl is the acoustic tensor, λ=ρv2 are eigenvalues, and δ=δik is the Kronecker delta function. The eigenvalues λ must satisfy

det(Qλδ)=0
(3)

or, equivalently, in terms of the characteristic polynomial

λ3λ2I1+λI2I3=0,
(4)

where the principle invariants of the acoustic tensor are

I1=trQ,I2=12[(trQ)2trQ2],I3=detQ.
(5)

Standard techniques for solving cubic equations can be used to solve Eq. (4) for the eigenvalues and, in turn, the phase velocities. The polarization ů is an eigenvector of Q.

The equation of motion given in Eq. (1) can be rewritten in terms of stresses only by using Hooke's law in the inverse form 12(ui,j+uj,i)=sijklσkl, where S=sijkl is the elastic compliance tensor. Thus10,11

12divσ+12(divσ)T=ρS:σ¨,
(6)

where is the gradient. Consider a traveling stress wave of the form σ=σf(n·xvt), where σ is a constant second rank tensor and f is twice differentiable in space and time. Substituting the stress wave σ into Eq. (6) and evaluating the derivatives leads to

(NλS):σ=0,
(7)

where

Nijkl=14(δiknjnl+δjkninl+δilnjnk+δjlnink)
(8)

has the same symmetry properties as the compliance and stiffness. To arrive at a characteristic equation of λ based on the stress formulation, Eq. (7) is written in an equivalent matrix form12 

(N̂λŜ):σ̂=0,
(9)

where notable hatted quantities are

σ̂=Kσ,ε̂=Kε,Ĉ=KCK,Ŝ=KSK,N̂=KNK,
(10)

with K being the 6 × 6 matrix

K=(I3×3002I3×3).
(11)

The matrix entries denoted as I3×3 are the elements of the 3 × 3 identity matrix. The quantities σ,ε, C, and S are written as the matrices

σ=(σ1σ2σ3σ4σ5σ6),ε=(ϵ1ϵ2ϵ3ϵ4ϵ5ϵ6),S=(s11s12s13s14s15s16s22s23s24s25s26s33s34s35s36s44s45s46SYMs55s56s66),C=(c11c12c13c14c15c16c22c23c24c25c26c33c34c35c36c44c45c46SYMc55c56c66),
(12)

which are formed from the Voigt index reduction, which maps pairs of indices into a single index according to 111,222,333,23or324,13or315, and 12or216. The propagation direction matrix N̂ is

N̂=KNK=(n12000n3n12n1n220n220n2n320n1n2200n32n2n32n3n1200n2n32n2n32n22+n322n1n22n3n12n3n120n3n12n1n22n32+n122n2n32n1n22n1n220n3n12n2n32n12+n222).
(13)

The wave speed parameter λ must satisfy the characteristic equation

det(N̂λŜ)=λ6detŜλ5a1+λ4a2λ3a3+λ2a4λa5+a6=0,
(14)

where

a1a2a3a4a5a6}=16!ϵijklmnϵpqrstu×{6ŜipŜjqŜkrŜlsŜmtN̂nu,15ŜipŜjqŜkrŜlsN̂mtN̂nu,20ŜipŜjqŜkrN̂lsN̂mtN̂nu,15ŜipŜjqN̂krN̂lsN̂mtN̂nu,6ŜipN̂jqN̂krN̂lsN̂mtN̂nu,N̂ipN̂jqN̂krN̂lsN̂mtN̂nu.
(15)

Under the matrix form, the acoustical tensor is given by Q=Ŝ1N̂=ĈN̂. The sextic equation (14) has six roots; however, a4, a5, and a6 are identically zero. To show this, we evaluate the constants aα in Eq. (15) with the twelfth rank ϵδ identity

ϵijklmnϵpqrstu=|δipδiqδirδisδitδiuδjpδjqδjrδjsδjtδjuδkpδkqδkrδksδktδkuδlpδlqδlrδlsδltδluδmpδmqδmrδmsδmtδmuδnpδnqδnrδnsδntδnu|,
(16)

where the right-hand side is a 6 × 6 determinant. Upon expanding the determinant in Eq. (16), substituting the result into Eq. (15), and considerable simplification, we find

a4=156![12(trN̂)2(trN̂Ŝ)224trN̂2ŜN̂2Ŝ48trŜ2N̂448trN̂3ŜN̂Ŝ+(trN̂)4(trŜ)2+3(trN̂2)2(trŜ)2+48trN̂trN̂3Ŝ2+48trN̂trN̂2ŜN̂Ŝ+24trN̂2trN̂2Ŝ2+12trN̂2trN̂ŜN̂Ŝ+48trN̂ŜtrN̂3Ŝ+16trN̂3trN̂Ŝ2+6trN̂4trŜ2+48trN̂4ŜtrŜ12trN̂2(trN̂Ŝ)2+8(trN̂)3trN̂Ŝ224(trN̂)2trN̂2Ŝ212(trN̂)2trN̂ŜN̂Ŝ(trN̂)4trŜ23(trN̂2)2trŜ26trN̂4(trŜ)2+24(trN̂2Ŝ)28(trN̂)3trN̂ŜtrŜ+8trN̂trN̂3(trŜ)2+6(trN̂)2trN̂2trŜ2+24(trN̂)2trN̂2ŜtrŜ6(trN̂)2trN̂2(trŜ)224trN̂trN̂2trN̂Ŝ248trN̂trN̂ŜtrN̂2Ŝ8trN̂trN̂3trŜ248trN̂trN̂3ŜtrŜ24trN̂2trN̂2ŜtrŜ16trN̂ŜtrN̂3trŜ+24trN̂trN̂2trN̂ŜtrŜ],
(17a)
a5=66![120trN̂trN̂4Ŝ120trN̂5Ŝ+60trN̂2trN̂3Ŝ+30trN̂ŜtrN̂4+40trN̂3trN̂2Ŝ+24trN̂5trŜ5(trN̂)4trN̂Ŝ15(trN̂2)2trN̂Ŝ+20(trN̂)3trN̂2Ŝ60(trN̂)2trN̂3Ŝ+(trN̂)5trŜ+30(trN̂)2trN̂2trN̂Ŝ+15trN̂(trN̂2)2trŜ10(trN̂)3trN̂2trŜ+20(trN̂)2trN̂3trŜ60trN̂trN̂2trN̂2Ŝ40trN̂trN̂ŜtrN̂330trN̂trN̂4trŜ20trN̂2trN̂3trŜ],
(17b)
a6=16![(trN̂)615(trN̂)4trN̂2+40(trN̂)3trN̂3+45(trN̂)2(trN̂2)290trN̂4(trN̂)2120trN̂trN̂2trN̂3+144trN̂5trN̂15(trN̂2)3+90trN̂4trN̂2+40(trN̂3)2120trN̂6],
(17c)

where I=I6×6. These three coefficients vanish due to the specific form of N̂. In order to see this we first note the identity

N̂(IN̂)(I2N̂)=0.
(18)

This implies trN̂n=32trN̂n112trN̂n2 for n3, which combined with trN̂=2,trN̂2=3/2 yields trN̂n=1+21n for n1 and hence a6 is zero. It also leads to the simplified expression

a5=1806!tr(IN̂)(I2N̂)2N̂2Ŝ,
(19)

which is zero on account of Eq. (18). Similarly, a4 can be reduced to

a4=906!tr[4N̂2Ŝ(N̂ŜN̂2Ŝtr(N̂ŜN̂2Ŝ))N̂Ŝ(N̂ŜtrN̂Ŝ)].
(20)

Consider, for instance, n=(1,0,0) for which N̂= diag (1,0,0,0,12,12) and it is easy to verify that the expression for a4 vanishes. This implies the general result for arbitrary n if we consider Ŝ as the compliance in the rotated coordinate system. In summary, a4=0,a5=0 and a6=0. Thus, a cubic equation for λ follows:

λ3λ2c1+λc2c3=0,wherecα=aα/detŜ,α=1,2,3
(21)

and using the properties of N̂ discussed above we find

a1=66![48trŜ5120trN̂Ŝ5+30trN̂ŜtrŜ4+40trN̂Ŝ2trŜ3+60trN̂Ŝ3trŜ2+120trN̂Ŝ4trŜ60trŜtrŜ440trŜ2trŜ35trN̂Ŝ(trŜ)415trN̂Ŝ(trŜ2)2+20trN̂Ŝ2(trŜ)360trN̂Ŝ3(trŜ)2+30trŜ(trŜ2)220(trŜ)3trŜ2+40(trŜ)2trŜ3+2(trŜ)5+30trN̂Ŝ(trŜ)2trŜ240trN̂ŜtrŜtrŜ360trN̂Ŝ2trŜtrŜ2],
(22a)
a2=156![96trN̂Ŝ448trN̂2Ŝ415trŜ424trŜ2N̂Ŝ2N̂48trŜ3N̂ŜN̂+12(trN̂Ŝ)2(trŜ)2+48trN̂ŜtrN̂Ŝ332trN̂ŜtrŜ348trN̂Ŝ2trŜ296trN̂Ŝ3trŜ+16trN̂2ŜtrŜ3+24trN̂2Ŝ2trŜ2+12trN̂ŜN̂ŜtrŜ2+20trŜtrŜ3+48trŜtrŜ2N̂ŜN̂+48trŜtrŜ3N̂216trN̂Ŝ(trŜ)3+8trN̂2Ŝ(trŜ)3+48trN̂Ŝ2(trŜ)212(trN̂Ŝ)2trŜ224trN̂2Ŝ2(trŜ)212trN̂ŜN̂Ŝ(trŜ)215(trŜ)2trŜ2+24(trN̂Ŝ2)2+52(trŜ)4+152(trŜ2)2+48trŜtrN̂Ŝtr(IN̂)Ŝ224trN̂2ŜtrŜtrŜ2],
(22b)
a3=206![tr(3I63N̂+126N̂2)Ŝ312tr(N̂Ŝ)3+72trŜ2N̂ŜN̂(IN̂)+9trŜtr(3N̂2N̂2)Ŝ2+9trN̂Ŝtr(52I8N̂+4N̂2)Ŝ2+18trN̂ŜtrN̂ŜN̂Ŝ+36trN̂2Ŝtr(N̂I)Ŝ2+18trN̂3ŜtrŜ292trŜtrŜ2+36trŜtr(N̂I)(N̂Ŝ)2+9(trŜ)2tr(N̂232N̂)Ŝ+36(trN̂Ŝ)2trŜ36trN̂ŜtrN̂2ŜtrŜ6(trN̂Ŝ)3+32(trŜ)3],
(22c)

while

detŜ=16![(trŜ)615(trŜ)4trŜ2+40(trŜ)3trŜ390(trŜ)2trŜ4+45(trŜ)2(trŜ2)2120trŜtrŜ2trŜ3+144trŜtrŜ515(trŜ2)3+90trŜ2trŜ4+40(trŜ3)2120trŜ6].
(23)

Equation (21) together with the constants cα completely define the characteristic equation for a completely stress-based formalism of elastic wave propagation and is the primary result of this letter.

The stress-based approach is consistent with the displacement formalism if it gives the same wave speed and stress polarization corresponding to each displacement solution. We first show that the speeds match and discuss polarizations in Sec. 4.

Comparison of Eq. (21) and Eq. (4) indicates that c1, c2, and c3 must be equal to the principle invariants I1, I2, and I3 seen in Eq. (5), respectively. To show this, consider the indentity13 

ϵijklmnŜipŜjqŜkrŜlsŜmtŜnu=ϵpqrstudetŜ.
(24)

Multiplying both sides of Eq. (24) by Ŝ1 and N̂ gives

ϵijklmnŜipŜjqŜkrŜlsŜmtN̂nw=ϵpqrstuŜuv1N̂vwdetŜ.
(25)

Next, multiply both sides of Eq. (25) by ϵpqrstw to give

ϵpqrstwϵpqrstuŜuv1N̂vwdetŜ=ϵpqrstwϵijklmnŜipŜjqŜkrŜlsŜmtN̂nw.
(26)

It can be shown that

ϵpqrstwϵpqrstu=5!δwu.
(27)

Thus,

Ŝuv1N̂vudetŜ=15!ϵpqrstuϵijklmnŜipŜjqŜkrŜlsŜmtN̂nu.
(28)

With the form of a1 in Eq. (15) together with Eq. (28), it is observed that

c1=Ŝij1N̂ji=ĈijN̂ji=trQ=I1.
(29)

The consistency relations for c2 and c3 follow a similar procedure. The results can be cast into a general formula for cα,

cα=1α!|δi1j1δiαj1δi1jαδiαjα|Ŝi1k11N̂k1j1Ŝi2k21N̂k2j2Ŝiαkα1N̂kαjα=1α!|δi1j1δiαj1δi1jαδiαjα|Ĉi1k1N̂k1j1Ĉi2k2N̂k2j2ĈiαkαN̂kαjα.
(30)

Then, the following consistency relations are easily obtained:

c1=Ŝ1:N̂=Ĉ:N̂=trQ=I1,
(31)
c2=12[(trŜ1N̂)2tr(Ŝ1N̂)2]=12[(trĈN̂)2tr(ĈN̂)2]=12[(trQ)2trQ2]=I2,
(32)
c3=16[(trŜ1N̂)33trŜ1N̂tr(Ŝ1N̂)2+2tr(Ŝ1N̂)3]=16[(trĈN̂)33trĈN̂tr(ĈN̂)2+2tr(ĈN̂)3]=16[(trQ)33trQtrQ2+2trQ3]=detQ=I3.
(33)

Equation (9) is a generalized eigenvalue problem that can be transformed into standard form

(ĈnλI):Σ̂=0,
(34)

where

Ĉn=Ĉ1/2N̂Ĉ1/2,Σ̂=Ŝ1/2σ̂
(35)

and Ĉ=Ŝ1 is the stiffness. The 6 × 6 symmetrix matrix Ĉn has three positive eigenvalues, the roots of the characteristic cubic equation, and three zero eigenvalues. The eigenvectors corresponding to the positive eigenvalues give the three propagating wave solutions. The stress states corresponding to the zero eigenvalues represent solutions that do not propagate because their speed is zero.

Elastic waves are, in practice, the result of some excitation. A common situation is that the flat surface of a solid is subject to prescribed motion. For instance, if displacements are excited uniformly across the surface, the displacements will propagate in the direction normal to the surface. Any displacement excitation will propagate since the three displacement polarization vectors of Eq. (2) span the space of three-dimensional vectors. The stress eigenvectors are elements of a six-dimensional space, which can be split into complementary three-dimensional subspaces of propagating and non-propagating stresses.

We examine these subspaces by considering the case of isotropy with two positive elastic constants, the bulk K and the shear μ moduli, for which

Ĉ=(3KA+2μB002μI3×3),whereA=13(111111111),B=I3×3A.
(36)

The matrix Ĉ1/2 follows from the properties A2=A,B2=B,AB=BA=0 as

Ĉ1/2=(3KA+2μB002μI3×3).
(37)

Taking n=(1,0,0) we find the matrix Ĉn of Eq. (35) is

Ĉn=(X00Y),
(38)

where

X=(2μ00000000)+2μ3(3K2μ)(211100100)+(3K2μ)23A,Y=(0000μ000μ).
(39)

The eigenvectors Σ̂ of Ĉn can then be determined, from which the stress vectors follow using σ̂=Ĉ1/2Σ̂. We find that the unnormalized propagating stresses are σ̂=σ̂(L), σ̂1(T),σ̂2(T), where

σ̂(L)=(3K+4μ3K2μ3K2μ000),σ̂1(T)=(000010),σ̂2(T)=(000001),
(40)

corresponding to λ=ρcL2 and λ=ρcT2, where cL=(K+43μ)/ρ and cT=μ/ρ. As expected, the propagating stresses correspond to longitudinal and degenerate transverse waves, respectively. The triply degenerate non-propagating stresses are

σ̂1(0)=(011000),σ̂2(0)=(011000),σ̂3(0)=(000100),
(41)

where the form of σ̂1(0) and σ̂2(0) are determined from the orthogonality condition discussed next.

Note that the stresses of Eqs. (40) and (41) are not orthogonal in the sense of 6-vectors. However, the associated Σ̂=Ŝ1/2σ̂ are orthogonal of the form

Σ̂=(3K+22μ3K2μ3K2μ000),(000010),(000001),(3K+2μ123K+2μ123K+2μ000),(011000),(000100),
(42)

where the first three are the propagating solutions of Eq. (40), and the last three are the non-propagating stresses of Eq. (41). It is noted that the solutions given by Eq. (41) are not unique as additional solutions can be formed by multiplying them by an arbitrary constant. In fact, the simplified forms seen in Eq. (41) were obtained by removing lengthy constant terms. The orthogonality condition for the 6-vectors Σ̂ of Eq. (42) is equivalent to the inner product σa:ϵb, where σa and σb are taken from the six stresses in Eqs. (40) and (41). Thus σa:ϵa>0 while σa:ϵb=0 for ab. For instance, σ(L):ϵj(T)=0,σ(L):ϵj(0)=0, σ1(0):ϵ2(0)=0, etc.

An elastic strain ε=Sσ is compatible if it is derived from a displacement u as εij=12(ui,j+uj,i). The necessary and sufficient conditions for compatibility14 are ϵikpϵjlqεkl,pq=0. Assuming uni-dimensional spatial dependence ε=εf(n·x,t) the compatibility conditions become Mijklεkl=0, where the fourth rank tensor

Mijkl=12(ϵikpϵjlq+ϵilpϵjkq)npnq
(43)

possesses the symmetries of fourth rank elastic tensors. The non-propagating stress solutions with zero velocity are therefore compatible if and only if they satisfy the simultaneous conditions

(44)
Nσ=0,
(44a)
Mε=0,
(44b)
where ε=Sσ. We now show that the only solution of Eq. (44) is the trival one ε=0,σ=0.

The simultaneous equations (44) can be written

ÑΣ̃=0,
(45a)
M̃Σ̃=0,
(45b)

where

Ñ=C1/2NC1/2,M̃=S1/2MS1/2,Σ̃=S1/2σ=C1/2ε.
(46)

It may be easily checked that M, like N is trimodal15 in the sense that three of its six Kelvin moduli16 are zero. Also, MN=NM=0 implying M̃Ñ=ÑM̃=0 and that the three-dimensional null spaces of M̃ and Ñ are distinct. A non-zero solution of one of the two conditions (45a) cannot be a solution of the other. The only solution of both is the trivial one Σ̃=0, and hence the non-propagating stress solutions are not compatible.

The same arguments prove that the propagating stress solutions are compatible. Thus, the stress satisfies Σ̃=λ1ÑΣ̃ and the compatibility condition (45b) is therefore satisfied since M̃Σ̃=λ1M̃ÑΣ̃=0.

Wave solutions can be superimposed by virtue of the linearity of the equations of motion. A longitudinal wave traveling in the x direction in an isotropic solid is any solution of the form σ=σ(L)f(L)(xcLt) for a sufficiently smooth but arbitrary function f(L). Similarly, a transverse wave is any solution of the form σ=σ1(T)f1(T)(xcTt)+σ2(T)f2(T)(xcTt).

These observations generalize to anisotropy. Thus, for a given direction of propagation in an anisotropic solid the propagating and zero velocity stress solutions partition the six-dimensional stress into two three-dimensional parts identified as propagating and non-propagating. The six states of stress may be defined in terms of an orthonormal set of stresses {σa} such that σa:ϵb=δab for a,b{1,,6}. Hence, the strain energy of any linear combination of these propagating and non-propagating solutions is simply the sum of the strain energies for each of the six elements. Non-propagating stress satisfies, from Eq. (7), N:σ(0)=0, which is equivalent to the zero traction condition σ(0)n=0. Propagating solutions satisfy the compatibility condition M:ε=0. It is interesting how the dual conditions yield the dual sets of stress and strain: null vectors of M and N are, respectively, propagating stresses and non-propagating strains. Further considerations of stress waves in anisotropic solids will be the subject of future work.

Considering both the stress and displacement formulation provides a complete treatment of stress waves in infinite elastic solids. Future analysis of the stress-based solutions are expected to provide new and exciting insights. In some cases, solving stress wave problems using the stress formulation might be more natural, for example, in handling certain boundary value problems or when a prestress is prescribed in the solid. Further applications of the stress formulation are under development by the authors.

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