aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Causality and Passivity in Elastodynamics
Ankit Srivastava ∗ Department of Mechanical, Materials, and Aerospace Engineering Illinois Institute of Technology, Chicago, IL, 60616 USA (Dated: July 24, 2018)What are the constraints placed on the constitutive tensors of elastodynamics by the requirementsthat the linear elastodynamic system under consideration be both causal (effects succeed causes) andpassive (system doesn’t produce energy)? The analogous question has been tackled in other areasbut in the case of elastodynamics its treatment is complicated by the higher order tensorial natureof its constitutive relations. In this paper we clarify the effect of these constraints on highly generalforms of the elastodynamic constitutive relations. We show that the satisfaction of passivity (andcausality) directly requires that the hermitian parts of the transforms (Fourier and Laplace) of thetime derivatives of the constitutive tensors be positive semi-definite. Additionally, the conditionsrequire that the non-hermitian parts of the Fourier transforms of the constitutive tensors be positivesemi-definite for positive values of frequency. When major symmetries are assumed these definitenessrelations apply simply to the real and imaginary parts of the relevant tensors. For diagonal andone-dimensional problems, these positive semi-definiteness relationships reduce to simple inequalityrelations over the real and imaginary parts, as they should. Finally we extend the results to highlygeneral constitutive relations which include the Willis inhomogeneous relations as a special case.
I. INTRODUCTION
Various aspects of nature are modeled as cause-effect relationships between different physical processes. Thesephysical processes are often functions of time and the relations between them, in some cases, can be more easilyanalyzed in the frequency or Laplace domains. A frequency dependent process is called dispersive and can be studiedby deriving the appropriate dispersion relations of the system. Physical systems often have an inherent assumptionof causality wherein effects are assumed not to precede causes. If the physical system is also linear and time-invariantthen certain sum/integral rules could be derived connecting the physical quantities involved . For eg. the Kramers-Kronig (K-K) relationships are integral relationships which connect the real part of the electromagnetic index ofrefraction to its imaginary part, thus connecting dispersion and loss in the medium. Since their introduction, the K-Krelationships have been used in the study of circuit theory and all forms of wave propagation .The K-K relations have recently attracted interest in the area of metamaterials where the goal is to create materialswith exotic electromagnetic, acoustic, and/or elastodynamic properties. The essential ideas emerge from early theo-retical works of Veselago and more recent experimental efforts by various research groups (See for a review).The possibility of creating materials with unprecedented material properties has led to far-reaching postulations oftheir applications, most visibly, in the area of cloaking . Since material properties can essentially be viewed astime-domain transfer functions which relate a cause to its effect (and, therefore, must be causal), K-K relations andtheir derivatives can be used to place some realistic constraints on the properties themselves. The K-K relations have,of late, been used as a tempering check on the optimism that has emerged in the area of metamaterials research. Thiscausality check includes, on one end of the spectrum, placing some realistic constraints on the application potential ofmetamaterials as cloaking devices to, on the other end, sobering realizations that a considerable amount of meta-materials research stands on shaky foundations, often proposing materials which violate such basic ideas as causalityand/or the second law of thermodynamics . This has led to a number of researchers advocating a need for improvedmodels for metamaterials .Closely connected to the idea of causality is the concept of passivity which refers to the assumption that the physicalprocess under consideration cannot produce energy . In fact, if the physical process (cause-effect relationship) canbe expressed in a convolution form in the time domain then its satisfaction of the passivity requirement automaticallymeans that it satisfies causality as well . A physical process can, in turn, be expressed in the convolution form ifit satisfies certain conditions such as linearity and time-invariance. It becomes interesting, therefore, to understandwhat constraints are placed upon a linear time-invariant cause-effect relationship (constitutive relationship) in electro-magnetic, acoustic, and/or elastodynamic areas by the requirement of passivity. Such knowledge can be used to placeconstraints on and understand the limitations of various metamaterial models which are used in these areas to arriveat such relationships. Considerable research in this direction has already taken place in the field of electromagneticswhere it is clear that passivity demands that the imaginary parts of the diagonal values of the Fourier transform of ǫ , µ be non-negative for all positive values of frequency (fields assumed to depend upon e − iωt ). However, it is notclear, to the author’s knowledge, what should be the equivalent constraints in elastodynamics for the most generalconstitutive cases. The case of 1-D longitudinal or shear wave propagation in an elastic medium is equivalent, in form,to the electromagnetic case. As such, it immediately follows that passivity should require that the corresponding1-D material properties (modulus and density) should behave analogously to the ǫ, µ . However, in 2- and 3-D, theelastodynamic constitutive tensor cannot, in general, be diagonalized. Moreover, recent advancements suggestthat the Willis constitutive relation , which is a coupled form of constitutive relation, is more appropriate for thedescription of inhomogeneous elastodynamics and, therefore, of elastodynamic metamaterials. It is not clear what theconstraints of passivity are on such highly general elastodynamic constitutive forms.In this paper we study the constraints which passivity places on highly general forms of the elastodynamicconstitutive relations. We use the passivity condition which is equivalent to the statement of passivity used inelectromagnetism and circuit theory and which is elaborated in subsequent sections. We also present our analysiswithin the context of distributions which is the proper space within which to describe the transfer functions of passivesystems. Furthermore, treating the constitutive tensors in the space of distributions ensures that the analysis appliesto the metamaterial cases of most interest and also to the static case (elastic case). II. BACKGROUND
Physical processes in the real world are often described as an interplay between physical variables and fields whichare dependent upon time. The relationships between the physical variables can be modeled as input-output relationswhere a time dependent variable v ( t ) is produced from another time dependent variable u ( t ) through some rule R , v ( t ) = R u ( t ). Although physical variables often satisfy certain continuity and differentiability conditions, it isdesirable to consider them as generalized functions for broader applicability. We identify four spaces at this point.Space D is the space of all complex valued functions φ ( t ) which are infinitely smooth and with compact support. Itis a subset of space S which consists of all infinitely smooth and complex-valued functions φ ( t ), called functions ofrapid descent, such that they and all their derivatives decrease to zero faster than every power of 1 / | t | as | t | → ∞ .Space S ′ of distributions of slow growth is the space of continuous linear functionals on S . Space D ′ is the space ofcontinuous linear functionals on D . It can be shown that D ⊂ S ⊂ S ′ ⊂ D ′ . Input-output relations can be completelyarbitrary but they reduce to a particularly simple form if the properties of single-valuedness, linearity, continuity,and time-translational invariance are assumed to hold for the operator R . These conditions are generally true forphysical processes and under these conditions the operator R reduces to a convolution operation: v ( t ) = R ∗ u = Z R d τ R ( t − τ ) u ( τ ) (1)where the last equality only holds if R, u are locally integrable distributions whose supports satisfy certain boundednessproperties (either
R, u have bounded supports, or both
R, u are either bounded on the left or on the right). Theoperator R is causal if it is not supported on t <
0. The final property of passivity can be stated by defining theenergy of the system. If the power absorbed by the system at time s is given by Re v ∗ ( s ) u ( s ) where ∗ denotes thecomplex conjugate, then define the energy absorbed by the system up to time t as: E ( t ) = Re Z t −∞ d s v ∗ ( s ) u ( s ) (2)The operator R is considered passive if E ( t ) ≥ t ∈ R . For operators in convolution passivity implies causalityand it also implies that R ∈ S ′ . Therefore, Eq. (2) is well defined for at least for all u ∈ S . For distributions in S ′ the Fourier transform, denoted by F , must satisfy hF f, φ i = h f, F φ i for f ∈ S ′ , φ ∈ S where h f, φ i is the value in C that f assigns to φ through the operation R ∞−∞ f ( t ) φ ( t )d t . Furthermore it can be shown that the Fourier transformof a distribution which is in S ′ is itself in S ′ . For distributions in S the Fourier transform is defined in the usual way( ω ∈ R ): F φ ≡ ˜ φ ( ω ) = Z R d t φ ( t ) e iωt ; φ ( t ) = 12 π Z R d ω ˜ φ ( ω ) e − iωt (3)In addition to R being a distribution of slow growth, causality implies that its support is in [0 , ∞ ). For such a caseits Laplace transform is given by ˆ R = h R, e − zt i and it can be shown that passivity directly implies that Re ˆ R ≥ z > R is analytic there . It is important to point out here that if R is causaland is in S ′ then its Laplace transform can be derived from its Fourier transform through its analytic extension in theright half of the complex plane. This is the point of departure for deriving K-K relations in electromagnetism whenthe constitutive tensors can be expressed diagonally. For transfer functions of higher order and more complexity which form the object of study of this paper, weneed to define some additional spaces. We will use bold symbols to denote tensors whose elements are distributions.If f ( t ) is a tensor of distributions, h f ( t ) , φ ( t ) i is the matrix of complex numbers obtained by replacing each elementof f ( t ) by the number that this element assigns to the testing function φ ( t ). We will use additional subscripts withthe spaces already defined above to denote the space in which all tensors of the relevant rank and distribution lie.For e.g. D ′ n × n × n × n is the space of all fourth order tensors whose elements are distributions in D ′ etc. For a tensorof even rank f we also define the operations f T which denotes a transpose over the major symmetry and f † whichdenotes a transpose over the major symmetry followed by conjugation. Now a single-valued, linear, time-invariant,and continuous input output relation can be written in the convolution form: v ( t ) = R ∗ u = Z R d τ R ( t − τ ) u ( τ ) (4)where appropriate boundedness of R , u are assumed and where v is a tensorial quantity derived from u through thelinear operator R . Total energy absorbed up to time t is given by: E ( t ) = Re Z t −∞ d s v † ( s ) u ( s ) (5)The operator R is considered passive if E ( t ) ≥ t ∈ R and to make sure that the integral above exists, we willrestrict the elements of the input u to be in D . If the operator R which is in the convolution form is passive thenit can also be shown to be causal and, furthermore, its elements are in S ′ . Moreover, certain important propertiesof the Laplace and Fourier transform of R follow. Its Laplace transform is given by ˆ R = h R , e − zt i and it is analytic(its elements are analytic) in the open right half of the complex plane. It can be derived from its Fourier transformthrough analytic continuation in its region of convergence. And finally, its hermitian part ( ˆ R h = . h ˆ R + ˆ R † i ) ispositive semi-definite for all Re z >
0. Note that these results follow from some fairly unrestrictive constraints on theinput field and transfer function which are easily satisfied in elastodynamics (and electromagnetism). Our effort hereis to apply and extend these results to the elastodynamic case.
III. CAUSALITY AND PASSIVITY IN ELASTODYNAMICS
We begin by considering a volume Ω within which the pointwise elastodynamic equation of motion and kinematicrelations are specified: σ ij,j + f i = ˙ p i ; ε ij = 12 ( u i,j + u j,i ) , (6)where σ , ε , p , u and f are the space and time dependent stress tensor, strain tensor, momentum vector, displacementvector, and body force vector respectively. These relations need to be supplied with appropriate constitutive relationswhich relate the various field variables to each other. For the current discussion we consider stress and velocity tobe independent fields (input fields) which lead to the emergence of strain and momentum fields (output/dependentfields) respectively. The relationships are expressed in terms of general constitutive operators whose properties needto be determined based upon the various subsequent assumptions about the system: ǫ ( x , t ) = D ( σ ( x , t )) , p ( x , t ) = P ( ˙ u ( x , t )) (7)Now we assume that the operators satisfy the conditions of single-valuedness, linearity, time-invariance, and conti-nuity and, furthermore, that the stress and velocity fields along with the operators satisfy appropriate boundednessconditions referred to in the last section. We also assume that the operator is real valued, i.e. it assigns a real outputfield to a real input field (but can assign a complex output field to a complex input field). Under these assumptionsthe constitutive relations of Eq. (7) can be specialized to the following form: ǫ ( x , t ) = D ∗ σ = Z ∞−∞ d s D ( x , t − s ) : σ ( x , s ) = Z ∞−∞ d s D ( x , s ) : σ ( x , t − s ) p ( x , t ) = ρ ∗ ˙ u = Z ∞−∞ d s ρ ( x , t − s ) · ˙ u ( x , s ) = Z ∞−∞ d s ρ ( x , s ) · ˙ u ( x , t − s ) (8)where the components of the constitutive tensors D , ρ are real valued distributions. D is a fourth order tensor fieldin S ′ n × n × n × n and possesses the usual minor symmetries that are associated with stiffness and compliance tensors. ρ is a second order tensor field in S ′ n × n . To make sure that energy, as defined later on, exists, we will take σ to be in D n × n and ˙ u to be in D n × . No major symmetries are assumed at this point. We also note a further result which willbe used later. For convolutions of distributions as appearing above we note that ˙ ǫ = ˙ D ∗ σ and ˙ p = ˙ ρ ∗ ˙ u . Causality : Causality refers to the requirement that an effect cannot precede its cause. With reference to theconstitutive relations in Eq. (8), it implies that the value of the strain and momentum fields at time t can onlydepend upon the values, respectively, of the stress and velocity fields at times prior to and including t . A necessaryand sufficient condition for a system to be causal is that its unit response function (constitutive operator in the presentcase) vanishes for t <
0. Specifically, causality implies the following for the constitutive tensors: D ( x , t ) = 0 , ρ ( x , t ) = 0 ∀ t < Passivity : Passivity refers to the requirement that the system cannot generate energy. For the elastodynamic casethe total energy at any time t contained in Ω comprises of the elastic energy contribution and the kinetic energycontribution: E ( t ) = Re 12 Z Ω d x [ σ ( x , t ) : ǫ ∗ ( x , t ) + p ( x , t ) · ˙ u ∗ ( x , t )] (10)The total energy can, therefore, be given by the following integral: E ( t ) = Z t −∞ d s ∂ E ( s ) ∂s = Re 12 Z t −∞ d s Z Ω d x ∂∂s [ σ ( x , s ) : ǫ ∗ ( x , s ) + p ( x , s ) · ˙ u ∗ ( x , s )] (11)Passivity requires that the total energy absorbed by the system be non-negative at all times: E ( t ) ≥ ∀ t (12)To calculate the total energy absorbed by the system in Eq. (11), we note that the time derivative of E ( t ) shouldequal the power input from the tractions, t ( x , t ), which are acting on ∂ Ω and the body forces, f , which are acting inΩ . This power input is given by: P ( t ) = Re (cid:20)Z ∂ Ω d x t i ( x , t ) ˙ u ∗ i ( x , t ) + Z Ω d x f i ( x , t ) ˙ u ∗ i ( x , t ) (cid:21) ≡ Re Z Ω d x [( σ ij ˙ u ∗ i ) ,j + f i ˙ u ∗ i ] (13)where the dependence on space and time is suppresed. The above is achieved through the application of the Gausstheorem and the relation t i = σ ij n j . By decomposing ˙ u i,j = ˙ ǫ ij + ˙ ω ij where ˙ ǫ and ˙ ω are the symmetric andantisymmetric parts of ˙ u i,j , respectively, and noting that the inner product of a symmetric tensor and an antisymmetrictensor goes to zero, we have: P ( t ) = Re Z Ω d x (cid:2) σ ij,j ˙ u ∗ i + σ ij ˙ ǫ ∗ ij + f i ˙ u ∗ i (cid:3) (14)Using the equation of motion (Eq. 6) and rearranging, we have: P ( t ) = Re Z Ω d x (cid:2) σ ij ˙ ǫ ∗ ij + ˙ p i ˙ u ∗ i (cid:3) (15)The above calculated power should equal the rate of change of the total energy stored in Ω: d E ( t ) dt = P ( t ) (16)Eqs. (11,12,16) together give: E ( t ) = Z t −∞ d s P ( s ) = Re Z t −∞ d s Z Ω d x (cid:2) σ ij ( x , s ) ˙ ǫ ∗ ij ( x , s ) + ˙ p i ( x , s ) ˙ u ∗ i ( x , s ) (cid:3) ≥ Z t −∞ d s (cid:2) σ ij ( s ) ˙ ǫ ∗ ij ( s ) + ˙ p i ( s ) ˙ u ∗ i ( s ) (cid:3) ≥ x ∈ Ω. Since the constitutive tensors inEq. (8) are real we note that ˙ ǫ ∗ = ˙ D ∗ σ ∗ and ˙ p = ˙ ρ ∗ ˙ u . Using these relations we have:Re Z t −∞ d s (cid:20) σ ij ( s ) Z ∞−∞ d v ˙ D ijkl ( v ) σ ∗ kl ( s − v ) + ˙ u ∗ i ( s ) Z ∞−∞ d v ˙ ρ ij ( v ) ˙ u j ( s − v ) (cid:21) ≥ D ( x , t ) = ˙ ρ ( x , t ) = 0 for all t <
0. Now we employ the distributional Laplace transform. This is done byfirst choosing σ ( s ) = σ φ ∗ ( s ) and ˙ u ( s ) = ˙ u γ ( s ) where σ , ˙ u are constant tensors and φ ( s ) , γ ( s ) are in D :Re Z t −∞ d s (cid:20) σ ij φ ∗ ( s ) Z ∞−∞ d v ˙ D ijkl ( v ) σ ∗ kl φ ( s − v ) + ˙ u ∗ i γ ∗ ( s ) Z ∞−∞ d v ˙ ρ ij ( v ) ˙ u j γ ( s − v ) (cid:21) ≥ φ ( s ) , γ ( s ) be equal to e zs for −∞ < s < a where t < a < ∞ and z ∈ C . With the requirement ofcausality on ˙ D , ˙ ρ the above inequality reduces to the following:Re (cid:20) σ ij σ ∗ kl Z ∞ d v ˙ D ijkl ( v ) e − zv + ˙ u ∗ i ˙ u j Z ∞ d v ˙ ρ ij ( v ) e − zv (cid:21) Z t −∞ e z r s d s ≥ (cid:20) σ ij σ ∗ kl Z ∞ d v ˙ D ijkl ( v ) e − zv + ˙ u ∗ i ˙ u j Z ∞ d v ˙ ρ ij ( v ) e − zv (cid:21) ≥ h σ : ˆ˙ D : σ ∗ + ˙ u · ˆ˙ ρ · ˙ u ∗ i ≥ D and ˆ˙ ρ into their hermitian and non-hermitian parts:ˆ˙ D = ˆ˙ D h + ˆ˙ D nh ; ˆ˙ D h = 12 h ˆ˙ D + ˆ˙ D † i ; ˆ˙ D nh = 12 h ˆ˙ D − ˆ˙ D † i ˆ˙ ρ = ˆ˙ ρ h + ˆ˙ ρ nh ; ˆ˙ ρ h = 12 h ˆ˙ ρ + ˆ˙ ρ † i ; ˆ˙ ρ nh = 12 h ˆ˙ ρ − ˆ˙ ρ † i (24)and note that only hermitian parts of the tensors contribute to the real part of Eq. (23). Furthermore, since thetensors σ , ˙ u in Eq. (23) are arbitrary the inequality E ( t ) ≥ φ : ˆ˙ D h : φ ∗ ≥ q · ˆ˙ ρ h · q ∗ ≥ φ is an arbitrary complex-valued second order symmetric tensor, q is an arbitrary complex-valued vector,and the relation holds for all x and z (in the region of convergence). Similar results can be derived for the Fouriertransform of the time derivatives of the constitutive tensors. Under the restriction that the support of a distribution f be bounded, its Fourier transform is given by h f ( t ) , e iωt i . Now we let φ ( t ) , γ ( t ) be equal to e − iωt in Eq. (20) andfollow the subsequent process to determine that the hermitian parts of the Fourier transforms of the time-derivativeconstitutive tensors must be positive semi-definite. However, the boundedness restrictions on the constitutive tensorsneed not be so severe for us to come to this conclusion. We merely assume that the constitutive tensors are distributionsof slow growth to come to the same conclusion. To do so we consider the following for a test function φ ( t ) ∈ S and adistribution f ( t ) ∈ S ′ : Z t −∞ d sφ ∗ ( s ) Z ∞−∞ d vf ( v ) φ ( s − v ) = 12 π Z t −∞ d sφ ∗ ( s ) Z ∞−∞ d ω ˜ f ( ω ) Z ∞−∞ φ ( s − v ) e − iωv d v = 12 π Z ∞−∞ d ω Z t −∞ d sφ ∗ ( s ) e − iωs ˜ f ( ω ) Z ∞−∞ d uφ ( u ) e iωu = 12 π Z ∞−∞ d ω ¯ φ ( ω ) ˜ f ( ω ) ¯ φ ∗ ( ω ) (26)where the distributional Fourier transform is used and the last step follows by choosing the test function φ ( τ ) such thatit vanishes for τ > t . It is clear from the above that under the much less restrictive conditions that the constitutivetensors be distributions of slow growth, Eq. (20) can be written, after some manipulations, in the following way:Re Z ∞−∞ d ω h σ ( ω ) : ˜˙ D ( ω ) : σ ∗ ( ω ) + ˙ u ( ω ) · ˜˙ ρ ( ω ) · ˙ u ∗ ( ω ) i ≥ σ , ˙ u are arbitrary, the above will be satisfied only if the following is true about the hermitian parts of the Fouriertransforms: φ : ˜˙ D h : φ ∗ ≥ q · ˜˙ ρ h · q ∗ ≥ . These relations can be used to place similar constraints on the transforms of constitutive tensors D , ρ . Forthis we need only consider the relations between the transforms of derivates as they apply to distributions in S ′ . Thefollowing relations are noted: ˜˙ f = − iω ˜ f ; ˆ˙ f = z ˆ f ; f ∈ S ′ (29)Since the elements of our constitutive tensors are assumed to be in S ′ the above relations apply to them. Specificallywe have, for instance for the Fourier transforms:˜ D ( ω ) = − ˜˙ D ( x , ω ) iω = i ˜˙ D ( x , ω ) ω ; ˜ ρ ( x , ω ) = i ˜˙ ρ ( x , ω ) ω (30)Eq. (30) explicitly relates the Fourier transforms of the constitutive tensors to the Fourier transforms of their timederivatives. The Laplace transforms can be similarly related. To extend the constraints which passivity applies tothe Fourier transforms of the constitutive tensors (analogous to Eq. 28) we use the hermitian transpose operation, † .It is clear that ˜ D h † = D h and ˜ D nh † = − D nh with similar relations holding for second order tensors. By expanding˜ D , ˜ ρ , ˜˙ D , ˜˙ ρ into their hermitian and non-hermitial parts in Eq. (30) and by applying the hermitian transpose operatorit becomes clear that the hermitian parts of the time derivative quantities are related to the non-hermitian parts ofthe original tensors in the following sense ( ω dependent implicit):˜˙ D h = ωi ˜ D nh ; ˜˙ ρ h = ωi ˜ ρ nh (31)Therefore, now Eq. (28) places the following constraints on the Fourier transforms of D , ρ : φ : ωi ˜ D nh ( x , ω ) : φ ∗ ≥ q · ωi ˜ ρ nh ( x , ω ) · q ∗ ≥ φ : ˜ D nh ( x , ω ) : φ ∗ and q · ˜ ρ nh ( x , ω ) · q ∗ result in purely imaginary numbers. Howeverthe factor i in the denominator ensures that the quantities in Eq. (32) are purely real. Passivity and causality of thesystem demand that these numbers also be non-negative for positive values of ω (and non-positive for negative valuesof ω ). We also note the corollary result that had we decided to represent Fourier transform through the exponential e − iωt instead of e iωt we would have arrived at the complementary result where the non-hermitian quantities abovewould have been required to be negative semi-definite instead of positive semi-definite. In the following sections wewill consider a specialization and a generalization of the above results. The specialization refers to cases where theconstitutive tensors possess major symmetries and the generalization refers to the above results in the context of moregeneral forms of constitutive relations such as the Willis kind of coupled relations. IV. WITH MAJOR SYMMETRIES
Up to now we have assumed no special forms for the compliance and density tensors beyond the minor symmetrieswhich ensure rotational stability of the system. We now consider the specialization of the above results to a casewhere the constitutive tensors possess major symmetries as well. For the density tensor we mean that its componentssatisfy ρ ij = ρ ji . Similarly we require that the fourth order compliance tensor D ijkl satisfy D ijkl = D klij . Since thecomponents of the Fourier and Laplace transforms of the constitutive tensors are only related to the correspondingtime domain components, it is clear that these major symmetries will extend to them as well. With these additionalrequirements Eqs. (24) imply that ˜˙ D h ∗ = ˜˙ D h and ˜˙ ρ h ∗ = ˜˙ ρ h essentially meaning that ˜˙ D h and ˜˙ ρ h are composedonly of the real parts of ˜˙ D and ˜˙ ρ respectively. Similarly, ˜˙ D nh and ˜˙ ρ nh are composed only of the imaginary parts of˜˙ D and ˜˙ ρ respectively. Furthermore, these characteristics should hold for all tensors of current interest (i.e. Fourierand Laplace transforms of D , ρ as well). Consideration of this specialization is of interest because for this case, thedefiniteness relations apply simply to the real and imaginary parts of the relevant tensors. Specifically, we have thefollowing relations for this case: φ : Re ˜˙ D : φ ∗ ≥ q · Re ˜˙ ρ · q ∗ ≥ ∀ φ = φ T , q φ : ωi Im ˜ D : φ ∗ ≥ q · ωi Im ˜ ρ · q ∗ ≥ ∀ φ = φ T , q (33)Restricting our attention for this section to the real parts (denoted by the subscript r) of the time derivative, fouriertransformed tensors and to the imaginary parts (denoted by the subscript i) of the fourier transformed tensors,and using the shorthand ≥ to imply positive semi-definiteness, the above relations are condensed to ( ω dependenceimplied): ˜˙ D r , ˜˙ ρ r , ω ˜ D i , ω ˜ ρ i ≥ ∀ x , ω Similar relations are derived by Milton and Willis in the context of minimum variational principles for time-harmonicwaves in a dissipative medium. Since the imaginary parts of the Fourier transformed constitutive relations, for thesimpler major-symmetric case as this one, corresponds to the dissipation in the system we note an interesting result asa corollary. For constitutive relations which do not necessarily possess major-symmetry, it is the non-hermitian partsof the Fourier transformed tensors which corresponds to dissipation in the system. In other words, a conservative system can be expected to be hermitian in the Fourier transform of its constitutive relations . One system whichimmediately corresponds to the major-symmetric specialization being considered here is the case of one dimensionalelastodynamics: ǫ ( x, t ) = Z t −∞ d s D ( x, t − s ) σ ( x, s ) p ( x, t ) = Z t −∞ d s ρ ( x, t − s ) ˙ u ( x, s )Causality and passivity results from the earlier sections can be immediately extended here to conclude that ˆ D, ˆ ρ areanalytic for Re z > D r , ˆ˙ ρ r , ˜˙ D r , ˜˙ ρ r , ω ˜ D i , ω ˜ ρ i ≥ ≥ actually means greater than or equalto in the present case and not just positive semi-definiteness. The static case, for which the constitutive tensorscan be represented through the delta distribution, is seen to trivially satisfy these conditions since ˜ δ = ˆ δ = 1. Thisensures that ω ˜ D i , ω ˜ ρ i = 0 etc. The symbol ≥ can be understood to mean greater than equal to, and not just positivesemi-definiteness, whenever a diagonal constitutive relation is being considered. In those cases passivity and causalitywould dictate that the ≥ relations apply to the diagonal elements individually. Diagonal relations for the densitytensor include those cases where ρ ij ∝ δ ij , and for the compliance tensor include those cases where D ijkl ∝ δ ik δ jl . V. GENERALIZATION TO OTHER CONSTITUTIVE RELATIONSHIPS
To derive the passivity relationships we required that energy could be expressed in a particular form (which itdoes automatically for the constitutive relations considered up to now). We will use this observation to generalize theresults from the previous sections to more general constitutive relations such as the Willis relations. In the subsequenttreatment we will understand the space dependence to be implicit in the sense of Eq. (14). Let w ( t ) , v ( t ) denotecolumn vectors consisting of n time dependent tensors. Elements of w ( t ) are assumed to be in D to ensure thatthe energy expression exists (elements of v ( t ) are also distributions but they need not be so restricted). Let v ( t ) bederivable from w ( x , t ) through a linear, real, time invariant, and causal relationship v = L ∗ w where L is a n × n matrix of real valued tensors: v ( t ) = Z ∞−∞ d s L ( t − s ) w ( s ) = Z ∞−∞ d s L ( s ) w ( t − s ) (34)Each element of every tensor in L is assumed to be a distribution of slow growth. Let us also assume that the energyabsorbed by the system up to a time t can be represented by: E ( t ) = Re Z t −∞ d s w † ( s ) ˙ v ( s ) = Re Z t −∞ d s w † ( s ) Z ∞−∞ d v ˙ L ( v ) u ( s − v ) (35)then some conclusions apply to the Fourier and Laplace transforms of L and ˙ L . For instance, decomposing ˜˙ L intoits hermitian and non-hermitian parts ˜˙ L = ˜˙ L h + ˜˙ L nh and noting that ( y † ˜˙ L h y ) † = y † ˜˙ L h y and, therefore, real and( y † ˜˙ L nh y ) † = − y † ˜˙ L nh y and, therefore, imaginary. This means that E ( t ) emerges from ˜˙ L h (or ˆ˙ L h ). Passivity requiresthat absorbed energy must be non-negative at all times or that E ( t ) ≥
0. This implies the following: y † ˜˙ L h y ≥ y † ˆ˙ L h y ≥ y † ωi ˜ L nh y ≥ x , t dependence implied): (cid:18) ǫ p (cid:19) = (cid:20) D S S ρ (cid:21) ∗ (cid:18) σ ˙ u (cid:19) (37)Power from Eq. (15) can be written as: P ( t ) = Re (cid:2) σ ij ˙ ǫ ∗ ij + ˙ p i ˙ u ∗ i (cid:3) ≡ Re (cid:2) σ ∗ ij ˙ ǫ ij + ˙ u ∗ i ˙ p i (cid:3) = Re w † ˙ v (38)with v = (cid:18) ǫ p (cid:19) ; w = (cid:18) σ ˙ u (cid:19) (39)Since energy can now be written as E ( t ) = Z t −∞ d s P ( s ) = Re Z t −∞ d s w † ( s ) ˙ v ( s ) (40)we immediately have from the earlier results in this section that passivity implies that the following tensors will bepositive semi-definite in the sense of Eq. (36):˜˙ L h = " ˜˙ D ˜˙ S ˜˙ S ˜˙ ρ + " ˜˙ D ˜˙ S ˜˙ S ˜˙ ρ † ; ˆ˙ L h = " ˆ˙ D ˆ˙ S ˆ˙ S ˆ˙ ρ + " ˆ˙ D ˆ˙ S ˆ˙ S ˆ˙ ρ † ; ˜ L nh = (cid:20) ˜ D ˜ S ˜ S ˜ ρ (cid:21) − (cid:20) ˜ D ˜ S ˜ S ˜ ρ (cid:21) † (41)It must be noted that the above results make no assumptions about the process by which Willis properties have beendefined or derived or whether the relations are hermitian or not. They are merely the constraints which must besatisfied if an elastodynamic system, represented in the Willis form, is required to be causal. In fact, the questionof whether the Willis constitutive relation displays self-adjointness (or are hermitian) or not has been addressed inliterature. In Ref. it has been shown that the property of self-adjointness is preserved at the level of the effectiveresponse. In other words, the effective relations are (not) self-adjoint if the constituting materials themselves are(not) self-adjoint. Ref. talks about the related question of hermiticity. The Willis relations exhibit several degreesof non-uniqueness and, to the author’s knowledge, it is not clear if some, many, or all sets of Willis properties (thatcan be assigned under any given case) satisfy the kinds of causality and passivity requirements discussed in this paper. VI. CONCLUSIONS
In this paper we clarify the constraints that causality and passivity place on the elastodynamic constitutive tensors.Analogous questions have been addressed in other fields but the elastodynamic case is generally more complicateddue to the higher order and non-diagonal nature of its constitutive relations. Here we deal with the problem inconsiderable generality wherein the elements of the constitutive tensors are assumed to be generalized functions intime. The treatment and conclusions presented here, therefore, apply to metamaterial applications which often involvesingular and coupled constitutive forms and also to the static limit where the constitutive tensors are in the form ofdelta distributions. Specifically we show that the satisfaction of passivity (and causality) directly requires that thehermitian parts, as defined later, of the transforms (Fourier and Laplace) of the time derivatives of the elastodynamicconstitutive tensors be positive semi-definite. Additionally, the conditions subsequently require that the non-hermitianparts of the Fourier transforms of the constitutive tensors be positive semi-definite for positive values of frequencyand negative semi-definite for negative values of frequency. We show that when major symmetries are assumed thesedefiniteness relations apply simply to the real and imaginary parts of the relevant tensors. For diagonal and one-dimensional problems, these positive semi-definiteness relationships reduce to simple inequality relations over the realand imaginary parts. Finally we extend the results to highly general forms of constitutive relations which include theWillis inhomogeneous relations as a special case.
VII. DATA ACCESSIBILITY
No data has been presented in this paper
VIII. COMPETING INTERESTS
We have no competing interests
IX. AUTHORS’ CONTRIBUTIONS
This author is the sole author of this paper
X. ACKNOWLEDGEMENTS
The author wishes to thank Prof. John R. Willis for his comments and suggestions.0
XI. FUNDING
The author acknowledges the support of the UCSD subaward UCSD/ONR W91CRB-10-1-0006 to the IllinoisInstitute of Technology (DARPA AFOSR Grant RDECOM W91CRB-101-0006 to the University of California, SanDiego). ∗ Corresponding Author; [email protected] Edward Charles Titchmarsh et al.
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