## Abstract

What are the constraints placed on the constitutive tensors of elastodynamics by the requirements that the linear elastodynamic system under consideration be both causal (effects succeed causes) and passive (system does not produce energy)? The analogous question has been tackled in other areas but in the case of elastodynamics its treatment is complicated by the higher order tensorial nature of its constitutive relations. In this paper, we clarify the effect of these constraints on highly general forms of the elastodynamic constitutive relations. We show that the satisfaction of passivity (and causality) directly requires that the hermitian parts of the transforms (Fourier and Laplace) of the time derivatives of the constitutive tensors be positive semi-definite. Additionally, the conditions require that the non-hermitian parts of the Fourier transforms of the constitutive tensors be positive semi-definite for positive values of frequency. When major symmetries are assumed these definiteness 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 real and imaginary parts, as they should. Finally, we extend the results to highly general constitutive relations which include the Willis inhomogeneous relations as a special case.

## 1. Introduction

Various aspects of nature are modelled as cause–effect relationships between different physical processes. These physical processes are often functions of time and the relations between them, in some cases, can be more easily analysed in the frequency or Laplace domains. A frequency-dependent process is called *dispersive* and can be studied by deriving the appropriate dispersion relations of the system. Physical systems often have an inherent assumption of causality wherein effects are assumed not to precede causes. If the physical system is also linear and time-invariant then certain sum/integral rules could be derived connecting the physical quantities involved [1,2]. For example, the Kramers–Kronig (K–K) relationships [3,4] are integral relationships which connect the real part of the electromagnetic index of refraction to its imaginary part, thus connecting dispersion and loss in the medium. Since their introduction, the K–K relationships have been used in the study of circuit theory [5] and all forms of wave propagation [6–13].

The K–K relations have recently attracted interest in the area of metamaterials where the goal is to create materials with exotic electromagnetic, acoustic and/or elastodynamic properties. The essential ideas emerge from early theoretical works of Veselago [14] and more recent experimental efforts by various research groups [15–17] (see [18] for a review). The possibility of creating materials with unprecedented material properties has led to far-reaching postulations of their applications, most visibly, in the area of cloaking [19–26]. As material properties can essentially be viewed as time-domain transfer functions which relate a cause to its effect (and, therefore, must be causal), K–K relations and their 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. This causality check includes, on one end of the spectrum, placing some realistic constraints on the application potential of metamaterials as cloaking devices [27] to, on the other end, sobering realizations that a considerable amount of metamaterials research stands on shaky foundations, often proposing materials which violate such basic ideas as causality and/or the second law of thermodynamics [28]. This has led to a number of researchers advocating a need for improved models for metamaterials [29–32].

Closely connected to the idea of causality is the concept of passivity which refers to the assumption that the physical process under consideration cannot produce energy [33]. In fact, if the physical process (cause–effect relationship) can be expressed in a convolution form in the time-domain then its satisfaction of the passivity requirement automatically means that it also satisfies causality [34]. A physical process can, in turn, be expressed in the convolution form if it satisfies certain conditions such as linearity and time-invariance. It becomes interesting, therefore, to understand what constraints are placed upon a linear time-invariant cause–effect relationship (constitutive relationship) in electromagnetic, acoustic and/or elastodynamic areas by the requirement of passivity. Such knowledge can be used to place constraints on and understand the limitations of various metamaterial models which are used in these areas to arrive at such relationships. Considerable research in this direction has already taken place in the field of electromagnetics where it is clear that passivity demands that the imaginary parts of the diagonal values of the Fourier transform of ** ϵ**,

**be non-negative for all positive values of frequency [35–39] (fields assumed to depend upon**

*μ**e*

^{−iωt}). However, it is not clear, to the author’s knowledge, what should be the equivalent constraints in elastodynamics for the most general constitutive cases. The case of one-dimensional 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 corresponding one-dimensional material properties (modulus and density) should behave analogously to the

*ϵ*,

*μ*. However, in two- and three-dimensional, the elastodynamic constitutive tensor cannot, in general, be diagonalized. Moreover, recent advancements [26,40–48] suggest that the Willis constitutive relation [49], which is a coupled form of constitutive relation, is more appropriate for the description of inhomogeneous elastodynamics and, therefore, of elastodynamic metamaterials. It is not clear what the constraints 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 elastodynamic constitutive relations. We use the passivity condition which is equivalent to the statement of passivity used in electromagnetism [38] and circuit theory [34] and which is elaborated in subsequent sections. We also present our analysis within the context of distributions which is the proper space within which to describe the transfer functions of passive systems. Furthermore, treating the constitutive tensors in the space of distributions ensures that the analysis applies to the metamaterial cases of most interest and also to the static case (elastic case).

## 2. Background

Physical processes in the real world are often described as an interplay between physical variables and fields which are dependent upon time. The relationships between the physical variables can be modelled as input–output relations where a time-dependent variable *v*(*t*) is produced from another time-dependent variable *u*(*t*) through some rule *ϕ*(*t*) which are infinitely smooth and with compact support. It is a subset of space *ϕ*(*t*), called functions of rapid descent, such that they and all their derivatives decrease to zero faster than every power of 1/|*t*| as *R*,*u* are locally integrable distributions whose supports satisfy certain boundedness properties (either *R*,*u* have bounded supports, or both *R*,*u* are either bounded on the left or on the right). The operator *R* is causal if it is not supported on *t*<0. The final property of passivity can be stated by defining the energy of the system. If the power absorbed by the system at time *s* is given by Re *v**(*s*)*u*(*s*), where * denotes the complex conjugate, then define the energy absorbed by the system up to time *t* as:
*R* is considered passive if *f*,*ϕ*〉 is the value in *f* assigns to *ϕ* through the operation *R* being a distribution of slow growth, causality implies that its support is in *z*>0 (passivity constraint) and that *R* is causal and is in

*For transfer functions of higher order* and more complexity which form the object of study of this paper, we need 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, 〈**f**(*t*),*ϕ*(*t*)〉 is the matrix of complex numbers obtained by replacing each element of **f**(*t*) by the number that this element assigns to the testing function *ϕ*(*t*). We will use additional subscripts with the spaces already defined above to denote the space in which all tensors of the relevant rank and distribution lie. For e.g. **f** we also define the operations **f**^{T} which denotes a transpose over the major-symmetry and **f**^{†} which denotes 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:
**R**,**u** are assumed and where **v** is a tensorial quantity derived from **u** through the linear operator **R**. Total energy absorbed up to time *t* is given by:
**R** is considered passive if **u** to be in **R** which is in the convolution form is passive then it can also be shown to be causal [34] and, furthermore, its elements are in **R** follow. Its Laplace transform is given by *z*>0. Note that these results follow from some fairly unrestrictive constraints on the input field and transfer function which are easily satisfied in elastodynamics (and electromagnetism). Our effort here is to apply and extend these results to the elastodynamic case.

## 3. Causality and passivity in elastodynamics

We begin by considering a volume *Ω* within which the point-wise elastodynamic equation of motion and kinematic relations are specified:
** σ**,

**,**

*ε***p**,

**u**and

**f**are the space- and time-dependent stress tensor, strain tensor, momentum vector, displacement vector and body force vector, respectively. These relations need to be supplied with appropriate constitutive relations which relate the various field variables to each other. For the current discussion, we consider stress and velocity to be independent fields (input fields) which lead to the emergence of strain and momentum fields (output/dependent fields), respectively. The relationships are expressed in terms of general constitutive operators whose properties need to be determined based upon the various subsequent assumptions about the system:

**D**,

**are real-valued distributions.**

*ρ***D**is a fourth-order tensor field in

**is a second-order tensor field in**

*ρ***to be in**

*σ**Causality*. Causality refers to the requirement that an effect cannot precede its cause. With reference to the constitutive relations in equation (3.3), it implies that the value of the strain and momentum fields at time *t*_{0} can only depend upon the values, respectively, of the stress and velocity fields at times prior to and including *t*_{0}. A necessary and sufficient condition for a system to be causal is that its unit response function (constitutive operator in the present case) vanishes for *t*<0. Specifically, causality implies the following for the constitutive tensors:

*Passivity*. Passivity refers to the requirement that the system cannot generate energy. For the elastodynamic case the total energy at any time *t* contained in *Ω* comprises of the elastic energy contribution and the kinetic energy contribution:
**t**(**x**,*t*), which are acting on ∂*Ω* and the body forces, **f**, which are acting in *Ω* [52]. This power input is given by:
*t*_{i}=*σ*_{ij}*n*_{j}. By decomposing *Ω*:
*Ω* is arbitrary, the above would be satisfied only if the inequality holds at each point in space. Going forward we, therefore, understand the passivity statement to be the following:
**x**∈*Ω*. As the constitutive tensors in equation (3.3) are real we note that *t*<0. Now we employ the distributional Laplace transform. This is done by first choosing ** σ**(

*s*)=

*σ**ϕ**(

*s*) and

*ϕ*(

*s*),

*γ*(

*s*) are in

*ϕ*(

*s*),

*γ*(

*s*) be equal to

*e*

^{zs}for

**is an arbitrary complex-valued second-order symmetric tensor,**

*ϕ***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 Fourier transform 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 〈

**f**(

*t*),

*e*

^{iωt}〉. Now we let

*ϕ*(

*t*),

*γ*(

*t*) be equal to

*e*

^{−iωt}in equation (3.15) and follow the subsequent process to determine that the hermitian parts of the Fourier transforms of the time-derivative constitutive tensors must be positive semi-definite. However, the boundedness restrictions on the constitutive tensors need not be so severe for us to come to this conclusion. We merely assume that the constitutive tensors are distributions of slow growth to come to the same conclusion. To do so, we consider the following for a test function

*ϕ*(

*τ*) such that it vanishes for

*τ*>

*t*. It is clear from the above that under the much less restrictive conditions that the constitutive tensors be distributions of slow growth, equation (3.15) can be written, after some manipulations, in the following way:

**D**,

**. For this we need only consider the relations between the transforms of derivatives as they apply to distributions in**

*ρ**ω*dependent implicit):

**D**,

**:**

*ρ**i*in the denominator ensures that the quantities in equation (3.27) are purely real. Passivity and causality of the system demand that these numbers also be non-negative for positive values of

*ω*(and non-positive for negative values of

*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 above would have been required to be negative semi-definite instead of positive semi-definite. In the following sections, we will consider a specialization and a generalization of the above results. The specialization refers to cases where the constitutive tensors possess major symmetries and the generalization refers to the above results in the context of more general forms of constitutive relations such as the Willis kind of coupled relations.

## 4. With major symmetries

Up to now, we have assumed no special forms for the compliance and density tensors beyond the minor symmetries which ensure rotational stability of the system. We now consider the specialization of the above results to a case where the constitutive tensors also possess major symmetries. For the density tensor, we mean that its components satisfy *ρ*_{ij}=*ρ*_{ji}. Similarly, we require that the fourth-order compliance tensor *D*_{ijkl} satisfy *D*_{ijkl}=*D*_{klij}. As the components of the Fourier and Laplace transforms of the constitutive tensors are only related to the corresponding time domain components, it is clear that these major symmetries will also extend to them. With these additional requirements equations (3.19) imply that **D**, ** ρ**). Consideration of this specialization is of interest because for this case, the definiteness relations apply simply to the real and imaginary parts of the relevant tensors. Specifically, we have the following relations for this case:

*ω*dependence implied):

*For constitutive relations which do not necessarily possess major-symmetry, it is the non-hermitian parts of 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 which immediately corresponds to the major-symmetric specialization being considered here is the case of one-dimensional elastodynamics:

*z*>0 and that

*ρ*

_{ij}∝

*δ*

_{ij}, and for the compliance tensor include those cases where

*D*

_{ijkl}∝

*δ*

_{ik}

*δ*

_{jl}.

## 5. Generalization to other constitutive relationships

To derive the passivity relationships we required that energy could be expressed in a particular form (which it does automatically for the constitutive relations considered up to now). We will use this observation to generalize the results from the previous sections to more general constitutive relations such as the Willis relations. In the subsequent treatment, we will understand the space dependence to be implicit in the sense of equation (3.9). Let **w**(*t*),**v**(*t*) denote column vectors consisting of *n* time-dependent tensors. Elements of **w**(*t*) are assumed to be in **v**(*t*) are also distributions but they need not be so restricted). Let **v**(*t*) be derivable 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:
**L** is assumed to be a distribution of slow growth. Let us also assume that the energy absorbed by the system up to a time *t* can be represented by:
**L** and **x**,*t* dependence implied):

## 6. Conclusion

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 complicated because of the higher order and non-diagonal nature of its constitutive relations. Here we deal with the problem in considerable generality wherein the elements of the constitutive tensors are assumed to be generalized functions in time. The treatment and conclusions presented here, therefore, apply to metamaterial applications which often involve singular and coupled constitutive forms and also to the static limit where the constitutive tensors are in the form of delta distributions. Specifically, we show that the satisfaction of passivity (and causality) directly requires that the hermitian parts, as defined later, of the transforms (Fourier and Laplace) of the time derivatives of the elastodynamic constitutive tensors be positive semi-definite. Additionally, the conditions subsequently require that the non-hermitian parts of the Fourier transforms of the constitutive tensors be positive semi-definite for positive values of frequency and negative semi-definite for negative values of frequency. We show that when major symmetries are assumed these definiteness 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 real and imaginary parts. Finally, we extend the results to highly general forms of constitutive relations which include the Willis inhomogeneous relations as a special case.

## Competing interests

I have no competing interests

## Funding

The author acknowledges the support of the UCSD subaward UCSD/ONR W91CRB-10-1-0006 to the Illinois Institute of Technology (DARPA AFOSR Grant RDECOM W91CRB-10-1-0006 to the University of California, San Diego).

## Acknowledgements

The author thanks Prof. John R. Willis for his comments and suggestions.

- Received April 19, 2015.
- Accepted June 23, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.