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The work deals with a mathematical model for real-time acoustic monitoring of material parameters of media in multi-state viscoelastic engineering systems continuously operating in irregular external environments (e.g., wind turbines in cold climate areas, aircrafts, etc.). This monitoring is a high-reliability time-critical task. The work consistently derives a scalar wave PDE of the Stokes type for the non-equilibrium part (NEP) of the average normal stress in a medium. The explicit expression for the NEP of the corresponding pressure and the solution-adequateness condition are also obtained. The derived Stokes-type wave equation includes the stress relaxation time and is applicable to gases, liquids, and solids.

One of the applications of the acoustic-sensing technology is monitoring of material parameters of engineering systems continuously operating in irregular external environments. This type of the operation indicates that the monitoring must be regular (e.g., periodic) and in the real-time mode. Many problems in this area deal with the systems that are multi-state and viscoelastic in the following sense. A system comprises at least two spatial domains, each of which is occupied with an isotropic medium that is spatially homogeneous at equilibrium and is at one of the three states of matter: gaseous, liquid, or solid. In addition to that, the states of at least two of these spatial components are different, and the components are generally visoelastic.

The features of the considered systems and available mathematical models used for the system material parameter monitoring are further specified and discussed in more detail below.

a) The regular monitoring of material parameters is implemented by the non-invasive sensing of acoustic signals in one or more components of the system. The subsequent identification of the parameters is performed by using the sensed signals and the corresponding medium-specific acoustic models (e.g., [

b) The regular real-time acoustic monitoring presents the sequence of the sensing cycles started at a series of time points and implemented with one or more sensors in the automatic mode. If, say,

Moreover, the aforementioned continuous automatic operation presumes zero user intervention. This makes high demands of reliability of the data processing. The above picture indicates that the regular real-time moni- toring is a high-reliability time-critical task.

c) Due to the above multi-state feature of the system, the models mentioned in Point (a) generally include acoustic models for fluids and solids. In each of these cases, they are not formulated for acoustic signals, i.e. non-equilibrium parts of the Cauchy stress matrix entries. The fluid acoustic models are formulated for the entries of the velocity vector, whereas the solid acoustic models are formulated for the entries of the displace- ment vector. This inevitably complicates the entire description necessary for the parameter identification. More- over, the diversity and complexity of the modeling are further contributed by the use of representations for the conjunction of the initial conditions and the boundary conditions at the interfaces between the system com- ponents, which are at different states of matter (see above).

d) Acoustic models for fluids and solids generally include a system of three scalar non-stationary partial differential equations (PDEs) in the three-dimensional physical space.

e) Common fluid mechanics acoustic models natively include not only elastic moduli of the medium but also its viscosities, and thereby they are applicable to the corresponding viscoelastic components of the system. Also, the mentioned models are consistently derived from a more general physical theory, kinetic theory (e.g., [

In contrast to that, common solid mechanics acoustic models include elastic moduli but do not include vis- cosities. There is an advanced model for visoelastic solids (e.g., [

However, as follows from the discussion in ( [

The features of common acoustic continuum mechanics models listed in Points (c)-(e) are not well suited for the time-critical nature indicated in Point (b). Consequently, a modeling basis for regular real-time acoustic monitoring of material parameters of multi-state engineering systems continuously operating in irregular exter- nal environments remains a research topic. The purpose of the present work is to contribute to this topic. More specifically, the work derives an acoustic PDE for an appropriate scalar component of the Cauchy stress matrix and explains why this PDE is applicable to gases, liquids, and solids, including viscoelastic media.

It should be noted that the idea of PDEs for the entries of the Cauchy stress matrix goes back to at least H. Grad who derives non-stationary spatially non-homogeneous PDE system for these entries ( [

The work is arranged as follows. Section 2 summarizes the basic facts on the Cauchy stress matrix and the key component of it, scalar and deviatoric stresses. The main result derived in the work is presented in Section 3 that also discusses the novelty of it and its connection to the related results of other authors. Section 4 concludes the work. The detailed derivation of the main result is carried out in Appendix A. It applies selected represent- ations associated with the coupling of Eulerian and Lagrangian coordinates outlined in Appendix B.

Acoustic signals present the spatiotemporal deviation,

The terms denoted with the sign “overline” are specified in the remark below.

Remark 2.1. As is well known, physical quantities at equilibrium are independent of time. The present work considers the media only such that, at equilibrium, they are independent of space as well. Consequently, the equilibrium versions of physical quantities do not depend on space either. These versions are denoted with the sign “overline” applied to the notation of the corresponding quantity (e.g., see (2.1)). ,

One usually represents matrix

where

The diagonal and off-diagonal entries of matrix S are known as the scalar normal and shear stresses, re- spectively. Since (2.3) determines P as the arithmetic mean of the total normal stresses, P is called the average normal stress (ANS).

As follows from (2.2) and (2.3), matrix

is called the deviatoric stress. Also, this stress is zero at equilibrium, i.e.

The relaxation of deviatoric stress Z to its equilibrium value (2.5) is usually described in terms of the stress relaxation time, say, θ, and according to asymptotic representation

In an isotropic medium, deviatoric stress

For the sake of simplicity, we also use θ in the expression

Value

This parameter is sometimes called the speed of bulk waves.

We note that ANS P completely determines not only scalar stress PI but also the entire stress S at equilibrium with expression

that follows from (2.2) and (2.5). In view of of (2.2), (2.8), and (2.5), Expression (2.1) is equaivalent to

where

Remark 2.2. If the medium is close to the equilibrium state sufficiently in order to neglect deviatoric stress

respectively. ,

Some of the above relations are used in the derivation of the main result of the present work (see Appendix A).

As is shown in Appendix A, under the assumptions listed in

The derivation of (3.1) also provides the corresponding description for the NEP

Equation (3.1) and any of (3.2) and (3.3) are linear. The corresponding solution-adequateness condition is (A.3.7) (see Proposition A.3.1).

In comparison with common wave PDE

Remark 3.1. The derivation of model (3.1)-(3.3) follows theory of viscous fluids but admits the terms native

1 | The medium is spatially homogeneous and isotropic at equilibrium. |
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2 | Elastic properties of the medium can be treated in terms of linear elasticity. |

3 | The medium is close to the equilibrium state sufficiently in order to neglect deviatoric component (2.4) of the total stress (2.2) (see also (2.5)). |

4 | The medium is assumed to be isothermal. As is well known (e.g., [ |

5 | There are no chemical reactions in the medium. |

6 | There are no body forces in the medium. |

7 | If the medium is not a linear solid, then inequality (A.1.13) holds. According to Proposition A.3.1, this inequality can be replaced with (A.3.7). |

8 | The medium is close to the equilibrium state sufficiently in order to replace |

9 | The medium is close to the equilibrium state sufficiently in order to neglect velocity |

in theory of inviscid solids (see Remarks A.1.1 and A.2.1). Thus, the derived model is suitable for gases, liquids, and solids. ,

We also note a connection of PDE (3.1) to a special wave equation that was introduced in 1845. Formally, PDE (3.1) for NEP

The Stokes-type wave PDEs for different variables are used in acoustic of viscoelastic solids since long ago. For example, Section 1 discusses the well-known Stokes-type wave PDE system for the displacement vector in a solid. Paper [

The present work considers material media, which are isothermal, spatially homogeneous and isotropic at equili- brium, with elastic properties treatable in terms of linear elasticity, and can be gaseous, liquid, or solid. The che- mical reactions and body forces in the media are neglected.

Under the assumptions listed in

Application of the derived equation allows to overcome the difficulties emphasized in Points (c)-(e) in Section 1 and thereby meet requirements resulting from the high-reliability and time-critical nature formulated in Point (b) in the mentioned section.

The authors express their gratitude to the Swedish Energy Agency Project 37286-1 for a partial support of the present work. The authors also thank Anders Boström, the Head of the Division of Dynamics, Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, for a stimulating discussion.

The purpose of this section is derivation of a description for the stress NEP

There are two approaches in continuum mechanics to modeling the space-time phenomena, Lagrangian and Eulerian (e.g., [

Models for linear inviscid solids are based on Lagrangian approach (e.g., [

Models for viscous fluids are based on Eulerian approach. They include volume and shear viscosities

According to Eulerian approach, a spatial point moving along a determinate trajectory is described with ODE

where

The total time derivative of a scalar variable, which depends on time and space, is, in view of (A.1), expressed as follows

where column vector

is the gradient with respect to the entries of vector

In view of Assumption 3 in

In view of Assumption 4 in

We consider the quasi-equilibrium versions of these equations. These are the topics of Appendixes A.1 and A.2, respectively. In each of the two cases, the related inviscid/elastic-solid representations are indicated.

A.1. Quasi-Equilibrium Version of the Mass Conservation LawThe present section derives the quasi-equilibrium version of the mass conservation law (A.5). Quantity

is the NEP of

then (A.5) is reduced to its linearized version

The rest of this section concentrates on the terms and conditions, which assure inequality (A.1.2).

The equation of state for a viscous fluid is usually available in the following form

where

Assumption 4 in

Note that the derivative of this function with respect to

and, thus (see (2.7)), the isothermal version of

We also note that

Equality (A.1.9) specifies (2.7) in terms of the equation of state (A.1.4).

The NEP of pressure

is, due to (A.1.5) and (A.1.9), coupled with (A.1.1) as shown

By virtue of (A.1.12) and (A.1.10), inequality (A.1.2) is equivalent to

Since (A.1.13) holds because of Assumption 7 in

Multiplying (A.1.3) by

Remark A.1.1. If the equation of state is unavailable for a solid, then representations (A.1.5)-(A.1.9) and (A.1.12) cannot be used. In this case, one can show that (A.1.13) is still valid (cf., Assumption 7 in

where

We also note that Equation (A.1.14) can be rewritten as the expression for

This equation is the quasi-equilibrium version of Equation (A.5), which is used below.

A.2. Quasi-Equilibrium Version of the Momentum Conservation LawThe present section derives the quasi-equilibrium version of the momentum conservation law (A.6). Owing to (A.5), equation (A.6) can be rewritten as

where

Under Assumption 7 in

Indeed, it is shown in Section A.1, that inequalities (A.1.13) and (A.1.2) are equivalent. Since, due to the aforementioned assumption, (A.1.13) is valid and inequality (A.1.2) is also valid. The latter fact and relation (A.1.1) enable one to replace

Remark A.2.1. Quasi-equilibrium viscous-fluid equation (A.2.2) is equivalent to the well-known equation of inviscid solid mechanics (e.g., [

By virtue of (B.4) and (B.5), equation (A.2.2) can be transformed into

In view of (B.3) and (B.7),

Equation (A.2.2) is the quasi-equilibrium version of Equation (A.5), which is used below.

A.3. Derivation of the PDEEquations (A.1.16) and (A.2.2) includes terms

More specifically, due to (2.10), (A.1.11), and the fact that the equilibrium value of velocity

Assumption 8 in

Applying (A.1.16) to (A.3.3) and taking into account expression in (2.6), one obtains

Application of operation

where

Assumption 9 in

and (3.2).

Proposition A.3.1. Let vector

interval,

Then inequality (A.1.13) is valid at the above

Proof. The proof is based on inequality

for the solution of ODE (3.2) with initial value

Since, as shown in Appendix A.1, (A.1.13) is equivalent (A.1.2) and the latter allows to reduce non-linear Equation (A.5) to its linearized version (A.1.3), the lenearization-enabling inequality (A.3.7) is in fact the solution-adequateness condition.

The obtained description for

It is possible to transform system (3.2), (A.3.6) into the explicit expression for

The closure is achieved in the following way. Applying (3.2) to (3.3), differentiating the resulting equality with respect to

As follows from the above derivation, Equations (3.2) and (3.3) are equivalent. Thus, they present the same equation in two different forms. Consequently, equation system (3.2), (A.3.6) is equivalently reduced to the following two relations: closed PDE (3.1) for

As noted in Appendix A, the position of a spatial point in Eulerian coordinates is described with ODE (see (A.1), (A.2))

Let the spatial point at time

Then the displacement of the point at time

where

Time t and position y constitute Lagrangian coordinates. Since y is independent of t (see (B.2)), Lagrangian version of Eulerian representation (A.3) is

In Lagrangian coordinates, the strain matrix is determined as follows

Note that

The rate of strain (B.6) is

that, after substitution of (B.4) into the right-hand side, becomes

In view of (B.7), relation (B.8) results in

The Lagrangian-coordinate expressions for the time derivative, displacement, strain, rate of displacement, and rate of strain are (B.5), (B.3), (B.6), (B.4), and (B.9), respectively.