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Multi-scale modelling of waves of porous media with applications to acoustic control and biomechanics

Periodic Reporting for period 1 - MUSAL (Multi-scale modelling of waves of porous media with applications to acoustic control and biomechanics)

Période du rapport: 2015-09-01 au 2017-08-31

The aim of the project is to develop new models and methods to predict the dynamic properties and to describe the propagation of elastic waves in heterogeneous porous media. This subject is widely multidisciplinary and the results have applications in different areas of high economic, social and and humanitarian importance.

The dynamic behaviour of heterogeneous media is characterised by a number of significant phenomena, such as phononic band gaps, negative refraction, dynamic anisotropy and wave focusing, acoustic diodes, acoustically invisible cloaks, wave localisation in structures with defects. These remarkable effects help to design new engineering devices in aerospace and automotive industries (acoustic absorbers, ultrasonic transducers and transmitters, lenses, wave guides, etc.)

The modelling of wave propagation in heterogeneous media has great importance for non-destructive testing of materials and structures in mechanical and civil engineering. The pattern of phononic bands represents a kind of “identification portrait”, which is unique for every material. The larger frequency range explored, the more accurate “portrait” can be compiled. This gives a possibility to detect even very small variations of the microstructure and to develop new, more precise methods of acoustic diagnostic.

In biomechanics, many live tissues (e.g. bones) can be modelled by porous media. Many people worldwide suffer from osteoporosis, a progressive bone disease that is characterized by a decrease in bone mass and density. Since osteoporosis itself has no symptoms, the detection of the bone structure using non-invasive testing is a challenge. The results of the project may ultimately be used for the early detection of the low bone density helping to stop the progress of osteoporosis by a prevention therapy.

The study of wave propagation in porous media is also important for geological explorations. Understanding how the internal texture of soils and rocks affects the characteristics of travelling elastic waves helps to develop new robust methods for the detection of gas and oil reserves.

The objectives of the project are as follows:

1. Development of new asymptotic methods to determine the dynamic properties of heterogeneous media.
2. Derivation of new homogenised models applicable in a wide range of frequencies.
3. Study of the propagation of elastic waves and determination of their dispersion properties.
4. Prediction of links between the properties of the microstructure and the characteristics of macroscopic waves.
5. Development of numerical procedures to verify the applicability of the proposed analytical approaches.
6. Clarification of phenomenological theories of wave propagation and suggesting new theoretical interpretations of their hypotheses.
We develop the method of high-frequency asymptotic homogenisation to derive effective constitutive equations that describe the dynamic response of heterogeneous media. The two-scale asymptotic procedure provides us with solutions applicable in the vicinity of the resonant frequencies of the unit cells. Using the method of Padé approximants, we match limiting asymptotic solutions and derive new composite macroscopic models shown to be valid in a wide range of frequencies and wave lengths.

The propagation of elastic waves is studied. Combining periodicity and anti-periodicity conditions in different directions of the translational symmetry of the structure, we detect specifically new types of waves that do not arise in the purely periodic case. Such waves can be interpreted as counterparts of the non-classical modes appearing in phenomenological theories, such as micro-rotational waves in Cosserat continuum and so called “second” wave in Biot’s theory. Dispersion curves evaluated and phononic band gaps identified.

Analysis of the dispersion surfaces allows us to predict the directions of energy propagation within heterogeneous media. As the frequency increases, the energy flows become focused in certain specific directions. This effect allows one to design phononic crystals and metamaterials with wave focusing capabilities, which is remarkably important for many engineering applications.

The dynamic response of non-linear heterogeneous solids is considered. Non-linearity leads to localisation of energy and its transfer from a low- to a high-frequency part of the spectrum. Heterogeneity involves scattering of the wave field, which compensates the influence of non-linearity. Studying a balance between non-linearity and heterogeneity, we estimate the areas of applicability of different theories of wave propagation.

The obtained macroscopic equations are applied to solving boundary value problems. We consider transient waves excited by pulse and harmonic loads. Due to a spatial redistribution of energy, the internal forces arising in a heterogeneous medium can be higher than the magnitude of the initial excitation. This effect is crucial for the dynamic failure of structures.The obtained analytical solutions demonstrate excellent agreement with the results of numerical simulations.

The new dynamic models proposed in the project encapsulate the explicit information about the microstructure. In contrast to phenomenological approaches, all the effective coefficients are determined on a rigorous theoretical basis in terms of microscopic properties of the medium. The results of the project give new theoretical interpretations of the classical Cosserat and Biot's models and clarify the limits of their practical applicability.

The dissemination activity includes 11 lectures delivered at international conferences and workshops; 5 papers published and 3 papers submitted for publication in peer-reviewed journals; 1 paper is being prepared for submission. 1 book is accepted for publication by Springer. The publications are deposited Open Access in Keele University research repository, see
The results of the project ensure considerable impact to European excellence and competitiveness in several different areas. The developed models, methods, and solutions can be directly employed for the purposes of non-destructive testing and structural health monitoring in aerospace, automotive and civil engineering industries enabling to reduce the weight and the costs of structures, to increase safety and prolong the lifetime period. The obtained results allow the acoustic optimisation of vehicles and buildings, when the sound field is predicted and controlled with the help of absorbing elements produced from heterogeneous metamaterials. This reduces noise pollution and creates a safe and healthy environment for people. Knowing the properties of ultrasonic waves and understanding the principles of their propagation in bone tissues provides medicine with new tools for the early diagnosis of osteoporosis, so that to prevent bone fractures and disability. The proposed methods and solutions are also applicable in geophysics to detect new gas and oil reserves.

The project leads to a greater integration across European scientists. We extended active and have established new links between the groups in the UK and the EU working in the field of mechanics and physics of heterogeneous media. This ensures increase in research mobility and communications, helps to avoid a duplication of research efforts and, on a large scale, will lead to a more efficient usage of the EU and national research funding.
Generation of higher-order modes and energy transfers in nonlinear vibrations of a homogeneous solid
Dynamic response of the monatomic lattice to the pulse load
Continuous model: composite materials; porous media; foams; bone tissues
Discrete model: cellular structures; phononic crystals; molecules; atomic chains; nanotubes
Contour plot of the dispersion surface of a 2D lattice showing anisotropy of the energy flow
Dispersion curves of locally periodic and anti-periodic waves in a fibrous heterogeneous medium
Suppression of modes interactions in a non-linear solid with microstructure