ACCELERATION WAVES IN COMPLEX MATERIALS

A framework for modeling acceleration waves propagation in complex materials is presented. Coupled propagation of elasto-acoustic, microstructural and thermal waves is investigated in the full three dimensional case. The presence of microstructure inside each material element is taken into account without introducing additional hypotheses on the physical nature of the microstructure itself, thus obtaining a general theory that is suitable for the whole class of complex bodies. In particular, jump conditions across the discontinuity interface that identifies the acceleration wave are obtained and the amplitude evolution equation is derived.

1. Introduction.Wave propagation in complex materials represents a very active research field thanks to the relevance of its technological implications.A material is defined as complex when it presents a fine structure that is capable of influencing prominently the gross mechanical response of the material itself.Complex materials are ubiquitous in current engineering applications, ranging from liquid crystals, to ferroelectrics, to porous materials, just to name a few.
Experimental results on heterogeneous subclasses of complex bodies show that the presence of microstructure greatly influences the features of wave propagation.For example, acoustic wave propagation in porous materials plays a prominent role in problems about noise suppression in building and structural stability.In fact, lighter bricks are employed in buildings situated in seismic regions for decreasing the overall building weight.However, lighter bricks are usually obtained by increasing the average porosity of them, i.e. by increasing the mean void volume fraction, but this induces severe drawbacks when dealing with sound propagation in this bodies.In fact, increasing the porosity induces amplification of sound propagation and therefore the material becomes more brittle and then it can crack if the building is subjected to earth movements.Therefore, studies on wave propagation in porous media can suggest ideas for improving the building structural stability, see for example [8].Several experiments have also been performed on porous materials such as aluminium foams [32] and polyester fiber materials [21], see also [39].
Implications of wave propagation in complex bodies are also important in the field of acoustic microscopy [14,25], where the sound propagation itself serves as probe for investigating material properties at small scales.
Several experimental evidences suggest that thermal (second sound) waves play a non negligible role in complex materials.For example, in [33] the author design a procedure to dry porous materials exploiting second sound waves, whereas in 638 PAOLO PAOLETTI [34] the authors have shown that thermal waves play a prominent role in heat transfer in biological tissues.Another class of materials where second sound wave propagation is of interest is represented by ceramics.In fact, ceramics are often used as light structures for thermal insulation, fire protection as well as in gas combustion burners and therefore they are often subjected to thermal stresses.Experiments aimed at investigating microcracking induced by compressive waves in ceramics are reported, for example, in [36].Nanofluids with small particle suspension also show non negligible influences of second sound waves, see for example [38].
All the above examples are special cases of complex materials and therefore developing a general modeling framework capable of describing wave propagation in complex bodies is of high interest.Here, we focus of propagation of acceleration waves, i.e. of discontinuity surfaces where the acceleration and the second time derivative of the substructural descriptor may suffer finite jumps, in the general case of materials with Lagrangian microstructure.Specialized models for some subclasses of complex bodies are present in the literature.The references are manifold, here we report only a partial list of them.For example, in [9,10,11,12,19,20,24] acceleration wave propagation in porous materials is investigated under several loading conditions and the authors derived balance equations describing both the wave propagation as well as the amplitude evolution.In [7] propagation in dielectric materials is analysed in the one-dimensional case, showing the effect of material memory on the wave features.A model for acceleration waves in dielectrics with polarization gradient can be found in [15].Acceleration wave propagation in thermo-elastic materials is analysed in [18,37], whereas in [1] the authors focus on acceleration waves in thermo-elastic micropolar media.A model for random materials can be found in [35].
In this paper a general framework for analysing acceleration wave propagation in complex materials is developed using the multifield theory of continua with microstructure [3,17,27].No additional hypotheses on the nature of the substructure are introduced in order to encompass the problem in full generality.The goal is obtaining the jump conditions that the relevant fields must satisfy across the discontinuity surface that identifies the propagating wave as well as deriving the differential equations describing the evolution of the wave amplitude.A similar approach has been presented in [28], however here we deal with a more general case, by allowing for the propagation of thermal waves.This implies modifying the constitutive assumptions introduced in [28] thus modifying the resulting model.Results presented in [28] can be then obtained as a special case by vanishing the coupling between thermal and elastic/substructural waves.Similarly, results presented in [10,11,12] for porous materials are recovered by considering a scalar substructural descriptor and restricting the analysis of the amplitude evolution to the one-dimensional case.Results presented in [19,20] for porous materials with large pores can be recovered limiting the analysis to the linearised setting and introducing a three-dimensional tensor as morphological descriptor.
The paper is organised as follows.The geometric description of complex materials is introduced in section 2, together with the relevant equations of balance in the bulk.Section 3 describes the notation employed for the description of jumps at the discontinuity interfaces that identify the acceleration waves.Jump conditions across the wave are derived for the relevant fields in section 4. Finally, the amplitude evolutionary equations are obtained in section 5. Some additional remarks are reported in section 6.
2. Geometric description and balance equations in the bulk.In standard continuum mechanics each material element is modeled only by its position in an Euclidean space of suitable dimension.This description has proven to be effective for Cauchy bodies, where no microstructure is present.On the other hand, in complex bodies interactions at substructural level, i.e. inside each material element, are capable of greatly influencing the gross mechanical response of the material.Thus the classical description of Cauchy bodies is no more appropriate and one needs to integrate the model with informations about the microstructure geometric properties, at least at a coarse level.Here, the approach of multifield theory [3,17,27,30] in the Lagrangian setting is exploited to deal with complex materials in full generality, i.e. without introducing any additional assumption about the microstructure prominent features.In particular, in this section we will follow a similar path of the one presented in [17,28,31].
Let then B identify a complex body in a configuration taken as reference.Here B is considered as an open, connected set in the three-dimensional ambient space E 3 , with surface-like boundary oriented by the normal at each point with at most a finite number of corners and edges.Subsequent macroscopic deformations of the body are modeled by a map that is assumed to be at least differentiable in space and time.Moreover, for each t ∈ [0, t] this map is assumed to be orientation preserving, i.e. its spatial derivative F = Dy satisfies detF > 0. The deformed configuration y (B) will be indicated with B a and it owns the same essential geometric properties of the original configuration B.
In complex materials, the minute scale characteristics influence the gross mechanical response of the body, and therefore a suitable model for this class of bodies must include a description of the substructure, at least at a coarse grained level.To this end, a field is introduced to model the geometrical features of the underlying microstructure.This map is assumed to be at least differentiable in space with spatial derivative N = Dν.In general the manifold M is only required to be finite-dimensional, paracompact and differentiable.The choice of M has physical implications on the model and it must be chosen on a case-by-case basis according to the material under analysis.Examples of possible choices for M are manifold.Dielectric materials presents a polarization vector associated to each material element, thus M is identified by a ball B pmax of radius p max equal to the maximum allowable polarization amplitude [13].Similarly, in magnetizable bodies the morphological descriptor would belong to the unitary sphere for describing the magnetic orientation, i.e. the spin, associated with each material point.When dealing with liquid crystals, one needs to take into account the orientation of the crystals themselves, thus M would be the projective space RP 2 [16].In this paper, the specific nature of M is left unspecified in order to construct a model for the general class of complex materials.
The macroscopic velocity and the microscopic rate of change of microstructure are given, respectively, by Macroscopic deformations, i.e. changes in the relative placement between material elements, generate standard actions represented by the first Piola-Kirchhoff stress tensor P ∈ Hom T * x B, T * y(x) B a and the body force b ∈ T * y(x) B a .Similarly, substructural actions can be divided into a contact part, modeled by the microstress S ∈ Hom T * x B, T * ν(x) M , and a bulk part β ∈ T * ν(x) M. Bulk substructural interactions can be given for example by the action of an external radiative field, such as the electric or the magnetic ones, whereas microstresses arise from inhomogeneous substructural changes.
Let us now denote as part a subset b of the body B sharing the same regularity properties and geometrical features of B itself.The set of the all parts of B will be indicated with P and V el will identify the set of the velocity pairs ( ẏ, ν) with compact support in B. Let us then represent the power of all the external actions over a generic part b of the body B along a motion (y, ν) by a real-valued map P ext b defined on P × V el.Natural requirements for this map are additivity over disjoint parts and linearity in the rates, thus the external power can be defined as [29,30] A crucial axiom for obtaining correct mechanical models of continua is the requirement of invariance of the external power with respect to isometric changes in observers, for example with respect to the action of SO(3) on space and the corresponding action of a Lie group G on M [30].In this specific case one only requires that observers related by a rigid motion in space evaluate the same power.Since microstructures are placed in space, depending on the nature of them changes in observers (atlas) in the ambient space imply changes in the atlas of M, enriching the notion of changes in observers.Such a point is presented and discussed in [31] where additional details can be found.Here we mention just that the immediate consequence of the invariance requirement for P ext b under (extended) changes in observers ruled by SO(3) and the arbitrariness of b imply the local balances under regularity assumptions for the stress fields (see [31] for further details): where z, an element of T * ν M, is called self force.More precisely, given the tensor A representing the effects on M of rigid changes in observers, z is such that A * z = eP F T −(DA * ) S. Note that this definition allows for an indeterminacy on z, namely any element of the null space of A * can be added to z without changing the balance equation ( 3).This indeterminacy can however be eliminated by imposing the stronger requirement of covariance, i.e. invariance with respect to the action of an arbitrary Lie group on M, see for example [5,29,17,31] for additional comments and the proof of the covariance.
In order to deal with relevant inertial effects in acceleration waves propagation, the bulk actions b and β are assumed to admit an additive decomposition between inertial and non inertial parts, namely b = b in + b ni and β = β in + β ni (see [27]).By using a generalized form of D'Alembert argument the inertial parts can be then identified by where κ (ν, ν) is the substructural kinetic co-energy, i.e. the Legendre transform of the substructural kinetic energy.Physical plausibility implies that κ (ν, 0) = 0. Within this setting, equations ( 2)-( 3) read (see [3]) Moreover, for the sake of simplicity, here we consider only a quadratic form for the substructural kinetic co-energy, namely κ (ν, ν) = 1  2 νI ν, so that ( 5) can be written as This choice for the substructural co-energy is obviously a special case for the kinetic co-energies that one can find in the whole class of complex bodies.However this hypothesis can be easily relaxed by considering additional terms on the right hand side of (6).For example, a possible extension will be adding a term C ν at the right hand side of ( 6) for taking into account possible inertial effects with respect to rotations.Note however that no additional effects will be present in the balance equations across the acceleration waves due to the continuity of ν across the surface, see section 4, and therefore we will limit the analysis to the quadratic substructural co-energy case.Finally, to complete the set of the field equations, balance of internal energy needs to be introduced and it is given by (see [28]) where ε indicates the internal energy density, dV is the internal power of the part b, Q is the heat flux across the boundary of such part and r is the internal heat source.Note that the choice of the part b is arbitrary and therefore (7) implies In order to obtain a priori restrictions on constitutive relations the second law of thermodynamics plays a fundamental role.In this respect, the Clausius-Duhem entropy inequality is the standard tool for obtaining these restrictions when the evolution occurs in thermodynamics equilibrium conditions.However, thermodynamic equilibrium can not be assumed to hold a priori when acceleration waves propagate across a medium and therefore a generalized version of the entropy inequality is required.Here, we follow the approach proposed by Green and Laws [2,22,23] to deal with out-of-equilibrium situations.It is based on the introduction of a generalised temperature φ > 0 that reduces to the absolute temperature θ at equilibrium.Of course this requires the introduction of an appropriate constitutive relation for this quantity.In this setting, the entropy inequality then reads [22] η Note that making the generalised temperature φ coincide with the absolute temperature θ allows for recovering the usual Clausius-Duhem entropy inequality, as expected.By introducing the generalized Helmoltz free energy ψ = ε − ηφ, the entropy inequality (9) reads Let us then assume that the generalised free energy ψ depends on the list of entries F, ν, N, θ, θ, Dθ, namely The standard argument for obtaining a priori constitutive relations then yields (see [3]) Moreover, the residual entropy inequality must hold and the generalised temperature φ must satisfy the constitutive restriction Note that, interestingly, the generalised temperature can depend not only on the absolute temperature and its derivative, but also on the morphological descriptor ν.Similar results have been obtained for materials with voids, see for example [10].
If the body presents a discontinuity surface where the first derivatives of the displacement field y and the morphological descriptor ν may suffer finite jumps, then balance equations ( 4), ( 6) and (8) are not sufficient to describe the body dynamics and additional jump conditions must hold.Such conditions can be obtained by taking the jump of ( 4), ( 6) and (8) and by applying the Kotchine's condition (see, for example, [4,19,26]).They can be then written as where the [•] indicates the jump across the discontinuity surface, n is the normal to that surface and U is the normal speed, see the following section for a formal definition of such quantities.Finally, by taking the jump of the entropy inequality (9), we get Similar equations have been obtained in [19] for porous materials.Note that if ẏ and ν are continuous, as in acceleration waves considered in the following sections, then (18) reduces to the Fourier condition [Q • n] = 0 and ( 19) is automatically satisfied because η is continuous.This observation allowed the authors in [19,20] to conclude that, within the linear setting, every acceleration wave is an homothermal wave.
Here, such conclusion can not be drawn because the residual entropy inequality (15) does not imply that Q is proportional to Dφ and therefore we will introduce the homothermality condition as an additional assumption of the model.

Discontinuity surfaces and acceleration waves.
Let us now consider a discontinuity surface moving in the Euclidean space and dividing the body into two not intersecting open pieces B + a and B − a .The surface is identified by Σ (x, t) = 0 (20) where the function Σ is continuous with its first derivatives.By computing the overall derivative of (20) one easily obtains the propagation velocity and the normal m = DΣ/ |∇Σ| (see also [26] for a detailed discussion about discontinuities surfaces).
Let us now consider a field a (y, t) defined on the whole body B a and continuous on both parts B + a and B − a , the jump across the surface is then defined as where and Σ a is the surface in B a corresponding to Σ.In the following, the jump [a] is always assumed to have finite value.Note also that [lg] = l [g] + [l] g holds for fields l and g admitting distributive product with respect to the sum.Within this notation, acceleration waves are then defined as discontinuity surfaces where [ν] = 0, [ ν] = 0, ( [θ] = 0. Moreover, in the following, only homothermal waves will be considered, i.e. θ = 0, [Dθ] = 0 (24) hold as well.

4.
Balance of interactions at the interfaces.The first step in analysing acceleration waves propagation consists in deriving jump conditions that the relevant fields have to satisfy across the discontinuity surface that identifies the acceleration wave itself.These balance equations can be obtained by developing the jumps of equations ( 4)- (6), see for example [6,26,40] in the case of simple bodies.Note that, thanks to jump conditions ( 21), ( 22), ( 23), (24) and constitutive relations (10), ( 11), (12), P , S and z are continuous across the surface, whereas their derivatives such as DivP and DivS can admit discontinuities.
A key tool for obtaining the balance equations across the discontinuity surface that identifies the acceleration wave is represented by the Hadamard's theorem.
Given a tensor field a (x, t), continuous with its first gradient in B + a and B − a , that presents a finite jump at the interface, the Hadamard's theorem states that the following relation hold see for example [26] for a detailed presentation of this result.An useful alternative formulation of this result states that, given a tensor field a as before, then there exists an other tensor field Λ of the same rank of a such that By applying Hadamard's theorem to the jump of equations ( 4) and ( 6) and taking into account the homothermality conditions (24), one directly obtains Moreover, computing the jump of equation ( 8) yields Some remarks are in order.Equations ( 25) and ( 26) have been already derived in [10] for the special case of porous materials, where the morphological descriptor is simply a scalar representing the void fraction inside each material element.An extension of such model, where the morphological descriptor is a three-dimensional tensor, is presented in [19,20].Here, no specific assumptions on the nature of the substructure have been made as well as no additional hypothesis on the rank of the tensor field ν has been introduced.On the other hand, in [28] the authors derived jump conditions for general complex bodies, but no jumps in the temperature field was allowed.Equations ( 25) and (26) should then be considered a generalization of previous results on jump conditions for acceleration waves in complex materials.Of course, allowing for jumps in the temperature field introduces an additional unknown in the problem, namely θ , and therefore equation ( 27) is included to close the system.
Moreover, note that constitutive relations (10) and (12) implies that T where the superposed T indicates major transposition.Let us now introduce the following fourth order tensors such that the jump conditions ( 25)-( 26) read Note that Q (m ⊗ m) represents the standard acoustic tensor, C (m ⊗ m) a sort of generalised acoustic tensor related to the presence of microstructure, whereas B (m ⊗ m) is the coupling term between gross and substructural acceleration wave propagation.Equations ( 27)-( 29) represent an homogeneous linear system of three equations in three unknowns, namely [ÿ], [ν] and θ .Let us then rewrite these three equations in a more compact form as where the following tensors have been introduced Assuming that the tensor D is not singular and that the scalar G − H • D −1 F does not vanish, a solution of this system is given by Note that equation (33) admits non trivial solutions iff the second order tensor that multiplies [ÿ] is singular.Some comments are in order.It is well known that in Cauchy materials, where no microstructure is present, the square of propagation velocity U of a purely elastic acceleration wave must be an eigenvalue of the acoustic tensor Q (m ⊗ m).This condition can be easily recovered from (33) by setting all the terms except A to zero, i.e. vanishing any influence of microstructure and thermal waves on the purely elastic wave associated with [ÿ].It is then clear that the presence of microstructure induces alterations in the propagation of acceleration waves even in absence of thermal waves, for example their propagation velocity (squared) must be an eigenvalue of A − BD −1 E, i.e. of the tensor Note that, in absence of thermal waves, elasto-acoustic and substructural waves can still decouple if B vanishes or alternatively [ÿ] and [ν] belongs, respectively, to the null spaces of B and B T .Similar results has been derived in [28], where the authors neglect the influence of thermal acceleration waves.Here, we generalise this condition taking into account the potential interplay between acoustic, microstructural and thermal waves.A simplified statement of this result, with scalar morphological descriptor and the additional hypothesis that the wave is advancing into an equilibrium region of the body, is also reported in [10].
5. Amplitude evolution.Until now, conditions for acceleration wave propagation in complex materials have been derived, but no informations about time evolution of wave amplitude have been obtained so far.This section aims at obtaining a set of ordinary differential equations for describing the time evolution of the wave amplitude.To this end, one needs to evaluate the jumps of the time derivatives of equations ( 4), ( 6) and ( 8), namely Once again, Hadamard's theorem plays a prominent role.In fact, by developing the divergence of the time derivative of the stress measure and applying Hadamard's theorem to equation (36), and taking into account the homothermal nature of the wave under analysis, one obtains Similarly, equation ( 37) yields Finally, equation (38) reads Note that, once the jumps in the second derivatives have been calculated as in the previous section, equations (39), ( 40) and (41) constitute a non homogeneous system of three equations in three unknowns, namely [ ... y ], [ ... ν ] and [ ... θ ].Again, it is useful to simplify the notation by writing equation ( 39)-(41) as Note that the coefficients of this system are equals to the ones in ( 30)- (32).Let us now write the terms M, N and O as where the tensors M 1 , N 1 and the scalar O 1 collect all the terms containing the differential operators Div and D, the tensors M 2 , N 2 and the scalar O 2 represent the terms that are linear in [ÿ], [ν] and θ , whereas the tensors M 3 , N 3 and the scalar O 3 collect all the quadratic terms.The complete expression of these quantities can be found in appendix B. This systems of equations ( 42)-( 44) can be then solved, thus obtaining The bicharacteristic approach described in [40] can be successfully exploited for taking into account the dependence between shape changes in the discontinuity surface and amplitude evolution.In fact the shape changes analysis remains unaltered and thus we will focus only on amplitude evolution.A brief description of the main results about surface shape changes is reported in appendix A (a more detailed discussion can be found, for example, in [28]).Let us now focus on the isolated wave speed case, i.e. on situations where only one direction of propagation corresponds to a given eigenvalue of (33) during the whole evolution.In this case, the jump in the acceleration field can be expressed as where r is the eigenvector corresponding to a given propagation velocity and σ is the amplitude of the macroscopic acceleration wave.The corresponding amplitudes of substructural and thermal waves can be then obtained from relations (35) and (34) as Note that the prefactor of [ ... y ] in (45) has the same form of the prefactor in (33) and therefore, it is useful to multiply both sides of (45) by the left eigenvector r L used in equation (48).Moreover the following relation holds By using the Euler's formula for the determinant (see appendix A) we can further simplify the equations by noticing that σ −1 D r L • ∂ m Rr R σ 2 is equal to (Jσ) −1 Jσ 2 where the prime indicates derivative along the bicharacteristic directions.Therefore equation (45) can be recasted in the following Bernoulli's type equation where The scalar coefficients A and B admit physical interpretation.In fact, A contains informations on the dynamical state of the material before the wave incidence, whereas B depends only on the constitutive nature of the material.
To better highlight the influence of the microstructure and the temperature jump on the acceleration wave amplitude evolution, let us now consider the case where the shape of the interface remains constant so that J is constant in (51).This happens, for example, in the one-dimensional setting analysed in [10,26] or when the material is homogeneous and the initial amplitude is constant on Σ as in [19].According to this hypothesis, equation (51) reduces to whose solution can be written as where σ 0 is the initial wave amplitude and s is the time coordinate along the bicharacteristic (see [26]).The values of the coefficients A and B depend on the constitutive relations that are typical of the material under analysis and on the state of the material before the wave front.Let us now consider for example a situation where the material is in equilibrium before the wave incidence, i.e.Ḟ + , ν, Ṅ + , θ, θ+ and Dθ + .In this case we have i.e.A contains only terms linked to the coupling between the temperature and the stress measures, both on macroscopic and substructural scale.If thermal effects were neglected so that the temperature did not enter in the entries of the free energy functional, then A would vanish.A simplified expression for B is obtained for decomposable free energies, i.e. energy functionals of the form ψ F, ν, N, θ, θ, Dθ = W el F, ν, θ, θ, Dθ + W sub ν, N, θ, θ, Dθ where W el is the elastic energy and W sub represent the contribution of the microstructure.For example, if the material has two ground states with ν = ±1, as media with spin-like microstructure, one can define an energy functional of the form where the first term represents the cost paid for deviating from the ground states and the second term penalizes inhomogeneities.If the energy admit such decomposition then the B vanishes and the macroscopic acceleration wave decouples from the microscopic wave.Moreover, most of the mixed derivative terms in M 3 , N 3 and O 3 vanish and therefore the expression for B is simplified.
The exact description of propagation of an acceleration wave in a given material will then depend on the constitutive relations that are typical of such material as well on the state of the body before wave incidence.However, we can obtain a qualitative description of the wave amplitude evolution by analysing the influence of A, B and σ 0 on the amplitude evolution (53).Three phenomena can occour: 1. the wave amplitude decays monotonically toward zero if A < 0, B > 0 and σ 0 > A/B or if A < 0, B < 0 and σ 0 > A/B; 2. the wave amplitude converge toward the constant value σ = A B if A > 0 and Bσ 0 > 0; 3. the wave amplitude blows up in a finite time if either A > 0 and σ 0 B < 0 or A < 0, B > 0 and σ 0 < A/B or if A < 0, B < 0 and σ 0 > A/B.The blow up time s b can be obtained by imposing the denominator of (53) equal to zero This phenomenon is usually associated with the development of shock waves in the material.Note that if A = 0 then the wave amplitude dynamics (52) becomes degenerate, meaning that it only presents a semi-stable equilibrium in zero.Note, however, that this degenerate condition is not structurally robust and every perturbation of the right hand side of (52) will restore the qualitative behaviour discussed above.This is the case, for example, when dissipative effects, both on macroscopic scale and substructural level, are included in the model, as in [9].
A qualitative picture of the wave amplitude evolution with B > 0 is reported in Figure 1.Note that if A < 0 then the wave amplitude relaxes toward zero if its initial value is below the threshold A/B.On the other hand, if A > 0 then the wave amplitude converges to the constant value A/B whenever its initial value is positive.Then the system dynamics undergoes a transcritical bifurcation as A goes through zero.
Similar phenomena occour also in the full three dimensional case where the wave amplitude evolution is governed by (51).
The richness of the family of solution of (51), together with the complex bodies ability of responding to stimuli that are not necessarily of mechanical nature, opens new possibilities in the design and control of such systems.One can then think about exploiting this additional degree of freedom for dynamically controlling the instantaneous value of the coefficients in the Bernoulli's equation (51) so that the wave propagates with a desired amplitude.This can allow, for example, the control of the evolution of acceleration waves in dielectric [15] or ferroelectric [13] bodies by means of an external electric field, without requiring any physical contact that could damage the material itself.For example, one can use an external electric field to drive the microstructure out of equilibrium ahead of the wave front so that the terms in A depending on ν+ and Ṅ + do not vanish.The ability of dynamically varying A can then allow for the control of the asymptotic value of the wave amplitude and prevent (or induce) the formation of shocks by modulating the threshold value A/B. 6.Additional remarks.In this paper a framework for analysing homothermal acceleration waves propagation in complex materials is presented.The described theory does not introduce any additional hypothesis on the nature of the microscopic substructure, thus allowing for obtaining results that encompass the whole class of complex bodies.Jumps on the second derivative of the temperature field are allowed.In particular, both jump conditions of displacement, substructural and temperature fields across the discontinuity interface and evolution of acceleration wave amplitude have been derived in the full three-dimensional case.
Results in the available literature can be obtained as special cases of the ones described in this paper.For example, results about porous media with tensorial microstructure presented in [19,20] can be recovered by limiting the analysis to a linear setting.Similarly, results reported in [10,11,12] can be obtained by considering a scalar morphological descriptor and imposing equilibrium conditions in the part of the body before the wave front.Finally, the general model presented in [28] can be recovered by neglecting the thermal waves, i.e. by imposing θ = 0.