Connecting Mesoscopic and Macroscopic Scale Lengths for Ultrasonic Wave Characterization of Micro-Cracked Material L

Connecting mesoscopic and macroscopic scale lengths

Connecting mesoscopic and macroscopic scale lengths for ultrasonic wave characterization of micro-cracked material L. R. Rakotomanana Institute of Mathematics, University of Rennes I -France

Abstract. Macroscopic failure of material is attributed mostly to the initial presence of micro-cracks and micro-voids and is governed by physical mechanisms at different length- scales. In order to include discontinuity mechanisms in the material deformation and its consequence on the energy dissipations during micro-crack kinetics, a theoretical model of micro-cracked continuum is derived in this paper. The model describes a micro-crack density in terms of CARTAN constants of structure and explicitly connects the macroscopic scale to the mesoscopic discontinuities. This approach contrasts to the usual method in continuum mechanics that seeks a phenomenological description by introducing an internal variable in the constitutive laws. An illustrative example of the model application is presented for the linear ultrasonic wave propagation test. The result highlights the importance of rigorously revisiting the dynamic equation in micro-cracked solid.

Keywords – A micro-cracking, B inhomogeneous material, B stress waves, C nondestructive evaluation.

Introduction

Brittle materials as glass, ceramics and polymers always contain more or less great amount of micro-cracks and crack-like flaws (~1µm to 10µm), which are unintentionally introduced during processing or surface machining. Toughness and strength of these materials are strongly dependent of the amount and structural orientation of internal micro-cracks. Although global failure of brittle material is usually attributed to a single macroscopic crack propagation, dense sets of micro-cracks appear around the single crack, resulting from dynamic instability e.g. (Sharon and Fineberg, 1996). Creation of micro-cracks surrounding the propagating macro-crack

L. Rakotomanana 1 Connecting mesoscopic and macroscopic scale lengths is thought to be responsible for limiting the crack speed to about 50% of the theoretical limit of Rayleigh surface wave speed, by dissipating energy. At the extreme, microcracking in the vicinity of macroscopic crack edge has been shown advantageous in controlling and even in arresting a single macroscopic crack propagation e.g. (Clegg, 1999). Material failure is thus simultaneously governed by different mechanisms on different length-scales.

Size effects in micro-cracked material

At least three length-scale levels are present to approach the micro-cracking phenomenon: macroscopic scale (~100µm), mesoscopic scale (~0.1µm to 10µm) and microscopic (atomistic) scale (~10-10 m). Until recently, the size scaling was neglected due to the early use of average stress (force per surface unit) and strain (no dimension). The limits of macroscopic approach are reached when facing the cause of true material weakness as micro-cracks. Indeed, new orientation of technology development has brought new interest in connecting mesoscopic scale to macroscopic scale beyond the macroscopic continuum description. Micro-engineering devices, electronic devices and micro-electromechanical systems, for which the entire size may be less than 10µm, may exhibit size dependence. For problems with crack lengths ranging from fraction of 1µm to 10µm, current macroscopic description misses the size effects. Connection between scale levels description is of central interest. This is particularly true for solid materials, because solids introduce a new length scale other than the lattice spacing (~1Å to 10Å), namely, the size of micro-cracks. Hierarchical modeling was suggested as one of the efficient method to connect continuum cracking, dislocation dynamics and atomic-scale simulation as molecular dynamics or lattice static e.g. (Tadmor et al., 2000). Mesoscopic scale has been proven to well connect with molecular dynamics approach by using very large-scale mechanistic simulations during crystal plasticity (100 million atoms) (Butalov et al., 1998). Very large-scale molecular dynamics also appears more and more able to bridge the atomistic scale to macroscopic experiments and description of continuum plasticity of material (10 million atoms) e.g. (Holian and Lomdahl, 1998). For more than forty years, the theory of strain gradient was proposed in various forms to bridge mesoscopic scale to macroscopic scale for elastic material deformation e.g. (Toupin, 1964) or when internal micro-slips occurred in crystal solids e.g. (Fleck and Hutchinson 1997). These theories involve strain (metric) for describing macroscopic deformation and strain gradient for mesoscopic mechanisms. Such an approach has been already intensively

L. Rakotomanana 2 Connecting mesoscopic and macroscopic scale lengths discussed in the past, namely for non-local elasticity due some basic conceptual flaws e.g. (Dunn and Serrin, 1985). In fracture mechanics the bridging of phenomenological approach (macroscopic) to mesoscopic physics that governs the dynamics of micro-cracks is far from clear either experimentally or theoretically e.g. (Blumenfeld, 1998). Seek of an efficient description of material whose dimensions fall between macroscopic continuum and dislocation mechanics remains a valuable motivation for developing an intermediate scale theory.

Ultrasonic techniques for micro-cracking detection

An immediate application of mesoscopic scale theory would be the non-destructive testing and monitoring of micro-devices with cracks. For most materials ultrasonic techniques have been developed to characterize the internal degradation by measuring the attenuation of ultrasonic waves. Various theoretical models have been developed for explaining and predicting empirical correlation found between attenuation and the presence of micro-cracks e.g. (Vary, 1988). Basically, attenuation is a collective effects of four contributions e.g. (Prosser, 1996). Diffraction is a beam spreading that is the dominant source near the crack (wavelength is same order as crack length). Far from the crack, absorption (conversion of sound energy to heat) has an exponential relationship of attenuation with distance. Scattering is the dissipation due to geometric dispersion of wave into adjacent media or into non-homogeneity within the material itself. Velocity dispersion induces a signal loss provoked by the different velocities for different frequencies involved in the wave. Despite its importance in ultrasonic measurement, most models do not account for attenuation in the initial wave equation. It is often assumed and added ad hoc for the sake of theory fitting with the experimental results e.g. (Breazeale et al., 1981; Vandenbossche et al., 1996). Furthermore, there is currently no consensus on the form of the wave equation that governs the combined macroscopic and mesoscopic mechanisms.

Numerous nonlinear ultrasonic techniques have been proposed to characterize the fatigue micro- cracking damage. Two basic nonlinear effects are usually proposed: acoustic-elastic effects (stress dependence on the attenuation) and higher harmonic generation. Wave attenuation has

2 been measured on the basis of Taylor expansion of the sound velocity cL = c0 + c1εext + c2εext +… where εext is the pre-strain level. This technique appeared to succeed in early detection of micro cracking for polymers and brittle polymers whereas failed for PVC and Nylon e.g. (Nagy, 1998).

L. Rakotomanana 3 Connecting mesoscopic and macroscopic scale lengths

Probably scattering losses were more important than adsorption losses for those materials. Similarly, higher harmonic generation was used to capture material degradation by assuming non-linear stress-strain law σ = Eε(1 + βε +…) e.g. (Jhang and Kim, 1999), E being the Young’s elastic modulus. The second harmonic parameter β was proposed to characterize the material degradation. However, β values augmented as the excitation frequency increased and there remained a doubt if this second harmonic parameter was an intrinsic effective parameter for the material degradation. Indeed, the stress-strain law of micro-cracked material could be linear, although with a lower modulus than intact material. In a same way, large strain theory and nonlinear stress-strain law were combined to derive nonlinear wave theory in order to measure non-homogeneous micro-cracked material. The material properties were determined on the basis of non-linear wave accounting for third-order elasticity e.g. (Ravasoo, 1999). The second-order coefficients of nonlinear terms (gradient of strain and cross terms) depended on the macroscopic LAMÉ elastic constants (λ, µ) and their spatial derivatives. Therefore these nonlinear and non- homogeneous approaches were not able to detect the presence of uniform distribution of micro- cracks with uniform macroscopic material density.

Experimental analysis of wave propagation in micro-porous ceramics (pores ~1µm) showed strong attenuation and cut-off of frequency e.g. (Craciun et al., 1998). A sudden decrease in the velocity at high porosity values was shown but could not be explained in the light of existing macroscopic models of wave propagation. The ability of classical wave propagation to model very micro-porous media was then questioned and the authors assumed that the strong attenuation was due to the wave scattering from the sample geometry disorder than due to the sound adsorption mechanisms in the porous ceramics. Cut-off frequency phenomenon was also observed in macroscopic fractured material, which exhibited distinct frequency bands with energy transmission (pass bands) and with near-zero energy transmission (stop- band with cut- off frequency) e.g. (Nakagawa et al., 2000). Corresponding discrete theory has been developed in 1D situation. Although experimental measurement techniques are becoming ever more accurate and sophisticated, the list of theoretical models describing microcracking detection is still far from complete. Due to the shortness of these micro-cracks characteristic length (~1µm to 10µm) than the usual wavelength used in ultrasonic techniques, homogeneous linear wave theory has often not sufficiently sensitivity to apprehend material degradation at the mesoscopic level.

L. Rakotomanana 4 Connecting mesoscopic and macroscopic scale lengths

Theoretical models should be developed not only for improving the measurement processes as for ultrasonic inspection techniques but also and mainly for better interpretation of the measured data.

The aim of this paper is twofold. First, a bridging theory between mesoscopic scale and macroscopic continuum scale is proposed for modeling the micro-cracks. The concept is based the presence of the mesoscopic discontinuities of scalar and vector fields. The second part of the paper is devoted to the derivation of macroscopic wave propagation equation accounting for microcracking mechanisms of material in order to provide supportive model for characterization of micro-cracked material.

Connection of macroscopic scale to mesoscopic scale

Macroscopic deformation At the macroscopic level, the deformation is described by the metric, which measures the length change of any small fiber embedded within material. Projected on a local vector basis (e1,e2,e3 )

a b deforming with the material, the metric tensor writes g = gab e ⊗ e . The strain components are

1 defined as εab = 2 (gab −δ ab ), where δab are the Krönecker symbols (equal to 1 if a = b and null otherwise). For small deformations, the strain tensor may be also defined in terms of the

1 displacements ua in the usual manner, that is εab = 2 (∇aub + ∇bua ) . Symbol ∇ denotes the connection used for calculating all derivatives with respect to space (gradient of tensor). The connection implicitly used in current macroscopic classical physics theory, including continuum mechanics, is basically the LEVI-CIVITA connection, which must a priori satisfy the fundamental equation ∇g ≡ 0 . It is worthwhile to notice that the nullity of the strain gradient should not be confused with homogeneity of strain. The macroscopic deformation theory is the same for the three bridging approaches below.

Mesoscopic mechanisms: Internal variable theory The first approach for bridging macroscopic scale to mesoscopic scale was historically the continuum damage mechanics, which introduced internal variables for describing the internal micro-cracking. Most continuum mechanics models of micro-cracked solids seek a phenomenological description by assuming macroscopic variables to capture the distribution of

L. Rakotomanana 5 Connecting mesoscopic and macroscopic scale lengths micro-cracks e.g. (Vakulenko and Kachanov, 1971, Chaboche, 1988, He and Curnier, 1995). Nevertheless, the existence of numerous theoretical models of brittle micro-cracked material based different choice of damage variables merely showed the missing of consensus in this domain e.g. (Rabier, 1989; He and Curnier, 1995).

It has been recently proposed that any stiffness degradation of elastic material could be described by a eight order damage tensor relating the intact eight order modulus tensor with the damaged one (Cauvin et al., 1999). This approach allowed to describe various damage symmetry as full anisotropy, orthotropy, tetragonal symmetry, hexagonal symmetry, transverse isotropy and isotropy. Up to now, the choice of internal variables was rather dictated by an inductive logic, in which generalization of particular observations remained the basic concept underlying the choice of internal variables. Most previous theories were based on effective stress (strain) e.g. (Chaboche, 1988) or on the hyper-elasticity energy approach e.g. (He et al, 1995). In those models, internal variables were used to simulate the change of material properties and did not have any influence on the formulation of the conservation laws.

Alternative method for internal variable theory proposes strong discontinuity across the crack lips. Physically, micro-cracks are displacement and/or velocity discontinuities in an otherwise intact material. Indeed, to be close to the physical phenomenon, some micro-mechanics models are based on the physical discontinuity of matter and then assume the description of micro-cracks with contacting lips with dry (or viscous if any) friction at these lips. Each micro-crack is then included into a cell, which is its direct neighborhood and is the smallest unit that allows bulk material properties to be quantified after homogenization. The most important properties of the basic cell is the ability to describe the relative translation of contacting lips (cohesion- decohesion) e.g. (Broberg, 1997). The crack opening modes (relative displacements of the crack lips) are the internal variables for these models e.g. (Maugin, 1992, Oliver et al., 1999). Numerous models are based on the discrete distribution of micro-crack within otherwise intact material e.g. (Capuani and Willis, 1999; Maugin, 1992; Oliver et al., 1999; Ramaniraka and Rakotomanana, 2000). Across each crack face, the displacement field u needs not be continuous and the internal variable is the displacement jump across the crack lips denoted by [u]. The displacement field is decomposed into a macroscopic field and a jump field u = u + HΩ [u].

Vector u is the continuum displacement in the absence of crack and HΩ the unit ramp function

L. Rakotomanana 6 Connecting mesoscopic and macroscopic scale lengths

(null at one lip of the crack Ω− , equal to 1 on the adjacent lip Ω+ and linear between them) e.g. (Oliver et al., 1999). The strain tensor may be extended the to account for the displacement jump

1 1 1 εab = 2 (∇aub + ∇bua ) + 2 HΩ(∇a[ub ]+ ∇b [ua]) + 2 δ(Ω)([ua]nb + na[ub ]), where δ(Ω) is the Dirac delta function placed in the crack Ω , equal to 1 for any point of the crack and null elsewhere. Vector n denotes the unit normal vector orthogonal to the crack surface. In this approach, mesoscopic mechanisms are connected to the macroscopic deformation by expanding the displacement and strain into a continuous part and unbounded part (discontinuous). For a finite size material piece, these models require the mathematical technique of homogenization, which may be cumbersome and practically difficult to apply in presence of randomly distributed micro- cracks.

Mesoscopic mechanisms: Strain gradient theory

The second approach defined a smooth strain gradient as supplementary variable. For bridging the macroscopic level to the mesoscopic level, continuum models were proposed, in which the strain gradient ηabc ≡ ∇a∇buc was implicitly assumed to be additional variables for describing the change of internal microstructure e.g. (Toupin, 1962; Gurtin, 1965; Fleck and Hutchinson, 1997). Numerous works proposed the second gradient of the displacement as internal variable for determining the material length scale during plastic deformation of small devices e.g. (Aifantis, 1992; Fleck and Hutchinson, 1997; Gao et al., 1999; Shizawa and Zbib, 1999; Huang et al., 2000). The effective strain measure of these recent theories was defined in e.g. (Fleck and Hutchinson, 1997) to include both the strain and its gradient:

2 2 Ee ≡ 3 ε ab′ ε ab′ + d1η1′ abcη1 ′abc + d2η2′ abcη2 ′abc + d3η3 ′abc η3 ′ abc .

The prime sign indicates the deviatoric part of the strain and the strain gradient. Tensors ηiabc′ are three mutually orthogonal third tensors according to ηabc′ = η1 ′abc + η2′ abc +η3 ′ abc e.g. (Fleck and

Hutchinson, 1997). The three lengths (d1,d2, d3) are the new constitutive parameters in the theory. Experimental indentation test, torsion and bending tests gave estimation of these new parameters e.g. (Fleck et al., 1994; Stölken and Evans, 1998; Begley and Hutchinson, 1998). Their value ranges from fraction of 1µm to 10µm. Strain gradient based theories provided the linking between the macroscopic cracking to the mesoscopic slipping which were not well

L. Rakotomanana 7 Connecting mesoscopic and macroscopic scale lengths described by conventional plasticity theories. The coherence of macroscopic continuum physics theory precluded the dependence of stress on higher gradients of strain e.g. (Dunn and Serrin, 1985) if corresponding higher order stress tensors were not introduced. High-grade material models, and therefore the specified theories deduced from them, may be incompatible with the usual form of macroscopic thermodynamics laws e.g. (Gurtin, 1965; Dunn and Serrin, 1985). The Fleck-Hutchinson theory of micron scale plasticity includes the higher order continuum theories of elasticity of sixties e.g. (Toupin, 1962; Gurtin, 1965) or more recent theories of plasticity e.g. (Gao et al., 1999; Popov and Kröner, 1999; Huang et al., 2000). To satisfy the Clausius-Duhem inequality, higher order stress was moreover defined by means of the work- conjugacy with the rotation and the stretch gradient, which are included in the strain gradient. The strain gradient plasticity is a description at the mesoscale length level of the more detailed dislocation dynamics occurring during crystal plastic deformation (Gao et al., 1999). However, an unclear point remains in the use of the gradient of the metric i.e. strain gradient as variable measuring the mesoscopic mechanisms. The connection implicitly used in strain gradient theory remains basically the LEVI-CIVITA connection, which has to satisfy the fundamental relation ∇g ≡ 0 .

Mesoscopic mechanisms: Connection theory As for crystal solid with defects (Kröner, 1981, Le and Stumpf, 1996), the deformation of micro- cracked material is not sufficiently described by only measuring length change of embedded fiber (strain) at the macroscopic level. Physically, each micro-crack results in a discontinuity of the deformation either between atoms (microscopic scale length ~10-10m) or between grains (mesoscopic level ~ 1µm to 10µm) in the material. Bridging between various length scale levels should account for this discontinuity. At the microscopic level (characteristic length 1Å to 10Å), typical approach for crystals was given by means of crystal connection ∇ e.g. (Bilby et al., 1955; Noll, 1965; Kröner, 1981), allowing relating a basic lattice cell to its neighborhood. Crystal connections were defined by considering regions of interest much greater than lattice spacing, and where the lattice has density of dislocations that is small compared with the lattice size, but large enough so over regions of interest the dislocations can be described by a continuous density (~ 10Å to 50Å). For bridging macroscopic level to mesoscopic level, we adopt a similar approach to seek additional variables by assuming physical mesoscopic discontinuity of vector

L. Rakotomanana 8 Connecting mesoscopic and macroscopic scale lengths and scalar fields, as temperature and displacement. A previous work permitted us to show that assuming scalar and vector fields discontinuity is more advantageous than assuming localized discontinuity of matter, that involves mechanics of contacting crack lips at the mesoscopic level (Ramaniraka and Rakotomanana, 2000).

The use of path integral-like method e.g. (Schouten, 1954), allowed obtaining mathematical variables to describe the mesoscopic jump fields within a continuum: torsion tensor of an affine connection for discontinuous scalar field, and the torsion and the curvature tensors (not null simultaneously) for discontinuous vector field (Rakotomanana, 1998). An affine connection is not a tensor variable but may be characterized by the torsion ℵ and curvature ℜ tensors, which are the additional variables for describing micro-cracked continuum:

c c c ea eb e ℵ= [(Γab − Γba) −ℵ0ab] ⊗ ⊗ c ,

e c e c e c e c e c ea eb ed e ℜ = [ b(Γda) − a (Γdb) + ΓdaΓeb − ΓdbΓea −ℵ0abΓed] ⊗ ⊗ ⊗ c ,

Where Γc = ec ∇ e and ℵc e ≡ e ,e are respectively the coefficients of the affine ab ( ea b ) 0ab c [ a b ] connection and the constants of structure of Cartan. Symbol [ , ] denotes the classical LIE- JACOBI bracket. Deformation of micro-cracked continuum includes the transformation of g (metric change for macroscopic mechanism) and the transformation of ∇ (topology change for mesoscopic mechanisms), the both deformations are projected onto an embedded basis

(e1,e2,e3 ), which deforms with the material.

In practice, the micro-crack disorder (1µm to 10µm) during deformation may be described by the

c CARTAN’s structural constants ℵ0ab , which measure the failure of the closure of an initial parallelogram designed on the material during the deformation. Failure of closure is due to

c micro-crack. Constants of structure defined by ℵ0abec ≡ [ea ,eb ] (a and b vary from 1 to 3) include the three modes of each crack opening. In the present study, instead of using a strain gradient variable, macroscopic deformation and mesoscopic physical mechanisms are described with

b a metric and constants of structure respectively, which may resumed into the co-vector ℵ0 =ℵ0 abe (a co-vector field can be assimilated as a normal vector of a surface). This co-vector, which was originally proposed in the general relativistic mechanics e.g. (Cartan, 1986) was applied in

L. Rakotomanana 9 Connecting mesoscopic and macroscopic scale lengths mechanics of material (Rakotomanana, 1998). The main difference with the strain gradient theory is that connection theory introduces separately the affine connection, which is not a gradient of the metric but rather independent variables.

Wave attenuation and dissipation in micro-cracked solids

The intent of this section is to present conservation laws formulation to highlight the difference between these approaches.

Macroscopic wave propagation

The material is assumed to have isotropic symmetry, undergoes elastic small strains and contains frozen micro-cracks e.g. (Noll, 1965; Wang, 1967). For material satisfying the HOOKE stress- strain law σ = λtr(ε)i + 2µε , NAVIER mathematical description of isotropic solid (1821) was corrected by CAUCHY (1822), by introducing the LAMÉ coefficients (λ, µ) to obtain the equation of motion in the absence of body force. This motion equation relates solid particle acceleration with the elastic internal force by using of longitudinal velocity cL = (λ + 2µ) ρ and transversal velocity cT = µ ρ of sound:

2 ∂ u 2 2 2 2 = (cL − cT )∇(divu) + cT Δu . ∂t

For the three approaches, the macroscopic wave propagation holds. The difference between the three methods would be in the definition of the divergence operator, in the averaging of the displacement vector and by the way the stress field and in the constitutive stress-strain law.

Mesoscopic wave scattering: internal variable theory

The internal variable theory is based on the use of a new macroscopic tensor field as for capturing the evolution of internal micro-cracks. The medium is assumed to be homogeneous, isotropic and linearly elastic with LAMÉ’s constants depending on the internal variable, reduced to a scalar field ℵ0 for the sake of simplicity. Therefore, the wave equation may be rewritten as follows:

L. Rakotomanana 10 Connecting mesoscopic and macroscopic scale lengths

2 ∂ u 2 2 2 2 = (cL 0 + cL1ℵ0 +…) −(cT 0 + cT1ℵ0 +…) ∇(divu) + (cT 0 + cT1ℵ0 +…) Δu ∂t ( )

Where the elastic constants are expressed on the basis of Taylor expansion of the sound velocity

2 cL = cL0 + c1ℵ0 + c2ℵ0 +…, in which ℵ0 denotes the damage level. Similar expansion holds for the transversal sound velocity. Such theoretical model had been used successfully in early detection of micro cracking for polymers and brittle polymers e.g. (Nagy, 1998). From this equation of motion, it is nevertheless evident that the wave characteristic is not fundamentally changed by the amount of micro-cracking but the overall stiffness. Moreover, the metric connection used for calculating the divergence and Laplacian operators is not modified by the presence of micro-cracks distribution. Characterization of materials at the mesoscopic scale seems to be beyond of the scope for such model.

Mesoscopic mechanisms: displacement jump method

Alternative version of internal variable theory considers the jump of displacement field at discrete distribution of crack as bridging variable. In a series of papers, (Capuani and Willis, 1999) introduces the wave propagation within discrete models of cracks within otherwise virgin matrix. By considering a random distribution of discrete micro-cracks, and by averaging the crack effects, the equation of wave propagation can be written as follows:

2 ∂ u 2 2 2 2 = (cL − cT )∇(divu) + cT Δu +κ ∂t Where an extra-body force κ in the virgin matrix, due to the presence of micro-cracks distribution appears. Details of the formulation of this extra-body force may be found elsewhere (Capuani and Willis, 1999). For one-dimensional propagating wave, in which unit normal vector of cracks is assumed to be along the dimension and where the displacement vector u = (u1,u2 ,u3 ) is depending only on x1 = x and on t , further simplification gives (Capuani and Willis, 1999):

∂ 2u ∂ 2u ∂ u ∂ 2u ∂ 2u ∂ 2u ∂ 2u 1 = c2 1 − c2νπr2 [ 1] , 2 = c2 2 , 3 = c2 3 . ∂t2 L ∂x 2 L Ω ∂x ∂t2 T ∂x2 ∂t2 T ∂x2

L. Rakotomanana 11 Connecting mesoscopic and macroscopic scale lengths

In which ν , rΩ are the Poisson’s ratio of the material and the radius of the micro-crack respectively. The second term connects the mesoscopic jump of displacement to the macroscopic wave propagation.

Mesoscopic wave scattering: strain gradient theory

Conservation laws associated to the strain gradient model have been mostly derived from the Principle of Virtual Power (Germain et al., 1983; Fleck and Hutchinson, 1997; Shizawa and Zbib, 1999) or Lagrangian method e.g. (Popov and Kröner, 1999). Based on the principle of virtual power, the wave motion equations take the form:

2 ∂ u 2 2 2 2 = (cL − cT )∇(divu) + cT Δu +κ(∇u,∇(∇u)). ∂t

In which, the extra-body force κ(∇u,∇(∇u)) depends both on the strain and on the strain gradient. The exact form of this force is derived from the Helmholtz free energy of the material φ(ε, ∇ε) from which the third order stress (“double stress”) due to mesoscopic mechanisms is obtained by derivative τabc ≡ ∂φ ∂ηabc . The extra-body force is deduced accordingly

2 κ a ≡ ∂ τ abc ∂xb ∂xc. Indeed, the strain gradient theory assumes that a material point carries a second order stress tensor σ and a third order stress tensor τ in order to satisfy the Clausium- Duhem inequality. Particular form of the wave propagation was not developed in those works since they were exclusively devoted to quasi-static plasticity and applied to indentation testing of materials. Alternatively, (Popov and Kröner, 1999) recently developed a dynamic theory of elastic-plastic model including microstructure, where they mainly considered dislocations at the microscopic level. The equations of motions they derived were based on the kinetic energy of macroscopic movements in the continuum and the microscopic mechanisms caused by dislocations along three mutually orthogonal sets of glide planes. The stress was decomposed into three components σ = σ macro + σ meso +σ micro . Then by considering the pseudo-gradient of displacement, which was a non integrable tensor when discontinuities occurred locally, the dislocation density was defined by αab ≡ ecda∇c (∇dub ) with ecda equal to 1 for cyclic permutation of 1, 2, 3, equal to -1 for anti-permutation and null otherwise. The wave propagation equations derived by (Popov and Kröner, 1999) may be rewritten to give the following:

L. Rakotomanana 12 Connecting mesoscopic and macroscopic scale lengths

2 ∂ u 2 2 2 2 = (cL − cT )∇(divu) + cT Δu +κ . ∂t

2 Where the extra-body force writes κ = −cTα , which shows that it is proportional to the density of dislocations. In terms of constants of structure, the equation further simplifies, when the inertial rotation effects are again neglected (Popov and Kröner, 1999):

2 ∂ u 2 2 2 2 2 = (cL − cT )∇(divu) + cT Δu + cTℵ0 . ∂t

For the particular case where the displacement vector u = (u1,u2 ,u3 ) is depending only on one coordinate x1 = x and on the time t , further simplification gives: ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u 1 = c2 1 + c2ℵ , 2 = c2 2 , 3 = c2 3 . ∂t2 L ∂x 2 T 0 ∂t2 T ∂x2 ∂t2 T ∂x2

Mesoscopic wave scattering: connection theory As for disordered crystal solids where the length-scale is of the order of several lattice constants (10Å – 50Å) (Blumenfeld, 1998), the divergence of a vector field should also account for the material structural flaw at the mesoscopic level. The divergence of a vector may be split into macroscopic contribution and mesoscopic contribution of micro-cracks divv = Divv +ℵ0 (v), in which Divv denotes the macroscopic divergence. The operator Divv reduces to the usual material divergence, which is extensively used in the framework of elastic large deformation of continua without micro-crack. The Laplacian is accordingly deduced Δv = Δ v + ∇v(ℵ0 ) where

Δv is the macroscopic Laplacian. The wave propagation equations in micro-cracked solid are directly obtained:

2 ∂ u 2 2 2 2 2 2 2 = (cL − cT )∇(Divu) + cT Δu + (cL − cT )∇[ℵ0 (u)] + cT ∇u(ℵ0 ) . ∂t The two last additional terms represent the configuration forces due to the mesoscopic mechanisms. The wave equation projected onto a Cartesian frame gives (summation for index a):

2 2 2 ∂ ub 2 2 ∂ ua 2 ∂ ub 2 2 ∂(ℵ0a ua ) 2 ∂ua 2 = (cL − cT ) + cT 2 + (cL − cT ) + cTℵ0a . ∂t ∂xb∂xa ∂xa ∂xb ∂xb

L. Rakotomanana 13 Connecting mesoscopic and macroscopic scale lengths

The continuous distribution of micro-cracks implies a coupling between the wave propagation along the three directions. For the particular case where the displacement vector u = (u1,u2 ,u3 ) is depending only on one coordinate x1 = x and on the time t , further simplification gives:

2 2 2 2 2 2 ∂ u1 2 ∂ u1 2 ∂ua 2 2 ∂ℵ0a ∂ u2 2 ∂ u2 ∂ u3 2 ∂ u3 2 = cL 2 + cLℵ0a +(cL − cT ) ua , 2 = cT 2 , 2 = cT 2 . ∂t ∂x ∂x ∂x ∂t ∂x ∂t ∂x The first equation governs the longitudinal wave propagation and the two last ones describe the transverse wave propagation. The existence of continuously distributed micro-cracks implies a coupling between the wave propagation along the three directions. The first equation looks like a linear damped KLEIN-GORDON waves equation e.g. (Kneubühl, 1997). Solving of the two last equations is straightforward under some boundary conditions and gives transverse waves. The first equation is more complicated but could be easily solved after separating the variables (valid only under some boundary conditions).

Examples of wave propagation with the connection method

The macroscopic-mesoscopic wave model could find its main application in measuring the loss in an ultrasonic signal due to propagation trough a sample device, which ideally is a plate of thickness d . Sensitivity analysis could be performed by means of non-dimensional ratio of wavelength to plate thickness λ d e.g. (Kautz, 1996). For the sake of simplicity, we consider a mono-dimensional problem where the displacement u is depending only on one coordinate x1 = x (perpendicular to the plate) and on the time t . The micro-crack distribution reduces to a scalar ℵ01 =ℵ0 (co-vector directed by the plate normal vector). If properties are furthermore not coordinate-dependent (uniform distribution of micro-cracks) in the plate, the wave equation drastically simplifies:

∂ 2u ∂ 2u ∂u = c2 + c2ℵ . ∂t2 L ∂x2 L 0 ∂x

For physical interpretation, a characteristic crack opening length is defined as dℵ ≡ 2 ℵ0 and the defect circular frequency as ωℵ ≡ cL dℵ , which allows giving the relation ω ωℵ = 2πdℵ λ . The quantity ℵ0 2 (inverse of defect length dℵ ) is the acoustic absorption coefficient e.g. (Breazeale et al., 1981). Suppose that the plate is subjected to the steady-state displacement boundary oscillatory condition u(0,t) = u cos(ωt) at the left boundary and the plate is bonded to a fixed

L. Rakotomanana 14 Connecting mesoscopic and macroscopic scale lengths support at the right boundary. The steady-state wave depends on the excitation frequency. When

ωℵ ≥ ω , the steady-state wave writes:

2 2 x sinh − 1 −ω ω (x − d) d − ℵ ℵ d u(x,t) = u ( ) e ℵ cos(ωt). sinh 1 −ω 2 ω2 d d ( ℵ ℵ ) This solution includes various contributions for attenuation. It is observed in all case that the wave attenuates exponentially with distance, which typically conforms to the usual absorption contribution e.g. (Breazeale et a., 1981; Posser, 1996). For very low frequency excitation

ω ωℵ << 1, the ratio of the crack length to the wavelength is much less than unity and we have small scattering-like effects. In principle, RAYLEIGH theory on wave scattering is applicable in this case. There is a resonance when ωℵ = ω , which occurs when wavelength and crack opening length is comparable in magnitude. Whenωℵ ≤ ω , the steady-state wave is given by:

2 2 x sin − ω ω −1(x − d) d − ℵ ℵ d u(x,t) = u ( ) e ℵ cos(ωt). sin ω2 ω 2 −1 d d ( ℵ ℵ )

2 2 2 2 2 Resonance peaks occur when ωn = n π cL L +ωℵ , n = 1,2,.... For the extreme case where the ratio ω ωℵ >> 1, the wave behavior should be comparable to reflection and refraction at locally planar interfaces. Moreover, energy peaks in this pass-band are transmitted through the medium. For all cases, attenuation of waves depends on the frequency and on the amount of micro-cracks parameters. This cut-off frequency would be a starting point for experimental measurement of the micro-cracks distribution. For fully three-dimensional samples, material is supporting compression and shear waves and the situation becomes much more complicated.

Considering another example of elastic micro-cracked plate subjected to the steady-state displacement boundary condition u(0,t) = u cos(ωt) at the left end and free at the right boundary, the steady-state wave take the following forms according to the value of frequency excitation.

Whenωℵ ≤ ω , the steady-state oscillatory wave is given by:

2 2 2 2 2 2 x ω ω −1cos ω ω −1(x − d) d + sin ω ω −1(x − d) d − ℵ ℵ ℵ ℵ ℵ d u(x,t) = u ( ) ( ) e ℵ cos(ωt). ω 2 ω 2 −1cos − ω 2 ω2 −1d d + sin − ω 2 ω 2 −1d d ℵ ( ℵ ℵ ) ( ℵ ℵ) The (infinite number) resonance frequencies are calculated with the relationship:

L. Rakotomanana 15 Connecting mesoscopic and macroscopic scale lengths

ω2 ω 2 −1 = tan ω 2 ω2 −1d d ℵ ( ℵ ℵ )

For low frequency ωℵ ≥ ω , the solution reduces to:

2 2 2 2 2 2 x 1 −ω ω cosh 1 −ω ω (x − d) d + sinh 1 −ω ω (x − d) d − ℵ ℵ ℵ ℵ ℵ d u(x,t) = u ( ) ( ) e ℵ cos(ωt). 1 −ω 2 ω2 cosh − 1 −ω 2 ω2 d d + sinh − 1− ω 2 ω 2 d d ℵ ( ℵ ℵ) ( ℵ ℵ ) Attenuation curves and frequency spectral curves of these steady-state waves are reported on Figures 3 and 4. The wave equation underlying these particular solutions is similar to the wave equation governing the longitudinal motion of the plate with a variable cross-section (linear variation) or with a variable Young’s modulus across the plate. Although starting with different physical situations, all of them result into scattering waves.

Nucleation and growth of micro-crack

For completeness, the evolution laws for micro-crack distribution is sketched below. The nucleation and growth of micro-cracks are characterized by the evolution law ℵ0 , which is considered as constitutive variable. Hypothesis of normal dissipation mechanism restricts the class of constitutive laws although remaining a relatively general framework to continuum models satisfying the second principle of thermodynamics. For normal dissipative materials, constitutive laws of the continuum with field discontinuity may be entirely reconstructed from a free energy and a potential of dissipation e.g. (Germain et al., 1983) φ = φ(ε,ℵ0 ) and

ψ = ψ ζ , where ζ is the objective rate of the micro-cracks density (Rakotomanana, 1998). ( ℵ0 ) ℵ0 Most mechanical behaviors are obtained by choosing special functions for the free energy and for the potential of dissipation. The simplest example is the linear isotropic elastic solid with

1 2 1 2 micro-crack density, defined by the quadratic potential φ = 2 λ(ℵ0 )tr (ε) + 2 µ(ℵ0 )tr(ε ) . For a plastic material characterized by the existence of a stress threshold, material response depends on the history of external applied forces. At first approximation, on can observe macroscopically that the behavior of such a material changes abruptly when the intensity of applied forces overpasses a certain critical value. This brusque variation requires a non-continuously differentiable model. The conjugate dissipation potential is defined by the partial Legendre-

Fenchel transform e.g. (Rakotomanana, 1998) ψ * Jℵ ≡ Supζ Jℵ ÷ζℵ −ψ ζℵ . The ( 0 ) ℵ0 [ 0 0 ( 0 )]

L. Rakotomanana 16 Connecting mesoscopic and macroscopic scale lengths evolution law of the density of micro-cracks is therefore calculated by means of the sub-gradient of the discontinuous dissipation potential:

* J . ζℵ0 ∈∂ψ ( ℵ0 ) This evolution law expresses the nucleation and the growth of region where dense sets of micro- cracks appear in the defected material. To compute the micro-crack density evolution, it is convenient to define first a set C , which is a convex set of the dual space J , where there is { ℵ0 } no evolution of the rates of micro-crack density. Set C contains the null tensor. For dual variables J in the interior of C , the density of micro-cracks remains constant whereas for { ℵ0 } those on the boundary, the density increases. Then, it is convenient to introduce the indicator function of the set C defined by:  0 if J ∈C I J ≡  { ℵ0 } C ( ℵ0 )  +∞ if not By analogy to classical dry friction and rate-independent plasticity theory (Moreau, 1970), the dissipation potential may be identified as the conjugate of the indicator function of the set C as ψ ζ = I * ζ . Details and proofs supporting this identification for evolution laws with ( ℵ0 ) C ( ℵ0 ) threshold may be found in e.g. (Maugin, 1992; Rakotomanana, 1998). By applying the LEGENDRE-FENCHEL transform, it is straightforward to derive:

I * ζ = Sup J ÷ζ − I ζ . C ( ℵ0 ) J ∈C ℵ0 ℵ0 C ( ℵ0 ) { ℵ0 } [ ] We then deduce the dissipation potential ψ ζ = I * ζ = Sup J ÷ ζ . The ( ℵ0 ) C ( ℵ0 ) C ( ℵ0 ℵ ) {ζℵ0 }∈ dissipation potential includes a homogeneous function of degree one in terms of micro-crack density rates ψ = Sup J ÷ζ . The last terms in brackets are positive and represent the C ( ℵ0 ℵ0 ) {ζℵ0 }∈ internal dissipation due to micro-cracks. Experimental investigations on the determination of practical form of the convex set C or alternatively the form of yield stress function should be undertaken for each material. In this way, bone biomechanics study e.g. (Zioupos et al., 1995) seems promising by discovering the classical Tsai-Wu criterion as candidate yield function for anisotropic material.

L. Rakotomanana 17 Connecting mesoscopic and macroscopic scale lengths

Discussion Macroscopic continuum mechanics has had great success in providing macroscopic information, which is essential for the mechanical shape design of engineering devices. The dislocation dynamics theory is particularly suited to understand the interaction of large numbers of dislocations and the pattern formation at the mesoscopic level, which is important for the structural design of materials. The atomistic theory is necessary to give the basic information at the lattice level of crystals at the microscopic level, in which the theory of molecular dynamics play a increasing role in the mechanistic simulation of dislocation motions. Technology development and constraint to design smaller and smaller devices suggest in many problems of material mechanics the use of multiple scales e.g. (Blumenfeld, 1998; Tadmor et al., 2000). The present work focuses on connecting mesoscopic and macroscopic scale levels. Starting with macroscopic continuum level, the mesoscopic level disorder (~ 1µm to 10µm) is accounted for by introducing the local discontinuity of field (micro-cracks effects). The model is constructed on the elementary mechanisms rather than “curve fitting” by merely imposing some non-linear behavior. This would facilitate the building of hierarchical models in the sense that models at one scale level would provide rational arguments to develop the models at another scale level. Namely, the use of strain gradient macroscopic theory does not have any implications in the basic wave propagation equation, whereas considering the physical mesoscopic mechanisms has a major consequence on the wave equation.

An immediate application of the connected macroscopic-mesoscopic model is the quantitative non-destructive evaluation of materials, which combines wave propagation theory and the technology as ultrasonic technique. Up to now, the basic science behind the more and more sophisticated ultrasonic technology remains the macroscopic wave propagation within linear or nonlinear stress-strain law. Experimental evidence of strong attenuation and cut-off of frequency in micro-porous ceramics (Craciun et al., 1998; Wanner, 1998) and pass band effects in fractured steels plates (Nagakawa et al., 2000) could not be fully explained in the light of only macroscopic models of wave propagation. Such phenomenon could be well approached with the model connecting macro-phenomenon to mesoscopic scale discontinuities. In a general manner, theoretical models should still be developed not only for improving the measurement processes as for ultrasonic inspection techniques but also and mainly for better interpretation of the

L. Rakotomanana 18 Connecting mesoscopic and macroscopic scale lengths experimental data. The present work proposes a macroscopic-mesoscopic model, which is linear and therefore closed-form solutions is easily obtained for simple situations. The reality of evanescent waves in a micro-cracked solids seems to be interesting in regards of controlling and better understanding internal damping in engineering material as ceramics or polymers, in earthquake propagation science and in characterizing hard biological material as bone tissue. Moreover, theory of wave propagation with mesoscopic mechanisms could be efficient for better helping the monitoring of in situ degradation of materials and the reliability of in situ micro- devices.

References 1. Aifantis EC. On the role of gradients in the locations of deformation and fracture, Int. J. Eng. Science 30(19), 1992, pp 1279-1299. 2. Begley MR, Hutchinson JW. The mechanics of size-dependent indentation, J. Mech. Phys. Solids 46 (10), 1998, pp2049-2068. 3. Bilby B.A., Bullough R., Smith E. Continuous ditributions of dislocations: a new application of the methods of non-Riemannian geometry, Proc. R. Soc. Lond. A231, 1955, pp 263-273. 4. Blumenfeld R. Dynamics of fracture propagation in the mesoscale: Theory, Theoretical and Applied Fracture mechanics 30, 1998, pp 209-223. 5. Butalov V, Abraham F. F., Kubin L., Devincre B, Yip S. Connecting atomistic and mesoscale simulations of crystal plasticity, Nature, 391, 1998, pp 669-672. 6. Breazeale M.A., Cantrell J.H. Jr., Heyman J.S. Ultrasonic wave velocity and attenuation measurements. In Methods of experimental physics vol 19: Ultrasonics, Edmonds P.D. Editor, Academic Press, Orlando, 1981, pp 67-135. 7. Broberg KB. The cell model of materials, Computational Mechanics 19, 1997, pp 447-452. 8. Capuani D, Willis JR. Wave propagation in elastic media with cracks. Part II: Transient nonlinear response of a cracked matrix, Eur. J. Mech. A/Solids 18, 1999, pp159-175. 9. Cartan E. On manifolds with affine connection and the theory of general relativity, Bibliopolis, edizioni di filosofia e science, Napoli, 1986. 10. Cauvin A, Testa RB. Damage mechanics: basic variables in continuum theories, Int J Solids Structures 36, 1999, pp 747-761.

L. Rakotomanana 19 Connecting mesoscopic and macroscopic scale lengths

11. Chaboche JL. Continuum damage mechanics: part I – general concepts. J Appl Mech 55, 1988, pp 59-64. 12. Clegg WJ. Controlling crack in ceramics, Science 286, 1999, pp 1097-1098. 13. Craciun F, Guidarelli G, Galassi C, Roncari E. Elastic wave propagation in porous piezoelectric ceramics, Ultrasonics 36, 1998, pp 427-430. 14. Dunn JE, Serrin J. On the thermodynamics of interstitial working, Arch. Rat. Mech. Anal., 17, 1985, pp 95-133. 15. Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42, 1994, pp 475-487. 16. Fleck NA, Hutchinson JW. Strain gradient plasticity. In Adv. In Applied mechanics 33, ed. Hutchinson JW & Wu TT, 1997, pp 295-361. 17. Frémond M, Nedjar B. Damage, gradient od damage and principle of virtual power, Int. J. Solids Structures 33(8), 1996, pp 1083-1103. 18. Gao H, Huang Y, Nix WD, Hutchinson JW. Mechanism-based strain gradient plasticity-I. Theory, J. Mech. Phys. Solids 47, 1999, pp 1239-1263. 19. Germain P, Nguyen QS, Suquet P. Continuum thermodynamics, J. Applied Mech., Trans ASME, vol 50, 1983, pp 1010-1020. 20. Gurtin ME. Thermodynamics and the possibility of spatial interaction in elastic materials. Arch. Ration. Mech. Anal. 19, 1965, pp 339-352. 21. He QC, Curnier A. A more fundamental approach to damaged elastic stress-strain relations, Int J Solids Structure 32 (10), 1995, pp 1433-1457. 22. Holian B.L., Lomdahl P.S. Plasticity induced by shock waves in nonequilibrium molecular- dynamics simulations, Science 280, 1998, pp 2085-2088. 23. Huang Y, Gao H, Nix WD, Hutchinson JW. Mechanism-based strain gradient plasticity-II. Analysis, J. Mech. Phys. Solids 48, 2000, pp 99-128. 24. Hutchinson JW. Plasticity at the micron scale, Int. J. Solids Structures 37, 2000, pp 225-238. 25. Jhang KY, Kim KC. Evaluation of material degradation using non-linear acoustic effect, Ultrasonics 37, 1999, pp 39-44. 26. Kautz H.E. Acousto-ultrasonic decay in metal matrix composite panels, Res. Nondestr. Eval. 8, 1996, pp 39-50. 27. Kneubühl FK. Oscillations and waves, Springer Verlag, Heidelberg, 1997.

L. Rakotomanana 20 Connecting mesoscopic and macroscopic scale lengths

28. Kröner E. Continuum theory of defects. In Physique des défauts, Nato series Ballian R et al. ed., North-Holland, 1981, pp 219-315. 29. Le KC, Stumpf H. On the determination of the crystal reference in nonlinear continuum theory of dislocations, Proc. R. Soc. Lond. A452, 1996, pp 359-371. 30. Maugin GA. The thermomechanics of plasticity and fracture, Cambridge University press, Cambridge, 1992. 31. Moreau JJ. Sur les lois de frottement, de plasticité et de viscosité, C. R. Acad. Sci. Paris A 271, 1970, pp 608-611. 32. Nagy PB. Fatigue damage assessment by nonlinear ultrasonic materials characterization, Ultrasonics 36, 1998, pp 375-381. 33. Nakagawa S, Nikei K.T., Myer L. R. Stop-pass behavior of acoustic waves in a 1D fractured system, J. Acoust. Soc. Am., 107(1), 2000, pp 40-50. 34. Noll W. Materially uniform simple bodies with inhomogeneities, Arch. Ration. Mech. Anal. 27, 1967, pp 1–32. 35. Oliver J, Cervera M, Manzoli O. Strong discontinuities and continuum plasticity models: the strong discontinuity approach, Int. J. Plasticity 15, 1999, pp 319-351. 36. Popov V, Kröner E. On the dynamic theory of elastoplastic medium with microstructure, Computational Materials Science 16, 1999, pp218-236. 37. Prosser W.H. Advanced AE Techniques in Composite Materials Research, J. of Acoustic Emission, vol 14(3-4), 1996, pp S1-S11. 38. Rabier PJ. Some remarks on damage theory, Int J Engng Sci 27, 1989, pp 29-54. 39. Rakotomanana L. Contribution à la modélisation géométrique et thermodynamique d’une classe de milieux faiblement continus, Arch. Ration. Mech. Anal. 141, 1998, pp 199-236. 40. Ramaniraka N, Rakotomanana L. Models of continuum with micro-crack distribution, Math. Mech. Solids (In press), 2000. 41. Ravasoo A. Nonlinear waves in characterization of inhomogeneous elastic material, Mechanics of Materials 31, 1999, pp 205-213. 42. Ravi-Chandar K, Yang B, On the role of microcracks in the dynamic fracture of brittle materials, J. Mech. Phys. Solids 45(4), 1997, pp 535-563. 43. Sharon E, Fineberg J. Microbranching instability and the dynamic fracture of brittle materials, Physical Review B 54(10), 1996, pp 7128-7139.

L. Rakotomanana 21 Connecting mesoscopic and macroscopic scale lengths

44. Schouten JA. Ricci calculus, Springer Verlag, Berlin, 1954. 45. Shizawa K, Zbib HM. A thermodynamical theory of gradient elastoplasticity with dislocation density tensor. I: Fundamentals, Int. J. Plasticity 15, 1999, pp 899-938. 46. Stölken JS, Evans AG. A microbend test method for measuring the plasticity length scale, Acta Materialia 46, 1998, pp 5109-5115. 47. Tadmor EB, Phillips R, Ortiz M. Hierarchical modeling in the mechanics of materials, Int. J. Solids Structures 37, 2000, pp 379-389. 48. Toupin RA. Elastic materials with couple-stress, Arch. Ration. Mech. Anal. 11, 1962, pp 385-414. 49. Vakulenko A, Kachanov M. Continuum theory of medium with cracks, Mechanics of Solids, 6(4), 1971, pp 145-151. 50. Vandenbossche B., Kriz R.D., Oshima T. Stress-wave displacement polarizations and attenuation in unidirectional composites: Theory and experiment, Res. Nondestr. Eval. 8, 1996, pp 101-123. 51. Vary, A. Concepts for interrelating ultrasonic attenuation, microstructure, and fracture toughness in polycrystalline solids, Materials Evaluation vol. 46 Issue 5, 1988, pp 642-649. 52. Wang CC. On the geometric structure of simple bodies, or mathematical foundation for the continuous distributions of dislocations, Arch. Ration. Mech. Anal. 27, 1967, pp 33-94. 53. Zioupos P, Currey JD, Mirza MS, Barton DC. Experimentally determined microcracking around a circular hole in a plate bone: comparison with predicted stresses, Phil. Trans. R. Soc. Lond. B 1995, pp 383-396.

L. Rakotomanana 22 Connecting mesoscopic and macroscopic scale lengths

Figure captions

1. Figure 1. The three scale length levels are represented. (a) At the lattice level (10Å – 50Å), the crystal connection represents a dislocation in a crystal lattice. A large circuit will have an error of closure proportional to the number of dislocations that it contains, that is, proportional to its area. (b) At the mesoscopic level (1 µm –10µm), the circuit crosses a micro-crack which induces a discontinuity of the field (displacement field). The discontinuity of scalar and vector field across the crack means a non null torsion and curvature tensors of the affine connection. (c) At the macroscopic level (≥ 100µm), only a metric tensor is considered to capture the deformation.

2. Figure 2. Wave amplitude ration through the plate vs. depth. The sample is a free-end plate (left and right ends) submitted to a sinusoidal wave displacement at the left side and free at the right side. The thickness of the plate is d = 100µm and the characteristic crack length

dℵ = 1µm. Graphics report the wave propagation within the plate at various excitation

frequency ω ωℵ = 2,3,6,7 . The micro-crack distribution characteristics are respectively the

6 −1 6 −1 9 crack length ℵ0 = 2 10 m ; circular frequency ωℵ ≈ 6000 10 rds ; fℵ ≈ 10 Hz . The

13 Debye frequency for this sample is fD ≈10 Hz .

3. Figure 3.Wave amplitude ratio through the plate vs. depth. The sample is a clamped-end plate (right end) submitted to a sinusoidal wave displacement at the left side and clamped at the right side. The thickness of the plate is d = 100µm and the characteristic length

dℵ = 1µm. Graphics report the wave propagation within the plate at various excitation

frequency ω ωℵ = 2,3,6,7 . (Same characteristics as previous Figure 2).

4. Figure 4. Amplitude ratio at a depth of y = 2µm vs. frequency ω for various samples. The

plate thickness is d = 100µm and the characteristic length dℵ = 1µm. Plates are excited at the left end with an imposed sinusoidal displacement at frequency ranging from ω = 6 109 Hz to ω = 7 109 Hz . (a) First column represents spectrum for free end plate at the right side: first row plate with micro-cracks, second row virgin plate and third row correlation function of

L. Rakotomanana 23 Connecting mesoscopic and macroscopic scale lengths

the micro-cracked and virgin plates. (b) Second column represents spectrum for clamped end plate at the right side: first row plate with micro-cracks, second row virgin plate and third row correlation function of the micro-cracked and virgin plates. (Same characteristics as previous Figure 2).

L. Rakotomanana 24