connecting mesoscopic and macroscopic scale lengths for ... · connecting mesoscopic and...
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Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 1
Connecting mesoscopic and macroscopic scale lengths forultrasonic wave characterization of micro-cracked material
L. R. RakotomananaInstitute of Mathematics, University of Rennes I -France
Abstract. Macroscopic failure of material is attributed mostly to the initial presence ofmicro-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 itsconsequence on the energy dissipations during micro-crack kinetics, a theoretical model ofmicro-cracked continuum is derived in this paper. The model describes a micro-crackdensity in terms of CARTAN constants of structure and explicitly connects themacroscopic scale to the mesoscopic discontinuities. This approach contrasts to the usualmethod in continuum mechanics that seeks a phenomenological description by introducingan internal variable in the constitutive laws. An illustrative example of the modelapplication is presented for the linear ultrasonic wave propagation test. The resulthighlights the importance of rigorously revisiting the dynamic equation in micro-crackedsolid.
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 stronglydependent 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, densesets 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
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 2
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 thevicinity 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 thussimultaneously 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 averagestress (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, neworientation 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 maybe 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. Connectionbetween 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 moleculardynamics 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 simulationsduring 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. (Holianand 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 deformatione.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
Connecting mesoscopic and macroscopic scale lengths
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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 ofmaterial 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 beendeveloped to characterize the internal degradation by measuring the attenuation of ultrasonic
waves. Various theoretical models have been developed for explaining and predicting empiricalcorrelation 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 ascrack 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 togeometric 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, mostmodels 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
been measured on the basis of Taylor expansion of the sound velocity cL = c0 + c1εext + c2εext2 +…
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).
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Probably scattering losses were more important than adsorption losses for those materials.
Similarly, higher harmonic generation was used to capture material degradation by assumingnon-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 andnonlinear 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 basisof 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 existingmacroscopic 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 strongattenuation 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 withenergy 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 developedin 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 to10µm) 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.
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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 measureddata.
The aim of this paper is twofold. First, a bridging theory between mesoscopic scale andmacroscopic 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 formicrocracking 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( )
deforming with the material, the metric tensor writes g = gab ea ⊗ eb . The strain components are
defined as εab = 12 (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
displacements ua in the usual manner, that is εab = 12 ∇aub +∇bua( ) . Symbol ∇ denotes the
connection used for calculating all derivatives with respect to space (gradient of tensor). Theconnection implicitly used in current macroscopic classical physics theory, including continuum
mechanics, is basically the LEVI-CIVITA connection, which must a priori satisfy thefundamental 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 thecontinuum damage mechanics, which introduced internal variables for describing the internal
micro-cracking. Most continuum mechanics models of micro-cracked solids seek aphenomenological description by assuming macroscopic variables to capture the distribution of
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 6
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 materialbased 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 fullanisotropy, orthotropy, tetragonal symmetry, hexagonal symmetry, transverse isotropy and
isotropy. Up to now, the choice of internal variables was rather dictated by an inductive logic, inwhich 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 thosemodels, 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 cracklips. 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 modelsare 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 bulkmaterial 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 intactmaterial 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 continuousand 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
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(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
εab =12 ∇au b +∇bu a( ) + 1
2 HΩ ∇a ub[ ]+∇b ua[ ]( ) + 12 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, whichmay 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 bridgingthe macroscopic level to the mesoscopic level, continuum models were proposed, in which thestrain 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:
Ee2 ≡ 2
3 ′ ε ab ′ ε ab + d1 ′ η 1abc ′ η 1abc + d2 ′ η 2abc ′ η 2abc + d3 ′ η 3abc ′ η 3abc .
The prime sign indicates the deviatoric part of the strain and the strain gradient. Tensors ′ η iabc are
three mutually orthogonal third tensors according to ′ η abc = ′ η 1abc + ′ η 2abc + ′ η 3abc 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
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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 theusual 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 ofplasticity 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 embeddedfiber (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 levelsshould 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, butlarge 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 similarapproach to seek additional variables by assuming physical mesoscopic discontinuity of vector
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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 localizeddiscontinuity 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 nullsimultaneously) 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, whichare the additional variables for describing micro-cracked continuum:
ℵ= Γabc − Γba
c( ) −ℵ0abc[ ] ea ⊗ eb ⊗ ec ,
ℜ = eb Γdac( ) − ea Γdbc( ) + Γdae Γebc − Γdbe Γeac −ℵ0ab
e Γedc[ ] ea⊗ eb⊗ ed ⊗ ec ,
Where Γabc = ec ∇ea eb( ) and ℵ0ab
c ec ≡ ea ,eb[ ] are respectively the coefficients of the affine
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
CARTAN’s structural constants ℵ0abc , which measure the failure of the closure of an initial
parallelogram designed on the material during the deformation. Failure of closure is due to
micro-crack. Constants of structure defined by ℵ0abc ec ≡ 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
metric and constants of structure respectively, which may resumed into the co-vector ℵ0 =ℵ0 abb ea
(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
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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 agradient 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:
∂ 2u∂t2
= cL2 − cT
2( )∇ divu( ) + cT2Δu .
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, reducedto a scalar field ℵ0 for the sake of simplicity. Therefore, the wave equation may be rewritten as
follows:
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∂ 2u∂t2 = cL 0 + cL1ℵ0 +…( )2
− cT 0 + cT1ℵ0 +…( )2( ) ∇ divu( ) + cT 0 + cT1ℵ0 +…( )2Δu
Where the elastic constants are expressed on the basis of Taylor expansion of the sound velocity
cL = cL0 + c1ℵ0 + c2ℵ02 +… , in which ℵ0 denotes the damage level. Similar expansion holds for
the transversal sound velocity. Such theoretical model had been used successfully in earlydetection 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 metricconnection used for calculating the divergence and Laplacian operators is not modified by the
presence of micro-cracks distribution. Characterization of materials at the mesoscopic scaleseems 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:
∂ 2u∂t2
= cL2 − cT
2( )∇ divu( ) + cT2Δu +κ
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):
∂ 2u1∂t2
= cL2 ∂
2u1∂x 2
− cL2νπrΩ
2 ∂ u1[ ]∂x
, ∂2u2∂t2
= cT2 ∂
2u2∂x2
, ∂ 2u3∂t2
= cT2 ∂
2u3∂x2
.
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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 ofvirtual power, the wave motion equations take the form:
∂ 2u∂t2
= cL2 − cT
2( )∇ divu( ) + cT2Δu +κ ∇u,∇ ∇u( )( ) .
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 materialφ 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
κ a ≡ ∂2τ 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 ofmaterials. Alternatively, (Popov and Kröner, 1999) recently developed a dynamic theory of
elastic-plastic model including microstructure, where they mainly considered dislocations at themicroscopic 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 decomposedinto 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 equationsderived by (Popov and Kröner, 1999) may be rewritten to give the following:
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L. Rakotomanana 13
∂ 2u∂t2
= cL2 − cT
2( )∇ divu( ) + cT2Δu +κ .
Where the extra-body force writes κ = −cT2α , 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):
∂ 2u∂t2
= cL2 − cT
2( )∇ divu( ) + cT2Δu + cT
2ℵ0 .
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:
∂ 2u1∂t2
= cL2 ∂
2u1∂x 2
+ cT2ℵ0 , ∂
2u2∂t2
= cT2 ∂
2u2∂x2
, ∂ 2u3∂t2
= cT2 ∂
2u3∂x2
.
Mesoscopic wave scattering: connection theoryAs 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 +∇vℵ0( ) where
Δ v is the macroscopic Laplacian. The wave propagation equations in micro-cracked solid are
directly obtained:
∂ 2u∂t2
= cL2 − cT
2( )∇ Divu( ) + cT2Δ u + cL
2 − cT2( )∇ℵ0 u( )[ ] + cT
2∇u ℵ0( ) .
The two last additional terms represent the configuration forces due to the mesoscopicmechanisms. The wave equation projected onto a Cartesian frame gives (summation for index a):
∂ 2ub∂t2
= cL2 − cT
2( ) ∂ 2ua∂xb∂xa
+ cT2 ∂
2ub∂xa
2 + cL2 − cT
2( )∂ ℵ0aua( )∂xb
+ cT2ℵ0a
∂ua∂xb
.
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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:
∂ 2u1∂t2
= cL2 ∂
2u1∂x 2
+ cL2ℵ0a
∂ua∂x
+ cL2 − cT
2( )∂ℵ0a
∂xua , ∂
2u2∂t2
= cT2 ∂
2u2∂x2
, ∂2u3∂t2
= cT2 ∂
2u3∂x2
.
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 acoupling 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. Thefirst 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 lossin 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 ofwavelength 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 coordinatex1 = 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 equationdrastically simplifies:
∂ 2u∂t2
= cL2 ∂
2u∂x2
+ cL2ℵ0
∂u∂x
.
For physical interpretation, a characteristic crack opening length is defined as dℵ ≡ 2ℵ0 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
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 15
support at the right boundary. The steady-state wave depends on the excitation frequency. Whenωℵ ≥ ω , the steady-state wave writes:
u x,t( ) = u sinh − 1 −ω 2 ωℵ
2 x − d( ) dℵ( )sinh 1 −ω 2 ωℵ
2 d dℵ( )e−
xdℵ cos ωt( ) .
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 excitationω 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 inthis 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:
u x,t( ) = u sin − ω 2 ωℵ
2 −1 x − d( ) dℵ( )sin ω2 ωℵ
2 −1 d dℵ( )e−
xdℵ cos ωt( ) .
Resonance peaks occur when ωn = n2π 2 cL2 L2 +ωℵ
2 , 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 supportingcompression and shear waves and the situation becomes much more complicated.
Considering another example of elastic micro-cracked plate subjected to the steady-statedisplacement 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:
u x,t( ) = u ω 2 ωℵ
2 −1cos ω 2 ωℵ2 −1 x − d( ) dℵ( ) + sin ω 2 ωℵ
2 −1 x − d( ) dℵ( )ω 2 ωℵ
2 −1cos − ω 2 ωℵ2 −1d dℵ( ) + sin − ω 2 ωℵ
2 −1d dℵ( )e−
xdℵ cos ωt( ) .
The (infinite number) resonance frequencies are calculated with the relationship:
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 16
ω2 ωℵ2 −1 = tan ω 2 ωℵ
2 −1d dℵ( )For low frequency ωℵ ≥ ω , the solution reduces to:
u x,t( ) = u 1 −ω 2 ωℵ
2 cosh 1 −ω 2 ωℵ2 x − d( ) dℵ( ) + sinh 1 −ω 2 ωℵ
2 x − d( ) dℵ( )1 −ω 2 ωℵ
2 cosh − 1 −ω 2 ωℵ2 d dℵ( ) + sinh − 1− ω 2 ωℵ
2d dℵ( )e−
xdℵ cos ωt( ) .
Attenuation curves and frequency spectral curves of these steady-state waves are reported onFigures 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 differentphysical 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. Thenucleation and growth of micro-cracks are characterized by the evolution law ℵ0 , which is
considered as constitutive variable. Hypothesis of normal dissipation mechanism restricts theclass 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 ζℵ0( ) , where ζℵ0is the objective rate of the micro-cracks density (Rakotomanana, 1998).
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
micro-crack density, defined by the quadratic potential φ = 12 λ ℵ0( )tr2 ε( ) + 1
2 µ ℵ0( )tr ε2( ) . 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 macroscopicallythat 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 * Jℵ0( ) ≡ Supζℵ0 Jℵ0 ÷ζℵ0 −ψ ζℵ0( )[ ] . The
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 17
evolution law of the density of micro-cracks is therefore calculated by means of the sub-gradient
of the discontinuous dissipation potential:
ζℵ0∈∂ψ * Jℵ0( ) .
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 Jℵ0{ } , where there is
no evolution of the rates of micro-crack density. Set C contains the null tensor. For dual
variables Jℵ0{ } in the interior of C , the density of micro-cracks remains constant whereas for
those on the boundary, the density increases. Then, it is convenient to introduce the indicator
function of the set C defined by:
IC Jℵ0( ) ≡ 0+∞
if Jℵ0{ }∈Cif 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 ζℵ0( ) = I *C ζℵ0( ) . Details and proofs supporting this identification for evolution laws with
threshold may be found in e.g. (Maugin, 1992; Rakotomanana, 1998). By applying theLEGENDRE-FENCHEL transform, it is straightforward to derive:
I *C ζℵ0( ) = Sup Jℵ0{ }∈C Jℵ0÷ζℵ0
− IC ζℵ0( )[ ] .We then deduce the dissipation potential ψ ζℵ0( ) = I *C ζℵ0( ) = Sup ζℵ0{ }∈C Jℵ0 ÷ ζℵ( ) . The
dissipation potential includes a homogeneous function of degree one in terms of micro-crack
density rates ψ = Supζℵ0{ }∈C Jℵ0
÷ζℵ0( ) . The last terms in brackets are positive and represent the
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 beundertaken 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 foranisotropic material.
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 18
DiscussionMacroscopic 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 ofdislocations 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 dynamicsplay a increasing role in the mechanistic simulation of dislocation motions. Technology
development and constraint to design smaller and smaller devices suggest in many problems ofmaterial 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 forby 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 onescale 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 thebasic 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 quantitativenon-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 moresophisticated 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 fracturedsteels 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 themodel 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
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 19
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 ofevanescent 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, inearthquake 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.
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L. Rakotomanana 23
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 amicro-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 ofthe 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 atthe 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
crack length ℵ0 = 2 106m −1 ; circular frequency ωℵ ≈ 6000 106 rds−1; fℵ ≈ 109Hz . The
Debye frequency for this sample is fD ≈1013Hz .
3. Figure 3.Wave amplitude ratio through the plate vs. depth. The sample is a clamped-endplate (right end) submitted to a sinusoidal wave displacement at the left side and clamped atthe 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: firstrow plate with micro-cracks, second row virgin plate and third row correlation function of
Connecting mesoscopic and macroscopic scale lengths
L. Rakotomanana 24
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 thirdrow correlation function of the micro-cracked and virgin plates. (Same characteristics as
previous Figure 2).