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Transcript
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 lengthscales. 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 microcracking 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 c L = c 0 + c1 ε ext + c2 ε 2ext + …
where ε ext 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
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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 = E ε (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 nonhomogeneous approaches were not able to detect the presence of uniform distribution of microcracks 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 cutoff 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,e 2 ,e3 )
deforming with the material, the metric tensor writes g = ga b ea ⊗ eb . The strain components are
defined as ε ab = 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 ε ab =
1
2
(∇a ub + ∇b ua ) .
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 (cohesiondecohesion) 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
ε ab =
1
2
(∇a ub + ∇b ua ) + 12 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, which
may be cumbersome and practically difficult to apply in presence of randomly distributed microcracks.
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:
Ee2 ≡ 23 ε ′abε ab′ + d1η1abc
′ η1abc
′ + d2 η′2abc η2abc
′ + d3η3abc
′ η3abc
′ .
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 ηabc
′ = η1abc
′ + η2abc
′ + η3abc
′ 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 workconjugacy 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 microcracked 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:
[
]
ℵ= ( Γacb − Γbc a) − ℵc0a b ea ⊗ eb ⊗ ec ,
[
]
ℜ = eb ( Γdca ) − ea (Γdc b) + ΓdeaΓebc − ΓdebΓeac −ℵe0a bΓedc e a ⊗ e b ⊗ e d ⊗ ec ,
(
Where Γac b = ec ∇ ea eb
)
and ℵc0a bec ≡ [ea ,eb ] are respectively the coefficients of the affine
connection and the constants of structure of Cartan. Symbol [ , ] denotes the classical LIEJACOBI 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,e 2 ,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 ℵc0ab , 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 ℵ0a bec ≡ [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 = ℵb0 abe a
(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 stressstrain law σ = λtr (ε )i + 2 µε , 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 c L =
(λ + 2 µ ) ρ
and transversal velocity cT = µ ρ of sound:
∂ 2u
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
∂ 2u
2
2
2
∇ (divu) + (cT 0 + cT1ℵ0 + …) Δu
2 = ( cL 0 + cL1ℵ0 + …) − ( cT 0 + cT1ℵ0 + …)
∂t
(
)
Where the elastic constants are expressed on the basis of Taylor expansion of the sound velocity
c L = c L0 + c1ℵ0 + c2ℵ20 + … , 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:
∂ 2u
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):
2
2
2
2
∂ 2 u3
∂ 2 u1
2 ∂ u2
2 ∂ u3
2 ∂ u1
2
2 ∂ [u1 ] ∂ u2
,
,
.
=
c
=
c
=
c
−
c
νπ
r
T
T
L
L
Ω
∂t 2
∂x 2 ∂t 2
∂x 2
∂t 2
∂x 2
∂x
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:
∂ 2u
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 τ abc ≡ ∂φ ∂ηabc . The extra-body force is deduced accordingly
κ a ≡ ∂ 2τ abc ∂x b ∂x c . 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 ClausiumDuhem 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 α ab ≡ ecda∇ c (∇ d ub ) 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
∂ 2u
= ( c2L − cT2 )∇(divu) + c2T Δu + κ .
∂t 2
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
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:
2
2
2
∂ 2 u1
∂ 2 u2
∂ 2 u3
2 ∂ u1
2
2 ∂ u2
2 ∂ u3
,
.
= c L 2 + cTℵ0 ,
= cT
= cT
∂t 2
∂x
∂t 2
∂x 2 ∂t 2
∂x 2
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:
∂ 2u
2
2
2
2
2
2
2 = ( cL − cT )∇( Divu) + cT Δu + (c L − 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 ub
∂ 2 ua
∂u
2
2
2 ∂ ub
2
2 ∂ (ℵ0a ua )
=
c
−
c
+
c
+ c2Tℵ0a a .
(
)
2
L
T
T
2 + (cL − cT )
∂t
∂xb ∂xa
∂xa
∂xb
∂x b
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 u1
∂ua
∂ 2 u2
∂ 2 u3
2 ∂ u1
2
2
2 ∂ℵ0a
2 ∂ u2
2 ∂ u3
,
,
.
=
c
+
c
ℵ
+
c
−
c
u
=
c
=
c
(
)
L
L 0a
L
T
T
T
∂t 2
∂x 2
∂x
∂x a ∂t 2
∂x 2 ∂t 2
∂x 2
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:
2
∂ 2u
∂u
2∂ u
2
.
=
c
2
L
2 + c Lℵ0
∂t
∂x
∂x
For physical interpretation, a characteristic crack opening length is defined as dℵ ≡ 2 ℵ0 and the
defect circular frequency as ωℵ ≡ c L dℵ , which allows giving the relation ω ωℵ = 2π dℵ 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:
u( x,t ) = u
(
sinh − 1 − ω 2 ωℵ2 ( x − d ) dℵ
(
sinh 1 − ω 2 ωℵ2 d dℵ
)
)e
−
x
dℵ
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 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. When ωℵ ≤ ω , the steady-state wave is given by:
u( x,t ) = u
(
sin − ω 2 ωℵ2 − 1 ( x − d ) dℵ
(
sin ω 2 ωℵ2 − 1 d dℵ
)
)e
−
x
dℵ
cos(ωt ) .
Resonance peaks occur when ω n = n 2π 2 c 2L L2 + ωℵ2 , 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.
When ωℵ ≤ ω , the steady-state oscillatory wave is given by:
u( x,t ) = u
(
) (
− 1d d ) + sin(−
ω 2 ωℵ2 − 1cos ω 2 ω ℵ2 − 1 (x − d ) dℵ + sin ω 2 ωℵ2 − 1( x − d ) dℵ
(
ω 2 ωℵ2 − 1cos − ω 2 ωℵ2
ℵ
ω 2 ωℵ2 − 1d dℵ
)
)e
The (infinite number) resonance frequencies are calculated with the relationship:
L. Rakotomanana
15
−
x
dℵ
cos(ωt ) .
Connecting mesoscopic and macroscopic scale lengths
ω 2 ω ℵ2 − 1 = tan
(
ω 2 ωℵ2 − 1d dℵ
)
For low frequency ωℵ ≥ ω , the solution reduces to:
u( x,t ) = u
(
) (
ω d d ) + sinh( −
1 − ω 2 ωℵ2 cosh 1 − ω 2 ωℵ2 ( x − d ) dℵ + sinh 1 − ω 2 ωℵ2 ( x − d ) dℵ
(
1 − ω 2 ωℵ2 cosh − 1 − ω 2
2
ℵ
ℵ
1− ω 2 ωℵ2 d dℵ
)
)e
−
x
dℵ
cos(ωt ) .
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 ζℵ0 , where ζℵ0 is 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 ) + 12 µ (ℵ0 )tr (ε 2 ) . 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 * Jℵ0 ≡ Supζ ℵ0 Jℵ0 ÷ ζℵ0 − ψ ζℵ0 . The
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:
ζℵ0 ∈∂ψ * ( Jℵ0 ) .
This evolution law expresses the nucleation and the growth of region where dense sets of microcracks 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
{ } in the interior of C , the density of micro-cracks remains constant whereas for
variables Jℵ0
those on the boundary, the density increases. Then, it is convenient to introduce the indicator
function of the set C defined by:
{ }
 0 if Jℵ ∈ C
0
IC Jℵ0 ≡ 
+∞ 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 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 the
LEGENDRE-FENCHEL transform, it is straightforward to derive:
[
( )
( )]
(ζ ) = Sup{
I *C ζℵ0 = Sup{ J }∈C Jℵ0 ÷ ζℵ0 − IC ζℵ0 .
ℵ0
( )
We then deduce the dissipation potential ψ ζℵ0 = I *C
ℵ0
}
ζℵ0 ∈C
(J
ℵ0
)
÷ ζℵ . The
dissipation potential includes a homogeneous function of degree one in terms of micro-crack
(
)
density rates ψ = Sup ζ ∈C Jℵ0 ÷ ζℵ0 . The last terms in brackets are positive and represent the
{ ℵ0 }
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 microdevices.
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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
crack length ℵ0 = 2 106 m −1 ; circular frequency ωℵ ≈ 6000 10 6 rds −1 ; fℵ ≈ 109 Hz . The
Debye frequency for this sample is f D ≈ 1013 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
9
left end with an imposed sinusoidal displacement at frequency ranging from ω = 6 10 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