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

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Connecting Mesoscopic and Macroscopic Scale Lengths for Ultrasonic Wave Characterization of Micro-Cracked Material L Connecting mesoscopic and macroscopic scale lengths Connecting mesoscopic and macroscopic scale lengths for ultrasonic wave characterization of micro-cracked material L. R. Rakotomanana Institute of Mathematics, University of Rennes I -France Abstract. Macroscopic failure of material is attributed mostly to the initial presence of micro-cracks and micro-voids and is governed by physical mechanisms at different length- scales. In order to include discontinuity mechanisms in the material deformation and its consequence on the energy dissipations during micro-crack kinetics, a theoretical model of micro-cracked continuum is derived in this paper. The model describes a micro-crack density in terms of CARTAN constants of structure and explicitly connects the macroscopic scale to the mesoscopic discontinuities. This approach contrasts to the usual method in continuum mechanics that seeks a phenomenological description by introducing an internal variable in the constitutive laws. An illustrative example of the model application is presented for the linear ultrasonic wave propagation test. The result highlights the importance of rigorously revisiting the dynamic equation in micro-cracked solid. Keywords – A micro-cracking, B inhomogeneous material, B stress waves, C nondestructive evaluation. Introduction Brittle materials as glass, ceramics and polymers always contain more or less great amount of micro-cracks and crack-like flaws (~1µm to 10µm), which are unintentionally introduced during processing or surface machining. Toughness and strength of these materials are strongly dependent of the amount and structural orientation of internal micro-cracks. Although global failure of brittle material is usually attributed to a single macroscopic crack propagation, dense sets of micro-cracks appear around the single crack, resulting from dynamic instability e.g. (Sharon and Fineberg, 1996). Creation of micro-cracks surrounding the propagating macro-crack L. Rakotomanana 1 Connecting mesoscopic and macroscopic scale lengths is thought to be responsible for limiting the crack speed to about 50% of the theoretical limit of Rayleigh surface wave speed, by dissipating energy. At the extreme, microcracking in the vicinity of macroscopic crack edge has been shown advantageous in controlling and even in arresting a single macroscopic crack propagation e.g. (Clegg, 1999). Material failure is thus simultaneously governed by different mechanisms on different length-scales. Size effects in micro-cracked material At least three length-scale levels are present to approach the micro-cracking phenomenon: macroscopic scale (~100µm), mesoscopic scale (~0.1µm to 10µm) and microscopic (atomistic) scale (~10-10 m). Until recently, the size scaling was neglected due to the early use of average stress (force per surface unit) and strain (no dimension). The limits of macroscopic approach are reached when facing the cause of true material weakness as micro-cracks. Indeed, new orientation of technology development has brought new interest in connecting mesoscopic scale to macroscopic scale beyond the macroscopic continuum description. Micro-engineering devices, electronic devices and micro-electromechanical systems, for which the entire size may be less than 10µm, may exhibit size dependence. For problems with crack lengths ranging from fraction of 1µm to 10µm, current macroscopic description misses the size effects. Connection between scale levels description is of central interest. This is particularly true for solid materials, because solids introduce a new length scale other than the lattice spacing (~1Å to 10Å), namely, the size of micro-cracks. Hierarchical modeling was suggested as one of the efficient method to connect continuum cracking, dislocation dynamics and atomic-scale simulation as molecular dynamics or lattice static e.g. (Tadmor et al., 2000). Mesoscopic scale has been proven to well connect with molecular dynamics approach by using very large-scale mechanistic simulations during crystal plasticity (100 million atoms) (Butalov et al., 1998). Very large-scale molecular dynamics also appears more and more able to bridge the atomistic scale to macroscopic experiments and description of continuum plasticity of material (10 million atoms) e.g. (Holian and Lomdahl, 1998). For more than forty years, the theory of strain gradient was proposed in various forms to bridge mesoscopic scale to macroscopic scale for elastic material deformation e.g. (Toupin, 1964) or when internal micro-slips occurred in crystal solids e.g. (Fleck and Hutchinson 1997). These theories involve strain (metric) for describing macroscopic deformation and strain gradient for mesoscopic mechanisms. Such an approach has been already intensively L. Rakotomanana 2 Connecting mesoscopic and macroscopic scale lengths discussed in the past, namely for non-local elasticity due some basic conceptual flaws e.g. (Dunn and Serrin, 1985). In fracture mechanics the bridging of phenomenological approach (macroscopic) to mesoscopic physics that governs the dynamics of micro-cracks is far from clear either experimentally or theoretically e.g. (Blumenfeld, 1998). Seek of an efficient description of material whose dimensions fall between macroscopic continuum and dislocation mechanics remains a valuable motivation for developing an intermediate scale theory. Ultrasonic techniques for micro-cracking detection An immediate application of mesoscopic scale theory would be the non-destructive testing and monitoring of micro-devices with cracks. For most materials ultrasonic techniques have been developed to characterize the internal degradation by measuring the attenuation of ultrasonic waves. Various theoretical models have been developed for explaining and predicting empirical correlation found between attenuation and the presence of micro-cracks e.g. (Vary, 1988). Basically, attenuation is a collective effects of four contributions e.g. (Prosser, 1996). Diffraction is a beam spreading that is the dominant source near the crack (wavelength is same order as crack length). Far from the crack, absorption (conversion of sound energy to heat) has an exponential relationship of attenuation with distance. Scattering is the dissipation due to geometric dispersion of wave into adjacent media or into non-homogeneity within the material itself. Velocity dispersion induces a signal loss provoked by the different velocities for different frequencies involved in the wave. Despite its importance in ultrasonic measurement, most models do not account for attenuation in the initial wave equation. It is often assumed and added ad hoc for the sake of theory fitting with the experimental results e.g. (Breazeale et al., 1981; Vandenbossche et al., 1996). Furthermore, there is currently no consensus on the form of the wave equation that governs the combined macroscopic and mesoscopic mechanisms. Numerous nonlinear ultrasonic techniques have been proposed to characterize the fatigue micro- cracking damage. Two basic nonlinear effects are usually proposed: acoustic-elastic effects (stress dependence on the attenuation) and higher harmonic generation. Wave attenuation has 2 been measured on the basis of Taylor expansion of the sound velocity cL = c0 + c1εext + c2εext +… where εext is the pre-strain level. This technique appeared to succeed in early detection of micro cracking for polymers and brittle polymers whereas failed for PVC and Nylon e.g. (Nagy, 1998). L. Rakotomanana 3 Connecting mesoscopic and macroscopic scale lengths Probably scattering losses were more important than adsorption losses for those materials. Similarly, higher harmonic generation was used to capture material degradation by assuming non-linear stress-strain law σ = Eε(1 + βε +…) e.g. (Jhang and Kim, 1999), E being the Young’s elastic modulus. The second harmonic parameter β was proposed to characterize the material degradation. However, β values augmented as the excitation frequency increased and there remained a doubt if this second harmonic parameter was an intrinsic effective parameter for the material degradation. Indeed, the stress-strain law of micro-cracked material could be linear, although with a lower modulus than intact material. In a same way, large strain theory and nonlinear stress-strain law were combined to derive nonlinear wave theory in order to measure non-homogeneous micro-cracked material. The material properties were determined on the basis of non-linear wave accounting for third-order elasticity e.g. (Ravasoo, 1999). The second-order coefficients of nonlinear terms (gradient of strain and cross terms) depended on the macroscopic LAMÉ elastic constants (λ, µ) and their spatial derivatives. Therefore these nonlinear and non- homogeneous approaches were not able to detect the presence of uniform distribution of micro- cracks with uniform macroscopic material density. Experimental analysis of wave propagation in micro-porous ceramics (pores ~1µm) showed strong attenuation and cut-off of frequency e.g. (Craciun et al., 1998). A sudden decrease in the velocity at high porosity values was shown but could not be explained in the light of existing macroscopic models of wave propagation. The ability of classical wave propagation to model very micro-porous media was then questioned and the authors assumed that the strong attenuation was due to the wave scattering from the sample geometry disorder than due to the sound adsorption mechanisms in the porous ceramics. Cut-off frequency phenomenon was also observed in macroscopic fractured material,
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