X1 Direction Research Articles

Degradation analysis of grinding machine spindle systems based on complexity

In order to monitor the degradation statuses of the spindle systems of machine tool during use, a degradation analysis method based on the complexity is proposed. In this article, the workpiece spindle systems of the grinding machine tool are taken as research object. First, the vibration signals of the workpiece spindle systems in X2 direction are monthly measured from April to October. The complexity values per month of the filtered vibration signals in X2 direction are calculated by Lempel–Ziv algorithm and are used as the index to monitor the degradation statuses of the workpiece spindle systems during 6 months running. The results show that the complexity values of the measured vibration signals from April to October gradually increase, and the status of the workpiece spindle systems of the grinding machine tool has a slight degradation. Finally, in order to verify the validity of the proposed method, the degradation bearing data published by Case Western Reserve University are used to verify the feasibility, and the result shows that the complexity is effective to the degradation analysis of the spindle systems of machine tool.

On the physical interpretation of the M-integral in nonlinear elastic defect mechanics

Several properties of the M-integral are clarified within the context of nonlinear elastic defect mechanics. A material damage expansion force vector associated with the integrand of the M-integral is derived by divergence vector operation of the Lagrangian energy density moment. It is concluded that the components of damage expansion force vector are indentified as the change of potential energy due to the self-similar expansion of one infinitesimal material element along the x1 or x2 direction. Subsequently, the physical interpretation of the M-integral over the contour enclosing all defects is explored in nonlinear elasticity containing multiple defects by an explicit analysis. The explicit analysis reveals that the M-integral is inherently related to the change of the total potential energy during the formation of defects from the initial non-damaged system in addition to the nonlinear strain energy within the area enclosed by integration contour.

Erlangen pipe flow: the concept and DNS results for microflow control of near-wall turbulence

The concept of a micropatterned surface morphology capable of producing self-stabilization of turbulence in wall-bounded flows is considered in pipes of non-circular cross-sections which act to restructure fluctuations towards the limiting state where these must be entirely suppressed. Direct numerical simulations of turbulence in pipes of polygon-shaped cross-sections with straight and profiled sides were performed at a Reynolds number \(Re_\tau \simeq {\mathrm 300}\) based on the wall shear velocity and the hydraulic diameter. Using the lattice Boltzmann numerical algorithm, turbulence was resolved with up to \({\mathrm 540\times 10^6}\) grid points (\({\mathrm 8,192\times 257 \times 256}\) in the x1, x2 and x3 directions). The DNS results show a decrease in the viscous drag around corners, resulting in a reduction of the skin-friction coefficient compared with expectations based on the well-established concept of hydraulic diameter and the use of the Blasius correlation. These findings support the conjecture that turbulence might be completely suppressed if the pipe cross-section is a polygon consisting of a sufficient number of profiled sides of the same length which intersect at right angles at the corners.

Persistence of turbulent flow in microchannels at very low Reynolds numbers

A fully resolved numerical simulation of a turbulent microchannel flow, with uniformly spaced two-dimensional obstruction elements mounted at the wall and normal to the flow direction, was carried out at a very low Reynolds number of Re ≃ 970 based on the centerline velocity and the microchannel height. Employing the lattice Boltzmann numerical technique, all energetic scales of turbulence were resolved with about 19 × 106 grid points (1261 × 129 × 128 in the x 1, x 2, and x 3 directions). The simulated results confirm the self-maintenance of turbulence at such a low Reynolds number. Turbulence persisted over more than 1,000 turnover times, which was sufficiently long to prove its self-maintenance. These findings support the conjecture that turbulence developing in microchannels having rough walls can not only be initiated but also maintained at very low Reynolds numbers.

충전제 함량과 형태에 따른 PP복합체의 모듈러스 변화에 대한 실증적 연구

The mechanical properties of polypropylene (PP) composites, especially the effects of the filler shapes on the modulus were studied. The experimental results were investigated and compared with the theoretical approaches proposed by Lee and Paul and based on Eshelby's principle, which three dimensional ellipsoids were filled as filler and analyzed in terms of aspect ratio, ρα=a1/a3 and ρβ=a1/a2. The shapes of fillers were observed by SEM and aspect ratios were statistically calculated. Young's moduli in the longitudinal and transverse directions for barium sulfate whose shape was sphere (ρα=ρβ=1) had the same values, as predicted values. The modulus in the x1 direction for a glass fibers increased as the filler content increased, while the modulus in the x3 direction was increased relatively small. Furthermore, mica was also used to investigate the effects of the primary and secondary aspect ratios on the mechanical properties.

Microscopic Multi-Directional Permeability Coefficient of Human Lumbar Vertebral Cancellous Bone Tissue

The mechanical performance of cancellous bone is characterized using experimental measure of parameters (elastic moduli, permeability, etc.) which apply poroelasticity theory. Poroelastic investigations of bone are being extended to investigations of micro-mechanisms of remodeling processes and transport of metabolic product in bone. To understand physiological and pathological behavior of human skeletal system, the accurate measurement of biomechanical properties for bone is the one of important works. Particularly, the microscopic measurement is very important since the biomechanical behavior at the small scale of bone could be closely related to the remodeling of bone. But microscopic measurement of permeability coefficient which is one of important parameter of poroelasticity theory was never measured. In this study, a small scale permeability testing machine was developed to measure accurate coefficient of microscopic specimen permeability. Total of twenty one cylindrical cancellous specimens with a diameter of 500 from seven fresh male and female lumbar vertebrae (14, 39, 59, 61, 69, 75, and 77 years) were fabricated using a micro-milling machine having a resolution of 10 . To determine the directional permeability, bone coordinate was set to be x1 (longitudinal axis) and others were set as x2 (posteroanterior axis) and x3 (latero-medial axis). The measured mean permeability (± SD) values of each direction from x1 to x3 were 9 5.56 10− × (S.D. ± 10 7.10 10− × ), 9 6.66 10− × (S.D. ± 10 6.56 10− × ), and 9 7.04 10− × (S.D. ± 10 8.47 10− × ) m2/Pa/sec, respectively. Mean value in the x1 direction of specimens were the smallest (p<0.01) in comparison to x2 and x3 directions.

In-plane static and dynamic properties of aluminium honeycombs

In-plane mechanical response of aluminium honeycombs subjected to both static and dynamic loads is studied in this paper. Quasi-static compressive tests were performed on seven types of HEXCEL aluminium honeycombs with different cell size and wall thickness. The deformed layers produced “X’shaped bands in honeycomb samples when crushed in the X1 direction, while the deformed layers were almost perpendicular to the loading direction when crushed in the X2 direction. The plateau stresses in the X direction agree well with Lu and Yu’s theory, while the plateau stresses in the X2 direction agree well with Gibson and Ashby’s theory. Finite element analysis (FEA) was also conducted by using ABAQUS/STANDARD on four types of the honeycombs and a comparison was made between quasi-static experimental and FEA results. Dynamic tests on six types of aluminium honeycombs were conducted by using a gas gun and a fixture. Impact velocities varied from 5 m/s to 14 m/s. The “X” deformation mode was observed in honeycombs when crushed in the X direction at these relatively low velocities. On the other hand, the deformed layers were developed parallel to the impact plate when honeycombs were crushed in the X2 direction. The dynamic plateau stresses were compared with the static tests as well as our previous FEA results.

Numerical experiments on wall turbulence at low Reynolds number

A direct numerical simulation of a turbulent channel flow, with regularly spaced two-dimensional roughness elements mounted at the wall and perpendicular to the flow direction, was performed at a very low Reynolds number of Re
940 based on the centerline velocity and the full channel height. Using the lattice Boltzmann numerical algorithm, all essential scales were resolved with about 19·106 grid points (1155 x 129 x 128 in the x1, x2 and x3 directions). The computed results confirm the existence of turbulence at such a low Reynolds number. Turbulence persisted over the entire computation time, which was sufficiently long to prove its self-maintenance. By examination of statistical features of the flow across the anisotropy-in variant map, it was found that these coincide with conclusions emerging from the analysis of transition and breakdown to turbulence in a laminar boundary layer exposed to small, statistically stationary, neutrally stable axisymmetric disturbances with the stream wise intensity component ( u'1) lower than the intensities in the normal ( u'1) and span wise directions u'1< u'2 = u'3. To further support the concept and the results of
theoretical considerations of the laminar to turbulent transition process in wall-bounded flows using statistical techniques and to demonstrate its great potential for engineering, an additional simulation was performed of a plane channel flow with regularly spaced riblet elements mounted at the wall and aligned parallel with the flow direction. The supplementary simulation was done at a Reynolds number of Re = 6584 using about 250·106 grid points (4096 x 257 x 240). Analysis of the simulation results carried out across the flow region located in the mid plane between the riblet elements confirms the central result which lies in the root of the statistical dynamics of the velocity fluctuations in wall-bounded flows: when the velocity fluctuations close to the wall tend towards the one-component state, so that the stream wise intensity component is much larger than the intensities in the normal and span wise directions, u'1< u'2 = u'3, the turbulent dissipation rate vanishes at the wall, leading to a significant reduction of the wall shear stress. For the simulated flow case the local value of wall shear stress reduction was found to exceed the wall shear stress reduction SR
92% which corresponds to a fully developed laminar channel flow with smooth walls at the same Reynolds number.

Dynamical (harmonic) interface stress field in the half-plane covered by the prestretched layer under a strip load

Within the framework of the piecewise homogeneous body model, by employing the three-dimensional linearized theory of elastic waves in initially stressed bodies the dynamical problem of the stress distribution in a half-plane covered with a prestretched layer is investigated. It is assumed that the free face plane of the covered layer is subjected to a uniformly distributed harmonic load acting on a strip extending to infinity in the x3 direction, which is perpendicular to the x1-x2 plane and is of width 2a in the x1 direction. The plane-strain state in the x1-x2 plane is analysed. The corresponding boundary-value problems are investigated by employing the exponential Fourier integral transformation. The numerical results regarding the interface normal stress distribution are presented. The influences of the problem parameters and pre-stretching of the covered layer on this distribution are analysed. Practical engineering application fields of the results are suggested.

Modelling the deformation mechanisms, structure–property relationships and applications of auxetic nanomaterials

AbstractAnalytical and Molecular Mechanics methods have been used to study the structure and deformation mechanisms acting at the molecular level in the auxetic polymorph of crystalline silica (α‐cristobalite). The Molecular Mechanics simulations indicate a stress‐induced phase transition from α‐cristobalite to ‘ordered’ β‐cristobalite occurs for uniaxial loading along the x3 direction. This is in reasonable agreement with the previous prediction from an analytical model assuming deformation is by concurrent dilation and cooperative rotation (about axes in the x1‐x2 plane, passing through the midpoints of opposing edges – the a‐axes) of the SiO4 tetrahedral molecular sub‐units, previously shown to predict the Poisson's ratio for loading in the x3 direction. The analytical models have been extended to include cooperative rotation of each tetrahedron about its axis (the c‐axis) mostly closely aligned with the principal unit‐cell x3‐axis. The new models enable significantly improved prediction of the Poisson's ratios of α‐cristobalite when loaded in one of the transverse (x1 or x2) directions. Parametric fitting of the analytical models indicate that the deformation mechanism for transverse uniaxial loading of α‐cristobalite is by concurrent dilation and cooperative rotation about the local a and c‐axes of the SiO4 tetrahedra. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Investigations of shear free turbulent diffusion in a rotating frame

AbstractWe reconsider the problem of shear free turbulent diffusion in a rotating frame, rotating about x1. Shear free turbulence is generated at a vibrating grid in the x2–x3 plane and diffuses away from the grid in x1 direction. An important property of this flow case is that there is no mean flow‐velocity. With the help of Lie‐group methods Reynolds‐stress transport models can be analyzed for this kind of flow in a rotating frame. From the analysis it can be found, that the turbulent diffusion only influences a finite domain. Implicating this solution in the model equations shows that even fully nonlinear Reynolds‐stress transport models (non‐linear in the Reynolds‐stresses for the pressure‐strain model) are insensitive to rotation for this type of flow. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The role of pressure in rubber elasticity.

We describe a series of molecular dynamics computations that reveal an intimate connection at the atomic scale between difference stress (which resists stretches) and pressure (which resists volume changes) in an idealized elastomer, in contrast to the classical theory of rubber elasticity. Our simulations idealize the elastomer as a "pearl necklace," in which the covalent bonds are stiff linear springs, while nonbonded atoms interact through a Lennard-Jones potential with energy epsilon(LJ) and radius sigma(LJ). We calculate the difference stress t(11)-(t(22)+t(33))/2 and mean stress (t(11)+t(22)+t(33))/3 induced by a constant volume extension in the x(1) direction, as a function of temperature T and reduced density rho(*)=Nsigma(IJ) (3)/nu. Here, N is the number of atoms in the simulation cell and nu is the cell volume. Results show that for rho(*)<1, the difference stress is purely entropic and is in good agreement with the classical affine network model of rubber elasticity, which neglects nonbonded interactions. However, data presented by van Krevelen [Properties of Polymers, 3rd ed. (Elsevier, Amsterdam, 1990), p. 79] indicate that rubber at standard conditions corresponds to rho(*)=1.2. For rho(*)>1, the system is entropic for kT/epsilon(LJ)>2, but at lower temperatures the difference stress contains an additional energy component, which increases as rho(*) increases and temperature decreases. Finally, the model exhibits a glass transition for rho(*)=1.2 and kT/epsilon(LJ) approximately 2. The atomic-scale processes responsible for generating stress are explored in detail. Simulations demonstrate that the repulsive portion of the Lennard-Jones potential provides a contribution sigma(nbr)>0 to the difference stress, the attractive portion provides sigma(nba) approximately 0, while the covalent bonds provide sigma(b)<0. In contrast, their respective contributions to the mean stress satisfy Pi(nbr)<0, Pi(nba)>0, and Pi(b)<0. Analytical calculations, together with simulations, demonstrate that mean and difference stresses are related by sigma(nbr)=-APi(nbr)P(2)(theta(b)), sigma(b)=BPi(b)P(2)(theta(b)), where P(2)(theta(b)) is a measure of the anisotropy of the orientation of the covalent bonds, and A and B are coefficients that depend weakly on rho(*) and temperature. For high values of rho(*), we find that [sigma(nbr)]>>[sigma(b)], and in this regime our model predicts behavior that is in good agreement with experimental data of D.L. Quested et al. [J. Appl. Phys. 52, 5977 (1981)] for the influence of pressure on the difference stress induced by stretching solithane.

The central role of excluded volume in the production of anisotropic stress in deformed polymer melts

Nonequilibrium molecular dynamics computations are used to calculate the stress relaxation in an idealized “pearl necklace” polymer melt. The covalent bonds in the melt are approximated by stiff linear springs with potential ub(r), while nonbonded atoms interact through a Lennard-Jones potential unb(r), which includes both a short-range repulsive portion, and a long-range attractive tail. The simulations are used to calculate the relaxation of difference stress t11−(t22+t33)/2 in the melt following a constant volume extension in the x1 direction. Results show that the dominant contribution to the difference stress arises from excluded volume interactions, through the repulsive portion of the Lennard-Jones potential, while the attractive tail gives rise to only a small portion of the difference stress. In contrast, attractive nonbonded interactions provide a dominant contribution to the mean stress. This behavior occurs because the difference stress is generated by deformation induced anisotropy in the orientation of nonbonded interactions acting on a representative atom in the melt. The anisotropy is localized to a small region surrounding the atom. The anisotropy in the orientation of these nonbonded pairs is generated by steric shielding: the deformation tends to orient the chain bonds parallel to the stretch direction, and so orients nonbonded pairs transverse to the stretch direction.

Stress relaxation in a diatomic liquid

By the use of nonequilibrium molecular dynamics we have studied the stress relaxation following imposition of a constant-volume elongation in the x1 direction on a model diatomic liquid. Three consecutive modes of relaxation of the stress difference τ=t11−12(t22+t33) are found, each governed by exponentials e−αit with α1&amp;gt;α2&amp;gt;α3. Each mode is shown to correspond to the return to isotropy of a different characteristic of the liquid structure that has been rendered anisotropic by the deformation, namely (1) r̄(θ), the mean distance from a generic atom of interacting atoms in the θ direction, with the angle θ measured from the stretch axis x1; (2) N̄(θ), the mean number density of interacting atoms in the θ direction; and (3) 〈〈P2(θb)〉〉, a measure of the orientation θb of molecular axes with respect to x1. The first two modes are identical in form to those studied previously [R. C. Picu and J. H. Weiner, J. Chem. Phys. 107, 7214 (1997)] for stress relaxation in a monatomic liquid, but their rates of decay differ because of differences in packing and in atomic mobility. During the third mode of relaxation it is found that τ=C〈〈P2(θb)〉〉, where C may be understood in terms of the intrinsic stress tensor, a tensor referred to a coordinate system fixed to the molecule [J. Gao and J. H. Weiner, J. Chem. Phys. 90, 6749 (1989)]. The relevance of these results to stress relaxation in polymer melts is discussed.

Aspects of the Failure and Postfailure of Fiber Composites in Compression

In this paper the analysis of the compressive failure of an AS4/PEEK composite started in [10] is continued. The compressive strength of this material is typically 60% of its tensile strength. It has been long suspected that this is related to fiber misalignment. Two types of fiber imperfections were found in the material analyzed: imperfections introduced during the manufacturing of the prepreg, and fiber waviness introduced during the lamination and final curing of the composite. Failure due to uniaxial compression takes the form of kink bands of distinct widths and inclinations. Kink bands were preserved by conducting the experiments on circular cross section cylindrical rods which were radially confined. Post-failure examination of the kink bands showed them to be at angles ranging from 11°-16° to the x2 direction of the material. Furthermore, all kink planes were found to have normals in the two orthogonal planes containing the imperfections. The problem was idealized as a two-dimensional solid with alternating layers of matrix and fibers. The models account for material and geometric nonlinearities and the two types of fiber imperfections seen in this composite. The response is characterized by a load maximum which represents the strength of the material. The strength is strongly influenced by the imperfection wavelength, amplitude and spatial distribution. The load maximum is followed by fiber bending confined to narrow inclined zones. The fibers eventually break into distinct kink bands because of excessive bending. The inclinations and widths of the highly deformed zones just before fiber breaking is expected to start were found to be similar to those of the kink bands measured in the experiments.

Aspects of the Failure and Postfailure of Fiber Composites in Compression

In this paper the analysis of the compressive failure of an AS4/PEEK composite started in [10] is continued. The compressive strength of this material is typically 60% of its tensile strength. It has been long suspected that this is related to fiber misalignment. Two types of fiber imperfections were found in the material analyzed: imperfections introduced during the manufacturing of the prepreg, and fiber waviness introduced during the lamination and final curing of the composite. Failure due to uniaxial compression takes the form of kink bands of distinct widths and inclinations. Kink bands were preserved by conducting the experiments on circular cross section cylindrical rods which were radially confined. Post-failure examination of the kink bands showed them to be at angles ranging from 11°-16° to the x2 direction of the material. Furthermore, all kink planes were found to have normals in the two orthogonal planes containing the imperfections. The problem was idealized as a two-dimensional solid with alternating layers of matrix and fibers. The models account for material and geometric nonlinearities and the two types of fiber imperfections seen in this composite. The response is characterized by a load maximum which represents the strength of the material. The strength is strongly influenced by the imperfection wavelength, amplitude and spatial distribution. The load maximum is followed by fiber bending confined to narrow inclined zones. The fibers eventually break into distinct kink bands because of excessive bending. The inclinations and widths of the highly deformed zones just before fiber breaking is expected to start were found to be similar to those of the kink bands measured in the experiments.

Thickness-shear and flexural vibrations of contoured crystal strip resonators

A system of two-dimensional equations of motion of successively higher-order approximations for contoured crystal plates are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions of the thickness coordinate of the plates. By removing the first-order thickness-stretch mode and the first-order x2x3 (or fast) thickness-shear mode and all the higher ones, a set of first-order equations of motion is obtained for contoured crystal plates and for frequencies up to and including those of the fundamental x1x2 (or slow) thickness-shear mode. The coupled thickness-shear and flexural vibrations are studied for contoured quartz strip resonators having the shape of (a) a double wedge and (b) a plate with beveled edges. Exact solutions in terms of infinite power series are obtained by Frobenius method for the plates with linearly varying thickness in the x1 direction. Frequency spectra and mode shapes are computed for both types of resonators. The effects of the contour on the frequencies and modes are examined.

Homogenization for thermoconductivity in a porous medium with periods of different orders in the different directions

We study here the of heat transfer in a porous medium consisting of a homogeneous isotropic material. We assume that the period of the medium is not the same in the two directions and that the ratio of these two periods tends to zero. We find the explicit representation of the coefficients of the averaged equations and prove strong convergence to the solution of the averaged problem. 1. Statement of the and simplest estimates By averaging processes in periodic structures appears the to solve the so-called cell problem (Sanchez-Palencia [7, 8], Bakhvalov [1], Bensoussan et al. [3]) in heterogeneous media. Except in the one-dimensional case this cannot be solved in exact form and for solving it we need to use numerical methods. The same arises in a porous medium with periodic holes (Cioranescu and Saint Jean Paulin [4]). In the case when the magnitude of the period is not the same in all the directions, some results are already known. For a heterogeneous material, the averaged equations in explicit form are obtained by Dubinskaya [6] who gives a whole asymptotic expansion without boundary conditons, when the coefficients are smooth. In a porous tall structure with rectangular holes, the limit equation and convergence results are obtained by Cioranescu and Saint Jean Paulin [5] using variational methods. In the present paper, we consider the thermoconductivity case in a two-dimensional isotropic porous medium with symmetric holes. The period is El in the Xl direction and E2 in the X2 direction, with E2 « El « 1. The perforations in the porous material are obtained from a model perforation KO 'The work of N.S. Bakhvalov was partially supported by the Russian Foundation for basic research, 93-01-01729, and most of it was done while he enjoyed the hospitality of the Mathematics Department of the University of Metz. 0921-7134/96/$8.00 © 1996 Elsevier Science B.Y. All rights reserved 254 N.S. Bakhvalov and J. Saint Jean Paulin / Homogenization for thermoconductivity ~Yl=1/2 (0,1) f----~' ___ ----,

Guided waves in a fluid–solid bilayered plate

Consider a fluid–solid bilayer of infinite extent with the x3 axis perpendicular to the bilayer and the material axes coinciding with the x1, x2, and x3 axes. The fluid is assumed to be elastic (and isotropic) while the solid is assumed to be elastic and orthotropic (i.e., with nine independent elastic constants). Consider a plane wave propagating in the x1 direction with the particle motion confined to the x1‐x3 plane and plane strain conditions in the x2 direction. There are six boundary conditions: zero normal and shear stresses at the traction‐free boundaries, zero shear stress and continuity of normal displacements, and normal stresses at the interface. The dispersion equation is obtained by setting the determinant of the resulting six‐by‐six matrix to zero. Numerical results in the form of phase velocity, group velocity, and mode shapes are presented for the case of a water/graphite‐epoxy bilayer. Several interesting features are observed and discussed. The physics underlying these phenomena has been explored by studying the mode shapes at (and in the vicinity of) the frequencies of interest. [Work supported by Texas Advanced Technology Program.]

Propagation of shear horizontal surface acoustic waves parallel to the grooves of a random grating

An elastic medium that occupies the region x3≳ζ(x1), where the surface-profile function ζ(x1) is assumed to be a stationary, stochastic process, is studied. The displacement field u(x;t) in this region is assumed to satisfy stress-free boundary conditions on the surface x3=ζ(x1). First the boundary conditions satisfied by the mean displacement field 〈u(x;t)〉 on the plane x3=0 are obtained, where the angle brackets 〈 〉 denote an average over the ensemble of realizations of ζ(x1). The results are then used to obtain the dispersion relation for surface acoustic waves of shear horizontal polarization propagating in the x2 direction, i.e., parallel to the grooves of the random grating defined by the surface-profile function ζ(x1). It is found that the surface acoustic waves of this polarization, which cannot exist on a planar, stress-free surface of an elastic medium, are trapped by the random roughness, but are bound to the surface for only a finite range of values of the wave number k2 characterizing their propagation in the x2 direction. The attenuation of these roughness-trapped surface acoustic waves is also determined.