X1 Direction Research Articles

Analysis of an interfacial crack between two nonhomogeneous piezoelectric materials using a new domain-independent interaction integral

Considering the difficulties introduced by piezoelectric material non-homogeneity and discontinuities to interfacial crack analysis, a domain-independent interaction integral (DII-integral) is proposed to derive the stress intensity factors (SIFs) and electric displacement intensity factor (EDIF) of an interfacial crack between nonhomogeneous piezoelectrics. For continuously varying material properties, traditional interaction integral (I-integral) needs to compute the derivatives of material parameters with respect to the x1 direction of the local coordinate system, while the proposed DII-integral is demonstrated to be unrelated to any derivatives of material parameters. Furthermore, a significant improvement is achieved that the new formulation of I-integral remains valid even when the integration domains include intricate and multiple interfaces, as long as the material interfaces are perfectly bonded. Firstly, the intensity factors (IFs) calculated by the combination of the DII-integral and the extended finite element method are compared with the analytical solutions, and it is found that the relative errors of the IFs are all less than 1.79 % with various crack lengths, which verifies the accuracy of the proposed method. For the piezoelectric bi-material model with continuously nonhomogeneous properties, the calculated IFs match each other well (relative deviation < 1.3 %) for different sizes of integration domains. In addition, for the perfectly bonded interface with discontinuous electromechanical properties, the IFs are almost identical (relative deviation < 0.85 %) whether or not the integration domain contains the material interface, which further numerically verifies the good domain-independence of the established DII-integral. For an interfacial crack near extra interfaces, the differences in elastic stiffness and piezoelectric coefficient between interface sides have an obvious influence on mode-I SIF, but mode-II SIF and EDIF are affected mainly by piezoelectric coefficient and dielectric permittivity.

Qualitative properties of solutions for dual fractional nonlinear parabolic equations

In this paper, we consider the dual fractional parabolic problem{∂tαu(x,t)+(−Δ)su(x,t)=f(u(x,t)),inR+n×R,u(x,t)=0,in(Rn﹨R+n)×R, where R+n:={x∈Rn|x1>0} is the right half space. We prove that the positive solutions are strictly increasing in x1 direction without assuming the solutions be bounded.So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂tα and the fractional Laplacian (−Δ)s. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂tα+(−Δ)s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or boundedness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions.We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems.

Energy absorption of re-entrant honeycombs in tension and compression

Energy absorption is essential for materials/structures in engineering application. It is related to their deformation mechanisms and force–displacement (stress–strain) relationships. Re-entrant honeycombs are expected to show better energy absorption performance in tension than in compression due to a longer stroke of plastic deformation. To examine this, the work aims to systematically examine the energy absorption performance of re-entrant honeycombs under both tensile and compressive loads in the two loading directions. Analytical models for re-entrant units are established based on an idealised deformation mechanism, which are validated using Finite element (FE) analysis and the experimental result in the reference. The results demonstrate that when loaded in the X2 direction, re-entrant honeycombs are more effective in tension rather than compression in terms of energy absorption, which means that one can take the full advantage of re-entrant honeycombs in tension-dominated energy dissipation applications. However, when loaded in the other direction, the performance depends on initial cell angles of the structures and θ0 = 45° is the divide.

Ultrasonic wave propagation in randomly layered heterogeneous media

This article considers the propagation of high frequency elastic waves in a layered material. Each layer is locally anisotropic and the layer thicknesses and slowness surface orientations are modelled by a (Markovian) process. This work is important in deepening our understanding of the ultrasonic non-destructive testing of carbon fibre reinforced polymer (CFRP) composites and polycrystalline materials. The paper focuses on monochromatic shear waves propagating in two-dimensional ((x1,x3) plane) heterogeneous media. The displacement is in the x2 direction and the model focuses on the reflection and transmission of the wave at layer interfaces. The rotation of the slowness surface in each layer lies in the (x1,x2) plane and varies with the wave propagation direction (x3) only. Expressions for the local and global coefficients for the reflected and transmitted wave amplitudes are derived and shown to satisfy energy conservation. The resulting stochastic differential equations lead to a self-adjoint infinitesimal generator which can be used to produce a Fokker–Planck equation to study the probability distribution of the transmission coefficient. Explicit expressions for the moments of the probability distributions of the power transmission and reflection coefficients are then derived. The dependency of the mean and standard deviation of the power transmission coefficient on the depth of wave penetration, the localisation length, and the direction of wave propagation is then reported.

In-plane dynamics of circular cell hexagonally packaged honeycombs in two principal axes

The in-plane deformation modes, stress–strain curves and energy absorption characteristics of circular cell hexagonally packaged honeycombs with the linear elastic and plastic base material with linear strain hardening are numerically investigated under crushing velocities in two principal axes. Three deformation modes are observed and the empirical formulas for critical velocities of deformation mode transition are derived. With increasing crushing velocities, the fluctuation of stress becomes more intense in the plateau stress phase of stress–strain curves, which appears from low-crushing velocities in x2 direction, while in x1 direction at high velocities. Densification strain is linear with t/R ratio for a given crushing velocity, and becomes larger with increasing velocities approximately in a linear relationship under a deformation mode for a given t/R ratio. The law of mean plateau stress is consistent with a one-dimensional shock wave model and static plateau stress is proportional to the square of relative density. Three types of empirical expressions of mean plateau stress are derived. Crushing force efficiency is approximately independent of t/R ratio and sensitive to crushing velocity. Crushing force efficiency becomes smaller and then reduces with increasing velocities, and fluctuates around a level under dynamic mode.

Nonexistence of solutions for indefinite fractional parabolic equations

We study fractional parabolic equations with indefinite nonlinearities∂u∂t(x,t)+(−Δ)su(x,t)=x1up(x,t),(x,t)∈Rn×R, where 0<s<1 and 1<p<∞. We first prove that all positive bounded solutions are monotone increasing along the x1 direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.

Numerical Investigation into In-Plane Crushing of Tube-Reinforced Damaged 5052 Aerospace Grade Aluminum Alloy Honeycomb Panels.

This paper aims to investigate the crashworthiness performance degradation of a damaged 5052 aluminum honeycomb panels under in-plane uniaxial quasi-static compression and the possibility of improving it using reinforcement tubes. The in-plane crushing behaviors and energy absorption capacities of the intact, damaged, and tube-reinforced damaged panels with different damage sizes in both X1 and X2 directions are numerically simulated by using the nonlinear FE method Abaqus/Explicit, and the crashworthiness performances are compared with each other. The validation of finite element model involves comparing the obtained simulation results with theoretical and experimental ones. Very good agreement between numerical, experimental, and theoretical results is achieved. The first maximum compressive load and the mean crushing load of the different honeycomb configurations are analyzed and compared through the load–strain curves. The energy absorption capacity of the damaged and the tube-reinforced damaged panels is calculated and then compared with their corresponding intact ones. The deformation modes are explained in detail. The obtained results show that the crashworthiness performance degradation is directly proportional to the damage size as well as the insertion of reinforcement tubes considerably improves in-plane crushing resistance of damaged honeycomb panels.

Energy absorption characteristics of circular-celled honeycombs under in-plane quasi-static compressive loadings

Abstract The energy absorption characteristics of hexagonally packaged circular-celled honeycombs and quadrilater-ally packaged circular-celled honeycombs are obtained under in-plane quasi-static compressive loadings through finite element analysis. The stress–strain curves, deformation modes, energy absorption efficiency, specific plateau stress, normalized energy absorption and energy absorption diagrams are discussed. The cell arrangement patterns influence the shapes of stress–strain curves and deformation modes. The densification strain is in linear relationship with the relative density and the specific plateau stress is proportional to relative density. The hexagonally packaged circular-celled honeycombs have the largest specific plateau stress in the x2 direction for a given relative density. The normalized energy absorption is nearly proportional to the strain before densification and increases with increasing relative density for a given strain in one compression direction. The envelope line in the energy absorption diagram is approximately a straight line tangent to the shoulder points through the origin. The hexagonally packaged circular-celled honeycombs outperform the quadrilaterally packaged circular-celled honeycombs in in-plane energy absorption.

A Liouville's theorem for a fractional elliptic system with indefinite nonlinearities

In this paper, we consider a fractional elliptic system with indefinite nonlinearities. We prove that every positive bounded solution of the system is monotone increasing along x1 direction by using the direct method of moving planes. Based on this monotonicity, we obtain the nonexistence of positive solutions and a Liouville type results for the fractional elliptic system. As an application, we get the Liouville type results for the fractional elliptic system with cubic nonlinearities.

Nonexistence of positive solutions for an indefinite fractional [formula omitted]-Laplacian equation

In this paper, we consider the following fractional p-Laplacian equation with indefinite nonlinearities (−Δ)psu(x)=x1uq(x),x∈Rn, where 0<s<1, p=q+1>2, we use the direct method of moving planes to prove the solution of above fractional p-Laplacian equation is monotone increasing along x1 direction, then obtain that the above equation possess no positive bounded solution on the whole space.There is no decay condition at infinity on solution for the fractional p-Laplacian, and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p-Laplacian here. We also cannot find a suitable auxiliary function for this fractional p-Laplacian. To circumvent these difficulties, we use the following two estimate to study the property of solution. (I) We estimate the singular integrals defining the fractional p-Laplacian along a sequence of approximate maxima; (II) We estimate the lower and upper bound of the fractional p-Laplacian equation.

Numerical Simulation on In-plane Deformation Characteristics of Lightweight Aluminum Honeycomb under Direct and Indirect Explosion.

Lightweight aluminum honeycomb is a buffering and energy-absorbed structure against dynamic impact and explosion. Direct and indirect explosions with different equivalent explosive masses are applied to investigate the in-plane deformation characteristics and energy-absorbing distribution of aluminum honeycombs. Two finite element models of honeycombs, i.e., rigid plate-honeycomb-rigid plate (RP-H-RP) and honeycomb-rigid plate (H-RP) are created. The models indicate that there are three deformation modes in the X1 direction for the RP-H-RP, which are the overall response mode at low equivalent explosive masses, transitional response mode at medium equivalent explosive masses, and local response mode at large equivalent explosive masses, respectively. Meanwhile, the honeycombs exhibit two deformation modes in the X2 direction, i.e., the expansion mode at low equivalent explosive masses and local inner concave mode at large equivalent explosive masses, respectively. Interestingly, a counter-intuitive phenomenon is observed on the loaded boundary of the H-RP. Besides, the energy distribution and buffering capacity of different parts on the honeycomb models are discussed. In a unit cell, most of the energy is absorbed by the edges with an edge thickness of 0.04 mm while little energy is absorbed by the other bilateral edges. For the buffering capacity, the honeycomb in the X1 direction behaves better than that in the X2 direction.

Effect of micromechanical parameters of composites with wavy fibers on their effective response under large deformations

The large deformation response of composites reinforced by continuous wavy fibers is investigated using three-dimensional Finite Element Analysis. The focus is placed on in-phase fibers with circular cross-sections following sinusoidal paths. The effects of the following micromechanical parameters are analyzed – relative fiber radius, fiber crimp ratio, fiber arrangement and matrix material compressibility. In addition, the responses predicted by three-dimensional and two-dimensional plane strain models are compared. The considered composite is modeled as a fully periodic wavy unit cell subjected to periodic boundary conditions and three load cases – elongations in the x1 (longitudinal) and x2 (transverse) directions, and simple shear in the x1–x2 plane. Both constituents of the composite, the fibers and the matrix, are modeled using an isotropic hyperelastic material formulation. The results are presented as plots of macroscopic Cauchy stress components versus applied stretch (or strain in the case of the shear loading) and of fiber undulation versus applied stretch for the longitudinal elongation.

Stress evaluation of metallic material under steady state based on nonlinear critically refracted longitudinal wave

Abstract This paper presents the study of applying the nonlinear ultrasonic wave to evaluate the stress state of metallic materials under steady state. The pre-stress loading method is applied to guarantee components with steady stress. Three kinds of nonlinear ultrasonic experiments based on critically refracted longitudinal wave are conducted on components which the critically refracted longitudinal wave propagates along x, x1 and x2 direction. Experimental results indicate the second and third order relative nonlinear coefficients monotonically increase with stress, and the normalized relationship is consistent with simplified dislocation models, which indicates the experimental result is logical. The combined ultrasonic nonlinear parameter is proposed, and three stress evaluation models at x direction are established based on three ultrasonic nonlinear parameters, which the estimation error is below 5%. Then two stress detection models at x1 and x2 direction are built based on combined ultrasonic nonlinear parameter, the stress synthesis method is applied to calculate the magnitude and direction of principal stress. The results show the prediction error is within 5% and the angle deviation is within 1.5°. Therefore the nonlinear ultrasonic technique based on LCR wave could be applied to nondestructively evaluate the stress of metallic materials under steady state which the magnitude and direction are included.

Energy Absorption Capability of Composite Hexagonal Array Systems: Numerical Analysis

In this paper, in-plane dynamic crushing test of E-Glass hexagonal array systems has been carried out numerically, using LS-DYNA FE software package. Two array systems of woven E-Glass/epoxy hexagonal were studied; single column array system and multi-column array system. The volume of material used in all models was constant. The significant effects of changing arrays’ sequences and number of E-Glass layers in the hexagonal cells on the energy absorption capability are investigated. To validate the FEM, a single 45º hexagonal was modelled and crushed in X1 direction; the obtained numerical results were compared with the experimental results in terms of energy absorption capability, deformation modes and load-displacement curves, showing good agreement. Results obtained from single column array system showed that 1×4 array exhibited the higher energy absorption capability, whereas in multi-column array system, 4×3 array had the maximum value of energy absorbing compared to the whole studied models in this paper. As a conclusion, the multi-column array system has a higher energy absorption capability compared to the single column array system.

Photoelastic and acousto-optic effects in 65GeS2-25Ga2S3-10CsCl glass

The peculiarities of photoelastic and acousto-optic effects are first tested in chalcohalide 65GeS2-25Ga2S3-10CsCl glass exploring the interferometry method. The determined piezo-optic coefficients occur to be (in Br) π11=2.8±0.6, π12=6.0±0.9 and π44=−3.2±1.1. In respect to maximal acousto-optic figure of merit М2=55.6·10−15s3/kg proper for longitudinal acoustic wave propagation and polarization along X2 (і=2) direction, this glass seems more perspective for photoelastic and acousto-optic modulation in visible and near IR ranges, than many alternative candidates, such as crystalline quartz, LiNbO3, CaWO4, PbMoO4, β-BaB2O4, etc. For the first time it has been shown that piezo-optic effect of isotropic solids is anisotropic, and the degree of anisotropy is the difference of π11–π12 coefficients.

A recipe for practical full-waveform inversion in orthorhombic anisotropy

Multiparameter full-waveform inversion (FWI) usually suffers from the inherent tradeoff in the multiparameter nature of the model space. In orthorhombic anisotropy, such tradeoff is magnified by the large number of parameters involved in representing the elastic or even the acoustic approximation of such a medium. However, using a new parameterization with distinctive scattering features, we can condition FWI to invert for the parameters to which the data are sensitive at different stages, scales, and locations in the model. Specifically, with a combination made up of a velocity and particular ratios of the elastic coefficients, the scattering potential of the anisotropy parameters has stationary scattering radiation patterns as a function of the type of anisotropy. With our new parameterization, P-wave data are mainly sensitive to the scattering potential of four parameters: the horizontal velocity in the x1 direction, vh; ∊, which provides scattering mainly near zero offset in the x1-x3 vertical plane; ∊d, which is the ratio of the horizontal velocity squared in the x1 and x2 directions; and δ3 describing the anellipticity in the horizontal plane. Since, with this parameterization, the radiation pattern for the horizontal velocity and ∊ is azimuthally independent, we can perform an initial VTI inversion for these two parameters, and then use the other two parameters to fit the azimuthal variation in the data. This can be done at the reservoir level or any region of the model. Including the transmission from reflections, the migration velocity analysis (MVA) component, into the picture, multiazimuth surface seismic data are mainly sensitive to the long-wavelength components of vh, two dimensionless parameters through the diving waves, and three other dimensionless parameters in the transmission to or from reflectors (especially, in the presence of large offsets). They are also sensitive to the short-wavelength component of vh and ∊.

Sensitivity of 3D Dose Verification to Multileaf Collimator Misalignments in Stereotactic Body Radiation Therapy of Spinal Tumor.

This study aimed to detect the sensitivity of Delt 4 on ordinary field multileaf collimator misalignments, system misalignments, random misalignments, and misalignments caused by gravity of the multileaf collimator in stereotactic body radiation therapy. (1) Two field sizes, including 2.00 cm (X) × 6.00 cm (Y) and 7.00 cm (X) × 6.00 cm (Y), were set. The leaves of X1 and X2 in the multileaf collimator were simultaneously opened. (2) Three cases of stereotactic body radiation therapy of spinal tumor were used. The dose of the planning target volume was 1800 cGy with 3 fractions. The 4 types to be simulated included (1) the leaves of X1 and X2 in the multileaf collimator were simultaneously opened, (2) only X1 of the multileaf collimator and the unilateral leaf were opened, (3) the leaves of X1 and X2 in the multileaf collimator were randomly opened, and (4) gravity effect was simulated. The leaves of X1 and X2 in the multileaf collimator shifted to the same direction. The difference between the corresponding 3-dimensional dose distribution measured by Delt 4 and the dose distribution in the original plan made in the treatment planning system was analyzed with γ index criteria of 3.0 mm/3.0%, 2.5 mm/2.5%, 2.0 mm/2.0%, 2.5 mm/1.5%, and 1.0 mm/1.0%. (1) In the field size of 2.00 cm (X) × 6.00 cm (Y), the γ pass rate of the original was 100% with 2.5 mm/2.5% as the statistical standard. The pass rate decreased to 95.9% and 89.4% when the X1 and X2 directions of the multileaf collimator were opened within 0.3 and 0.5 mm, respectively. In the field size of 7.00 (X) cm × 6.00 (Y) cm with 1.5 mm/1.5% as the statistical standard, the pass rate of the original was 96.5%. After X1 and X2 of the multileaf collimator were opened within 0.3 mm, the pass rate decreased to lower than 95%. The pass rate was higher than 90% within the 3 mm opening. (2) For spinal tumor, the change in the planning target volume V18 under various modes calculated using treatment planning system was within 1%. However, the maximum dose deviation of the spinal cord was high. In the spinal cord with a gravity of -0.25 mm, the maximum dose deviation minimally changed and increased by 6.8% than that of the original. In the largest opening of 1.00 mm, the deviation increased by 47.7% than that of the original. Moreover, the pass rate of the original determined through Delt 4 was 100% with 3 mm/3% as the statistical standard. The pass rate was 97.5% in the 0.25 mm opening and higher than 95% in the 0.5 mm opening A, 0.25 mm opening A, whole gravity series, and 0.20 mm random opening. Moreover, the pass rate was higher than 90% with 2.0 mm/2.0% as the statistical standard in the original and in the 0.25 mm gravity. The difference in the pass rates was not statistically significant among the -0.25 mm gravity, 0.25 mm opening A, 0.20 mm random opening, and original as calculated using SPSS 11.0 software with P > .05. Different analysis standards of Delt 4 were analyzed in different field sizes to improve the detection sensitivity of the multileaf collimator position on the basis of 90% throughout rate. In stereotactic body radiation therapy of spinal tumor, the 2.0 mm/2.0% standard can reveal the dosimetric differences caused by the minor multileaf collimator position compared with the 3.0 mm/3.0% statistical standard. However, some position derivations of the misalignments that caused high dose amount to the spinal cord cannot be detected. However, some misalignments were not detected when a large number of multileaf collimator were administered into the spinal cord.

Physical interpretation of the J 2 integral by identifying the associated crack translation

The physical interpretation of the J integral (i.e., J1 integral) is clear. J1 integral is the energy release rate associated with crack extension (i.e., translation of the crack in the x1 direction). However, the physical interpretation of the J2 integral remains unclear. In this study, different crack translations in the x2 direction were selected and tested by calculating the energy release rates associated with the crack translations and comparing them with the theoretical value of the J2 integral.

Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion

In this paper we prove the existence of the unique global classical solution with small initial data to the Cauchy problem of a scalar conservation law with degenerate diffusion by establishing the uniform a priori decay estimates of solutions. In order to compensate the degeneracy on the x1 direction by the diffusion on other directions, we introduce the frequency decomposition method and obtain the low frequency estimate and the high frequency estimate of the solution by the Green's function method and energy method respectively.

SU‐E‐T‐100: An Approach to Improving the Dynamic Delivery Accuracy for Breast IMRT

Purpose:A method is established to improve the accuracy of the IMRT dose delivery for the treatment of breast and chest wall tumors when the higher dose gradient is delivered at the end of the leaf sequence.Methods:Dynamic MLC deliveries on Varian Linacs are achieved through the motion of the leaves from X1 to X2 direction. If the higher dose gradient is at the end of the leaf motion sequence, this can Result in an increased error in the overall dose delivery. Such errors have been observed in Lateral beams for Left‐sided and Medial beams for Right‐sided treatments. To evaluate and resolve this issue we adopted an approach where the fluence for such beams was geometrically flipped (mirrored) to treat the higher end of the dose gradient first.Results:Using this method, it was possible to deliver the optimized dose map to the area of interest still using only the X1–X2 direction of the leaf motion. The accuracy of this method was tested on different beam as part of our pre‐treatment QA program on both Varian delivery systems. With this approach we found that there was significant improvement in delivery accuracy on both 21EX and TrueBeam systems. Beams of initial Gamma index (3% and 3mm) 89–93% were increased to 98–99%. We also observed superior delivery accuracy with TB compared to the 21EXConclusion:This work demonstrate the need for a delivery sequence option from X2–X1 in situations where the MLC sequence indicates higher dose gradient component is being delivered at the end of the sequence. Results from this work can be considered in the IMRT beam optimization in the treatment planning systems. Further work will be required to establish the application of this approach in clinical setting.