Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

E-Cervix imaging displaying cervical elasticity characteristics of healthy adult nulliparous women at different age groups and different menstrual cycle stages

Objective To observe the cervical elasticity of healthy adult nulliparous women at different age groups and different stages of menstrual cycle with E-Cervix imaging technology.Methods A total of 218 healthy adult nulliparous women who underwent transvaginal ultrasound examination for routine physical examination were retrospectively enrolled,including 103 in follicular phase,78 in ovulation phase and 37 in luteal phase.Cervical canal length(CL)and E-Cervix elasticity parameters were compared among different age groups and different stages of menstrual cycle,including elasticity contrast index(ECI),hardness ratio(HR),cervical internal and external orifice strain values(IOS and EOS)and IOS/EOS ratio.Results No significant difference of CL nor cervical elasticity parameters was detected among healthy adult nulliparous women at different age groups(all P>0.05).There were significant differences of ECI,HR and IOS among different menstrual cycle stages(all P<0.05),among which women in follicular phase had higher ECI and IOS but lower HR than those in luteal phase(all P<0.05).Conclusion No significant difference of cervical elasticity existed among healthy adult nulliparous women at different age groups.Meanwhile,cervical elasticity of healthy adult nulliparous women changed during menstrual cycle,in follicular phase had higher ECI and IOS but lower HR than in luteal phase.

Monitoring the change in horizontal stress with multi-wave time-lapse seismic response based on nonlinear elasticity theory

Monitoring the change in horizontal stress from the geophysical data is a tough challenge, and it has a crucial impact on broad practical scenarios which involve reservoir exploration and development, carbon dioxide (CO_(2)) injection and storage, shallow surface prospecting and deep-earth structure description. The change in in-situ stress induced by hydrocarbon production and localized tectonic movements causes the changes in rock mechanic properties (e.g. wave velocities, density and anisotropy) and further causes the changes in seismic amplitudes, phases and travel times. In this study, the nonlinear elasticity theory that regards the rock skeleton (solid phase) and pore fluid as an effective whole is used to characterize the effect of horizontal principal stress on rock overall elastic properties and the stress-dependent anisotropy parameters are therefore formulated. Then the approximate P-wave, SV-wave and SH-wave angle-dependent reflection coefficient equations for the horizontal-stress-induced anisotropic media are proposed. It is shown that, on the different reflectors, the stress-induced relative changes in reflectivities (i.e., relative difference) of elastic parameters (i.e., P- and S-wave velocities and density) are much less than the changes in contrasts of anisotropy parameters. Therefore, the effects of stress change on the reflectivities of three elastic parameters are reasonably neglected to further propose an AVO inversion approach incorporating P-, SH- and SV-wave information to estimate the change in horizontal principal stress from the corresponding time-lapse seismic data. Compared with the existing methods, our method eliminates the need for man-made rock-physical or fitting parameters, providing more stable predictive power. 1D test illustrates that the estimated result from time-lapse P-wave reflection data shows the most reasonable agreement with the real model, while the estimated result from SH-wave reflection data shows the largest bias. 2D test illustrates the feasibility of the proposed inversion method for estimating the change in horizontal stress from P-wave time-lapse seismic data.

Ni-P-SBR composite-electroless-plating enables Si anode with high conductivity and elasticity for high performance Li-ion batteries application

Silica-based anode is widely employed for high energy density Li-ion batteries owing to their high theoretical specific capacity(4200 m A h g-1).However,it is always accompanied by a huge volume expansion(300%)and shrinks during the lithiation/delithiation process,further leading to low cycle stability.Efforts to mitigate the adverse effects caused by volume expansion such as robust binder matrix,Coreshell structure,etc.,inevitably affect the electronic conductivity within the electrode.Herein,a high conductivity and elasticity Si anode(Ni-P-SBR(styrene-butadiene rubber)@Si)was designed and fabricated via the Ni-P-SBR composite-electroless-plating process.In this design,the Si particles are surrounded by SBR polymer and Ni particles,where the SBR can adapt to the volume change and Ni particles can provide the electrode with high electronic conductivity.Therefore,the Ni-P-SBR@Si delivers a high initial capacity of 3470 m A h g-1and presents capacity retention of 49.4%within 200 cycles at 600 m A g-1.Additionally,a high capacity of 1153 m A h g-1can be achieved at 2000 m A g-1and can be cycled stably under bending conditions.This strategy provides feasible ideas to solve the key issues that limit the practical application of Si anodes.

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.

Data envelopment analysis for scale elasticity measurement in the stochastic case:with an application to Indian banking

In the nonparametric data envelopment analysis literature,scale elasticity is evaluated in two alternative ways:using either the technical efficiency model or the cost efficiency model.This evaluation becomes problematic in several situations,for example(a)when input proportions change in the long run,(b)when inputs are heterogeneous,and(c)when firms face ex-ante price uncertainty in making their production decisions.To address these situations,a scale elasticity evaluation was performed using a value-based cost efficiency model.However,this alternative value-based scale elasticity evaluation is sensitive to the uncertainty and variability underlying input and output data.Therefore,in this study,we introduce a stochastic cost-efficiency model based on chance-constrained programming to develop a value-based measure of the scale elasticity of firms facing data uncertainty.An illustrative empirical application to the Indian banking industry comprising 71 banks for eight years(1998–2005)was made to compare inferences about their efficiency and scale properties.The key findings are as follows:First,both the deterministic model and our proposed stochastic model yield distinctly different results concerning the efficiency and scale elasticity scores at various tolerance levels of chance constraints.However,both models yield the same results at a tolerance level of 0.5,implying that the deterministic model is a special case of the stochastic model in that it reveals the same efficiency and returns to scale characterizations of banks.Second,the stochastic model generates higher efficiency scores for inefficient banks than its deterministic counterpart.Third,public banks exhibit higher efficiency than private and foreign banks.Finally,public and old private banks mostly exhibit either decreasing or constant returns to scale,whereas foreign and new private banks experience either increasing or decreasing returns to scale.Although the application of our proposed stochastic model is illustrative,it can be potentially applied to all firms in the information and distribution-intensive industry with high fixed costs,which have ample potential for reaping scale and scope benefits.

Reconfigurable mechanism-based metamaterials for ternary-coded elastic wave polarizers and programmable refraction control

Elastic metamaterials with unusual elastic properties offer unprecedented ways to modulate the polarization and propagation of elastic waves.However,most of them rely on the resonant structural components,and thus are frequency-dependent and unchangeable.Here,we present a reconfigurable 2D mechanism-based metamaterial which possesses transformable and frequency-independent elastic properties.Based on the proposed mechanism-based metamaterial,interesting functionalities,such as ternarycoded elastic wave polarizer and programmable refraction,are demonstrated.Particularly,unique ternary-coded polarizers,with 1-trit polarization filtering and 2-trit polarization separating of longitudinal and transverse waves,are first achieved.Then,the strong anisotropy of the proposed metamaterial is harnessed to realize positive-negative bi-refraction,only-positive refraction,and only-negative refraction.Finally,the wave functions with detailed microstructures are numerically verified.

Hepatic Elastometry in the Management of Hepatitis B at National Hospital of Niamey

Introduction: Viral hepatitis B is a public health problem in Niger which is classified as a high endemicity area with a prevalence ranging from 8 to 17% depending on the studies [1] and that of HBV-related cirrhosis is about 40.26% in 2024. The decision to treat is based on a combination of three parameters: viral load, ALT values and the degree of hepatic fibrosis [2]. The latter is assessed by hepatic elastography (Fibroscan), which is a decisive factor in treatment. In Niger, until 2024, the decision to treat or not to treat a patient with HBV was based on the determination of viral load B and transaminases, and no work evaluating the contribution of this third element, liver elasticity, has been done, hence the interest of our study. Objective: To study the contribution of Fibroscan in measuring hepatic elasticity in the management of patients with HBV. Methodology: This was a prospective descriptive study conducted from January 05 to November 30, i.e. a period of 11 months, on clinically asymptomatic HBsAg-positive patients who had undergone FibroScan liver elasticity measurement. The examination was carried out by a hepatogastroenterologist who had received training in the Fibroscan. The median of ten measures of liver elasticity at the same point with an IQR of less than 30% was considered the valid measure and no or minimal fibrosis was defined as a value ˂7 Kpa, moderate fibrosis as a value between 7 and 10 kpa, severe fibrosis as a value greater than 10 Kpa, and the existence of cirrhosis as a value greater than 14 Kpa were analyzed using SPSS version 20 software. Results: Out of 398 patients monitored for HBV, 60 cases met the inclusion criteria, i.e., a frequency of 15.07%. The mean age of the patients was 35.63 years, with extremes of 18 and 70 years. They were predominantly male, with a sex ratio of 3.2. Married patients accounted for 61.67% (n = 37). Jaundice was absent in 91.67% (n = 55). The circumstances of discovery of HBV were the routine health check-up, followed by blood donation with 50% and 46.67%, respectively. The viral load was >2000 UI/ml in 32.7% (n = 17). HBeAg was negative in 93.33% of cases (n = 56). ALT levels were normal in 47 patients (78.33%). Mean liver elasticity was 6.7 KPa. Fibrosis was classified as F0 - F1 in 75% (n = 45), F1 - F2 in 18.33% (n = 11) and F3 - F4 in 6.67% (n = 4) of patients. There was no significant relationship between viremia value, liver activity, degree of fibrosis and quantitative HBsAg. Conclusion: Measurement of hepatic elasticity has made it possible to diagnose cases of compensated cirrhosis and significant fibrosis in patients considered to be inactive carriers (viral load ˂2000 IU/ml and normal transaminases) in asymptomatic HBV+ patients. This made it possible to put these patients on Tenofovir in order to avoid decompensation for the first group and for the second the progression to cirrhosis. It is an excellent tool to aid in the decision to start treatment.

Publisher Correction to:Highly Elastic,Bioresorbable Polymeric Materials for Stretchable,Transient Electronic Systems

Due to typesetting mistake,Hanul Min was missed to be denoted as a corresponding author in the article.The type-setter apologizes for this.The original article has been corrected.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,which permits use,sharing,adaptation,distribution and reproduction in any medium or format,as long as you give appropriate credit to the original author(s)and the source,provide a link to the Creative Commons licence,and indicate if changes were made.The images or other third party material in this article are included in the article’s Creative Commons licence,unless indicated otherwise in a credit line to the material.If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use,you will need to obtain permission directly from the copyright holder.

A novel relationship between elastic modulus and void ratio associated with principal stress for coral calcareous sand

Elastic moduli,e.g.shear modulus G and bulk modulus K,are important parameters of geotechnical materials,which are not only the indices for the evaluation of the deformation ability of soils but also the important basic parameters for the development of the constitutive models of geotechnical materials.In this study,a series of triaxial loading-unloading-reloading shear tests and isotropic loading-unloadingreloading tests are conducted to study several typical mechanical properties of coral calcareous sand(CCS),and the void ratio evolution during loading,unloading and reloading.The test results show that the stress-strain curves during multiple unloading processes are almost parallel,and their slopes are much greater than the deformation modulus at the initial stage of loading.The relationship between the confining pressure and the volumetric strain can be defined approximately by a hyperbolic equation under the condition of monotonic loading of confining pressure.Under the condition of confining pressure unloading,the evolution of void ratio is linear in the e-lnp0 plane,and these lines are a series of almost parallel lines if there are multiple processes of unloading.Based on the experimental results,it is found that the modified Hardin formulae for the elastic modulus estimation have a significant deviation from the tested values for CCS.Based on the experimental results,it is proposed that the elastic modulus of soils should be determined by the intersection line of two spatial surfaces in the G/K-e-p’/pa space(pa:atmosphere pressure).“Ye formulation”is further proposed for the estimation of the elastic modulus of CCS.This new estimation formulation for soil elastic modulus would provide a new method to accurately describe the mechanical behavior of granular soils.

Theoretical analysis of the elastic Kelvin-Helmholtz instability in explosive weldings

By considering the joint effects of the Kelvin-Helmholtz(KH) and Rayleigh-Taylor(RT) instabilities, this paper presents an interpretation of the wavy patterns that occur in explosive welding. It is assumed that the elasticity of the material at the interface effectively determines the wavelength, because explosive welding is basically a solid-state welding process. To this end, an analytical model of elastic hydrodynamic instabilities is proposed, and the most unstable mode is selected in the solid phase. Similar approaches have been widely used to study the interfacial behavior of solid metals in high-energy-density physics. By comparing the experimental and theoretical results, it is concluded that thermal softening,which significantly reduces the shear modulus, is necessary and sufficient for successful welding. The thermal softening is verified by theoretical analysis of the increase in temperature due to the impacting and sliding of the flyer and base plates, and some experimental observations are qualitatively validated.In summary, the combined effect of the KH and RT instabilities in solids determines the wavy morphology, and our theoretical results are in good qualitative agreement with experimental and numerical observations.

Determining the effective elastic thickness through cross-correlation between isostatic disturbances

The elastic thickness parameter was estimated using the mobile correlation technique between the observed isostatic disturbance and the gravity disturbance calculated through direct gravimetric modeling. We computed the vertical flexure value of the crust for a specific elastic thickness using a given topographic dataset. The gravity disturbance due to the topography was determined after the calculation. A grid of values for the elastic thickness parameter was generated. Then, a moving correlation was performed between the observed gravity data(representing actual surface data) and the calculated data from the forward modeling. The optimum elastic thickness of the particular point corresponded to the highest correlation coefficient. The methodology was tested on synthetic data and showed that the synthetic depth closely matched the original depth, including the elastic thickness value. To validate the results, the described procedure was applied to a real dataset from the Barreirinhas Basin, situated in the northeastern region of Brazil. The results show that the obtained crustal depth is highly correlated with the depth from known models. Additionally, we noted that the elastic thickness behaves as expected, decreasing from the continent towards the ocean. Based on the results, this method has the potential to be employed as a direct estimate of crustal depth and elastic thickness for any region.

Linearized waveform inversion for vertical transversely isotropic elastic media:Methodology and multi-parameter crosstalk analysis

Seismic migration and inversion are closely related techniques to portray subsurface images and identify hydrocarbon reservoirs.Seismic migration aims at obtaining structural images of subsurface geologic discontinuities.More specifically,seismic migration estimates the reflectivity function(stacked average reflectivity or pre-stack angle-dependent reflectivity)from seismic reflection data.On the other hand,seismic inversion quantitatively estimates the intrinsic rock properties of subsurface formulations.Such seismic inversion methods are applicable to detect hydrocarbon reservoirs that may exhibit lateral variations in the inverted parameters.Although there exist many differences,pre-stack seismic migration is similar with the first iteration of the general linearized seismic inversion.Usually,seismic migration and inversion techniques assume an acoustic or isotropic elastic medium.Unconventional reservoirs such as shale and tight sand formation have notable anisotropic property.We present a linearized waveform inversion(LWI)scheme for weakly anisotropic elastic media with vertical transversely isotropic(VTI)symmetry.It is based on two-way anisotropic elastic wave equation and simultaneously inverts for the localized perturbations(ΔVp_(0)/Vp_(0)/Vs_(0)/Vs_(0)/,Δ∈,Δδ)from the long-wavelength reference model.Our proposed VTI-elastic LWI is an iterative method that requires a forward and an adjoint operator acting on vectors in each iteration.We derive the forward Born approximation operator by perturbation theory and adjoint operator via adjoint-state method.The inversion has improved the quality of the images and reduces the multi-parameter crosstalk comparing with the adjoint-based images.We have observed that the multi-parameter crosstalk problem is more prominent in the inversion images for Thomsen anisotropy parameters.Especially,the Thomsen parameter is the most difficult to resolve.We also analyze the multi-parameter crosstalk using scattering radiation patterns.The linearized waveform inversion for VTI-elastic media presented in this article provides quantitative information of the rock properties that has the potential to help identify hydrocarbon reservoirs.

Microwave-induced thermoacoustic elastic imaging:A simulation study

Microwave-induced thermoacoustic imaging(MTI)has the advantages of high resolution,high contrast,non-ionization,and non-invasive.Recently,MTI was used in the¯eld of breast cancer screening.In this paper,based on the¯nite element method(FEM)and COMSOL Multiphysics software,a three-dimensional breast cancer model suitable for exploring the MTI process is proposed to investigate the in°uence of Young's modulus(YM)of breast cancer tissue on MTI.It is found that the process of electromagnetic heating and initial pressure generation of the entire breast tissue is earlier in time than the thermal expansion process.Besides,compared with normal breast tissue,tumor tissue has a greater temperature rise,displacement,and pressure rise.In particular,YM of the tumor is related to the speed of thermal expansion.In particular,the larger the YM of the tumor is,the higher the heating and contraction frequency is,and the greater the maximum pressure is.Di®erent Young's moduli correspond to di®erent thermoacoustic signal spectra.In MTI,this study can be used to judge di®erent degrees of breast cancer based on elastic imaging.In addition,this study is helpful in exploring the possibility of microwave-induced thermoacoustic elastic imaging(MTAE).

On Klein tunneling of low-frequency elastic waves in hexagonal topological plates

Incident particles in the Klein tunnel phenomenon in quantum mechanics can pass a very high potential barrier.Introducing the concept of tunneling into the analysis of phononic crystals can broaden the application prospects.In this study,the structure of the unit cell is designed,and the low frequency(<1 k Hz)valley locked waveguide is realized through the creation of a phononic crystal plate with a topological phase transition interface.The defect immunity of the topological waveguide is verified,that is,the wave can propagate along the original path in the cases of impurities and disorder.Then,the tunneling phenomenon is introduced into the topological valley-locked waveguide to analyze the wave propagation,and its potential applications(such as signal separators and logic gates)are further explored by designing phononic crystal plates.This research has broad application prospects in information processing and vibration control,and potential applications in other directions are also worth exploring.

Acoustic radiation force on a cylindrical composite particle with an elastic thin shell and an internal eccentric liquid column in a plane ultrasonic wave field

A model with three-layer structure is introduced to explore the acoustic radiation force(ARF)on composite particles with an elastic thin shell.Combing acoustic scattering of cylinder and the thin-shell theorem,the ARF expression was derived,and the longitudinal and transverse components of the force and axial torque for an eccentric liquid-filled composite particle was obtained.It was found that many factors,such as medium properties,acoustic parameters,eccentricity,and radius ratio of the inner liquid column,affect the acoustic scattering field of the particle,which in turn changes the forces and torque.The acoustic response varies with the particle structures,so the resonance peaks of the force function and torque shift with the eccentricity and radii ratio of particle.The acoustic response of the particle is enhanced and exhibits higher force values due to the presence of the elastic thin shell and the coupling effect with the eccentricity of the internal liquid column.The decrease of the inner liquid density may suppress the high-order resonance peaks,and internal fluid column has less effects on the change in force on composite particle at ka>3,while limited differences exist at ka<3.The axial torque on particles due to geometric asymmetry is closely related to ka and the eccentricity.The distribution of positive and negative force and torque along the axis ka exhibits that composite particle can be manipulated or separated by ultrasound.Our theoretical analysis can provide support for the acoustic manipulation,sorting,and targeting of inhomogeneous particles.

Research on the flow stability and noise reduction characteristics of quasi-periodic elastic support skin

To enhance flow stability and reduce hydrodynamic noise caused by fluctuating pressure,a quasiperiodic elastic support skin composed of flexible walls and elastic support elements is proposed for fluid noise reduction.The arrangement of the elastic support element is determined by the equivalent periodic distance and quasi-periodic coefficient.In this paper,a dynamic model of skin in a fluid environment is established.The influence of equivalent periodic distance and quasi-periodic coefficient on flow stability is investigated.The results suggest that arranging the elastic support elements in accordance with the quasi-periodic law can effectively enhance flow stability.Meanwhile,the hydrodynamic noise calculation results demonstrate that the skin exhibits excellent noise reduction performance,with reductions of 10 dB in the streamwise direction,11 dB in the spanwise direction,and 10 dB in the normal direction.The results also demonstrate that the stability analysis method can serve as a diagnostic tool for flow fields and guide the design of noise reduction structures.

Real-time Shear Wave Elastography Assessment of Muscle Elasticity in Patients with Renal Failure

Objective:To explore the value of real-time elastic shear wave in evaluating muscle elasticity in patients with renal failure.Methods:50 patients with chronic renal failure from January 2019 to December 2022 were randomly selected as the experimental group,and 50 healthy patients aged 21-61 during the same period were selected as the control group,and the basic information of the patients,including age,gender,body mass index,etc.,were collected.Besides,the Young's modulus of the two groups of patients were observed and analyzed.Results:The Young's modulus values of left and right gastrocnemius muscles in the experimental group were significantly lower than those in the control group(P<0.05),and there was no statistical difference between the left and right sides of the experimental group and the control group(P>0.05).Conclusion:Real-time shear wave elastography provides a non-invasive,real-time and effective tool for the assessment of muscle elasticity in patients with renal failure.Through further research and optimization,real-time shear wave elastography will play a greater role in the prevention and treatment of patients with renal failure,improving the quality of life and prognosis of patients.

Dynamic characteristics of multi-span spinning beams with elastic constraints under an axial compressive force

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier series,which ensures the continuity of the derivative at the boundary and enhances the convergence.The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation.The efficiency and accuracy of the present method are validated in comparison with the finite element method(FEM)and other methods.The effects of the boundary spring stiffness,the number of spans,the spinning velocity,and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied.The results show that the present method can freely simulate any boundary constraints without modifying the solution process.The elastic range of linear springs is larger than that of torsion springs,and it is not affected by the number of spans.With an increase in the axial compressive force,the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger,while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary.

Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines,for which the rock physics model is indispensable.However,up to now,multitudinous analytical models focus primarily on the cracked rock with the isotropic background,while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth.Hence,we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy,respectively,by forty groups of numerical experiments.Based on the conclusion from the experiments,the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background.Further,we comprehensively analyzed,according to the developed model,effects of the crack dip angle,background anisotropy,filling fluid and crack density on the effective elastic properties of the cracked rock.The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P-and SH-wave velocities,whereas they have relatively small effects on the SV-wave velocity anisotropy.Moreover,the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters,while its effects on shear coefficients depend on the crack dip angle.The increasing crack density reduces velocities of the dry rock,and decreasing rates of the velocities are affected by the crack dip angle.By comparing with exact numerical results and experimental data,it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients.Moreover,the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.