Effect of boundary conditions on shakedown analysis of heterogeneous materials

The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field(SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force(NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements(RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions(DBCs), uniform traction boundary conditions(TBCs), and periodic boundary conditions(PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit(EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs.Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.

Least Square Finite Element Model for Analysis of Multilayered Composite Plates under Arbitrary Boundary Conditions

Laminated composites are widely used in many engineering industries such as aircraft, spacecraft, boat hulls, racing car bodies, and storage tanks. We analyze the 3D deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions on one edge and simply supported on other edge. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. This study presents the elastic analysis of laminated composite plates subjected to sinusoidal mechanical loading under arbitrary boundary conditions. Least square finite element solutions for displacements and stresses are investigated using a mathematical model, called a state-space model, which allows us to simultaneously solve for these field variables in the composite structure’s domain and ensure that continuity conditions are satisfied at layer interfaces. The governing equations are derived from this model using a numerical technique called the least-squares finite element method (LSFEM). These LSFEMs seek to minimize the squares of the governing equations and the associated side conditions residuals over the computational domain. The model is comprised of layerwise variables such as displacements, out-of-plane stresses, and in- plane strains, treated as independent variables. Numerical results are presented to demonstrate the response of the laminated composite plates under various arbitrary boundary conditions using LSFEM and compared with the 3D elasticity solution available in the literature.

Boundary Conditions for Momentum and Vorticity at an Interface between Two Fluids

Boundary conditions for momentum and vorticity have been precisely derived, paying attention to the physical meaning of each mathematical expression of terms rigorously obtained from the basic equations: Navier-Stokes equation and the equation of vorticity transport. It has been shown first that a contribution of fluid molecules crossing over a conceptual surface moving with fluid velocity due to their fluctuating motion is essentially important to understanding transport phenomena of momentum and vorticity. A notion of surface layers, which are thin layers at both sides of an interface, has been introduced next to elucidate the transporting mechanism of momentum and vorticity from one phase to the other at an interface through which no fluid molecules are crossing over. A fact that a size of δV, in which reliable values of density, momentum, and velocity of fluid are respectively defined as a volume-averaged mass of fluid molecules, a volume-averaged momentum of fluid molecules and a mass-averaged velocity of fluid molecules, is not infinitesimal but finite has been one of the key factors leading to the boundary conditions for vorticity at an interface between two fluids. The most distinguished characteristics of the boundary conditions derived here are the zero-value conditions for a normal component of momentum flux and tangential components of vorticity flux, at an interface.

Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter

In this paper, the inverse spectral problem of Sturm-Liouville operator with boundary conditions and jump conditions dependent on the spectral parameter is investigated. Firstly, the self-adjointness of the problem and the eigenvalue properties are given, then the asymptotic formulas of eigenvalues and eigenfunctions are presented. Finally, the uniqueness theorems of the corresponding inverse problems are given by Weyl function theory and inverse spectral data approach.

Dynamic characterization of tailing dam using fully coupled dynamic analysis with different boundary conditions-a case study

The seismic response analysis of a tailing dam is studied using a fully coupled effective stress approach in conjunction with an advanced multi yield surface plastic constitutive model for tailing material.Strain controlled static and cyclic triaxial tests were carried out to obtain the constitutive model for the tailing material.The tailing materials were collected from the Rampura Agucha tailing dam(Rajasthan State,India).A 2D nonlinear finite element(FE)model was then developed using different boundary conditions from the tailing embankment constructed using the downstream and upstream method of rising using OpenSees software.In first case,the model boundary was fixed in both the X and Y directions,and in the second case,viscous dashpots were introduced for both side and horizontal boundaries.The model was validated with experimental results on tailing material.Analyses were carried out considering five different earthquake motions,which were applied at the base.Comparisons of the different boundary conditions in terms of displacement flow vectors,pore pressure and stress-strain curves during shaking are presented.From the analysis,it was observed that the viscous boundary condition replicates the actual field conditions more accurately than the fixed boundary condition.In addition,it was found that the tailing embankment constructed by the downstream and upstream method of rising is not susceptible to liquefaction and lateral spreading for earthquake motions,even for a magnitude>5.5.

Effect of spin on the instability of THz plasma waves in field-effect transistors under non-ideal boundary conditions

Terahertz(THz) radiation can be generated due to the instability of THz plasma waves in field-effect transistors(FETs). In this work, we discuss the instability of THz plasma waves in the channel of FETs with spin and quantum effects under non-ideal boundary conditions. We obtain a linear dispersion relation by using the hydrodynamic equation, Maxwell equation and spin equation. The influence of source capacitance, drain capacitance, spin effects, quantum effects and channel width on the instability of THz plasma waves under the non-ideal boundary conditions is investigated in great detail. The results of numerical simulation show that the THz plasma wave is unstable when the drain capacitance is smaller than the source capacitance;the oscillation frequency with asymmetric boundary conditions is smaller than that under non-ideal boundary conditions;the instability gain of THz plasma waves becomes lower under non-ideal boundary conditions. This finding provides a new idea for finding efficient THz radiation sources and opens up a new mechanism for the development of THz technology.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method,the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method(ICCM).The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate.The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method.From the continuity condition of the vibration displacement function at the cutout,the transition matrix between the two coordinate systems is constructed,and the mass and stiffness matrices are completely obtained.As a result,the calculation is simplified and the computational efficiency of the solution is improved.In this paper,numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods,and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

One-and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes,focusing in particular on the cells close to the boundaries of the domain.In fact,most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that,taking into account the boundary conditions,fills the ghost cells with appropriate values,so that a standard reconstruction can be applied also in the boundary cells.In Naumann et al.(Appl.Math.Comput.325:252–270.https://doi.org/10.1016/j.amc.2017.12.041,2018),motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network,a different technique was explored that avoids the use of ghost cells,but instead employs for the boundary cells a different stencil,biased towards the interior of the domain.In this paper,extending that approach,which does not make use of ghost cells,we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids.In several numerical tests,we compare the novel reconstruction with the standard approach using ghost cells.

SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

Stability Scrutinization of Agrawal Axisymmetric Flow of Nanofluid through a Permeable Moving Disk Due to Renewable Solar Radiation with Smoluchowski Temperature and Maxwell Velocity Slip Boundary Conditions

The utilization of solar energy is essential to all living things since the beginning of time.In addition to being a constant source of energy,solar energy(SE)can also be used to generate heat and electricity.Recent technology enables to convert the solar energy into electricity by using thermal solar heat.Solar energy is perhaps the most easily accessible and plentiful source of sustainable energy.Copper-based nanofluid has been considered as a method to improve solar collector performance by absorbing incoming solar energy directly.The goal of this research is to explore theoretically the Agrawal axisymmetric flow induced by Cu-water nanofluid over a moving permeable disk caused by solar energy.Moreover,the impacts of Maxwell velocity and Smoluchowski temperature slip are incorporated to discuss the fine points of nanofluid flow and characteristics of heat transfer.The primary partial differential equations are transformed to similarity equations by employing similarity variables and then utilizing bvp4c to resolve the set of equations numerically.The current numerical approach can produce double solutions by providing suitable initial guesses.In addition,the results revealed that the impact of solar collector efficiency enhances significantly due to nanoparticle volume fraction.The suction parameter delays the boundary layer separation.Moreover,stability analysis is performed and is found that the upper solution is stable and physically trustworthy while the lower one is unstable.

Coseismic site response and slope instability using periodic boundary conditions in the material point method

This paper proposed the explicit generalized-a time scheme and periodic boundary conditions in the material point method(MPM)for the simulation of coseismic site response.The proposed boundary condition uses an intuitive particle-relocation algorithm ensuring material points always remain within the computational mesh.The explicit generalized-a time scheme was implemented in MPM to enable the damping of spurious high frequency oscillations.Firstly,the MPM was verified against finite element method(FEM).Secondly,ability of the MPM in capturing the analytical transfer function was investigated.Thirdly,a symmetric embankment was adopted to investigate the effects of ground motion arias intensity(I_(a)),geometry dimensions,and constitutive models.The results show that the larger the model size,the higher the crest runout and settlement for the same ground motion.When using a Mohr-Coulomb model,the crest runout increases with increasing I_(a).However,if the strain-softening law is activated,the results are less influenced by the ground motion.Finally,the MPM results were compared with the Newmark sliding block solution.The simplified analysis herein highlights the capabilities of MPM to capture the full deformation process for earthquake engineering applications,the importance of geometry characterization,and the selection of appropriate constitutive models when simulating coseismic site response and subsequent large deformations.

Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

Oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions via Prufer transformation

A class of Sturm-Liouville problems with discontinuity is studied in this paper.The oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions are obtained.The main method used in this paper is based on Prufer transformation,which is different from the classical ones.Moreover,we give two examples to verify our main results.

Experimental investigation on the permeability of gap-graded soil due to horizontal suffusion considering boundary effect

The boundary condition is a crucial factor affecting the permeability variation due to suffusion.An experimental investigation on the permeability of gap-graded soil due to horizontal suffusion considering the boundary effect is conducted,where the hydraulic head difference(DH)varies,and the boundary includes non-loss and soil-loss conditions.Soil samples are filled into seven soil storerooms connected in turn.After evaluation,the variation in content of fine sand(ΔR_(f))and the hydraulic conductivity of soils in each storeroom(C_(i))are analyzed.In the non-loss test,the soil sample filling area is divided into runoff,transited,and accumulated areas according to the negative or positive ΔR_(f) values.ΔR_(f) increases from negative to positive along the seepage path,and Ci decreases from runoff area to transited area and then rebounds in accumulated area.In the soil-loss test,all soil sample filling areas belong to the runoff area,where the gentle-loss,strengthened-loss,and alleviated-loss parts are further divided.ΔR_(f) decreases from the gentle-loss part to the strengthened-loss part and then rebounds in the alleviated-loss part,and C_(i) increases and then decreases along the seepage path.The relationship between ΔR_(f) and Ci is different with the boundary condition.Ci exponentially decreases with ΔR_(f) in the non-loss test and increases with ΔR_(f) generally in the soil-loss test.

CONVEXITY OF THE FREE BOUNDARY FOR AN AXISYMMETRIC INCOMPRESSIBLE IMPINGING JET

This paper is devoted to the study of the shape of the free boundary for a threedimensional axisymmetric incompressible impinging jet.To be more precise,we will show that the free boundary is convex to the fluid,provided that the uneven ground is concave to the fluid.

Development of D_α band symmetrical visible optical diagnostic for boundary reconstruction on EAST tokamak

To investigate the potential of utilizing visible spectral imaging for controlling the plasma boundary shape during stable operation of plasma in future tokamak, a D_α band symmetric visible light diagnostic system was designed and implemented on the Experimental Advanced Superconducting Tokamak(EAST). This system leverages two symmetric optics for joint plasma imaging. The optical system exhibits a spatial resolution less than 2 mm at the poloidal cross-section, distortion within the field of view below 10%, and relative illumination of 91%.The high-quality images obtained enable clear observation of both the plasma boundary position and the characteristics of components within the vacuum vessel. Following system calibration and coordinate transformation, the image coordinate boundary features are mapped to the tokamak coordinate system. Utilizing this system, the plasma boundary was reconstructed, and the resulting representation showed alignment with the EFIT(Equilibrium Fitting) results. This underscores the system's superior performance in boundary reconstruction applications and provides a diagnostic foundation for boundary shape control based on visible spectral imaging.

Analytical solutions of turbulent boundary layer beneath forward-leaning waves

As a typical nonlinear wave,forward-leaning waves can be frequently encountered in the near-shore areas,which can impact coastal sediment transport significantly.Hence,it is of significance to describe the characteristics of the boundary layer beneath forward-leaning waves accurately,especially for the turbulent boundary layer.In this work,the linearized turbulent boundary layer model with a linear turbulent viscosity coefficient is applied,and the novel expression of the near-bed orbital velocity that has been worked out by the authors for forward-leaning waves of arbitrary forward-leaning degrees is further used to specify the free stream boundary condition of the bottom boundary layer.Then,a variable transformation is found so as to make the equation of the turbulent boundary layer model be solved analytically through a modified Bessel function.Consequently,an explicit analytical solution of the turbulent boundary layer beneath forward-leaning waves is derived by means of variable separation and variable transformation.The analytical solutions of the velocity profile and bottom shear stress of the turbulent boundary layer beneath forward-leaning waves are verified by comparing the present analytical results with typical experimental data available in the previous literature.

The Boundary Element Method for Ordinary State-Based Peridynamics

The peridynamics(PD),as a promising nonlocal continuum mechanics theory,shines in solving discontinuous problems.Up to now,various numerical methods,such as the peridynamic mesh-free particlemethod(PD-MPM),peridynamic finite element method(PD-FEM),and peridynamic boundary element method(PD-BEM),have been proposed.PD-BEM,in particular,outperforms other methods by eliminating spurious boundary softening,efficiently handling infinite problems,and ensuring high computational accuracy.However,the existing PD-BEM is constructed exclusively for bond-based peridynamics(BBPD)with fixed Poisson’s ratio,limiting its applicability to crack propagation problems and scenarios involving infinite or semi-infinite problems.In this paper,we address these limitations by introducing the boundary element method(BEM)for ordinary state-based peridynamics(OSPD-BEM).Additionally,we present a crack propagationmodel embeddedwithin the framework ofOSPD-BEM to simulate crack propagations.To validate the effectiveness of OSPD-BEM,we conduct four numerical examples:deformation under uniaxial loading,crack initiation in a double-notched specimen,wedge-splitting test,and threepoint bending test.The results demonstrate the accuracy and efficiency of OSPD-BEM,highlighting its capability to successfully eliminate spurious boundary softening phenomena under varying Poisson’s ratios.Moreover,OSPDBEMsignificantly reduces computational time and exhibits greater consistencywith experimental results compared to PD-MPM.