Dynamic Boundary Value Problem Research Articles

Cluster modeling of the interaction of statyonary SH-waves with a system of curvilinear cracks in a half-space

A method is proposed for solving problems of mathematical physics for semi-infinite media containing systems of curvilinear cuts. Based on singular integral equations (SIE), a unified approach to solving the problem has been developed. Numerical implementation was carried out using parallelisation. A graph of the duration of the initial cluster hour is given for calculating an array of searched functions in the context of a parabolic form as a function of the number of processes for one variant of implementation. It is shown that the entire algorithm scales well and has an efficient number of processes. The expected result was obtained, because the structure of the calculation process in models based on SIE is mostly unchanged and built on well-defined procedures. It turned out that 150–200 processes are effective. An accuracy of 10–12 was achieved with the number of collocation points of the contour of each section N = 300, because the algorithm for numerical solution of the problem uses interpolation by Chebyshev polynomials in accordance with the fact that the unknown function has a key feature at the ends of the section, which causes a higher speed of convergence of the algorithm. The study of the question of the further increase in the accuracy of the result was not conducted. The corresponding dynamic boundary value problems for a restrained and force-free half-plane are studied. The influence of the curvature of defects, their interaction and proximity of the boundary on the magnitude and nature of the change in the dynamic stress intensity coefficients was studied. To check the reliability of the algorithm, two tests were carried out: removing three inhomogeneities at a distance of 106 of their length from each other and saturating the system by increasing the number of geometrically equal reflectors to 13–15. When removed, the characteristics of each reflector tend to the characteristics of a single one, and when saturated, they tend to the results of the corresponding periodic problem. The agreement between the results showed good reliability of the algorithm. The proposed method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

Study on coupled problems of thermoelasticity in Strains

In the work, within the framework of the strain compatibility conditions of Saint-Venant, two equivalent dynamic boundary value problems of thermoelasticity with respect to strains are formulated. In the case of the first boundary value problem, the dynamic equations of thermoelasticity are obtained from the compatibility conditions, in the second case, instead of the first three equations of thermoelasticity, the equations of motion expressed with respect to deformations are considered. Discrete analogues of boundary value problems are constructed using the finite-difference method in the form of explicit and implicit schemes. The solution of explicit schemes is reduced to recurrent relations with respect to deformations and temperature. Implicit schemes are solved by sequential application of the elimination method. The validity of the formulated thermoelastic boundary value problems is substantiated by comparing the numerical results of the problem of a thermoelastic parallelepiped obtained by different methods, as well as solving a similar problem in displacements.

A Study of Deformation and Fracture of the Eutectic in an Additively Manufactured Al-Si Composite Alloy

Experimental and numerical studies were conducted on AlSi12 alloy fabricated by wire-feed electron beam additive manufacturing to examine the structure, thermomechanical behavior and fracture of a eutectic microvolume at the scale of several microns. Dynamic boundary value problems were solved under plane strain conditions. The composite structure of the eutectic phase consisting of an aluminum matrix and silicon particles was taken into account explicitly in the calculations. Isotropic models of the thermoelastoplastic matrix and elastic-brittle particles were implemented in ABAQUS/Explicit. Composite deformation was calculated both with and without allowance for residual stresses caused by cooling of the composite after its fabrication. It was shown that after the cooling of the eutectic, silicon particles are compressed, and the aluminum matrix is under both bulk compressive and tensile as well as under pure shear stresses. It was found that residual stresses play a negative role at the stages of intense deformation of the composite. The fracture strain of the eutectic strongly depends on the yield point of the matrix, while the ultimate fracture stress varies but only slightly. Favorable morphology of silicon particles was determined which prevents early fracture of the eutectic.

Elastic drifted Brownian motions and non-local boundary conditions

We provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a functional associated with non-local boundary conditions of tempered type. By exploiting such connections we write some functionals of the drifted Brownian motion in terms of a simple (positive and non-decreasing) process, the inverse of a tempered subordinator. In our view, such a representation is useful in many applications and brings new light on dynamic boundary value problems.

Two-Scale Computational Analysis of Deformation and Fracture in an Al-Si Composite Material Fabricated by Electron Beam Wire-Feed Additive Manufacturing

Numerical simulation of deformation and fracture of an AlSi12% alloy additively fabricated by layer-by-layer electron beam melting of a wire is carried out. The microstructure of the alloy is studied by scanning and transmission electron microscopy at different resolutions. The experimental study at a length scale of several dozens of microns reveals a dendritic structure, which can be treated as a composite material consisting of aluminum arms separated by a eutectic network. The volume fraction of dendrites varies with the distance from the base plate in the build direction. The eutectics can also be thought of as a composite with an aluminum matrix reinforced by silicon particles at a scale of a few microns. Particles of different shapes are nearly equally spaced in the matrix. The eutectic and dendritic structures are taken into account explicitly in the calculations. The dynamic boundary-value problems are solved by ABAQUS/Explicit. The isotropic elastic-plastic and elastic models are used to simulate the response of aluminum and silicon. The fracture model includes a maximum distortion energy criterion formulated for the particle and matrix materials in terms of the equivalent stress and plastic strain. A two-scale approach is proposed to investigate deformation and fracture of the AlSi12% alloy. On the eutectic scale, the thermomechanical behavior of the Al matrix-silicon particle two-phase composite is simulated to obtain the homogenized properties of the eutectic composite material, which is then used at a higher scale to investigate the deformation and fracture of a two-phase dendritic structure. Residual stresses formed during cooling of the additively manufactured material were found to decrease the strength of the composite, while the strength increases with the volume fraction of dendrites.

Frequency‐dependent response of a rigid disk on a transversely isotropic soil layer subjected to horizontal motion

AbstractThe transverse isotropy of soil induced by geological processes over a long period has attracted considerable attention, which can lead to significant differences in the origination, propagation and attenuation of the stand waves compared with purely isotropic soil. Considering that the circular geometry is most common in many engineering, an analytical investigation of the dynamic behavior of a rigid disk resting on the transversely isotropic soil layer overlying the rigid bedrock under forced horizontal displacements is presented in this paper. Firstly, the governing equations of soil motion are established based on the elasticity theory of cylindrical coordinate system. Secondly, a set of partial differential equations for this class of problems are decoupled to ordinary differential forms by virtue of complete potential functions, Fourier expansion method and Hankel transform technique. Furthermore, the displacements, strains and stresses formulation in term of the spatial domain under general boundary conditions are obtained with success. For validation, dynamic boundary value problems in the finite‐layered soil containing transversely isotropic materials are degenerated to the isotropic solution corresponding to elastic half‐space. After that, various examples are also included to investigate the influence of the elastic modulus ratio, shear modulus ratio and thickness of soil layer on the dynamic response of the system. The numerical results suggest that the soil anisotropy and soil layer thickness have a non‐neglectful effect on the horizontal dynamic vibration of the rigid disk.

Computational Analysis of the Influence of Residual Stress on the Strength of Composites with Different Aluminum Matrices and Carbide Particles

A numerical study of the mechanical behavior of aluminum matrix–carbide particle composites subjected to combined thermomechanical loading is carried out. The composite structure, corresponding to that observed experimentally, is explicitly taken into account in the calculations. The mechanical response of the aluminum matrix and carbide particles is described using the isotropic elastic–plastic and elastic–brittle models. A fracture criterion of the maximum equivalent stress acting in the local regions of volumetric tension is used to study the crack initiation and propagation in the particles. The dynamic plane stress boundary value problems of cooling and tension of the composites are solved by the finite element method ABAQUS/Explicit. The influence of the cooling-induced residual stress and thermomechanical properties of the matrix and particle materials on the strength of the composites is investigated. A positive or negative effect of the residual stress is found to depend on the ratio between the particle strength and the matrix yield stress. Compressive residual stress formed in the particle after the cooling increases the strength of composites with hard matrices and low-strength particles. A decrease in the matrix–particle interfacial curvature results in a change in the fracture mechanism from in-particle cracking to debonding, which increases the composite strength. Composite elongation upon the fracture onset decreases with the volume fraction of the particles.

Second Order Iterative Dynamic Boundary Value Problems with Mixed Derivative Operators with Applications

Second Order Iterative Dynamic Boundary Value Problems with Mixed Derivative Operators with Applications

МАТЕМАТИЧНА МОДЕЛЬ ВЗАЄМОДІЇ СТАЦІОНАРНИХ SH-ХВИЛЬ З СИСТЕМОЮ КРИВОЛІНІЙНИХ ТРІЩИН У НАПІВПРОСТОРІ

A method for solving mathematical physics problems is proposed for semi-infinite media containing systems of curved crack-slits. Most of the studies known from the literature relate to the problems of diffraction of elastic waves on straight and circular cuts. However, in reality, the crack is usually not straight or circular. Studies have shown that the curvature of a crack significantly affects the value of the dynamic stress intensity coefficients. The value of this parameter also depends on the proximity of the defects to each other, since they always fall within the range of the reflected wave. The stress-strain state of media with complex properties can be effectively modeled by computing complexes in combination with software systems. Most studies are devoted to the development of the finite element method. However, the method of integral equations is very effective for solving anti-planar problems of diffraction theory. The advantage of this method is the reduction in the number of spatial variables, the high speed of convergence, and the possibility of using various efficient numerical solution methods. The method also has the ability to build efficient parallel computing schemes. A unified approach to solving the problem is developed on the basis of singular integral equations (SIEs). The corresponding dynamic boundary value problems for a clamped and force-free half-plane are investigated. The influence of the defect curvature, their interaction, and the proximity of the boundary on the magnitude and nature of the dynamic stress intensity coefficients is studied. Parallel algorithms allow to significantly reduce the computation time and analyse the characteristics of the wave field in more detail. The combination of the SIE method, which reduces the dimensionality of the problem by one, and provides significant savings in computing time due to the parallelization of computational procedures, leads to a significant increase in the efficiency of the proposed algorithm. The method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

Optimal trajectory smoothing for direct energy deposition additive manufacturing processes

In free form additive manufacturing processes (e.g., metal blown powder, glass filament, ceramic paste extrusion), part quality depends on both the rate at which the part is built and where material is deposited. This creates a natural conflict between these two criteria at locations where the curvature of the programmed path is small, such as corners, because at these locations it is not possible to precisely follow the path without changing acceleration and, hence, varying the part build rate due to the variation in path velocity. In some additive manufacturing processes it is not possible to easily vary the material feed rate to maintain a constant part build rate when path velocity varies; thus, uneven material deposition occurs. In this work an optimal trajectory smoothing methodology is created that formulates the path following problem as an optimal dynamic system boundary value problem. The methodology naturally allows a trade-off between path following error and velocity error and is experimentally applied in a blown powder direct metal deposition additive manufacturing process. The results show that path and velocity errors increase as the corner angle decreases and the velocity entering the corner increases; however, they can be decreased by increasing the corner's leg length. Further, as the weight on velocity error increases, the velocity error, excess deposited material, and total print time decrease asymptotically while path error increases asymptotically.

The Influence of the Volume Fraction of Ceramic Particles on the Plastic Strain Localization and Fracture of Metal Matrix Composites. Computational Analysis

In this paper, the mechanical behavior of the metal matrix composite under the action of different thermomechanical loading is investigated. The inhomogeneous structure of composites is explicitly considered in the calculations. Isotropic elastoplastic and elastic-brittle constitutive models are used to describe the mechanical response of the aluminum matrix and ceramic particles, respectively. A representative volume of the composite material containing several boron carbide particles located in the microvolume of aluminum alloy 6061T6 is considered. Two structures are created with different volume fractions of ceramic particles. It is found that the strength of the structure with lower volume fraction of particles is higher than the strength of the structure with larger volume fraction. Two types of problems are solved: 1) tension with considering residual stresses, and 2) tension without considering residual stresses. The Duhamel-Neumann relations are used to take into account residual stresses in the calculations. The influence of residual stresses on the fracture of particles in mesovolumes is investigated. Dynamic boundary value problems are solved by the finite element method in the ABAQUS / Explicit software package.

Numerical Study of the Influence of Residual Thermal Stresses on the Fracture of Metal Matrix Composites

A numerical study of the mechanical behavior of the «aluminum-boron carbide» composite material under different thermomechanical loading is performed. The complex structure of the composites is explicitly taken into account in the calculations. Isotropic elastoplastic and elastic-brittle constitutive models are used to describe the mechanical response of the aluminum matrix and ceramic particles, respectively. Huber type fracture criterion taking into account the local stress state: bulk tension, or compression, is chosen to study the origination and propagation of cracks in ceramic particles. The composite material microvolume with a single boron carbide particle embedded into the aluminum matrix is considered. The influence of the residual thermal stresses arising during cooling of the composite material from temperature close to the temperature of recrystallization of aluminum to the room temperature on the ceramic particle fracture is investigated. Dynamic boundary value problems are solved numerically by the finite element method in the ABAQUS / Explicit software package.

The influence of bi-layer metal-matrix composite coating on the strength of the coated material

Deformation and fracture in aluminum with a bi-layer composite coating are studied numerically. Dynamic boundary-value problems in the plane-stress formulation are solved by the finite element method, using ABAQUS/Explicit. Isotropic elastoplastic and elastic-brittle constitutive models are used to simulate the mechanical response of the aluminum matrix and carbide ceramic particles, respectively. Microstructure of the composite coatings takes into account the complex shape of particles explicitly. To investigate the crack initiation and propagation in ceramic particles, a Huber type fracture criterion was chosen that takes into account the type of local stress state: bulk tension or compression. The influence of the arrangement of the coating layers on the fracture of ceramic particles and on the macroscopic strength of the coated materials is studied. Plastic strain localization, crack patterns and residual stress formation are numerically investigated during cooling followed by tension of the coated material.

Solvability of a boundary value problem of chaotic dynamics of polymer molecule in the case of bounded interaction potential

Solvability of a boundary value problem of chaotic dynamics of polymer molecule in the case of bounded interaction potential

A mesoscopic analysis of a localized shear band propagation effect on the deformation and fracture of coated materials

The numerical simulations of the deformation and fracture in an iron boride coating – steel substrate composition are presented. The dynamic boundary-value problem is solved numerically by the finite-difference method. A complex geometry of the borided coating – steel substrate interface is taken into account explicitly. To simulate the mechanical behavior of the steel substrate, use is made of an isotropic strain hardening model including a relation for shear band propagation. Local regions of bulk tension are shown to arise near the interface even under simple uniaxial compression of the composition and in so doing they determine the mesoscale mechanisms of fracture. The interrelation between plastic deformation in the steel substrate and cracking of the borided coating is studied. Stages of shear band front propagation attributable to the interface complex geometry have been revealed. The coating cracking pattern, location of the fracture onset regions and the total crack length are found to depend on the front velocity in the steel substrate.

Microstructure-based analysis of residual stress concentration and plastic strain localization followed by fracture in metal-matrix composites

A numerical study is performed to investigate the deformation and fracture in metal matrix – ceramic particle composites using Al6061T6 and ZrC as an example of the compound materials. The step-by-step packing method is adopted for generating three-dimensional model microstructures of metal-matrix composites, with the irregular shape of particles and polycrystalline structure of the matrix being taken into account. The dynamic boundary-value problems are solved by the finite-element method. Constitutive models describing the mechanical behavior of the composite compounds include anisotropic elasticity, crystal plasticity with the strain hardening and strain fracture criterion for the matrix and isotropic elastic-brittle formulation with the stress fracture criterion for particles. A user-defined subroutine is developed to combine the anisotropic matrix and isotropic particle responses and implemented in the finite element calculations of the composite thermomechanical behavior. The interrelated plastic strain localization in the aluminum matrix and crack origination and growth in the matrix and ceramic particle are investigated under compression and tension, including the influence of the cooling-induced residual stresses. A detailed analysis is performed to evaluate the residual stress concentration in local regions of bulk tension formed under all-round and uniaxial compression of the composite due to the concave and convex interfacial asperities.

Numerical Solution of Coupled Thermo-Elastic-Plastic Dynamic Problems

The article considers a numerical method for solving a two-dimensional coupled dynamic thermoplastic boundary value problem based on deformation theory of plasticity. Discrete equations are compiled by the finite-difference method in the form of explicit and implicit schemes. The solution of the explicit schemes is reduced to the recurrence relations regarding the components of displacement and temperature. Implicit schemes are efficiently solved using the elimination method for systems with a three diagonal matrix along the appropriate directions. In this case, the diagonal predominance of the transition matrices ensures the convergence of implicit difference schemes. The problem of a thermoplastic rectangle clamped from all sides under the action of an internal thermal field is solved numerically. The stress-strain state of a thermoplastic rectangle and the distribution of displacement and temperature over various sections and points in time have been investigated.

Bulk-surface coupling: Derivation of two models

Motivated by various physical, cellular and ecological applications, there has been a recent resurgence of interest in studying the boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface, by using “bulk-surface models” (involving volumetric densities and surface densities) or by using models involving dynamical boundary conditions for volumetric densities. The surface in the models has no thickness. However, in some real world applications, the “surface” is actually a thin membrane, having positive, albeit small thickness δ, as in the cases of thermal barrier coatings for turbine engine blades and cell membranes. In such situations, rigorous derivations for these models seem lacking and desirable. In this paper, we start with two full models each of which contains reaction-diffusion equations in the bulk and the thin membrane, respectively, with two types of reasonable transmission conditions linking the two. Then in the limit of δ→0, we obtain two effective models, with one being a bulk-surface model and the other being the dynamical boundary value problem model, from which we can also recover the surface density of the other substance. Our analysis reveals that to have such effective models, it is crucial that (i) appropriate transmission conditions are satisfied on the interface between the bulk and the thin membrane, and (ii) the diffusion tensor in the thin membrane is “optimally aligned”, which means the principal axes of the tensor are either normal or tangential to the interface.

Infinitely Many Positive Solutions for an Iterative System of Conformable Fractional Order Dynamic Boundary Value Problems on Time Scales

In this paper, we establish infinitely many positive solutions for the iterative system of conformable fractional order dynamic equations on time scales $$ \begin{aligned} &\mathcal{T}_α^{\Delta}\big[\mathcal{T}_β^{\Delta}\big(\vartheta_\mathtt{n}(t)\big)\big]=\varphi(t)\mathtt{f}_\mathtt{n}\left(\vartheta_{\mathtt{n}+1}(t)\right),~t\in(0,1)_\mathbb{T},~1

A method for calculating the Green's function of spatial dynamic boundary value problems of the anisotropic elasticity theory taking into account asymptotic components

A method for calculating the Green's function of spatial dynamic boundary value problems of the anisotropic elasticity theory taking into account asymptotic components