Boundary Value Problems Of Elasticity Research Articles

Boundary influence on the fields and arrangement of the needle domains in barium titanate single crystals

The superposition theory is adopted to study the boundary value problem of needle domains in barium titanate single crystals, which decomposes into an infinite-medium solution and a complementary solution. The model of an equivalent edge dislocation and line charge is adopted to calculate the fields due to the discrete needle domains in an infinite medium. The complementary solutions are obtained from a linear elastic boundary value problem using the finite element method. The influence of boundaries on the fields and arrangement of the needles in thin barium titanate single crystals is analysed, by considering traction-free boundaries and using the plane stress approximation. According to the principle of minimum potential energy, it is found that pairs of parallel needle domains can be stabilized if the effective charge associated with the needle tips is reduced to 48% of the material’s full polarization charge. The evolution of comb-like arrays of needle domains under compressive mechanical load is simulated, and the results are found to be qualitatively similar to experimental observations which have not previously been explained theoretically.

Modeling the boundary shape of the problems described by Navier–Lamé equations using NURBS curves in parametric integral equations system method

This paper is an extended version of the ICCS conference paper (Kapturczak et al., 2020, [1]) and presents a way to improve the boundary shape modeling process in solving boundary value problems in elasticity. The inclusion of NURBS curves into the mathematical formalism of the parametric integral equations system method (PIES) is proposed. The advantages of such an application are widely discussed. Recently, the Bezier curves, mainly the cubic curves (of third-degree), were used. The segments of the boundary shape were modeled by such curves (with ensuring continuity at the connection points). Using NURBS curves, the boundary shape can be modeled with only one curve. So, continuity is automatically ensured. Additionally, the second degree NURBS curve is enough to obtain the shape with high accuracy (better than cubic Bezier curves). The NURBS curve is defined by points, their weights, and the knots vector. Such parameters significantly improve the shape modification process, which can directly improve e.g. the shape identification process. The examples of shape modification using such parameters are presented. The boundary shapes of the examples (even defined by both linear and curvilinear segments) can be defined using only one closed NURBS curve. The impact of modeling accuracy on the final PIES solutions is examined on examples described by the Navier–Lamé equations. To improve calculations, the PIES method using NURBS curves was implemented as a computer program. Then, it was decided to verify the accuracy of the obtained solutions. For comparison, the solutions were also obtained using analytical solutions, boundary element method, and PIES method (with the Bezier curves). An improvement in the boundary shape modeling was noticed. It significantly affects the accuracy of solutions. As a result, the consumption of computer resources was reduced, while the process of boundary shape modeling and the accuracy of the obtained results were improved.

Parametric integral equation system (PIES) for solving problems with inclusions and non-homogeneous domains using Bézier surfaces

The paper presents the approach for solving 2D elastic boundary value problems defined in domains with inclusions with different material properties using the parametric integral equation system (PIES). The main feature of the proposed strategy is using Bézier surfaces for global modeling of inclusions. Polygonal inclusions are defined by bilinear surfaces, while others by bicubic surfaces. It is beneficial over other numerical methods (such as FEM and BEM) due to the lack of discretization. Integration over inclusions defined by surfaces is also performed globally without division into subareas. The considered problem is solved iteratively in order to simulate different material properties by applying initial stresses within the inclusion. This way of solving avoids increasing the number of unknowns and can also be used for elasto-plastic problems without significant changes. Some numerical tests are presented, in which the results obtained are compared with those calculated by other numerical methods. This paper is an extended version of author's conference paper [1]. It has been enriched with, among others, the description of modeling more complex inclusions, as well as additional results obtained by PIES compared to other numerical methods.

A finite element implementation of the stress gradient theory

In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.

Topology optimization of hyperelastic structures with anisotropic fiber reinforcement under large deformations

Fiber-reinforced soft materials have emerged as promising candidates in various applications such as soft robotics and soft fibrous tissues. To enable a systematic approach to design fiber-reinforced materials and structures, we propose a general topology optimization framework for the computational optimized design of hyperelastic structures with nonlinear and anisotropic fiber reinforcements under large deformations. This framework simultaneously optimizes both the material distribution in the matrix phase and the orientations of the underlying fiber reinforcements, by parameterizing matrix and fiber phases individually using two sets of design variables. The optimized distribution of fiber orientations is chosen from a set of discrete orientations defined a priori, and several fiber orientation interpolation schemes are studied. In addition, this work proposes a novel anisotropic material interpolation scheme, which integrates both matrix and fiber design variables (both with material nonlinearity) into the stored-energy function. To improve the computational efficiency of both optimization and nonlinear structural analysis, we derive a fully decoupled fiber-matrix update scheme that performs parallel updates of the matrix and fiber design variables and employ the virtual element method (VEM) together with a tailored mesh adaptivity scheme to solve the finite elasticity boundary value problem. Design examples involving three objective functions are presented, demonstrating the efficiency and effectiveness of the proposed framework in designing anisotropic hyperelastic structures under large deformations.

Elastic Field of a Dilatational Cylindrical Inclusion in an Elastically Isotropic Half-Space

In this article, a new solution to the elasticity boundary-value problem for a dilatational cylindrical inclusion embedded in an elastically isotropic half-space is presented. To solve this problem, the results for the infinitesimally thin dilatational disk in an elastically isotropic half-space, are explored. For displacements, strains, and stresses of a dilatational cylindrical inclusion, the analytical expressions are obtained with Lipschitz-Hankel integrals. The comparison of the found solution with previously known one, is given.

Elasticity Boundary-Value Problems for Straight Wedge Disclinations. A Review on Methods and Results

Elasticity Boundary-Value Problems for Straight Wedge Disclinations. A Review on Methods and Results

Analysis of a mixed boundary value problem for an orthotropic elasticity using a mapping function

A solution is derived for a mixed boundary orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped hole problems can be solved by changing the mapping function. Once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, an infinite plane with a square hole subjected to uniform tension is analyzed. The stress distributions are shown for Cases I and III problems corresponding to two characteristic roots of the fundamental equation. In a Case III problem, the symmetry of stress distributions is lost. A solution of an external force boundary value problem can be derived from that of the mixed boundary value problem.

Variational Statements and Discretization of the Boundary-Value Problem of Elasticity Where Stress at the Boundary is Known

Variational Statements and Discretization of the Boundary-Value Problem of Elasticity Where Stress at the Boundary is Known

Inversion Symmetry of the Solutions of Boundary-Value Problems of Elasticity for a Half-Space

The inversion symmetry of the components of the displacement vector and stress tensor in the solution of the first boundary-value problem of elasticity for a half-space is studied. The case where one component of loading on the half-space boundary has inversion symmetry and the other two components are equal to zero is considered. Inversion symmetry is also studied for the mixed problem where normal forces act and tangential forces are equal to zero one part of the half-space boundary, while the conditions of smooth contact are prescribed on the other part, and the problem of the torsion of an elastic half-space with tangential stresses given on its boundary.

Explicit solution of the Dirichlet boundary value problem of elasticity for porous infinite strip

The object of the present paper is to consider the Dirichlet boundary value problem of the coupled linear quasi-static theory of elasticity for porous isotropic elastic infinite strip. The general representations of a regular solutions of a system of considered equations for a homogeneous isotropic medium are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, the Dirichlet BVP is solved effectively (in quadratures) for the infinite strip.

A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity

We present the analytical formulation and the finite element solution of a fractional-order nonlocal continuum model of a Euler-Bernoulli beam. Employing consistent definitions for the fractional-order kinematic relations, the governing equations and the associated boundary conditions are derived based on variational principles. Remarkably, the fractional-order nonlocal model gives rise to a self-adjoint and positive-definite system accepting a unique solution. Further, owing to the difficulty in obtaining analytical solutions to this boundary value problem, a finite element model for the fractional-order governing equations is presented. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the nonlocal response of a Euler-Bernoulli beam subjected to various loading and boundary conditions. The fractional-order positive definite system will be used here to address some paradoxical results obtained for nonlocal beams through classical integral approaches to nonlocal elasticity. Although presented in the context of a 1D Euler-Bernoulli beam, the f-FEM formulation is very general and could be extended to the solution of any general fractional-order boundary value problem.

Uniqueness for the inverse boundary value problem of piecewise homogeneous anisotropic elasticity in the time domain

We consider the inverse boundary value problem of recovering piecewise homogeneous elastic tensor and piecewise homogeneous mass density from a localized lateral Dirichlet-to-Neumann or Neumann-to-Dirichlet map for the elasticity equation in the space-time domain. We derive uniqueness for identifying these tensor and density on all domains of homogeneity that may be reached from the part of the boundary where the measurements are taken by a chain of subdomains whose successive interfaces contain a curved portion.

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

By introducing the elastic boundary value condition, the elastic boundary value problem of extended modified Bessel equation is proposed, we can use the following method to solve it. First, two linear independent solutions of extended modified Bessel equation are obtained. Second, the generating function of solution is constructed. Third, the kernel function of solution is constructed using the elastic right value condition. Finally, the solution is obtained by assembling coefficients with the left boundary value condition. As for its application, a fractal homogeneous reservoir seepage model under the elastic outer boundary condition is established, and solution of the model is obtained. Influences of reservoir parameters on characteristic curves corresponding to dimensionless bottom hole pressure and its derivative are analyzed, which provide a new theoretical basis for exploring the flow law of oil. It can be found that seepage model under the elastic outer boundary condition regards three idealized outer boundary conditions (infinite, constant pressure and closed) and homogeneous reservoir seepage model without considering fractal as special cases, so it can reflect real situation of reservoir better and it is helpful to the development of related well test analysis software.

Trapping Elastic Waves by a Semi-Infinite Cylinder with Partly Fixed Surface

We consider the three-dimensional mixed boundary value problem in elasticity about time harmonic oscillations of a semi-infinite anisotropic cylinder. We show that for certain position and shape of the clamping zone of the surface the elastic wave is trapped; i.e., the problem admits a nontrivial solution with exponential decay at infinity or, conversely, the absence of the trapped wave is guaranteed on all frequencies. We state some open questions that concern similar spectral problems.

Explicit solutions of boundary value problems of elasticity for circle with a double-voids structure

The present paper is devoted to the explicit solutions of the equilibrium boundary value problems (BVPs) for an elastic circle and for full plane with circular hole with a double-voids structure. The regular solution of the system of equations for an isotropic material with a double-voids structure is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The Dirichlet-type BVPs for a circle and for a plane with a circular hole are solved explicitly. The obtained solutions are presented as absolutely and uniformly convergent series.

Convergence behaviour of the enriched scaled boundary finite element method

SummaryIn this work, a very efficient numerical solution of three‐dimensional boundary value problems of linear elasticity including stress singularities is discussed, focussing on its convergence behaviour. For the employed scaled boundary finite element method, a discretization is only needed at the boundary, while the solution is considered analytically in a scaling coordinate. This presents a major advantage for two‐dimensional problems, when the scaling center is placed at a stress singularity. Unfortunately, three‐dimensional problems usually do not only include point singularities but also line singularities, which results in singular gradients in the boundary coordinates and thereby diminishes the method's original advantages. To alleviate this drawback, this work discusses an enrichment of the separation of variables representation with analytical asymptotic near fields of the line singularities. In contrast to previous works, besides the near‐field functions with λ=0.5, also those with λ=1.5 were determined and used for enrichment. This leads to a high accuracy and it is shown that this approach is required to recover the convergence properties of smooth boundary value problems without singularities when using quadratic Lagrange shape functions. In order to recover the convergence rates for higher order shape functions, near‐field functions with higher singularity exponent have to be included for enrichment.

Study of stress–strain state of elastic body with hyperbolic notch

The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two-dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems are presented.

Spherical harmonics method for computing the image stress due to a spherical void

We develop an efficient numerical method for calculating the image stress field induced by spherical voids in materials, and applied the method to dislocation-void interactions. The method is constructed based on a complete set of basis functions for the displacement potential of the elastic boundary value problem for a spherical hole, as well as the corresponding displacement, stress, and traction fields, all in terms of linear combinations of spherical harmonics. Using the fast transformation between the real and spherical-harmonics spaces provided by the SHTOOLS package, the method is more efficient than other image stress solvers such as the finite-element method. This method can be readily extended for solving elasticity problems involving inclusions and inhomogeneities, as well as contact between spheres. The tools developed here can also be useful for fast solution of differential equations with spherical boundaries beyond elasticity.

Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity

We consider a linearized inverse boundary value problem for the elasticity system. From the linearized Dirichlet-to-Neumann map at zero frequency, we show that a transversely isotropic perturbation of a homogeneous isotropic elastic tensor can be uniquely determined. From the linearized Dirichlet-to-Neumann map at two distinct positive frequencies, we show that a transversely isotropic perturbation of a homogeneous isotropic density can be identified at the same time.