Equilibrium Boundary Value Problem Research Articles

Non-standard interface conditions in flexure of mixture unified gradient Nanobeams

Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.

Analytical Solution of Thermo-Mechanical Properties of Functionally Graded Materials by Asymptotic Homogenization Method.

In this work, a general mathematical model for functionally graded heterogeneous equilibrium boundary value problems is considered. A methodology to find the local problems and the effective properties of functionally graded materials (FGMs) with generalized periodicity is presented, using the asymptotic homogenization method (AHM). The present models consist of the matrix metal Mo and the reinforced phase ceramic ZrC, the constituent ratios and the property gradation profiles of which can be described by the designed volume fraction. Firstly, a new threshold segmentation method is proposed to construct the gradient structure of the FGMs, which lays the groundwork for the subsequent research on the properties of materials. Further, a study of FGMs varied along a certain direction and the influence of the varied constituents and graded structures in the behavior of heterogeneous structures are investigated by the AHM. Consequently, the closed–form formulas for the effective thermo–mechanical coupling tensors are obtained, based on the solutions of local problems of FGMs with the periodic boundary conditions. These formulas provide information for the understanding of the traditional homogenized structure, and the results also be verified the correctness by the Mori–Tanaka method and AHM numerical solution. The results show that the designed structure profiles have great influence on the effective properties of the present inhomogeneous heterogeneous models. This research will be of great reference significance for the future material optimization design.

Mathematical modelling and design of isolation diaphragms for pressure gauges

The paper discusses some questions of equilibrium, stability and shape optimization of circular membranes with arbitrary profile along the meridian. Such membranes can be used as a flexible barrier to isolate the media on one side of a diaphragm from media on the other side, whilst allowing the transmission of pressure with minimal loss. The main goal of the analysis is to propose techniques for designing such shape of corrugated membranes that satisfy the requirements of so call “slow growing linear or almost linear” behavior of the pressure-volume diagram. The applied Kirchhoff-Love theory for non-shallow shells was used as a base for deriving nonlinear as well as linearized boundary value problems of equilibrium. A genetic algorithm was used to evaluate and optimize the characteristics of corrugated membranes. The presented analysis of the membrane with the maximum linear stroke showed its efficiency and reliability. At the same time, special attention should be paid to the basic parametrization used in modelling, since the mathematically optimal solution is not always suitable from the point of view of membrane production technology.

Effective predictions of heterogeneous flexoelectric multilayered composite with generalized periodicity

In this work, the general mathematical statements for flexoelectric heterogeneous equilibrium boundary value problems are reported. A methodology to find the local problems and the effective properties of flexoelectric composites with generalized periodicity is presented, using the two-scales asymptotic homogenization method. The statement of the homogenized boundary values problem is given. A procedure to solve the local problems of stratified multilayered composites with complex geometry and perfect contact at the interface is proposed. Consequently, the analytical expressions of the effective coefficients are obtained. The piezoelectric limit case for rectangular bi-laminated composites is validated. Finally, numerical analysis to illustrate the behavior of the effective properties for rectangular and wavy flexoelectric bi-layered structures are shown.

Face Stability Analysis of Shield Tunnel Using Slip Line Method

The sufficient support pressure is essential to guarantee the safe construction of shield tunnel. Thus, it is necessary to analyze the stability and assess the limit support pressure of the tunnel face. The main methods for face stability analysis mostly focused on finite element method, limit equilibrium method, and numerical simulation method. In this paper, the slip line method is applied to analyze the stability of the tunnel face. The soil is supposed as ideal isotropic, homogeneous, and incompressible continuous material, which obeys the Mohr–Coulomb yield criterion. A mathematical model of the limit equilibrium boundary value problem is established. The slip line method is used to solve the slip line field and stress field of the soil behind the tunnel face. Limit support pressure and failure mechanism of the tunnel face are then obtained. In addition, comparisons between the results of this study and those of existing approach are performed, and the influence factors are also discussed. The results show that the slip line method is proven to be reliable for the evaluation of limit support pressure of the tunnel face stability.

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.

Effect of induced temperature field on development of curvilinear crack with bonds between the faces in end zones

Temperature changes near the end of a curvilinear cohesive crack and their influence on crack growth are investigated. The problem of local temperature changes consists in a delay or retardation of the cohesive crack growth. The bonds between the curvilinear crack faces in the end zones are modeled by application to the crack surface cohesive forces caused by the presence of bonds. The boundary value problem of equilibrium of the curvilinear crack with interfacial bonds in the end zones under action of external tensile loads, induced temperature field and tractions in the bonds preventing to its opening, is reduced to a system of singular integral equations with a Cauchy-type kernel. From the solution of this equation system, normal and tangential tractions in the bonds are found. Analysis of the limit equilibrium of the crack using the end zone model is performed on the basis of a criterion of bonds limiting stretching and includes: 1) establishment of tractions depending on opening of the crack faces; 2) evaluation of the stress state near the curvilinear crack with taking into account tensile loads, induced temperature field, tractions in the bonds; 3) determination of the critical external tensile loads.

Control of crack propagation by shape-topological optimization

An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional. &nbsp The control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological derivatives of the elastic energy functional are evaluated. &nbsp The domain decomposition technique [42] is applied to the shape [56] and topological [54,55] sensitivity analysis of variational inequalities. &nbsp The nonlinear crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.

Equilibrium configurations and stability of a damaged body under uniaxial tractions

This paper deals with the equilibrium problem in nonlinear dissipative inelasticity of damaged bodies subject to uniaxial loading. To model the damage effects, a damage function, affecting the stored energy function, is defined. In the framework of the continuum thermodynamics theory, the constitutive law for damaged hyperelastic materials and an inequality for the energy release rate are derived. By means of an energy-based damage criterion, the irreversible evolution law for the damage function is obtained. After formulating the equilibrium boundary value problem, explicit expressions governing the global development of the equilibrium paths are written. Successively, the stability of the equilibrium solutions are assessed through the energy criterion. For a damaged body under uniaxial loading, seven inequalities are derived. These conditions, if fulfilled, ensure the stability of the solutions under each type of small perturbation. Finally, a number of applications for compressible neo-Hookean and Mooney–Rivlin materials are performed.

Spatial resolution of wrinkle patterns in thin elastic sheets at finite strain

Koiter's nonlinear plate theory is used to simulate the wrinkling patterns observed in stretched thin elastic sheets. The phenomenon considered is associated with wrinkle patterns distributed over the interior of the sheet, in regions where the stretching and bending energies are of the same order of magnitude. Numerical solutions to several equilibrium boundary-value problems are obtained by the method of dynamic relaxation based on a dissipative dynamical system and compared with existing experimental, numerical, and analytical results.

Cracks with interfacial bonds in the hub of a friction pair

The contact deformation of the hub of a plunger pair is considered. It is assumed that, during the repeated reciprocating motion of the plunger, fracture of the materials of the elements of the contact pair occurs. The friction surface of the bushing contains rectilinear cracks with the end zones characterized by the presence of interaction between the cracks faces. The boundary-value problem of equilibrium of the hub of a friction pair with a crack reduced to solving system of nonlinear singular integro-differential equations with a Cauchy type kernel.

Magnetoelasticity of highly deformable thin films: Theory and simulation

A non-linear two-dimensional theory is developed for thin magnetoelastic films capable of large deformations. This is derived directly from the three-dimensional theory. Significant simplifications emerge in the descent from three dimensions to two, permitting the self-field generated by the body to be computed a posteriori. The model is specialized to isotropic elastomers and numerical solutions are obtained to equilibrium boundary-value problems in which the membrane is subjected to lateral pressure and an applied magnetic field.

Estimation of vascular open configuration using finite element inverse elastostatic method

This paper presents a new method for predicting the open configuration of vascular organs. The method utilizes finite element inverse elastostatic formulations. The equilibrium boundary value problem is formulated on the homeostatic configuration, and is solved inversely to find the open, stress-free configuration. The method is non-invasive, and enables us to estimate the open configuration based on information that is readily available form in vivo measurements. Examples involving both axisymmetric and asymmetric geometries are presented to demonstrate the utility of the method.

A nonlinear finite element model of cartilage growth

The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MATLAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

Cracks with bonding between the lips in bushings of friction couples

We study the problem of contact deformation of a bushing of friction couple. The friction surface of the bushing contains rectilinear cracks with end zones characterized by the presence of interaction between the crack lips. In the case of repeated reciprocal motion of a plunger, the material of the bushing finally fails as a result of contact interaction. The boundary-value problem of equilibrium of the bushing of friction couple with cracks whose lips are interacting is reduced to the solution of a system of nonlinear singular integro-differential equations with Cauchy-type kernel. The solution of this system is used to find the normal and tangential forces in the bonds. The condition of limiting equilibrium of a crack with end zone is formulated with regard for the criterion of ultimate length of the bonds in the material.

Equilibria and vibration of a heavy pinched loop

A thin strip is bent such that the two ends are brought together and clamped (pinched) to form a teardrop shape. The clamped end is held at various angles with the loop either upright, horizontal, downward, or halfway between these positions. The length of the loop is increased, and the resulting equilibrium shapes, as well as small in-plane vibrations about equilibrium, are investigated analytically and experimentally. When the loop is held upright, in-plane buckling occurs at a critical length, and subsequent postbuckling deflections can be large. For the other orientations, except the hanging one, deflections also become large as the length is increased. In the analysis, the strip is assumed to be an inextensible elastica which is unstrained when straight, and its self-weight is included. A shooting method is applied to obtain numerical solutions to the nonlinear equilibrium boundary value problem and the linear vibration boundary value problem. Polycarbonate strips are used in the experiments, and data are acquired with a laser vibrometer. The experimental deflections, frequencies, and mode shapes exhibit excellent agreement with the analytical solutions.

Equilibrium analysis of finitely deformed elastic networks

A specialized membrane theory is used to analyze equilibrium configurations of finitely deformed elastic networks. The effects of wrinkling of the network are incorporated by using a certain relaxed strain energy function derived from minimum energy considerations. The stresses derived from this function are non-compressive at all values of the strain. In particular, a fibre strain associated with vanishing fibre stress may be viewed as resulting from fine-scale wrinkling of the fibre. In this way destabilizing compressive stresses are automatically excluded from the solution of an equilibrium boundary value problem. The properties of the relaxed strain energy are used to show that all equilibrium configurations are absolute minimizers of the total potential energy, for certain classes of boundary data. The equilibrium equations are discretized by a differencing method derived from Green's theorem, and artificial mass, damping and time are incorporated. Equilibrium configurations are then obtained in the long-time limit of a damped dynamical problem. Several examples of two- and three-dimensional deformations are presented, and comparisons with analytical solutions are made wherever possible.

Nonlinear Spatial Equilibria and Stability of Cables Under Uni-axial Torque and Thrust

Low tension cables subject to torque may form complex three-dimensional (spatial) equilibria. The resulting nonlinear static deformations, which are dominated by cable flexure and torsion, may produce interior loops or kinks that can seriously degrade the performance of the cable. Using Kirchhoffrod assumptions, a theoretical model governing cable flexure and torsion is derived herein and used to analyze (1) globally large equilibrium states, and (2) local equilibrium stability. For the broad class of problems described by pure boundary loading, the equilibrium boundary value problem is integrable and admits closed-form elliptic integral solutions. Attention is focused on the example problem of a cable subject to uni-axial torque and thrust. Closed-form solutions are presented for the complex three-dimensional equilibrium states which, heretofore, were analyzed using purely numerical methods. Moreover, the stability of these equilibrium states is assessed and new and important stability conclusions are drawn.

The use of displacement potentials in second order elasticity

This paper is concerned with the use of a representation in terms of displacement potentials in second order elasticity for equilibrium problems of homogeneous and isotropic materials. After justifying the adoption of an existing representation for linear elasticity for the purpose at hand, appropriate representations for solutions of second order elasticity problems in terms of displacement potentials (for both compressible and incompressible materials) are discussed. The use of the representations in obtaining complete solutions for equilibrium boundary-value problems is then illustrated by application to two examples of plane strain problems of compressible materials.

On positive solutions of a nonlinear equilibrium boundary value problem

We give necessary and sufficient conditions for the existence of positive solutions to a nonlinear elliptic system arising from mathematical ecology with the Dirichlet boundary data assigned by the equilibria of the corresponding ODE system. The main result is Theorem 1. The tools are Dancer fixed point index and degree theory.