Nonlinear Elastic Beam Research Articles

Longitudinal fracture analysis of inhomogeneous beams with continuously varying sizes of the cross-section along the beam length

Analyses of longitudinal fracture behavior of inhomogeneous beams which have continuously varying sizes of the cross-section along the beam length are carried-out. Beams of a rectangular cross-section are studied. It is assumed that beams exhibit continuous (smooth) material inhomogeneity along the width and height of the cross-section. A longitudinal crack located arbitrary along the beam height is analyzed. First, a cantilever beam with linearly varying width and height along the beam length is considered. The material of the beam has non-linear elastic mechanical behavior. The external loading consists of one bending moment applied at the free end of the lower crack arm. The fracture behavior is analyzed in terms of the strain energy release rate assuming that the modulus of elasticity is distributed continuously in the beam cross-section. The balance of the energy is considered in order to derive the strain energy release rate. A solution to the strain energy release rate is obtained also by considering the complementary strain energy for verification. The longitudinal fracture behavior of the inhomogeneous non-linear elastic cantilever beam configuration is studied also for the cases when the variation of the width and height of the cross-section is described by sine and power laws.

Inhomogeneous beams of circular cross-section under axial forces: a longitudinal fracture analysis

This paper deals with longitudinal fracture analysis of inhomogeneous non-linear elastic beams with continuously varying radius of the cross-section along the beam length. The external loading consists of axial forces. Beams with a longitudinal crack presenting a circular cylindrical surface are considered. The beams exhibit continuous material inhomogeneity in radial direction. A general approach for analyzing the strain energy release rate is developed by considering the energy balance. The approach is used to analyze the longitudinal fracture of an inhomogeneous non-linear elastic cantilever beam configuration. The strain energy release rate in the cantilever beam is derived also by considering the complementary strain energy for verification. The influence of the continuously varying radius of the cross-section along the length of the cantilever beam on the strain energy release rate is evaluated. The effects of the material inhomogeneity, location of the crack in radial direction, crack length and the material non-linearity on the strain energy release rate are also evaluated.

Inhomogeneous Beams with Different Behaviour in Tension and Compression: A Longitudinal Crack Study

An approach for investigating the strain energy release rate for longitudinal cracks in nonlinear elastic inhomogeneous beams is derived. The Ramberg-Osgood stress-strain relation is applied to describe the unsymmetrical mechanical behaviour of the material with respect to tension and compression. It is assumed that the beams have continuous material inhomogeneity in the height direction. Cracks can be located arbitrarily along the beam height. The approach is applied to investigate a longitudinal crack in a cantilever beam. The J-integral is used for verification of the solution.

The bending-stretching effect of a nonlinear elastic beam under finite deformation

Second-order solutions for the pure bending of a beam under finite deformation are obtained in this paper. The equilibrium equations are solved analytically for the second-order displacements, from which the first Piola–Kirchhoff stresses, the curvatures and a bending-stretching parameter are calculated for a cross section with a transverse axis of symmetry. The displacements are dependent on the square of the inverse of elastic constant, whereas the stresses on the inverse of elastic constant. The curvatures and anticlastic curvatures are gradated in the transverse direction. The longitudinal stress is nonlinearly distributed across the section, and exhibits anomalous characteristics in materials with a negative Poisson's ratio. The shear stresses have vanishing force and moment resultants, and bending-twisting coupling does not occur. The normal stresses give rise to force-couple resultants. The longitudinal force resultant representing the bending-stretching effect is dependent on a reduced elastic parameter ξ. This effect is present as long as there is finite deformation, even if the material is linearly elastic. A small shear modulus μ and/or a large third-order elastic constant l, m or n will result in a large ξ. This study may be applicable to the elasticity design of soft medical and wearable robots, where elastic compatibility with tissues is a primary consideration.

Common Fixed Points of(α,η)−(θ,ϝ)Rational Contractions with Applications

We present the notion of set valued(α,η)-(θ,ϝ)rational contraction mappings and then some common fixed point results of such mappings in the setting of metric spaces are established. Some examples are presented to support the concepts introduced and the results proved in this paper. These results unify, extend, and refine various results in the literature. Some fixed point results for both single and multivalued(θ,ϝ)rational contractions are also obtained in the framework of a space endowed with partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

Effects of Three-Axial Material Inhomogeneity on Lengthwise Cracks in Non-Linear Elastic Beams

The present paper investigates lengthwise crack behaviour in three-axial inhomogeneous non-linear elastic cantilever beams, i.e., beams which exhibit inhomogeneous material properties along the width, height and length. The non-linear mechanical behaviour of the inhomogeneous material is described by the Ramberg-Osgood equation assuming that the modulus of elasticity varies linearly along the width, height and length of beams. A solution to the strain energy release rate is derived by considering the balance of the energy. The results obtained in the present paper indicate that the three-axial inhomogeneous material properties have a significant influence on the strain energy release rate in non-linear elastic beams.

Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading

This study presents the elastica of non-prismatic cantilever beams with rectangular cross-sections that are subjected to combined loading. The considered beams are nonlinearly elastic and obey Ludwick’s constitutive law. The combined loading system used in this study provides uniform loading, tip point loading, and tip couple loading individually or in combination. This loading system can create a total of seven loading cases that have not been covered in the literature. Ordinary differential equations governing the large deformed shapes of the elastica of the beams are derived and solved numerically. The effects of beam parameters on elastica behavior, including tip responses and strains and stresses loaded onto the cross-sections, were studied. It was observed that the exponential constant of the mechanical properties is very sensitive to elastica behavior. The results also demonstrate that the stresses near the neutral axis are larger with a larger exponential constant.

Influence of sine material gradients on delamination in multilayered beams

The present paper deals with delamination fracture analyses of the multilayered functionally graded non-linear elastic Symmetric Split Beam (SSB) configurations. The material is functionally graded in both width and height directions in each layer. It is assumed that the material properties are distributed non-symmetrically with respect to the centroidal axes of the beam cross-section. Sine laws are used to describe the continuous variation of the material properties in the cross-sections of the layers. The delamination fracture is analyzed in terms of the strain energy release rate by considering the balance of the energy. A comparison with the J-integral is performed for verification. The solution derived is used for parametric analyses of the delamination fracture behavior of the multilayered functionally graded SSB in order to evaluate the effects of the sine gradients of the three material properties in the width and height directions of the layers and the location of the crack along the beam width on the strain energy release rate. The solution obtained is valid for two-dimensional functionally graded non-linear elastic SSB configurations which are made of an arbitrary number of lengthwise vertical layers. A delamination crack is located arbitrary between layers. Thus, the two crack arms have different widths. Besides, the layers have individual widths and material properties.

Material optimization for nonlinearly elastic planar beams

We consider the problem of an optimal distribution of soft and hard material for nonlinearly elastic planar beams. We prove that under constant vertical load the optimal distribution involves no microstructure and is ordered, and we provide numerical simulations confirming and extending this observation.

Longitudinal cracks in non-linear elastic beams exhibiting material inhomogeneity

Longitudinal fracture behavior of non-linear elastic beam configurations is studied in terms of the strain energy release rate. It is assumed that the beams exhibit continuous material inhomogeneity along the width as well as along the height of the cross-section. The Ramberg-Osgood stress-strain relation is used for describing the non-linear mechanical behavior of the inhomogeneous material. A solution to strain energy release rate is derived that holds for inhomogeneous beams of arbitrary cross-section under combination of axial force and bending moments. Besides, the solution may be applied at any law of continuous distribution of the modulus of elasticity in the beam cross-section. The longitudinal crack may be located arbitrary along the beam height. The solution is used to investigate a longitudinal crack in a beam configuration of rectangular cross-section under four-point bending. The crack is located symmetrically with respect to the beam mid-span. It is assumed that the modulus of elasticity varies continuously according a cosine law in the beam cross-section. The longitudinal fracture behavior of the inhomogeneous beam is studied also by applying the J-integral approach for verification of the non-linear solution to the strain energy release rate derived in the present paper. Effects of material inhomogeneity, crack location along the beam height and non-linear mechanical behavior of the material on the longitudinal fracture behavior are evaluated. Thus, the solution derived in the present paper can be used in engineering design of inhomogeneous non-linear elastic structural members to assess the influence of various material and geometrical parameters on longitudinal fracture.

Two positive solutions for nonlinear fourth-order elastic beam equations

Two positive solutions for nonlinear fourth-order elastic beam equations

Nonlinear elastic beam problems with the parameter near resonance

AbstractIn this paper, we consider the nonlinear fourth order boundary value problem of the form$$ \begin{array}\text \left\{ \begin{aligned} &u^{(4)}(x)-\lambda u(x)=f(x, u(x))-h(x), \ \ x\in (0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{aligned}\right. \end{array} $$which models a statically elastic beam with both end-points cantilevered or fixed. We show the existence of at least one or two solutions depending on the sign of λ−λ1, where λ1 is the first eigenvalue of the corresponding linear eigenvalue problem and λ is a parameter. The proof of the main result is based upon the method of lower and upper solutions and global bifurcation techniques.

Eigenvalue Interval for Singular Fractional-Order Elastic Beam Equation

The purpose of this paper, is to give the eigenvalue interval for nonlinear elastic beam equation and according to the range of the eigenvalue λ, establish sufficient conditions for the existence of positive solutions.

Analytical Study of Delamination in Multilayered Two-Dimensional Functionally Graded Non-Linear Elastic Beams

AbstractThe present paper is focused on the delamination fracture in a multilayered two-dimensional functionally graded beam configuration which exhibits non-linear behavior of the material. The beam is loaded by two longitudinal forces applied at the beam free ends. The beam contains a delamination crack which is located symmetrically with respect to the beam mid-span. The delamination is studied analytically in terms of the strain energy release rate. TheJ-integral approach is applied for verification of the analysis of the strain energy release rate. The solution derived is valid for a beam made of an arbitrary number of layers. It is assumed that each layer has individual thickness and material properties. Also, the material is two-dimensional functionally graded in the cross-section of each layer. The solution obtained can be applied for a delamination crack located arbitrary along the height of the beam cross-section. It is shown that the solution is very convenient for investigating the influences of material gradients and crack location on the delamination fracture behavior. The results obtained can be used for optimization of multilayered two-dimensional functionally graded structural members and components with respect to their delamination fracture performance.

Оценка собственных частот колебаний чистого изгиба композиционных нелинейно-упругих балок и круглых пластин

Оценка собственных частот колебаний чистого изгиба композиционных нелинейно-упругих балок и круглых пластин

Seismic Design of a Single Bored Tunnel: Longitudinal Deformations and Seismic Joints

The large diameter bored tunnel passing through rock and alluvial deposits subjected to seismic loading is analyzed for estimating longitudinal deformations and member forces on the segmental tunnel liners. The project site has challenges including high hydrostatic pressure, variable ground profile and high seismic loading. To ensure the safety of segmental tunnel liner from the seismic demands, the performance-based two-level design earthquake approach, Functional Evaluation Earthquake and Safety Evaluation Earthquake, has been adopted. The longitudinal tunnel and ground response seismic analyses are performed using a three-dimensional quasi-static linear elastic and nonlinear elastic discrete beam–spring elements to represent segmental liner and ground spring, respectively. Three components (longitudinal, transverse and vertical) of free-field ground displacement–time histories evaluated from site response analyses considering wave passage effects have been applied at the end support of the strain-compatible ground springs. The result of the longitudinal seismic analyses suggests that seismic joint for the mitigation measure requiring the design deflection capacity of 5–7.5 cm is to be furnished at the transition zone between hard and soft ground condition where the maximum member forces on the segmental liner (i.e., axial, shear forces and bending moments) are induced. The paper illustrates how detailed numerical analyses can be practically applied to evaluate the axial and curvature deformations along the tunnel alignment under difficult ground conditions and to provide the seismic joints at proper locations to effectively reduce the seismic demands below the allowable levels.

Eigenvalue problem for nonlinear elastic beam equation of fractional order

In this study, under some suitable assumptions, we determine an explicit eigenvalue interval for the existence of positive solution of singular fractional-order nonlinear elastic beam equation with bending term. Our analysis rely on cone theoretic techniques. Moreover, we consider some special cases and an example to affirm the applicability of the main result.

Solution of contact problems for Gao beam and elastic foundation

This paper presents mathematical formulations and a solution for contact problems that concern the nonlinear beam published by Gao (Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech Res Commun 1996; 23: 11–17) and an elastic foundation. The beam is subjected to a vertical and also axial loading. The elastic deformable foundation is considered at a distance under the beam. The contact is modeled as static, frictionless and using the normal compliance contact condition. In comparison with the usual contact problem formulations, which are based on variational inequalities, we are able to derive for our problem a nonlinear variational equation. Solution of this problem is realized by means of the so-called control variational method. The main idea of this method is to transform the given contact problem to an optimal control problem, which can provide the requested solution. Finally, some results including numerical examples are offered to illustrate the usefulness of the presented solution method.

Volumetric locking suppression method for nearly incompressible nonlinear elastic multi-layer beams using ANCF elements

The analysis and solution of many modern flexible multibody dynamic problems require formulations that are able to effectively model bodies with nonlinear materials undergoing large displacements and deformations. The absolute nodal coordinate formulation (ANCF) in connection with a continuum-based approach is one way to deal with these systems. The main objective of this work is to extend an existent approach for the modelling of slender structures within the ANCF framework with nonlinear, nearly incompressible materials using the volumetric energy penalty technique. The main part of the study is devoted to the evaluation of multi-layer beam models and simplifications in the locking suppression method based on F-bar projection. The results present significantly better agreement with the reference solution for multi-layer structures built with the standard ANCF beam element as compared with the earlier implementation.

Existence of positive solutions for the nonlinear elastic beam equation via a mixed monotone operator

In this paper, we use a mixed monotone operator method to investigate the existence and uniqueness of positive solution to a nonlinear fourth-order boundary value problem which describes the deflection of an elastic beam with the left extreme fixed and the right extreme is attached to a bearing device given by a known function. Moreover, we present a concrete example illustrating the result obtained.