scispace - formally typeset
Search or ask a question

Showing papers in "Archive of Applied Mechanics in 1998"


Journal ArticleDOI
TL;DR: In this article, a frequency domain analysis of railway track on an elastic halfspace or on a layered halfspace is performed, and the results are compared with those for a simpler model, where ballast and subgrade are considered as a viscoelastic foundation.
Abstract: Vertical dynamic behaviour of a railway track on an elastic halfspace or on a layered halfspace is investigated by a frequency domain analysis. The results are compared with those for a simpler model, where ballast and subgrade are considered as a viscoelastic foundation. In the low- and medium-frequency range up to 250 Hz, great differences are observed between the results of the halfspace model and the results of the viscoelastic foundation model. This is because the damping due to wave propagation and coupling between sleepers cannot be modelled correctly by a viscoelastic foundation. Contradictions observed in the past between measured and calculated results can be explained with the new halfspace model. For frequencies higher than 250 Hz, the influence of the subgrade is negligible, so that here the simpler viscoelastic foundation model can be used.

119 citations


Journal ArticleDOI
TL;DR: In this article, the authors discuss the role and applications of the Hill principle in modern micromechanics of industrial composite materials, including homogeneization of heterogeneous media, definition of effective properties and size effect in heterogeneous materials.
Abstract: We discuss the Hill principle's role and applications in modern micromechanics of industrial composite materials. Uniform boundary conditions, fundamental in micromechanics, are introduced as a class of Hill solutions. Mixed uniform conditions, basic for experimental testing, are analysed. Domains of application of the Hill principle are reviewed, like homogeneization of heterogeneous media, definition of effective properties and size effect in heterogeneous materials. Generalization of the Hill condition is realized for arbitrary materials, in particular for nonlinear inelastic composites with imperfect interfaces.

98 citations


Journal ArticleDOI
TL;DR: In this article, a PEM truss beam coupled with a transmission electrical line is considered, where the coupling is obtained by piezoelectric actuators which act as bars in the module and as capacitances in the electrical line.
Abstract: We call piezoelectromechanical (PEM) truss beam a truss modular beam coupled with a transmission electrical line when the coupling is obtained by piezoelectric actuators which act as bars in the module and as capacitances in the electrical line. The truss module length is assumed negligible with respect to the considered wave lengths. The transmission electrical line is assumed continuously distributed along the truss beam. Applying the method of virtual power as expounded in [2] we formulate a continuum model for PEM truss beams and we prove that there exists a critical value for the transmission electrical impedance in the neighborhood of which the electromechanical modal coupling is maximum and the possible electrical dissipation of mechanical energy is relevant.

84 citations


Journal ArticleDOI
TL;DR: In this article, a dimensionless response number is proposed for the dynamic plastic response of beams and plates made of rigid-perfectly plastic materials subjected to dynamic loading, which is obtained at dimensional reduction of the basic governing equations.
Abstract: A dimensionless number, termed response number in the present paper, is suggested for the dynamic plastic response of beams and plates made of rigid-perfectly plastic materials subjected to dynamic loading. It is obtained at dimensional reduction of the basic governing equations of beams and plates. The number is defined as the product of the Johnson's damage number and the square of the half of the slenderness ratio for a beam; the product of the damage number and the square of the half of the aspect ratio for a plate or membrane loaded dynamically. Response number can also be considered as the ratio of the inertia force at the impulsive loading to the plastic limit load of the structure. Three aspects are reflected in this dimensionless number: the inertia of the applied dynamic loading, the resistance ability of the material to the deformation caused by the loading and the geometrical influence of the structure on the dynamic response. For an impulsively loaded beam or plate, the final dimensionless deflection is solely dependent upon the response number. When the secondary effects of finite deflections, strain-rate sensitivity or transverse shear are taken into account, the response number is as useful as in the case of simple bending theory. Finally, the number is not only suitable to idealized dynamic loads but also applicable to dynamic loads of general shape.

83 citations


Journal ArticleDOI
TL;DR: In this paper, the dynamics of one-mode approximation of an axially moving continuum such as a moving magnetic tape is studied, where the system is modeled as a beam moving with varying speed, and the transverse vibration of the beam is considered.
Abstract: Nonlinear dynamics of one-mode approximation of an axially moving continuum such as a moving magnetic tape is studied. The system is modeled as a beam moving with varying speed, and the transverse vibration of the beam is considered. The cubic stiffness term, arising out of finite stretching of the neutral axis during vibration, is included in the analysis while deriving the equations of motion by Hamilton's principle. One-mode approximation of the governing equation is obtained by the Galerkin's method, as the objective in this work is to examine the low-dimensional chaotic response. The velocity of the beam is assumed to have sinusoidal fluctuations superposed on a mean value. This approximation leads to a parametrically excited Duffing's oscillator. It exhibits a symmetric pitchfork bifurcation as the axial velocity of the beam is varied beyond a critical value. In the supercritical regime, the system is described by a parametrically excited double-well potential oscillator. It is shown by numerical simulation that the oscillator has both period-doubling and intermittent routes to chaos. Melnikov's criterion is employed to find out the parameter regime in which chaos occurs. Further, it is shown that in the linear case, when the operating speed is supercritical, the oscillator considered is isomorphic to the case of an inverted pendulum with an oscillating support. It is also shown that supercritical motion can be stabilised by imposing a suitable velocity variation.

76 citations


Journal ArticleDOI
TL;DR: In this article, the contribution of a regular honeycomb core to the effective in-plane stiffnesses of a sandwich structure is investigated by means of an appropriate closed-form approach, and the resultant effective stiffnesses are derived as a function of the total core thickness.
Abstract: The subject of the consideration is the contribution of a regular honeycomb core to the effective in-plane stiffnesses of a sandwich structure. Due to the coupling of the core displacements with those of the sandwich face sheets, the stiffness contribution of the core is not proportional to its total thickness, as could be expected. The corresponding thickness effect is investigated by means of an appropriate closed-form approach. In doing so, the total elastic core strain energy is calculated based on an adequately chosen displacement representation. Further on, the resultant effective stiffnesses are derived as a function of the total core thickness. A comparative computation of the effective stiffnesses by finite element analysis gives good agreement.

66 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed to couple the beam with a fourth-order transmission line, obtained from the standard one by adding a voltage-driven current generator, thus electrically paralleling the structure of the bending wave equation.
Abstract: A new device to damp mechanical waves in modular truss beams has been proposed in [1]. It is based on the electro-mechanical coupling of the truss beam with an electrical transmission line by a line distribution of PZT actuators. It has been proved in [1] that extensional and torsional waves can be damped using a standard second-order transmission line, and that such a line is not suitable to damp bending waves. In the present paper, we propose to couple the beam with a fourth-order transmission line, obtained from the standard one by adding a voltage-driven current generator, thus electrically paralleling the structure of the bending wave equation. As a detailed description of the system would require huge numerical programming, to test qualitatively the efficiency of the proposed electro-mechanical coupling we consider a coarse continuum model of PiezoElectro-Mechanical (PEM) beams, using an identification procedure based on the principle of virtual power [4]. We define the critical value for line impedance maximizing the electro-mechanical energy exchange for every wave frequency, thus proving that the electric damping of bending waves by distributed PZT control is technically feasible.

61 citations


Journal ArticleDOI
TL;DR: In this paper, the thermoelectroelastic Green's functions for bimaterials subjected to a temperature discontinuity are presented by way of Stroh formalism, and the problem of a crack of arbitrary orientation near a bimmaterial interface between dissimilar thermopiezoelectric material is analyzed.
Abstract: For a two-dimensional piezoelectric plate, the thermoelectroelastic Green's functions for bimaterials subjected to a temperature discontinuity are presented by way of Stroh formalism. The study shows that the thermoelectroelastic Green's functions for bimaterials are composed of a particular solution and a corrective solution. All the solutions have their singularities, located at the point applied by the dislocation, as well as some image singularities, located at both the lower and the upper half-plane. Using the proposed thermoelectroelastic Green's functions, the problem of a crack of arbitrary orientation near a bimaterial interface between dissimilar thermopiezoelectric material is analysed, and a system of singular integral equations for the unknown temperature discontinuity, defined on the crack faces, is obtained. The stress and electric displacement (SED) intensity factors and strain energy density factor can be, then, evaluated by a numerical solution at the singular integral equations. As a consequence, the direction of crack growth can be estimated by way of strain energy density theory. Numerical results for the fracture angle are obtained to illustrate the application of the proposed formulation.

52 citations


Journal ArticleDOI
TL;DR: In this article, the average stress fracture criterion (ASFC) was compared with three nonlocal fracture criteria (NLFC) in application to plane problems: the ASFC, the minimum stress fracture (MSFC) and the fictitious crack fracture (FCFC), each of them may be considered as an equality for a particular form of the general nonlocal strength functional.
Abstract: Comparative analysis has been carried out for three nonlocal fracture criteria (NLFC) in application to plane problems: the average stress fracture criterion (ASFC), the minimum stress fracture criterion (MSFC) and the fictitious crack fracture criterion (FCFC). Each of them may be considered as an equality for a particular form of the general nonlocal strength functional. The criteria contain two material parameters: a characteristic length and the tensile strength (ASFC and MSFC) or the critical stress intensity factor (FCFC). The criteria have been used for a strength description of a plate containing a smooth stress concentrator (circular hole) or a singular stress concentrator (central straight crack). It has been ascertained that ASFC and FCFC lead to identical results for the symmetrically loaded central straight crack. ASFC and MSFC may be successfully used for the description of strength of bodies with smooth as well as singular concentrators, while FCFC gives incorrect predictions for large smooth concentrators and for some other cases. A comparison of the predicted and experimental data has shown that ASFC is preferable in most cases; nevertheless, there exists a systematic deviation of experimental points from the criterion predictions.

41 citations


Journal ArticleDOI
TL;DR: In this article, a gradient-enhanced smeared crack model and bond-slip interface elements are utilized in finite element simulations of reinforced concrete to obtain finitely sized fracture process zones and realistic crack spacings.
Abstract: A gradient-enhanced smeared crack model and bond-slip interface elements are utilized in finite element simulations of reinforced concrete. The crack model is rooted in an enhanced plasticity theory. It uses the Rankine failure surface dependent on an equivalent inelastic strain measure as well as on its Laplacian. As a result, finitely sized fracture process zones and realistic crack spacings are obtained. A reinforced concrete bar in uniaxial tension is analyzed to demonstrate the regularizing influence of the internal length parameter in the model and to evaluate the influence of the model parameters on the energy dissipation in multiple cracks. A comparison of numerical simulations with experimental results for a beam without shear reinforcement in four-point bending concludes the analysis.

37 citations


Journal ArticleDOI
TL;DR: In this paper, a fully saturated two-phase solid or structure subjected to variable, in particular cyclic, external actions is described as a nonhardening poroelastoplastic material with piecewise linearized yield loci.
Abstract: A fully saturated two-phase solid or structure subjected to variable, in particular cyclic, external actions is described as a nonhardening poroelastoplastic material with piecewise linearized yield loci. With reference to a multifield finite element model, sufficient and necessary conditions for shakedown are established by the static Melan's approach. Shakedown analysis by linear programming is briefly discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors focused on the problem of a moving load on a Timoshenko beam-half plane system and analyzed both the subcritical and the supercritical states via a FE-simulation.
Abstract: In this contribution, attention is focused on the problem of a moving load on a Timoshenko beam-half plane system. Both the subcritical and the supercritical state will be analysed via a FE-simulation. The character of the response is explained by the analytical derivation and the elaboration of the eigen-value problem that follows from the characteristic wave equations together with the boundary conditions. It will be demonstrated that also transcritical states can occur. The total number of critical states and the values of the corresponding critical velocities are determined by the beam-half plane stiffness properties as well as the contact conditions.

Journal ArticleDOI
TL;DR: In this article, a hierarchical approach using several mechanical models with different complexities and modeling depths to describe a single engineering system is presented, where the mechanical models are derived from (but not limited to) multibody dynamics.
Abstract: In this paper a hierarchical approach using several mechanical models with different complexities and modeling depths to describe a single engineering system is presented. The mechanical models are derived from (but not limited to) multibody dynamics. The computer power available and improvements in theoretical understanding allow today not only to perform analyses but also to attack the problem of multimodel synthesis. Therefore, hierarchical modeling is used as a basis to analyze simultaneously models with different complexities and different excitations, and to optimize the performance with the most appropriate model for an investigated mechanical effect. Since only one single engineering system is investigated, its different models must be coupled by shared parameters, and the different criteria have to be combined with multicriteria optimization algorithms in order to obtain a single feasible design. An example taken from vehicle dynamics demonstrates the application of the approach.

Journal ArticleDOI
TL;DR: In this paper, the problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem.
Abstract: The problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem. The exact electric boundary conditions at the rim of the hole are introduced in the analysis. Expressions for the elastic and electric variables induced within and outside the cavity are derived in closed form. Hoop stress around the hole and electric fields in the hole are obtained. The limit situation when the hole is reduced to a slit crack is discussed, and the intensity factors for the problem are obtained.

Journal ArticleDOI
TL;DR: In this article, a fast numerical algorithm for calculating the response of a halfspace under any surface loads is presented, which can reduce the calculation time significantly, thus allowing the computation of complex problems.
Abstract: The article presents a fast numerical algorithm for calculating the response of a halfspace under any surface loads. Under certain conditions there exists an analytical solution to the problem in the Fourier domain. To get the desired response, a numerical inverse Fourier transform of this analytic solution has to be made. By using a wavelet decomposition, the proposed algorithm can reduce the calculation time significantly, thus allowing the computation of complex problems. As an example, the response of the beam-halfspace coupled system under moving load is presented.

Journal ArticleDOI
TL;DR: The concept of nonlocal interface residual (NIR) is introduced in nonlocal theory as discussed by the authors, and the nonlocal constitutive equation is used to calculate nonlocal stresses due to edge dislocation on the interface of bi-materials.
Abstract: The basic theory of nonlocal elasticity is stated with emphasis on the difference between the nonlocal theory and classical continuum mechanics. The concept of Nonlocal Interface Residual (NIR) is introduced in nonlocal theory. With the concept of NIR and the nonlocal constitutive equation, we calculate nonlocal stresses due to an edge dislocation on the interface of bi-materials. The nonlocal stress distribution along an interface is quite different from the classical one. Instead of the singularity in the dislocation core, nonlocal stress gives a finite value in the core. A maximum of the stress is also found near the dislocation core.

Journal ArticleDOI
TL;DR: In this paper, the authors investigate the influence of the parametric excitation from the track as well as static and dynamic unbalances of wheels and brake disks on the dynamic response of the wheelset running at various speeds on the track of various average vertical stiffness.
Abstract: In the paper, dynamic interaction between elastic high-speed-train car wheelset and track is studied using a discrete-continuous mechanical model. The model enables to investigate the influence of the parametric excitation from the track as well as static and dynamic unbalances of wheels and brake disks on the dynamic response of the wheelset running at various speeds on the track of various average vertical stiffness. From the results of numerical simulation it follows that particularly severe periodic resonances occur for the track of large stiffness, yielding high vertical dynamic contact forces between the wheels and rails. The maximum dynamic response has been obtained for parameters corresponding to the conditions at which the phenomenon of grumbling noise generation is usually observed in reality.

Journal ArticleDOI
TL;DR: In this paper, the stability of thin films of fluids subject to gravity along inclined planes obeying a power-law constitutive relation of the Ostwald-de Waele type was studied.
Abstract: We study the stability of thin films of fluids subject to gravity along inclined planes, obeying a power-law constitutive relation of the Ostwald-de Waele type A first analysis, in which the inertia terms are ignored, shows such flow to be stable against small, linear perturbations; a second analysis, in which the inertia terms are included, proves that there are stable and unstable regimes that are separated by a critical Ostwald-de Waele number O Numerical computations for selected values of O demonstrate the decay and growth rate behavior of some finite amplitude disturbances

Journal ArticleDOI
TL;DR: In this article, it is shown that not only the shape, but also the ratio of shear-to-bending rigidity of the beams do influence the apparent (phenomenological) Poisson's ratio.
Abstract: Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This fact has been shown to be valid for certain mechanisms, composites with voids and frameworks and has recently been verified for microstructures optimally designed by the homogenization approach. For microstructures composed of beams, it has been postulated that nonconvex shapes (with reentrant corners) are responsible for this effect. In this paper, it is numerically shown that mainly the shape, but also the ratio of shear-to-bending rigidity of the beams do influence the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids, or for composites with irregular shapes of inclusions, even if the constituents are quite usual materials, provided that their porosity is strongly manifested. Elements of the numerical homogenization theory and first attempts towards an optimal design theory are presented in this paper and applied for a numerical investigation of such types of materials.

Journal ArticleDOI
TL;DR: In this article, an integral-type solution to the equilibrium equation is expressed in terms of the eigencurvature, and closed-form solutions of the displacement and corresponding resultant moment are obtained for interior points as well as for exterior points of the ellipse.
Abstract: An infinite plate containing an elliptic subregion in which a uniform eigencurvature is prescribed is analyzed. The problem is formulated by using the classical plate theory. Employing the Maysel's relation, an integral-type solution to the equilibrium equation is expressed in terms of the eigencurvature. Closed-form solutions of the displacement and corresponding resultant moment are obtained for interior points as well as for exterior points of the ellipse. An infinite plate containing an elliptic inhomogeneity in which a uniform eigencurvature is prescribed is also considered. The disturbance of the displacement and corresponding resultant moment due to the inhomogeneity is determined by the equivalent eigencurvature method. Solutions of a circular finite plate with uniform eigencurvature in a circular zone are also obtained analytically.

Journal ArticleDOI
TL;DR: In this paper, the authors developed a rational method for the analysis of an arbitrarily laminated elastic or transversely isotropic hollow sphere under internal and/or external pressure, and solved the problem of a periodically layered sphere consisting of many equal groups of n different thin layers.
Abstract: The aim of this paper is (1) to develop a rational method for the analysis of an arbitrarily laminated elastic, isotropic or transversely isotropic hollow sphere under internal and/or external pressure, (2) to solve the problem of a periodically layered sphere consisting of many equal groups of n different thin layers. The transfer matrix method is used, and exact closed-form solutions are worked out, supplemented by a numerical example. It turns out that by means of the proposed homogenization an originally (periodically) inhomogeneous isotropic sphere is replaced by a homogeneous anisotropic one belonging to the type of spherical symmetric anisotropy.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the influence of the anisotrophy and plastic spin effects on criteria for adiabatic shear band localization of plastic deformation in the presence of the plastic spin.
Abstract: The main objective of the paper is the investigation of the influence of the anisotrophy and plastic spin effects on criteria for adiabatic shear band localization of plastic deformation. A theory of thermoplasticity is formulated within a framework of the rate-type covariance material structure with a finite set of internal state variables. The theory takes into consideration such effects as plastic non-normality, plastic-induced anisotropy (kinematic hardening), micro-damage mechanism, thermomechanical coupling and plastic spin. The next objective of the paper is to focus attention on cooperative phenomena in presence of the plastic spin, and the discussion on the influence of synergetic effects on localization criteria. A particular constitutive law for the plastic spin is assumed. The necessary condition for a localized plastic deformation region to be formed is obtained. This condition is accomplished by the assumption that some eigenvalues of the instantaneous adiabatic acoustic tensor vanish. A procedure has been developed which allows us to discuss two separate groups of effects on the localization phenomenon along a shear band. Plastic spin, spatial covariance and kinematic hardening effects are investigated at an isothermal process in an undamaged solid. In the second case, an adiabatic process in a damaged solid is discussed when the spatial covariance terms and the plastic spin are neglected. Here the thermomechanical coupling, micro-damage mechanism and kinematic hardening effects are examined. For both cases, the criteria for adiabatic shear band localization are obtained in an exact analytical form. Particular attention is focused on the analysis of the following effects: (i) plastic non-normality; (ii) plastic spin; (iii) covariant terms; (iv) plastic strain-induced anisotropy; (v) micro-damage mechanism; (vi) thermomechanical couplings. Cooperative phenomena are considered, and synergetic effects are investigated. A discussion of the influence of the plastic spin, kinematic hardening and covariant terms on the shear band localization conditions is presented. A numerical estimation of the effects discussed is given.

Journal ArticleDOI
TL;DR: In this paper, the stress singularity around the bond edge of a cylindrical joint with two dissimilar materials is analyzed using the Love stress function approach, and an asymptotic description consisting of one singular-stress term and one constant stress term is presented.
Abstract: The stress singularity around the bond edge of a cylindrical joint with two dissimilar materials is analysed using the Love stress function approach. The order of the stress singularity is proved to be the same as that under plane strain deformation. Emphasis is placed on the asymptotic description of the stress field because, in axisymmetric deformation problems, the singular-stress term alone cannot correctly describe the stress field near the singular point, even under mechanical loading at a very small range. An asymptotic description consisting of one singular-stress term and one constant stress term is presented. The constant stress term depends on the r-direction displacement of the singular point, in which it differs from the plane deformation problem's solution.

Journal ArticleDOI
TL;DR: In this paper, a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property is defined by the relation, i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate by the power product form, arbitrarily.
Abstract: The paper deals with a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property, which is defined by the relation \(\), i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate \(\) by the power product form, arbitrarily. Fundamental differential equation for such nonhomogeneous medium has been already proposed in [5]. It is given by a second-order partial differential equation. However, it was found that the fundamental equation is not sufficient in general to solve several kinds of boundary-value problems. On the other hand, it is shown in the present paper making use of the fundamental equations system for a nonhomogeneous medium, which has been proposed in our previous paper [7], it is possible to solve axisymmetric problems for a thick plate (layer) subjected to an arbitrarily distributed load or a concentrated load on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacements stress and components are shown in graphical form.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear finite element procedure for the macroscopic rate-independent analysis of micromechanics of crystalline solids undergoing finite elastic-plastic deformations including plastic volume expanding effects is presented.
Abstract: The present paper deals with a nonlinear finite element procedure for the macroscopic rate-independent analysis of micromechanics of crystalline solids undergoing finite elastic-plastic deformations including plastic volume expanding effects. Particular attention is focused on deviations from the Schmid law of the critical resolved shear stress to model different stress-deflection behavior in uniaxial tension and compression tests. Numerical simulations of elastic-plastic boundary-value problems should demonstrate the efficiency of the algorithm and discuss the influence of different model parameters on stress-carrying and deformation behavior observed experimentally.

Journal ArticleDOI
TL;DR: In this paper, a neural network model is applied to optimization problems of material compositions for a functionally graded material plate with arbitrarily distributed and continuously varied material properties in the thickness direction, and the optimal material composition is determined by taking into account the effect of temperature-dependence of material properties.
Abstract: A neural network model is applied to optimization problems of material compositions for a functionally graded material plate with arbitrarily distributed and continuously varied material properties in the thickness direction. Unsteady temperature distribution is evaluated by taking into account the bounds of the number of the layers. Thermal stress components for an infinite functionally graded material plate are formulated under traction-free mechanical conditions. As a numerical example, a plate composed of zirconium oxide and titanium alloy is considered. In the optimization problem of minimizing the thermal stress distribution, the numerical calculations are carried out making use of the neural network. The optimum material composition is determined by taking into account the effect of temperature-dependence of material properties. The results obtained by neural network and ordinary nonlinear programming method are compared.

Journal ArticleDOI
Baisheng Wu1
TL;DR: In this article, the postbuckling behavior of an elastic column with spring supports of equal stiffness of extensional type at both clamped ends is studied, focusing on those of spring stiffnesses near the critical value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously.
Abstract: The postbuckling behavior of an elastic column with spring supports of equal stiffness of extensional type at both clamped ends is studied. Attention is focused on those of spring stiffnesses near the critical value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. By using the Liapunov-Schmidt-Koiter approach, we show that there are precisely two secondary bifurcation points on each primary postbuckling state for the spring stiffness greater than the critical value. The bifurcation takes place at one of the two least buckling loads. The corresponding secondary postbuckling states connect all the secondary bifurcation points in a loop. For the spring stiffness less than the critical value, no secondary bifurcation occurs. Asymptotic expansions of the primary and secondary postbuckling states are constructed. The stability analysis indicates that the primary postbuckling state for the spring stiffness greater than the critical value is bifurcating from the first buckling load and becomes unstable from a stable state via the secondary bifurcation, i.e., secondary buckling occurs.

Journal ArticleDOI
TL;DR: In this article, an analytical method for computing normal and shear stresses generated in a curved laminated beam under bending loads is presented. But the authors assume that each cross section is symmetrical and loads are applied in the plane of symmetry.
Abstract: This paper outlines an analytical method for computing normal and shear stresses generated in a curved laminated beam under bending loads. Each cross section is assumed to be symmetrical and loads are applied in the plane of symmetry. We build a statically admissible stress field in order to plot normal and shear stress distributions.

Journal ArticleDOI
TL;DR: In this paper, a constitutive model is derived for the isothermal nonlinear viscoelastic response in polymers, which do not possess the separability property and treats a polymer as a system of nonlinear elastic springs (adaptive links), which break and emerge due to micro-Brownian motion of chains.
Abstract: A constitutive model is derived for the isothermal nonlinear viscoelastic response in polymers, which do not possess the separability property. The model is based on the concept of transient networks, and treats a polymer as a system of nonlinear elastic springs (adaptive links), which break and emerge due to micro-Brownian motion of chains. The breakage and reformation rates for adaptive links are assumed to depend on some strain energy density. The viscoelastic behavior is described by an integral constitutive equation, where the relaxation functions satisfy partial differential equations with coefficients depending on the strain history. Adjustable parameters of the model are found by fitting experimental data for a number of polymers in tension at strains up to 400 per cent. To validate the constitutive relations, we consider loading with different strain rates, determine adjustable parameters at one rate of strains, and compare prediction of the model with observations at another rate of strains. Fair agreement between experimental data and results of numerical simulation is demonstrated when the rates of strains differ by more than a decade.

Journal ArticleDOI
TL;DR: In this paper, the authors show that crack kinking and branching may also occur in the quasi-static regime when an isotrophic or equi-biaxial tensile state of stress arises at the tip of a cohesive crack, and may represent alternative itineraries (i.e., path bifurcation) of the fracture process.
Abstract: Experimental, theoretical and numerical investigations show that crack kinking and crack branching can be observed and simulated in brittle solids and in the fast dynamical propagation of quasi-brittle fractures. The present study shows that kinking and branching may also occur in the quasi-static regime when an isotrophic or equi-biaxial tensile state of stress arises at the tip of a cohesive crack, and may represent alternative itineraries (i.e. path bifurcation) of the fracture process. Specific reference is made to the common but meaningful case of the three-point-bending test. Various numerical techniques apt to capture the above occurrence are comparatively presented, and the influence of path bifurcation on the overall behaviour of the specimen is discussed.