Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

A viscoelastic constitutive modeling of rubber-like materials with the Payne effect

This paper aims to present a viscoelastic constitutive model of rubber-like materials, which can capture the Payne effect under dynamic cyclic loadings. The Payne effect is induced by a damage process of bond rupture inside the rubber-like materials, which leads to the storage and loss moduli changing with the dynamic strain amplitude. A viscoelastic constitutive relation is established based on the nonequilibrium thermodynamics for the rubber-like materials by constructing the Helmholtz free energy as the superposition of a hyperelastic model and a convolution viscous model. The neo-Hookean hyperelastic model and the convolution viscous model in terms of the Prony series are then employed in a modification that the material parameters concerned are treated as internal variables and can be identified through a simple but effective approach. At last, the Payne effect is effectively predicted in a good agreement with experimental results.     

On the free‐edge effect in piezoelectric laminated plates

The present paper considers the effect of electromechanical coupling on the interlaminar stresses and the electric field strengths at free edges of laminated plates with piezoelectric material properties. The results of coupled and uncoupled piezoelectric analyses performed by use of the finite element method are compared. Exemplarily, a symmetric cross‐ply and a symmetric angle‐ply laminate are investigated under uniaxial tension and without any electrical loading. It is shown that the interlaminar stresses at the free edge are significantly higher in the coupled case for the symmetric cross‐ply laminate, whereas the coupling effect for the symmetric angle‐ply laminate is of minor significance. In addition, the occurrence of electric field strengths with singular character is revealed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformation criterion of the strength of plastics in biaxial compression

The deformation at break under uniaxial loading is taken as a criterion of the strength of an anisotropic glass-reinforced plastic in biaxial compression. Formulas are obtained for determining the breaking stresses and their limits of applicability are investigated. The calculated values of the breaking stresses are in agreement with the experimental values for both instantaneous and long-term loading.Mekhanika Polimerov, Vol. 4, No. 2, pp. 276–281, 1968     

Experimental Investigations on and Modelling of the Transient Phenomena of the Payne-Effect.

The wide majority of industrially–used rubber is filled with a considerable amount of active fillers like carbon black or silica. Due to this, the material is strengthend and mechanical key features like stiffness and strength are significantly increased. In contrast to unfilled rubber, filled elastomers show a pronounced amplitude dependence, which is widely known as Fletcher–Gent or Payne effect. Besides that, some recently published works show a significant history dependence of this effect with distinctive relaxation phenomena. In the present work, some experiments on typical tyre rubber compounds with focus on these amplitude dependent phenomena are presented. On this basis, an appropriate thermodynamic consistent phenomenological material model of finite viscoelasticity is introduced. In order to incorporate the history dependent phenomena of the amplitude dependence, this model is generalized with intrinsic time scales on the basis of inner structure variables, which are a measure of the materials microstructure. The performance of the model is critically demonstrated by a few simulation results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.     

Reasons for the lowering of the lifetime of polymers under cyclic loading

A model is proposed which explains the lowering of the lifetime of polymers on transition from static to cyclic loading by the interaction of local stresses from an external load with residual stresses which arose after preceding loading cycles. The results predicted on the basis of the model have been checked experimentally on polymethyl methacrylate, and a good agreement has been shown.M. I. Kalinin Leningrad Polytechnic Institute. Translated from Mekhanika Polimerov, No. 2, pp. 279–283, March–April, 1976.     

The effect of overloads on the residual strength and life of laminated GRP

The effect of a short duration cyclic overload on the residual life and strength of laminated glass-fiber reinforced polyester is studied. A uniaxial tensile fatigue loading with the stress ratio 0.1 is considered. The residual life of the composite decreases due to the overload, while the residual strength is almost unaffected. A reasonable agreement of experimental data with the prediction by a residual strength model and by Miner's rule is observed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 701–706, May–June, 1999.     

On the simulation of nonlinear hardening in metallic materials using the concept of representative directions

We generalize a uniaxial model of finite strain viscoplasticity using the concept of representative directions. As a result, a new phenomenological material model is obtained, which can describe the mechanical behavior under arbitrary loading conditions. The original uniaxial model takes the nonlinear isotropic and kinematic hardening into account, but it does not cover the distortional hardening. We show that the isotropic and kinematic hardening is completely retained during the process of generalization. Moreover, the distortional hardening effects are naturally induced by the concept. The resulting material model is validated by a comparison with real experimental data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A network evolution model for the anisotropic Mullins effect

In this contribution, a micro-mechanical constitutive model for filled rubberlike materials is proposed. Rubber network is decomposed into two parts, elastic rubber network and polymer-filler network. As a consequence of filler contribution, the last one is subjected to damage. A non-Gaussian strain energy function for a single chain with a constant valence angle has been postulated. Damage in different directions is governed by the corresponding maximal micro-stretch and is considered as a result of network evolution. Directional network rearrangement as a consequence of network evolution has been employed in order to describe induced anisotropy and permanent set. The model shows good agreement with experimental data on uniaxial tension tests in two orthogonal directions. (© 2008 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model to study the soil arching effect in stone column-supported embankment resting on soft foundation soil

Soil arching is a common phenomena in pile or columnar (vibroconcrete columns, soil–cement columns by mixing or grouting, stone columns) supported geosynthetic-reinforced or unreinforced embankments resting on soft soil. Due to soil arching, stress acting on soft soil or geosynthetic reinforcement decreases and stress on piles or columns increases. In this paper, using mechanical elements (such as spring, dashpot), a generalized mathematical model has been developed to study the soil arching effect in stone column-supported geosynthetic-reinforced and unreinforced embankments resting on soft soil. Pasternak model concept has been used to model the embankment soil. The soft soil has been idealized by spring-dashpot system to include the time-dependent behavior. The stone columns and geosynthetic reinforcement are idealized by stiffer nonlinear springs and rough elastic membrane, respectively. The consolidation effect of soft soil due to inclusions of stone columns has also been included in the model to study its effect on soil arching. Plane strain condition has been considered in the analysis. A finite difference scheme has been used to solve the governing differential equations and results are presented in non-dimensional form. It has been observed that the height of embankment, degree of consolidation of soft soil, stiffness of the stone column material, spacing between the stone columns, use of geosynthetic reinforcement and properties of soft and embankment soils (such as ultimate bearing capacity of soft soil, shear modulus and ultimate shearing resistance of embankment soil) significantly influence the degree of soil arching.     

Numerical analysis of the effect of membrane preloads on the low-speed impact response of composite laminates

This article presents a comprehensive study on the mechanical behaviour of composite laminated plates undergoing a low-speed impact of an external body while they are subjected to in-plane preloads. The effect of such preloading was investigated by means of finite-element analysis of several impact events on laminates with three different span-to-thickness ratios. Tensile and compressive preloads, both uniaxial and biaxial, were considered; in the case of compression, the impact on buckled specimens was also studied. The results obtained show that the span-to-thickness ratio is a fundamental parameter in determining the effect of initial strains. Under a tensile preload, the impact-caused peak stresses were higher than in the case of no preload, and their increment was higher in thicker laminates. Under compression, the most dangerous influence of initial stresses was found at medium span-to-thickness ratios for preloads comparable with the buckling load, whereas, in other cases, negligible or even beneficial effects were observed. These results can justify some experimental findings from the existing literature, even if they were obtained without modelling the material degradation due to damage. Also, they allow us to conclude that the explanation of other phenomena strictly related to damage, as well as an accurate prediction of the extent of damage, requires a failure model.     

Vibration of submerged floating tunnels due to moving loads

The concept of submerged floating tunnel (SFT) has become an increasingly attractive idea to cross the straits. The structural solution in this scheme includes buoyancy force on tunnel body plus tension in mooring tethers. This paper investigates the effect of submergence on the dynamic response of SFT due to moving load. The inertial effect of the fluid is accounted for by evaluating the added mass of tunnel using two and three dimensional models. It is found that fluid–structure interaction increases dynamic amplification of the tunnel deflection (in some cases very significantly). The results show that although the 3D model predicts lesser inertial contribution for surrounding fluid, it is not always possible to associate the larger response with the 2D or 3D models. The discrepancy between the results of the two models decreases as the tether stiffness increases. This indicates that the adoption of Morison’s equation for evaluating the fluid loading on the tunnel is a reasonable assumption when the tether stiffness is high. It is also found that by increasing the tether stiffness, it is possible to introduce a major reduction in the dynamic amplification of the response and by this way control the dynamic response of the SFT.     

Stress Distribution in an Elastic Body with a Periodically Curved Row of Fibers

Within the framework of a piecewise homogeneous body model, with the use of exact three-dimensional equations of elasticity theory for anisotropic bodies, a method is developed for investigating the stress distribution in an infinite elastic matrix containing a periodically curved row of cophasal fibers. It is assumed that fiber materials are the same and fiber midlines lie in the same plane. The self-balanced stresses arising in the interphase in uniaxial loading the composite along the fibers are investigated. The influences of problem parameters on these stresses are analyzed. The corresponding numerical results are presented.     

Microcrack Evolution in Functionally Graded Material under Dynamic Loading

The damage process of metal-ceramic functionally graded material (FGM) is investigated. The microcrack evolution in a layered structure is analyzed using a numerical simulation of stresses and configurational forces. The modelling of an FGM of alumina ceramic and a metallic phase with gradually changing volume fraction of alumina is performed. A structure of two different layers bonded to a substrate is simulated. The stiffness and density of the three materials are varying. The evolution of configurational forces is simulated. The influence of the crack length on the crack driving force is studied for the case of a stress wave loading. The stress loading is applied in the horizontal direction as a dead load. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuous model for investigation of physically nonlinear deformation of orthotropic sandwich plates

A phenomenological yield condition for quasi-brittle and plastic orthotropic materials with initial stresses is suggested. All components of the yield tensor are determined from experiments on uniaxial loading. The reliability estimates of the criterion suggested is discussed. For a plastic material without initial stresses, the given condition transforms into the Marin—Hu criterion. The defining equations of the deformation theory of plasticity with isotropic and “anisotropic” hardening, associated with the yield condition suggested, are obtained. These equations are used as the basis for a highly accurate nonclassical continuous model for nonlinear deformation of thick sandwich plates. The approximations with respect to the transverse coordinate take into account the flexural and nonflexural deformations in transverse shear and compression. The high-order approximations allow us to model the occurrence of layer delamination cracks by introducing thin nonrigid interlayers without violating the continuity concept of the theory. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. pp. 329–340, May–June, 2000.     

Constitutive Modelling of Resins in the Stiffness Domain

An analytic method for inverting the constitutive compliance equations of viscoelasticity for resins is developed. These equations describe the HWKK/H rheological model, which makes it possible to simulate, with a good accuracy, short-, medium- and long-term viscoelastic processes in epoxy and polyester resins. These processes are of first-rank reversible isothermal type. The time histories of deviatoric stresses are simulated with three independent strain history functions of fractional and normal exponential types. The stiffness equations are described by two elastic and six viscoelastic constants having a clear physic meaning (three long-term relaxation coefficients and three relaxation times). The time histories of axiatoric stresses are simulated as perfectly elastic.The inversion method utilizes approximate constitutive stiffness equations of viscoelasticity for the HWKK/H model. The constitutive compliance equations for the model are a basis for determining the exact complex shear stiffness, whereas the approximate constitutive stiffness equations are used for determining the approximate complex shear stiffness. The viscoelastic constants in the stiffness domain are derived by equating the exact and approximate complex shear stiffnesses. The viscoelastic constants are obtained for Epidian 53 epoxy and Polimal 109 polyester resins. The accuracy of the approximate constitutive stiffness equations are assessed by comparing the approximate and exact complex shear stiffnesses. The constitutive stiffness equations for the HWKK/H model are presented in uncoupled (shear/bulk) and coupled forms. Formulae for converting the constants of shear viscoelasticity into the constants of coupled viscoelasticity are given as well.     

On the feasibility of using thermal gradients for active control of interlaminar stresses in laminated composites

In this paper, a method is proposed to actively control interlaminar stresses near the free edges of laminated composites by though-thickness thermal gradients. Theoretical solutions are given for optimal steady-state through-thickness temperature distributions under uniaxial loading that are required to eliminate or reduce the interlaminar stresses below a prescribed level. The optimal solutions are obtained by minimizing appropriate performance indices that are functions of the far-field properties, with respect to the through-thickness temperature differences. In the second part, an experimental investigation is conducted on a glass/epoxy cross-ply laminate with embedded piezoelectric sensors and a thermal heater. Through the experiment, the feasibility of the thermal control of interlaminar stresses is demonstrated.     

Durability of polymers as a function of loading regime during repeated loading with a small number of cycles

Specimens of oriented polycaproamide and unoriented polymethyl methacrylate subjected to uniaxial extension were used to study durability as a function of the number of loading cycles and the intervals between them in different segments of the creep curve. It was shown that the deformation hardening during the initial segment of the creep curve governs the ability of the specimen to withstand subsequent loading cycles. With a relatively low loading rate, deloading-loading cycles during the initial segment of the creep curve lead primarily to an increase in local stresses. During the second stage, repeated loading causes principally an increase in local heating. The interval between loading cycles has only a weak influence on durability.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 252–256, March–April, 1972.     

Vibrational creep of polymeric materials

An experimental study of the effect of vibration on the creep process has been carried out in the case of the rigid porous polyurethane PPU-3, as a function of the magnitude of the vibrational loading and the level of basic static stresses. It has been shown that with increase in the velocity amplitude of the dynamic stresses, the creep process is accelerated, without being accompanied thereupon by vibrational heating of the material. The possibility has been established of approximating vibrational creep curves by the integral equation of Volterra, using a discrete series of relaxation times transformed by the vibro-time analogy method.For Communication No. 3, see [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 223–232, March–April, 1970.