Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

Analytical models for analysis of automated warehouse material handling systems

Automated storage and retrieval systems (AS/RSs) and autonomous vehicle storage and retrieval systems (AVS/RSs) are two competing technologies for the handling, storage and retrieval of unit-loads in the reserve section of an automated warehouse. In this paper, we model variants of the two systems as an open queuing network (OQN) and use an existing tool for the analysis of OQNs, called the manufacturing system performance analyser (MPA), for analysing the performance of the AS/RS and AVS/RS. Experimental results are provided to show that MPA is a better choice than simulation to quickly evaluate alternate configurations of the two systems. We use MPA to answer a series of design questions for AS/RS and AVS/RS design conceptualisation.     

A nonlinear fractional viscoelastic material model for polymers

In this contribution a test scheme based on tensile tests at different velocities, relaxation experiments and deformation controlled loading and unloading processes with intermediate relaxations has been used to experimentally characterize the nonlinear, inelastic material behavior. Based on the experimental observations a small strain nonlinear fractional viscoelastic material model is derived. In order to use the model within a finite element analysis, the constitutive equations have been generalized for the multiaxial case. The experimental test scheme and the fractional viscoelastic material model are subsequently applied to characterize and compute the mechanical behavior of the thermoplastic Polypropylene. After the identification of the material parameters several uniaxial and multiaxial simulations have been carried out and compared with experimental results.     

Application of fractional derivative models in linear viscoelastic problems

Appropriate knowledge of viscoelastic properties of polymers and elastomers is of fundamental importance for a correct modelization and analysis of structures where such materials are present, especially when dealing with dynamic and vibration problems. In this paper experimental results of a series of compression and tension tests on specimens of styrene-butadiene rubber and polypropylene plastic are presented; tests consist of creep and relaxation tests, as well as cyclic loading at different frequencies. Experimental data are then used to calibrate some linear viscoelastic models; besides the classical approach based on a combination in series or parallel of standard mechanical elements as springs and dashpots, particular emphasis is given to the application of models whose constitutive equations are based on differential equations of fractional order (Fractional Derivative Model). The two approaches are compared analyzing their capability to reproduce all the experimental data for given materials; also, the main computational issues related with these models are addressed, and the advantage of using a limited number of parameters is demonstrated.     

On the dynamic stochastic response of FE models

A numerical procedure to compute the mean and covariance matrix of the response of nonlinear structures modeled by large FE models is presented. Non-white, non-zero mean, non-stationary Gaussian distributed excitation is represented by the well known Karhunen–Loéve expansion, which allows to describe any type of non-white Gaussian excitation in contrast to filtered white noise which might not be easily adjusted to available statistical data. The solution procedure differs considerably from standard methodologies using a state space representation. In the proposed approach, step-by-step integration procedures developed for deterministic FE analysis are applied to compute the first two moments of the stochastic response.     

Universal linear viscoelastic approximation property of fractional viscoelastic models with application to asphalt concrete

The paper presents a comprehensive linear viscoelastic characterization of asphalt concrete using fractional viscoelastic models. For this purpose, it is shown that fractional viscoelastic models are universal approximators of relaxation and retardation spectra. This essentially means that any spectrum can be mathematically represented by fractional viscoelastic models. Characterization of asphalt concrete is performed by constructing the dynamic modulus master curve and determining the parameters of the generalized fractional Maxwell model (GFMM). This procedure is similar to the widely used one of determining the master curve of asphalt concrete using a statistical function such as the sigmoidal model. However, from the GFMM, the relaxation modulus, creep compliance, continuous relaxation spectrum, and Prony series parameters can be determined analytically. A further advantage of the GFMM is that unlike the sigmoidal model, which only gives a representation of either the dynamic modulus or the storage modulus, the GFMM gives a representation of both the storage modulus and loss modulus (and therefore also the dynamic modulus and phase angle). The procedure was successfully applied to ten different mixes used in the State of Virginia.     

Estimation of model errors in the calibration of viscoelastic material models

In this paper we discuss a posteriori error estimation in the context of parameter identification problems. In particular, we pay attention to the errors arising from the discretization of the parameters, which is perceived as a model error. A previously developed method for goal‐oriented a posteriori error estimation is employed for two fundamentally different types of model hierarchies. The numerical results show rather good agreement between estimated and actual errors. Copyright © 2008 John Wiley & Sons, Ltd.     

State-dependent models of material handling systems in closed queueing networks

A comprehensive algorithmic analysis of finite state-dependent queueing models and exponentially distributed workstations is formulated and presented. The material handling system is modeled with finite state-dependent queueing network M/G/c/c models and the individual workstations are modeled with exponentially distributed single and multi-server M/M/c queueing models. The coupling of these queueing models is unique via the material handling structure. The performance modeling of the systems for series, merge, and split and other complex network topologies are included so as to demonstrate the type of topological network design that is possible with these incorporated material handling systems. Of some importance, it is shown that these integrated M/M/c and M/G/c/c networks have a product form when the population arriving at the M/G/c/c queues is controlled. Numerous experimental results demonstrate the efficacy of our approach for a variety of contexts and situations.     

An analytical and numerical study of a dynamically accelerating semi-infinite crack in a linear viscoelastic material

In a previous paper, a general method was presented for constructing the solution to the problem of a semi-infinite, mode III crack propagating dynamically through an infinite, general linear viscoelastic body. The only restrictions placed upon the crack tip speed were that it have constant sign and in magnitude not exceed the glassy shear wave speed. In the present contribution, those previous analytical results are applied to a study of dynamic unsteady crack growth in a linear viscoelastic body. In particular, a numerical algorithm for computing the stress intensity factor is given along with example simulations of running cracks using the Achenbach-Chao viscoelastic model and a stress intensity factor (SIF) fracture criterion. We also compare the transient SIF with the dynamic steady state SIF, and examine the transition to constant crack speed for a dynamically accelerating crack in a viscoelastic material.     

Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models

The dynamic response of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. The mechanical behavior of the viscoelastic material is described by differential constitutive equations with fractional order derivatives. The governing equations, which are derived by considering the dynamic equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacements, whose order is in general greater than two with respect to time derivatives. A method is presented to establish the additional required initial conditions beside the described initial displacements and velocities. Using the Analog Equation Method (AEM) in conjunction with the Domain Boundary Element Method (D/BEM) the governing equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using the numerical method for multi-term FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the dynamic response of plane bodies described by different viscoelastic models.     

On the thermodynamically consistent fractional wave equation for viscoelastic solids

Since the well-known methods for the computation of harmonically induced LAMB waves in elastic plates cannot be applied directly to viscoelastic material, an enhanced material model is developed. It is based on fractional time derivatives and incorporates a fractional KELVIN?CVOIGT model. A proof of the thermodynamic consistency of this model is given. On the basis of the developed material model, the fractional wave equation is derived which allows for the calculation of LAMB waves in viscoelastic solids.     

Marginal analysis guided design justification: a material handling example

The design and justification of an integrated manufacturing system and all of the supporting sub-systems (e.g. material handling system) is a difficult task. One of the most difficult aspects in the design process is determining how and where to proceed. This paper proposes the use of a marginal analysis-directed branch and bound approach as the basis for directing the design process so that a combined economic and performance justification path results. The background and rationale for a marginal analysis guided design approach is given and an example of a material handling system design is presented to illustrate the development of a design justification decision path.     

Experimental and numerical analysis of a bus component in composite material

The aim of this work is the fatigue design of a structural component manufactured in composite materials. The use of composite materials in structural applications implies that the reliability and safety requirements are met, in particular from a point of view of the fatigue life. A bus component is considered: the composite material is obtained by means of the pultrusion technique: the longitudinal reinforcement is glass fibre, while the matrix is polyester resin. At first, a static and fatigue characterization of the pultruded composite is performed to identify the mechanical behaviour along the fibres and in the normal direction, following an extensive testing programme on specimens. During the fatigue tests, the variation of the elastic modulus and the residual strength are monitored to characterize the damage; further experimental tests are performed to fit parameters required for models of fatigue life prevision. Fatigue bending tests are performed on the component, showing crack propagation through the section: a check at the SEM is performed to identify the main internal type of damage. A final comparison with a numerical simulation is proposed: this numerical model will be useful for the optimal fatigue design of the section.     

On the numerical integration of a class of pressure-dependent plasticity models

An unconditionally stable algorithm for the numerical integration of elastoplastic pressure-dependent constitutive relations is analysed in detail in this paper. The application of the method to plane stress problems, in which the out-of-plane strain component is not defined kinematically, is discussed. The tangent moduli resulting from this integration algorithm are obtained by consistent linearization of the elastoplastic constitutive equations. The algorithm is applied to Gurson's constitutive model, some one-dimensional problems are solved, and comparisons with exact solutions are made. The paper closes with a numerical study of the necking of an axi-symmetric specimen using Gurson's plasticity model to describe the constitutive behaviour of the material.     

Experiments on crack propagation in a viscoelastic material

The problem of plane stress crack growth in a rectangular linearly viscoelastic plate under a pair of equal and opposite forces applied to the crack surface is studied experimentally. A fatigue crack is generated at the end of a slot cut on the specimen. After application of two opposite forces to the specimen with a fatigue crack, the crack grows gradually. It is found that under sufficiently small load, the speed of crack growth decreases as time increases, and crack arrest occurs finally. However when the applied load is sufficiently large, the crack grows with increasing speed until a catastrophic failure of the specimen takes place. Between these two extreme cases, the crack grows with an oscillatory speed. A theory based on the models of an infinite plate and double cantilevers is proposed for the explanation of the behaviour of crack growth under various loads. Our experimental results check fairly well with theoretical predictions.