Finite element analysis of viscoelastic structures using Rosenbrock-type methods

The consistent application of the space-time discretisation in the case of quasi-static structural problems based on constitutive equations of evolutionary type yields after the spatial discretisation by means of the finite element method a system of differential-algebraic equations. In this case the resulting system of differential-algebraic equations with the unknown nodal displacements and the evolution equations at all spatial quadrature points of the finite element discretisation are solved by means of a time-adaptive Rosenbrock-type methods leading to an iteration-less solution scheme in non-linear finite element analysis. The applicability of the method will be studied by means of a simple example of a viscoelastic structure.     

Buckling analyses of viscoelastic structures considering ageing and damage effects

The durability of structures made of composite materials is affected by diverse factors, among which creep deformation, ageing and damage. Ageing is related to elapsed time (with different effects in different environmental conditions); damage is related to microstructural modifications due stress and/or strain inputs.The formal definition of these effects and the corresponding constitutive formulations are presented. These formulations are used to obtain analytical solutions and also to implement a Finite Element programme for viscoelastic analysis of laminar structures in a large displacement with small strains context. Then, the effects of ageing and damage on both the short and the long term stability of beam columns are analyzed. Some examples for verification of the formulation and for the comparison of effect are presented.     

Finite element simulations of poly-crystalline shape memory alloys based on a micromechanical model

We present a finite element implementation of a micromechanically motivated model for poly-crystalline shape memory alloys, based on energy minimization principles. The implementation allows simulation of anisotropic material behavior as well as the pseudo-elastic and pseudo-plastic material response of whole samples. The evolving phase distribution over the entire specimen is calculated. The finite element model predicts the material properties for a relatively small number of grains. For different points of interest in the specimen the model can be consistently evaluated with a significantly higher number of grains in a post-processing step, which allows to predict the re-orientation of martensite at different loads. The influence of pre-texture on the material’s properties, due to some previous treatment like rolling, is discussed.     

Finite element analysis of micromechanical failure modes in a heterogeneous ceramic material system

A micromechanical model that provides explicit accounts for arbitrary microstructures and arbitrary fracture patterns is developed and used. The approach uses both a constitutive law for the bulk solid constituents and a constitutive law for fracture surfaces. The model is based on a cohesive surface formulation of Xu and Needleman and represents a phenomenological characterization for atomic forces on potential crack/microcrack surfaces. This framework of analysis does not require the use of continuum fracture criteria which assume, for example, the existence of K-fields. Numerical analyses carried out concern failure in the forms of crack propagation and microcrack formation. Actual microstructures of brittle alumina/titanium diboride (Al₂O₃/TiB₂) composites are used. The results demonstrate the effects of microstructure and material inhomogeneities on the selection of failure modes in this material system. For example, the strength of interfaces between the phases is found to significantly influence the failure characteristics. When weak interfacial strength exists, interfacial debonding and microcrack initiation and growth are the principal mode of failure. When strong interfacial strength is derived from material processing, advancement of a dominant crack and crack branching are observed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Limit analysis of composite materials with anisotropic microstructures: A homogenization approach

A nonlinear mathematical programming approach together with the finite element method and homogenization technique is developed to implement kinematic limit analysis for a microstructure and the macroscopic strength of a composite with anisotropic constituents can be directly calculated. By means of the homogenization theory, the classical kinematic theorem of limit analysis is generalized to incorporate the microstructure - Representative Volume Element (RVE) chosen from a periodic composite/heterogeneous material. Then, using an associated plastic flow rule, a general yield function is directly introduced into limit analysis and a purely-kinematic formulation is obtained. Based on the mathematical programming technique, the finite element model of microstructure is finally formulated as a nonlinear programming problem subject to only one equality constraint, which is solved by a direct iterative algorithm. The calculation is entirely based on a purely-kinematical velocity field without calculation of stress fields. Meanwhile, only one equality constraint is introduced into the nonlinear programming problem. So the computational cost is very modest. Both anisotropy and pressure-dependence of material yielding behavior are considered in the general form of kinematic limit analysis. The developed method provides a direct approach for determining the macroscopic strength domain of anisotropic composites and can serve as a powerful tool for microstructure design of composites.     

A three-dimensional micromechanical model to predict the viscoelastic behavior of woven composites

Polymer matrix based cloth composites are increasingly used in engineering applications. For such composites, significant viscoelastic behavior can be observed for dynamic load conditions. The viscoelastic effect is primarily due to the polymeric matrix used as most of the fibers used in structural applications are elastic. Matrix does not show a major contribution in the axial properties of composites, thus in the traditional modeling its viscoelastic nature is often ignored. However, the effective out of plane properties are influenced by the matrix material and exhibit significant damping characteristics. Therefore, a complete three-dimensional (3-D) model considering the viscoelastic nature of matrix is needed for better understanding of cloth composites. An analytical 3-D micromechanical model based on classical laminate theory (CLT) is verified, in this paper for the prediction of effective elastic and viscoelastic properties of a cloth composite. The method is shown to be accurate. This model is extended to the viscoelastic regime with the use of Laplace transform and correspondence principle. Prony series coefficients for composite cloth are obtained for different volume fractions of fibers in yarn. It is observed from the hysteresis plots that dissipation in out of plane normal and shear modes is significantly higher than the normal directions.     

Finite element analysis of creep fracture initiation in a model superalloy material

Detailed finite element analyses were performed for a single edge-cracked specimen geometry under both plane stress and plane strain constraint for a superalloy material that obeys a power-law creep relationship. The objectives of these analyses were to elucidate the stationary creep crack-tip fields and to provide guidance for the experimental measurement of crack-tip deformations. New results demonstrate that, for both plane stress and plane strain, the angular variations in the creep strain fields do not agree with HRR-type predictions, although the radial variations are in agreement with HRR-type creep strain field predictions in a zone very near the crack tip. Thus, the use of experimental measurement of surface displacement and/or strain data for the location of HRR-type fields may not be possible, unless modifications to the existing HRR-type theory are made. It is also noted that the size of the stress-based HRR-dominance zone is only a fraction of the creep zone size in plane stress, and is very small (especially along =0°) compared to the creep zone size in plane strain. Furthermore, the dominance of the singular strain fields are at least two orders of magnitude smaller than the corresponding stress dominance zones. As such, unless the microstructural features of the material are smaller than the dimensions of the dominance zones, the basis for using stress or strain-based fracture parameters derived from the HRR-type fields for prediction of creep fracture initiation is unclear.     

Viscoelastic failure analysis of composite plates and shells

The present paper describes a numerical method for modeling the failure behavior of composite laminates in the presence of large displacements and creep. Geometrical nonlinear finite elements are used for the discretization. The modeling of material behavior includes thermal, hygroscopic and viscoelastic effects, using an efficient state variable representation. Incremental damage is determined and used to calculate a modified stiffness matrix. Thus, the procedure can be used to analyze, for example, buckling, creep buckling and creep buckling including damage. A detailed formulation, computational procedures and examples to check the accuracy of the code are presented.     

Implementation of a constitutive micromechanical model for damage analysis in glass mat reinforced composite structures

A micromechanical model based on a probabilistic approach is implemented in the finite element code CASTEM 2000 to develop numerical simulations that efficiently predict the overall damaged behaviour of random oriented fibre composites. The proposed damage constitutive model is based upon the generalised Mori and Tanaka scheme and Eshelby's equivalence theory. Damage mechanisms occurring at each composite constituent (fibres, matrix and interface) are associated to Weibull probabilistic functions to model their onset and progressive growth at the microscopic scale level. It is obvious that the damaged behaviour of the composite material depends widely on the microscopic material parameters (fibre length, fibre volume fraction, fibre orientation, …). On one hand, the micromechanical model uses homogenisation techniques which enabled us to link these microscopic parameters to the material behaviour and to evaluate explicitly their influences. On the other hand, the implementation of the derived behaviour law into a finite element code enabled us to reflect on the effect of these microscopic parameters on the overall response of a simple composite structure presenting heterogeneous stress fields. In fact, the damage evolution in each constituent (local scale) and the related stiffness reduction are estimated at any material point (integration point) or node of the considered structure subject to a specific loading. Numerical simulations of a composite plate with a hole under in-plane tension were performed to validate the implementation of the behaviour law. Numerical results have been compared to experimental curves and damage evolutions monitored by acoustic emission techniques. Simulations agree well with experimental results in terms of damage onset and growth.     

Effective mechanical properties of EM composite conductors: an analytical and finite element modeling approach

An analytical model and numerical approach to predict the effective mechanical properties of a composite conductor consisting of metallic core and insulation layers are presented in this paper. The analytical model was developed based on a two-step homogenizations and mechanics analysis for composite unit cell. The Step 1 homogenization derives the effective properties of the out-wrapped composite insulation layers. The Step 2 homogenization further smears the metallic core and the effective composite insulation layers to develop homogenized mechanical properties for composite conductor according to appropriate homogenization sequences. The procedure of using numerical approach and finite element method to determine the unit cell effective constants were also described and the results of the FEA prediction were presented. The analytical predictions were compared well to the numerical results for the nine material constants that characterize the effective mechanical properties of the composite conductor.     

Nonlinear hygro-thermo-elastic vibration analysis of doubly curved composite shell panel using finite element micromechanical model

Nonlinear free vibration behavior of laminated composite curved panel under hygrothermal environment is investigated in this article. The mathematical model of the laminated panel is developed using Green–Lagrange-type geometrical nonlinearity in the framework of higher-order mid-plane kinematics. The corrugated composite properties are evaluated through the micromechanical model and all the nonlinear higher-order terms are included in the present model for the sake of generality. The equation of vibrated panel is obtained using Hamilton's principle and discretized with the help of the finite element steps. The solutions are computed numerically using the direct iterative method. The effect of parameters on the nonlinear vibration responses is examined thoroughly by solving the wide variety of numerical examples.     

Finite element analysis of textile composite preform stamping

The forming or draping of a textile composite preform may result in large changes in the fibrous microstructure of the preform. This change in the local fiber orientation leads to significant changes in the fabric permeability as well as the mechanical properties of the ensuing composite structure. Therefore, this change in orientation of the tows of the preform needs to be known accurately to calculate the various effective properties of the composite. A new finite element approach for stamping analysis of a plain-weave textile composite preform has been developed. This model is simple, efficient and can be used in the existing finite element codes. The model represents the preform as a mesh of 3-D truss elements and 3-D shell elements. The truss elements model the tows, which are allowed to both scissor and slide relative to one another. The shell elements represent a fictitious material that accounts for inter-tow friction and fiber angle jamming. The model takes into account large strains and large deformations. In-plane uniaxial tension tests have been performed on plain-weave specimens for determining the constitutive law of the transforming medium and to show the inter-tow sliding. Application of the model is demonstrated by simulating the stamping of a preform by a spherical punch. The results from the simulation show good correlation with results from the experiments.     

Finite element analysis of composite revolution structures wound by wide plies

An isoparametric axisymmetrical finite element is developed for the analysis of multilayer composite structures wound by wide plies. This element takes into account the variation of angle along the width of a layer for a given parallel of a revolution structure. Evaluation of strain and stress tensors is performed in mean and extreme fibre directions of a ply. Moreover, this model can represent geometrical nonlinear behaviour and material nonlinearities. The extended possibilities of this model are illustrated for two aerospace structures: a spherical wound satellite tank and the composite head of a rocket motor.     

Finite element analysis and optimization of composite wheelchair wheels

A methodology for the design of Paralympics wheelchair wheels has been developed and provided a framework for comparison between design solutions. Finite element analysis was used as a tool to develop an understanding of wheel design, provide a basis for evaluation of design solutions, and as a mean of design optimisation. A finite element model of an existing wheel design was created and analysed. It was found that the pushrim was significantly over designed for the application. As such, wheel improvement focused on redesign of this component. A number of design solutions were analysed and compared based on weight, rotational moment of inertia, side profile surface area, and wheel stiffness. From this, it was decided that a thin pushrim attached via spokes to the rim and the hub in a tangential spoking pattern would offer the best solution.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

Finite element dynamic stability analysis of laminated composite skew plates containing cutouts based on HSDT

The finite element dynamic stability analysis of laminated composite skew structures subjected to in-plane pulsating forces is carried out based on the higher-order shear deformation theory (HSDT). The two boundaries of the instability regions are determined using the method proposed by Bolotin. The numerical results obtained for square and skew plates with or without central cutout are in good agreement with those reported by other investigators. The new results for laminated skew plate structures containing cutout in this study mainly show the effect of the interactions between the skew angle and other various parameters, for example, cutout size, the fiber angle of layer and thickness-to-length ratio. The effect of the magnitude of the periodic in-plane load on the dynamic instability index is also investigated.     

The performance of unreinforced masonry walls subjected to low-velocity impacts: Finite element analysis

A simplified discrete-crack finite element modelling approach has been developed to model the performance of unreinforced brickwork and blockwork masonry walls subject to out-of-plane impacts. The approach involves the use of linear elastic solid elements for masonry units in conjunction with a specially formulated contact interface model for masonry joints. Key features of the latter include: (i) a Mohr–Coulomb failure criterion; (ii) a cohesive crack model for initial fracture; (iii) inclusion of dilatancy. The contact interface model has been implemented in LS-DYNA, a three-dimensional non-linear explicit finite element program. The modelling approach was used to simulate the behaviour of a series of unreinforced walls tested previously in the laboratory. It was found that the dynamic response of full-scale masonry walls could be predicted with reasonable accuracy. However, parametric studies showed that wall response was highly dependent on small changes in loading impulse, base friction, fracture energy, joint failure stress and angle of dilatancy.     

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler–Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.     

Time-dependent boundary element analysis for an interface crack in a two-dimensional unidirectional viscoelastic model composite

Interfacial stress singularities in a unidirectional two-dimensional laminate model consisting of an elastic fiber and a viscoelastic matrix have been investigated using the time-domain boundary element method. First, the interfacial singular stresses between the perfectly bonded fiber and the matrix of a unidirectional laminate subjected to a uniform transverse tensile strain have been investigated near the free surface, but without any edge crack. Such stress singularity might lead to fiber-matrix debonding or an edge crack. Then, the overall stress intensity factor for the case of a small interfacial edge crack of length a has been computed. The numerical procedure does not permit calculation of the limiting case for which the edge crack length vanishes.     

Implicit iterative finite element scheme for a strain gradient crystal plasticity model based on self-energy of geometrically necessary dislocations

In this study, an implicit iterative finite element scheme is developed for the strain gradient theory of single-crystal plasticity that accounts for the self-energy of geometrically necessary dislocations (GNDs). This strain gradient theory belongs to the Gurtin framework for viscoplastic single-crystals. The self-energy of GNDs gives a specific form of energetic higher-order stresses. An implicit finite element equation is obtained for solving a set of homogenization equations. The developed scheme is employed to analyze a model grain, and is verified by comparison with the analytical estimation derived by Ohno and Okumura (2007) [4]. The computational efficiency of the scheme and the incremental stability are discussed. Furthermore, it is shown that the developed scheme is available and applicable to different types of higher-order stresses including energetic and dissipative terms.