Asymptotic Analysis of Free-Edge and Free-Corner Effects in Laminate Structures

Stress fields in the vicinity of free edges and corners of composite laminates exhibit singular characteristics and may lead to premature interlaminar failure modes like delamination fracture. It is of practical interest to investigate the nature of the arising free-edge and free-corner stress singularities - i.e. the singularity orders and modes - closely. The present investigations are performed using the Boundary Finite Element Method (BFEM) which in essence is a fundamental-solution-less boundary element method employing standard finite element formulations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Simple Model for the Simulation of Delamination in Fiber-ReinforcedComposite Laminates under Mixed-Mode Loading Conditions

For a reliable prediction of the mechanical behavior of unidirectional fiber-reinforced composite laminates (FRCL), it is inevitable to take into account various damage and fracture mechanisms. In this work, delamination under arbitrary mixedmode loading conditions is examined in the framework of the finite element method. Delamination is assumed to be caused by failure of the resin-rich area in the interface between two layers of FRCL's. In this work, a cohesive interface elementin terms of natural stress-strain relationships which allows to describe the interlaminar mechanical behavior of FRCL's is introduced. The proposed model prevents the restoration of cohesion in the interface. The interpenetration of the crack faces is avoided by incorporating a simple contact algorithm. A representative numerical example shows the applicability of the proposed concept. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of the Three-Dimensional Stress State in Composite Shell Structures with Mixed Finite Elements

Being able to compute the complete three-dimensional stress state in layered composite shell structures is essential in order to examine complicated interlaminar failure modes such as delamination. We lay out a mixed finite element formulation with independent displacements, rotations, stress resultants and shell strains. A mixed hybrid shell element with 4 nodes and 5 or 6 nodal degrees of freedom is developed, so that the element formulation can also be used for problems with shell intersections. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Higher-order theories for symmetric and unsymmetric fiber reinforced composite beams with C finite elements

A simple C⁰ isoparametric finite element formulation based on a set of higher-order displacement models for the analysis of symmetric and asymmetric multilayered composite and sandwich beams subjected to sinusoidal loading is presented. These theories do not require the usual shear correction coefficients which are generally associated with the Timoshenko theory. The four-noded Lagrangian cubic element with kinematic models having four, five and six degrees of freedom per node is used. A computer algorithm is developed which incorporates realistic prediction of transverse interlaminar stresses from equilibrium equations. By comparing the results obtained with the elasticity solution and the CPT (classical laminated plate theory) it is shown that the present higher-order theories give a much better approximation to the behaviour of laminated composite beams, both thick and thin. In addition numerical results for unsymmetric sandwich beams are presented which may serve as benchmark for future investigations.     

Analysis of the free-edge effect in composite laminates by the boundary finite element method

Because of the risk of delamination due to high interlaminar stresses in the vicinity of free edges of composite laminates, there is a strong interest in efficient methods for the analysis of this free-edge effect. By the example of a symmetric [0°/90°]_s cross-ply laminate, the Boundary Finite Element Method is presented as a very efficient numerical method, which combines the advantages of the finite element method and the boundary element method. Analogously to the boundary element method, only the boundary is discretized, while the element formulation is finite element based. The resultant stress field is shown to be in very good agreement qualitatively and quantitatively with the comparative finite element analysis. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 355–366, March–April, 2000.     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.     

Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.     

Delamination growth of laminated circular plates under in-plane loads and movable boundary conditions

An investigation is made on interlaminar delamination growth of composite laminated circular plates under in-plane loads and movable delamination boundary conditions. A four-dissociated-region model is developed on the basis of von-Karman plate theory. The model is geometrically nonlinear and the laminated circular plate considered is subjected to axisymmetrical delamination. The effects of transverse shear deformation and contact effect of the delamination on the laminated plates are taking into account in the development of the governing equations of the laminated circular pates with random axisymmetrical delamination. The formulas for describing the total energy release rate and its individual mode components along the delamination front are also derived with considerations of Griffith criterion for fracture. Based on the model established, the delamination growth is numerically studied; and the influences of the parameters such as delamination radii and depths, together with material properties of the plates on the energy release rate are analyzed in detail.     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II： TIME DISCRETIZATION

In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.     

Iterative solution of the drift-diffusion equations

The drift-diffusion model can be described by a nonlinear Poisson equation for the electrostatic potential coupled with a system of convection-reaction-diffusion equations for the transport of charge. We use a Gummel-like process [10] to decouple this system. Each of the obtained equations is discretised with the finite element method. We use a local scaling method to avoid breakdown in the numerical algorithm introduced by the use of Slotboom variables. Solution of the discrete nonlinear Poisson equation is accomplished with quasi-Newton methods. The nonsymmetric discrete transport equations are solved using an incomplete LU factorization preconditioner in conjunction with some robust linear solvers, such as (CGS), (BI-CGSTAB) and (GMRES). We investigate the behaviour of these iterative methods to define the effective strategy for this class of problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical Analysis of a Problem Involving a Viscoelastic Body with Double Porosity

We study from a numerical point of view a multidimensional problem involving a viscoelastic body with two porous structures. The mechanical problem leads to a linear system of three coupled hyperbolic partial differential equations. Its corresponding variational formulation gives rise to three coupled parabolic linear equations. An existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are performed in one and two dimensions to show the accuracy of the approximation and the behaviour of the solution.     

Interlaminar fracture and low-velocity impact of carbon/epoxy composite materials

The interlaminar fracture and the low-velocity impact behavior of carbon/epoxy composite materials have been studied using width-tapered double cantilever beam (WTDCB), end-notched flexure (ENF), and Boeing impact specimens. The objectives of this research are to determine the essential parameters governing interlaminar fracture and damage of realistic laminated composites and to characterize a correlation between the critical strain energy release rates measured by interlaminar fracture and by low-velocity impact tests. The geometry and the lay-up sequence of specimens are designed to probe various conditions such as the skewness parameter, beam volume, and test fixture. The effect of interfacial ply orientations and crack propagation directions on interlaminar fracture toughness and the effect of ply orientations and thickness on impact behavior are examined. The critical strain energy release rate was calculated from the respective tests: in the interlaminar fracture test, the compliance method and linear beam theory are used; the residual energy calculated from the impact test and the total delamination area estimated by ultrasonic inspection are used in the low-velocity impact test. Results show that the critical strain energy release rate is affected mainly by ply orientations. The critical strain energy release rate measured by the low-velocity impact test lies between the mode I and mode II critical strain energy release rates obtained by the interlaminar fracture test. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 195–214, March–April, 2000.     

Numerical elastic-plastic simulation of interlaminar stresses in a notched angle-ply thermoplastic composite laminate

A thermoplastic angle-ply AS4/PEEK laminate with a hole is considered. The interlaminar stresses along the hole edge at different interfaces under uniaxial extension are investigated. According to the symmetries of the structure and loading, a suitable finite-element model is developed. Utilizing a three-dimensional elastic-plastic finite-element procedure elaborated previously, a finite-element modeling of the interlaminar stresses in a thick angle-ply composite laminate is carried out. Based on the interlaminar stresses obtained, the dangerous locations of delamination initiation are predicted. The results obtained indicate that there is some relationship between the dangerous locations and fiber orientations in the adjacent layers, and it maybe inferred that the critical locations are near the regions where the hole edge is tangent to the fiber orientation in the layers adjacent to the interface. The interlaminar stresses at the same interfaces are not sensible to distances from the midplane of the laminate. Very high interlaminar tensile stresses are found to exist on the hole edge at the +25°/+25° or –25°/–25° interfaces, and delaminations can initiate there first. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 427-440, May-June, 2009.     

A Positivity-preserving Finite Element Method for Quantum Drift-diffusion Model

In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.     

Finite element construction for simulating delamination of sandwich composite plates and masses

Displacements and transverse normal stresses in sandwich plates and masses have been approximated by the Ambartsumyan iterative approach to constructing mathematical models of the stress-strain state of sandwich structures. A linear distribution of the displacements in the sandwich structure is set up as the first step of the iterative process, while in the subsequent steps the displacement approximations with higher-order polynomials are obtained. The approximation of the compression stresses is based on Hooke's law using the expression of the tangential displacements in the second step and the normal displacements in the third step of the iterative process. Two shear functions are introduced. The finite element is rectangular and has four nodes. The number of degrees of freedom of finite elements is independent of the quantity of the layers that may be orthotropic. The finite element allows us to simulate delamination by a thin low-modulus interlayer. In doing so, the quantity of the layers increases, while the order of the resolving set of equations does not grow. A number of numerical experiments were carried out. It has been shown that the delamination can greatly increase the level of the stresses in the structure. This effect is especially significant for thin structures. The stresses are somewhat lower when taking into account the interlaminar friction.Submitted to the 10th International Conference on Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 251–263, March–April, 1998.     

A porous thermoviscoelastic mixture problem: numerical analysis and computational experiments

In this paper, we study, from the numerical point of view, a porous thermoviscoelastic mixture problem. The mechanical problem is written as a linear coupled system of two hyperbolic partial differential equations for the porosities and a parabolic partial differential equation for the temperature field. An existence and uniqueness result and an energy decay property are stated. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. A priori error estimates are proved from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations in an academical one-dimensional example and the behaviour of the solutions in one- and two-dimensional problems.     

层合板壳脱层屈曲的有限元分析

通过建立一类新的参考面有限单元,得到适用于分析层合板壳脱层屈曲问题的有限元方法。指出了利用Mindlin假设意义下的变形协调条件,可以将大多数胜任层合板壳分析的一般板壳单元改造为相应的参考面单元。这一方法确保了位移场的合理性和协调条件的满足。为验证参考面单元的有效性,还对壳体脱层屈曲的几个算例作了数值分析。     

On the h-adaptive coupling of FE and BE for viscoplastic and elasto-plastic interface problems

This paper presents a posteriori error estimates for the symmetric finite element and boundary element coupling for a nonlinear interface problem: A bounded body with a viscoplastic or plastic material behaviour is surrounded by an elastic body. The nonlinearity is treated by the finite element method while large parts of the linear elastic body are approximated using the boundary element method. Based on the a posteriori error estimates we derive an algorithm for the adaptive mesh refinement of the boundary elements and the finite elements. Its implementation is documented and numerical examples are included.     

Interlaminar fracture toughness of graphite fiber-reinforced polyimide composites

Conclusion The present study has proved the effectiveness of the application of viscoelastic polymers with increased fracture toughness to graphite/polyimide composites interlaminar fracture toughness improvement. Thermoplastic polysulphone film and thermoresistant structural adhesive have proved to be inherently more effective for composites' delamination resistance growth than maleimide resin toughening and structural modification. The former inevitably results in increase of the honeycomb delamination resistance (Fig. 1) and its durability.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 6, pp. 848–852, November–December, 1994.     

Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion

Under consideration is a 2D-problem of elasticity theory for a body with a thin rigid inclusion. It is assumed that there is a delamination crack between the rigid inclusion and the elastic matrix. At the crack faces, the boundary conditions are set in the form of inequalities providing mutual nonpenetration of the crack faces. Some numerical method is proposed for solving the problem, based on domain decomposition and the Uzawa algorithm for solving variational inequalities.We give an example of numerical calculation by the finite element method.