Reconstruction of multidirectional interferometric data using an isoparametric finite-element method

The spatial resolution of tomographic reconstructions is critical when the object field contains large- and small-scale features. Simply increasing the number of elements used in the reconstruction process throughout the domain is generally an unsatisfactory method to achieve higher resolution because additional multiview data are required. Here a new series-expansion reconstruction procedure, based on isoparametric finite-element concepts, is described. This procedure permits the shape and size of the reconstruction elements to be arbitrarily specified. The method is demonstrated by the use of an analytic function and is directly compared with results obtained from other series-expansion methods on a uniform grid. Given identical input data and reconstruction grids, the absolute error of reconstruction is improved by the use of the new method. The advantages of performing the reconstruction of a complex field on a nonuniform grid is also demonstrated.     

A three-dimensional multilayer composite finite element for stress analysis of composite laminates

A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains ε_x, ε_x, ε_xy and three transverse stresses σ_z, σ_yz, σ_xz as the basic variables. The continuity of the transverse stresses σ_z, σ_yz, σ_xz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.     

Representation of singularities with isoparametric finite elements

The development of a generalized quadrilateral finite element that includes a singular point at a corner node is presented. Inter-element conformability is maintained so that monotone convergence is preserved. The global-local concept of finite elements is used to formulate the complete set of equations. Examples of crack tip singularities are given.     

A drafting program for isoparametric finite elements

Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.     

Fracture strength of composite laminates with an elliptical opening

The ultimate strength of composite laminates containing elliptical openings can be predicted reasonably well using two fracture models which utilize the first ply failure strength of the notched and corresponding unnotched laminates. These models have the capability to predict the fracture strength of anisotropic laminates with an opening of general construction and subjected to general in-plane loading. Although the characteristic lengths for the present models are determined empirically, it is found that the characteristic lengths for an elliptical opening of any aspect ratio can be expressed in a closed-form function. These parameters are determined using three (e.g., two circular holes and one crack) or more data points. The experimental result shows that the notched strength of the graphite/epoxy cross-ply laminate is quite sensitive to the opening aspect ratio.     

Multiscale finite element modeling of failure process of composite laminates

A multiscale nonlinear finite element modeling technique is developed in this paper to predict the progressive failure process for composite laminates. A micromechanical elastic–plastic bridging constitutive model, which considers the nonlinear material properties of the constituent fiber and matrix materials and their interaction and the damage and failure in fibrous composites at the fiber and matrix level, is proposed to represent the material behavior of fiber-reinforced composite laminates. The micromechanics constitutive model is employed in the macroscale finite element analysis of structural behavior especially progressive failure process of the fiber-reinforced composites based on a 4-node 24-DOF shear-locking free rectangular composite plate element.     

Formulation of a triangular finite element with an embedded interface via isoparametric mapping

 This paper presents the formulation of a triangular finite element with an embedded interface, designed for the simulation of discrete crack propagation processes. The element is developed within a displacement-based framework. Linear interpolation of the displacement discontinuities along the internal interface is assumed in order to ensure compatibility across inter-element boundaries. The proper representation of the rigid body motions and the solvability of the discretised version of the mechanical problem in point are specifically addressed. Finally, the element performance is illustrated through comparison with some alternative proposals. Received 24 January 2001     

Multi-layer higher-order finite elements for the analysis of free-edge stresses in composite laminates

The analysis of the free-edge stress distributions in composite laminates under uniaxial tension is approached by a finite element technique based on a multi-layer higher-order laminate theory. Several finite elements corresponding to different through-thickness assumed distributions of the displacement unknowns are developed. Numerous stacking sequences are examined in the applications. The results are compared with the ones obtained by various investigators with other modelling approaches. The use of the proposed technique is demonstrated to be simple and effective both for the analysis of in-plane and out-of-plane distributions of intralaminar and, noticeably, interlaminar stress components. © 1998 John Wiley & Sons, Ltd.     

An improved isoparametric transformation for finite element analysis

The standard isoparametric transformation for finite element analysis is shown to be only a special case of a more general transformation which allows for flexibility in locating side and interior element nodes. Side nodes positioned at the same relative distance from corner nodes in both local and global space, and interior nodes positioned such that they do not affect the Jacobian of the co-ordinate transformation, provide improved accuracy over nodes positioned locally without regard for the global configuration. This concept is demonstrated in the context of two-dimensional quadratic quadrilateral isoparametrics, but is also applicable to higher orders and dimensions. In the test cases examined, the new formulation provides a reduction in error of up to three orders of magnitude and also provides accurate solutions for nodal placements which are not allowed using the standard transformations.     

Analytical integration formulae for linear isoparametric finite elements

Exact and approximate analytical expressions can be derived for integrals arising in finite element methods, employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions. The formulae associated with rectangular elements, arbitrarily oriented in space, can be shown to be a special case. The proposed method provides considerable savings in computational effort, in comparison with a numerical method that employs Gaussian quadrature procedures. In addition, the method, when applied to a quadrilateral inscribable in a circle, can be shown to produce better accuracy than the associated (2 × 2) Gaussian quadrature formulae.     

On matrix crack saturation in composite laminates

This paper presents a comparative experimental investigation on matrix crack saturation in the constrained laminae of glass–fibre resin composite laminates. Fatigue tests under load and strain control were conducted on [±θ/90₃]_s and [0/±θ₄/0]_T laminates, respectively. The crack density was periodically measured by using digital image method and optical microscope examination during fatigue tests, to determine crack saturation. It is found that matrix crack saturation occurs in all laminates of different lay-ups under both stress and strain control cyclic loading. The crack density at saturation in the constrained plies is dependent on the cyclic loading type, the off-axis angle of the laminates, as well as the constraining ply. Based on the strain energy release rate, a crack saturation criterion is developed and the experimental findings are qualitatively analysed.     

On refined computational models of composite laminates

Finite element models of the continuum-based theories and two-dimensional plate/shell theories used in the analysis of composite laminates are reviewed. The classical and shear deformation theories up to the third-order are presented in a single theory. Results of linear and non-linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes. Computational modelling issues related to composite laminates, such as locking, symmetry considerations, boundary conditions, and geometric non-linearity effects on displacements, buckling loads and frequencies are discussed. It is shown that the use of quarter plate models can introduce significant errors into the solution of certain laminates, the non-linear effects are important even at small ratio of the transverse deflection to the thickness of antisymmetric laminates with pinned edges, and that the conventional eigenvalue approach for the determination of buckling loads of composite laminates can be overly conservative.     

Sensitivity analysis for dynamic mechanical systems with finite rotations

This paper presents a sensitivity analysis for dynamic systems with large rotations using a semi‐analytical direct differentiation technique. The choice of a suitable time integration strategy for the rotation group appears to be critical for the development of an efficient sensitivity analysis. Three versions of the generalized‐α time integration scheme are considered: a residual form, a linear form, and a geometric form. Their consistency is discussed, and we show that the residual form, which is the most direct extension of the generalized‐α algorithm defined in structural dynamics, should not be used for problems with large rotations. The sensitivity analysis is performed and close connections are highlighted between the structure of the sensitivity equations and of the linearized dynamic equations. Hence, algorithms developed for the original problem can be directly reused for the sensitivities. Finally, a numerical example is analysed in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

On the use of isoparametric finite elements in linear fracture mechanics

Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three-dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three-dimensional problems.     

Compressive failure of finite size unidirectional composite laminates with a region of fibre waviness

The compressive behaviour of finite unidirectional composites with a region of misaligned reinforcement is investigated via finite element analyses. Models with and without fibre bending stiffness are compared, confirming that compressive strength is accurately predicted without modelling fibre bending stiffness for real composite components which typically have waviness defects of several millimetres wavelength. Various defect parameters are investigated. Results confirm the well-known sensitivity of compressive strength to misalignment angle, and also show that compressive strength falls rapidly with the proportion of laminate width covered by the wavy region. A simple empirical equation is proposed to model the effect of a single patch of waviness in finite specimens. Other parameters such as length and position of the wavy region are found to have a smaller effect on compressive strength. The modelling approach is finally adapted to model distributed waviness and thus determine the compressive strength of composites with realistic waviness defects.     

Finite-rotation theories for composite laminates

Summary For the finite-element analysis of arbitrary composite laminates various finite-rotation theories are presented in a single formulation. First a refined theory is derived which allows a quadratic shear deformation distribution across the thickness. The so-called difference vector appearing in the kinematic relations is expressed in terms of rotational degrees of freedom permitting a clear determination of the deformed normal vector in every nonlinear range. The constitutive relations derived are applicable to orthotropic material properties varying arbitrarily across the thickness and to curvilinear laminate coordinates as well. This refined theory is then transformed into simplified theories of Kirchhoff-Love and Mindlin-Reissner types. Finally, the last formulation is used to formulate a layer-wise theory able to grasp the shear deformations very accurately. Kinematic relations and the corresponding constraints are presented in two alternative forms suitable for the application of isoparametric and classical finite-element formulations.     

Stress analysis of imperfect composite laminates with an interlaminar bonding theory

Delamination is a major damage mode in laminated composites since it can cause severe structural degradation. Based on an interlaminar shear stress continuity theory and a linear shear slip theory, a so-called Interlayer Shear Slip Theory was presented in a previous study. This theory was verified to be feasible for shearing-mode delamination analysis. However, in order to account for opening-mode delamination in laminated composites, the continuity of interlaminar normal stress and the modelling of normal separation on the composite interface should also be considered. The present study gives a complete discussion on the Interlaminar Bonding Theory. The effects of interlaminar bonding condition on the laminate deformation and stress distribution are also presented. It is concluded from numerical results that the present theory is suitable for analysis of composite laminates with imperfect interfaces.     

Mesh design in finite element analysis of post-buckled delamination in composite laminates

The role of mesh design in the post-buckling analysis of delamination in composite laminates is addressed in this paper. The determination of the strain energy release rate (SERR) along the crack front is central to the analysis. Frequently, theoretical analysis is limited to treatment of the problem in two dimensions, since considerable complexity is encountered in extending the analysis to three dimensions. However, many practical problems of embedded delamination in composite laminates are inherently three-dimensional in nature. Although in such cases, the finite element (FE) method can be employed, there are some issues that must be examined more closely to ensure physically realistic models. One of these issues is the effect of mesh design on the determination of the local SERR along the delamination front. There are few studies that deal with this aspect systematically. In this paper, the effect of mesh design in the calculation of SERR in two-dimensional (2D) and three-dimensional (3D) FE analyses of the post-buckling behavior of embedded delaminations is studied and some guidelines on mesh design are suggested. Two methods of calculation of the SERR are considered: the virtual crack closure technique (VCCT) and crack closure technique (CCT). The 2D analyses confirm that if the near-tip mesh is symmetric and consists of square elements, then the evaluation of the SERR is not sensitive to mesh refinement, and a reasonably coarse mesh is adequate. Despite agreement in the global post-buckling response of the delaminated part, the SERR calculated using different unsymmetrical near-tip meshes could be different. Therefore, unsymmetrical near-tip meshes should be avoided, as convergence of the SERR with mesh refinement could not be assured. While the results using VCCT and CCT for 2D analyses agree well with each other, these techniques yield different quantitative results when applied to 3D analyses. The reason may be due to the way in which the delamination growth is modeled. The CCT allows simultaneous delamination advance over finite circumferential lengths, but it is very difficult to implement and the results exhibit mesh dependency. Qualitatively, however, the two sets of results show similar distributions of Mode I and Mode II components of the SERR. This is fortunate, since the VCCT is relatively easy to implement.