Equivalent homogeneous finite element for composite materials via reissner principle. Part I: Finite element for plates

The stiffness matrix in the finite element method for multi-layered materials is generally computed by expressing the strain energy in each layer and adding them together. In order to lower the computing time, which may be prohibitive if the number of layers is high, and to get accurate information on the stresses, especially on transverse shear stresses, we present a new finite element using the Reissner principle. In the first part the case of plates will be detailed: extensions to shell problems will be presented in the second part. The efficiency of the method is tested on a special analytic solution, and some examples are given.     

An assumed strain finite element model for large deflection composite shells

A nine node finite element model has been developed for analysis of geometrically non-linear laminated composite shells. The formulation is based on the degenerate solid shell concept and utilizes a set of assumed strain fields as well as assumed displacement Two different local orthogonal co-ordinate systems were used to maintain invariance of the element stiffness matrix. The formulation assumes strain and the determinant of the Jacobian matrix to be linear in the thickness direction. This allows analytical integration in the thickness direction regardless of ply layups. The formulation also allows the reference plane to be different from the shell midsurface. The results of numerical tests demonstrate the validity and the effectiveness of the present approach.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite element criteria for some shells

Simple solutions are extracted from the exact compatibility equations of inextensional bending for a special class of shell shapes, and it is speculated that these solutions provide useful criteria towards evaluating finite elements for these and closely related shapes. Curved finite elements of a quadratic parametric representation, which do not belong to the special class, are then studied. By using a series of exact inextensional bending displacements, it is concluded from numerical evidence that the criteria apply to finite element shapes in this representation when they are shallow, as is usually the case in practice. Criteria for homogeneous membrane actions follow from Gol'denveizer's static–geometric analogue.     

Elastic postbuckling analysis via finite element and perturbation techniques part II: Application to shells of revolution

The general theory developed in Part 1 of this paper for the finite element stability analysis of structural systems, using perturbation expansions in the vicinity of a critical point, is applied here to the analysis of shells of revolution. The discretization of the shell is performed by means of a semianalytical approximation, and the matrices required for the evaluation of critical points and postcritical equilibrium paths are obtained. Two cases are presented: bifurcation in axisymmetric and in asymmetric buckling modes. The derivatives required for an imperfection analysis are also obtained. A technique of switching between two paths using continuation methods is also discussed, in which the switch is performed using derivatives of the perturbation expansion. Results are presented for bifurcation in axisymmetric and in non-axisymmetric modes, and compared with known solutions or with results from changing the path using continuation methods; good correlation is shown. For structures displaying unstable bifurcation, the influence of load and geometric imperfections is evaluated.     

A mixed finite element formulation for shells through a reduced Reissner's principle in elastodynamics

Summary The use of Mixed models based in Reissner's principle in statics has been found to lead to some desirable simplifications in Finite Element formulations, in particular in plates and shells. Reduced formulations of Reissner's principle such as the one used by Prato have proved to be even more successful. In this paper, a reduction similar to that of Prato is attempted on a mixed elastodynamic variational principle by Karnopp.

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Eine gemischte finite Elemente-Formulierung für Schalen durch ein reduziertes Reissnersches Prinzip der Elastodynamik

Zusammenfassung Die Verwendung von gemischten Modellen basiert auf Reissners Prinzip der Statik führt zu erwünschten Vereinfachungen bei der Formulierung von finiten Elementen im speziellen bei Untersuchungen von Platten und Schalen. Reduzierungen des Reissnerschen Prinzips, wie sie von Prato angewendet worden sind, haben sich sogar als noch erfolgreicher erwiesen. In dieser Untersuchung wird eine Reduktion, ähnlich der von Prato, für ein gemischtes elastodynamisches Variationsprinzip nach Karnopp, vorgenommen.

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Symbol Table A Domain of integration of the Functional. Also area of the triangle - b Second fundamental form of the shell middle surface - C ^ijkl Elastic Constants - E ₁,E ₁ ^* Strain Energy and Co-Energy density - e _ij Elastic strain tensor - f _i Body force density tensor - I _ks Karnopp's functional, specialized to shells - I _ksc Contracted Karnopp's functional, specialized to shells - i, j, k Index 1, 2, 3 - K ₁,K ₁ ^* Kinetic Energy and Co-Energy density - K ^* Kinetic co-energy density for shell - m Moment tensor defined at the mid-surface - n In-plane stress tensor defined at the middle surface - n Qualifier for the boundary normal - p ,p ³ Boundary forces - Prescribed boundary forces - p Shear force tensor defined at the mid-surface - R Position vector of a point in the volume of the shell - r Position vector of a point on the mid-surface - r _i Net impulse density tensor - S _u Portion of the boundary where displacements are preseribed - S Portion of the boundary where forces are prescribed - s Qualifier for the direction tangent to the boundary - t Time variable - t ^ij Stress tensor - u ,u ₃ Mid-surface displacements - Mid-surface velocities - V Volume - v _i Displacement tensor - , Indices. Range 1, 2 - Shear strain tensor for the middle surface - Variation operator - Mid-surface strain tensor - Mid-surface curvature strain tensor - Direction cosine tensor for boundary normal - Mid-surface rotation tensor - Mid-surface angular velocity tensor - _M Strain energy density - _M * Strain co-energy density - _B * Bending strain co-energy density - _TS ^* Transverse shear strain co-energy density - | Covariant differentiation with respect tox , etc - Partial differentiation with respect tox , etc - .(dot) Time differentiation - -(bar) Prescribed quantities     

Nonlinear finite element analysis of laminated composite shells in hygrothermal environments

The large deflection bending behaviour of composite cylindrical shell panels subjected to hygrothermal environments is investigated in this paper. The present finite element formulation considers doubly curved thick shells and includes large deformations with Green–Lagrange strains. The analysis is carried out using quadratic eight-noded isoparametric element and the problem is solved using the incremental modified Newton–Raphson scheme. A parametric study is carried out varying the curvature ratios of composite cylindrical shell panels with simply supported and clamped support conditions.     

Mixed finite element method for axisymmetric shells

A mixed variational formulation is used as basis for developing a mixed finite element method for axisymmetric shell. The independent unknowns of the method are the axial and radial displacement components, the rotation of the normal to the middle surface and the meridional bending stress couple. The basic element is a frustrum of curved meridian. General advantages of the mixed method are presented, one of which is the possibility of using piece-wise linear functions of the meridional arclength to represent the basic unknowns. Test results are presented for plate bending, transverse shear deformation, membrane behaviour, edge-zone bending, bending near the junction of two shells, convergence of the method and accuracy of middle surface curvature interpolation. Shell geometries considered are cylindrical, conical, spherical and ellipsoidal. Good results are obtained which should increase interest in the relatively less known and less tested mixed method as compared to the stiffness method.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part II: Pressure-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.     

An augmented finite element method for modeling arbitrary discontinuities in composite materials

An augmented finite element method (“A-FEM”) is presented that is a variant of the method of Hansbo and Hansbo (Comput Methods Appl Mech Eng, 193: 3523–3540, 2004), which can fully account for arbitrary discontinuities that traverse the interior of elements. Like the method of Hansbo and Hansbo, the A-FEM preserves elemental locality, because element augmentation is implemented within single elements and involves nodal information from the modified element only. The A-FEM offers the additional convenience that the augmentation is implemented via separable mathematical elements that employ standard finite element nodal interpolation only. Thus, the formulation is fully compatible with standard commercial finite element packages and can be incorporated as a user element without access to the source code. Because possible discontinuities include both elastic heterogeneity and cracks, the A-FEM is ideally suited to modeling damage evolution in structural or biological materials with complex morphology. Elements of a multi-scale approach to analyzing damage mechanisms in laminated or woven textile composites are used to validate the A-FEM and illustrate its possible uses. Key capabilities of the formulation include the use of meshes that need not conform to the surfaces of heterogeneities; the ability to apply the augmented element recursively, enabling modeling of multiple discontinuities arising on different, possibly intersecting surfaces within an element; and the ease with which cohesive zone models of nonlinear fracture can be incorporated.     

Elastic postbuckling analysis via finite element and perturbation techniques. Part 1: Formulation

The main equations for the equilibrium, stability and critical state analysis of discrete elastic systems are presented following the works of Thompson, but in such a way that the original set of generalized coordinates and loads are preserved in the Total Potential Energy. This introduces differences in the resulting equations in bifurcation analysis but does not introduce any new feature regarding the physics of the problem. The new formulation is approximated by means of a standard finite element approach based on interpolation of displacements, in which the derivatives of the potential energy are approximated. The terms retained are those of moderately large rotation theory. The energy analysis is finally related to the more conventional finite element notation in terms of stiffness matrices, and it is shown how in such a way it can be included in present day codes. Part 2 of the paper deals with applications to the analysis of shells of revolution using a semi-analytical approximation. Two cases are presented in detail: bifurcation in axisymmetric and in asymmetric modes, and the results show good correlation with those of other authors. The influence of load and geometric imperfections is evaluated.     

An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells

In this work, we present a new efficient four-node finite element for shallow multilayered piezoelectric shells, considering layerwise mechanics and electromechanical coupling. The laminate mechanics is based on the zigzag theory that has only seven kinematic degrees of freedom per node. The normal deformation of the piezoelectric layers under the electric field is accounted for without introducing any additional deflection variables. A consistent quadratic variation of the electric potential across the piezoelectric layers with the provision of satisfying the equipotential condition of electroded surfaces is adopted. The performance of the new element is demonstrated for the static response under mechanical and electric potential loads, and for free vibration response of smart shells under different boundary conditions. The predictions are found to be very close to the three dimensional piezoelasticity solutions for hybrid shells made of not only single-material composite substrates, but also sandwich substrates with a soft core for which the equivalent single layer (ESL) theories perform very badly.     

Finite element modeling of low-velocity impact on laminated composite plates and cylindrical shells

In this study, composite laminates and shell structures subjected to low-velocity impact are investigated by numerical analysis using ABAQUS finite element code. In order to model the impact phenomena by commercial finite element codes, various procedures are available. Accurate modeling requires the appropriate selection of element type, solution method, impactor modeling method, meshing pattern and contact modeling. In this investigation, by considering several case studies with various conditions, validity of the existed modeling processes is examined. In each case, by comparing the results of various methods with the related available experimental test results in existing literature, the best procedure is proposed which can serve as benchmark method in low-velocity impact modeling of composite structures for future investigations.     

Fixed-point iteration to nonlinear finite element analysis. Part II: Formulation and implementation

The finite element formulation and implementation of the Fixed-Point Iteration (FPI) to linear/nonlinear structural static or dynamic analysis are developed. The direct and tangent formulations are presented and compared with the Newton–Raphson method (NRM). ‘Modified’ FPI algorithms have also been derived. A graphical interpretation of the method is introduced and suggested to call the FPI ‘the Saw method’. Mixing both the FPI and NRM is shown to be possible and may be useful in some applications. The overall strategies, iterative algorithms, and the appropriate norm convergence criteria necessary to implant the FPI into existing finite element programs are also included in the development. The superiority of the FPI over the NRM as seen from the development and the formulation lies in three major factors. First, the existing assembly process of element matrices is eliminated completely from the nonlinear finite element analysis. Secondly, the Gauss elimination or Crout's method is also eliminated. In the finite element terminology, this means that nonlinear structural static or dynamic responses can he obtained without recourse to the inverse of the structural stiffness matrix. Thirdly, the FPI can also be applied equally to linear structural analysis. Hence, the assembly process and the programming and storage associated with it can be removed from the existing finite element programs. While the FPI can solve problems that the NRM can, it will also be able to handle some engineering problems where the latter cannot. Buckling problems and problems where the force–displacement curve changes curvature are examples where the FPI is expected to be efficient.     

Free vibration of composite circular cylindrical shells with random material properties. Part II: Applications

In the second part of this study the approach developed in Part I has been used to analyse free vibration of three composite circular cylindrical shells with random scatter in the material properties. The cases considered are – specially orthotropic symmetric shells in axisymmetric and asymmetric oscillations, and antisymmetric cross ply laminated shell in axisymmetric oscillations. With known statistics of the material properties the mean and the variance of the natural frequencies have been obtained. Numerical results have been presented for graphite–epoxy composite shells.     

Finite element simulation of delamination growth in composite materials using LS-DYNA

In this paper, a modified adaptive cohesive element is presented. The new elements are developed and implemented in LS-DYNA, as a user defined material subroutine (UMAT), to stabilize the finite element simulations of delamination propagation in composite laminates under transverse loads. In this model, a pre-softening zone is proposed ahead of the existing softening zone. In this pre-softening zone, the initial stiffness and the interface strength are gradually decreased. The onset displacement corresponding to the onset damage is not changed in the proposed model. In addition, the critical energy release rate of the materials is kept constant. Moreover, the constitutive equation of the new cohesive model is developed to be dependent on the opening velocity of the displacement jump. The traction based model includes a cohesive zone viscosity parameter (η) to vary the degree of rate dependence and to adjust the maximum traction. The numerical simulation results of DCB in Mode-I is presented to illustrate the validity of the new model. It is shown that the proposed model brings stable simulations, overcoming the numerical instability and can be widely used in quasi-static, dynamic and impact problems.     

A rectangular finite element for analysing composite multilayered shallow shells in statics,vibration and buckling

This paper presents a new 32-degree-of-freedom finite element of multilayered composite, moderately thick, shallow shells. The element is a four-node C¹ rectangular element and is built from standard interpolations but with a new kind of kinematics which allows us to exactly ensure the continuity conditions for displacements and stresses at the interfaces between the layers of a laminated structure and the boundary conditions at the upper and lower surfaces of the shell. The transverse shear deformation which is represented by cosine functions is of a higher order and allows us to avoid shear correction factors. The element is evaluated on standard problems and in comparison with exact three-dimensional and analytical solutions for multilayered plates and shells in statics, vibrations and stability.     

A mixed variational principle for finite element analysis

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of modified potential and complementary energy principles. Compatibility and equilibrium are satisfied throughout the domain a priori, leaving only the boundary conditions to be satisfied by the variational principle. This leads to a finite element model capable of relaxing troublesome interelement continuity requirements. The nodal concept is also abandoned and, instead, generalized parameters serve as the degrees-of-freedom. This allows for easier construction of higher order elements with the displacements and stresses treated in the same manner. To illustrate these concepts, plane stress and plate bending analyses are presented.