Finite Element Displacement Formulation Research Articles

To ℘ or not to p – the mixed displacement–pressure p, versus the higher order ℘ displacement finite element formulation, for nearly incompressible linear elasticity

To ℘ or not to p – the mixed displacement–pressure p, versus the higher order ℘ displacement finite element formulation, for nearly incompressible linear elasticity

Geometrically nonlinear Euler–Bernoulli and Timoshenko micropolar beam theories

Two ways of incorporating moderate rotations of planes normal to the axis of a straight beam into the Euler–Bernoulli and the Timoshenko micropolar beam theories are presented. In the first case, the von Karman nonlinear strains are used to incorporate the moderate rotations of normal planes into the beam theories. In the second case, appropriate approximations are made on the nonlinear Cosserat deformation gradient to reflect the condition of moderate rotations of the normal planes. The governing nonlinear differential equations and corresponding natural boundary conditions in both cases are derived using the principle of virtual displacements. A weak-form Galerkin displacement finite element formulation is presented for the developed nonlinear beam theories. The phenomenon of locking usually encountered in beam displacement finite elements is eliminated using higher-order finite elements with nodes located at spectral points. Finally, numerical examples are presented to illustrate the effect of coupling number and bending characteristic length scale on deflections and microrotations when a micropolar beam is modeled with the developed nonlinear beam theories.

Mechanical Characterization and Finite Element Investigation on Properties of PLA-Jute Composite

materials are on high demand in recent days amongst high performance products that need to be strong enough in severe loading conditions. It can be developed with a suitable combination of matrix and reinforcement materials to meet the requirement of specific applications. The present work attempts to form composite plates using a combination of natural polymers and fibers to improve certain properties. The focus is to develop a biodegradable composite of comparable properties with those available commercially. The formed plates are tested to determine different mechanical properties such as tensile strength, impact resistance, hardness etc. The thermal parameters are also determined by DSC (Differential scanning calorimetry) test. A finite element analysis is conducted and the results have been compared with Graphite epoxy composite for checking the effective use of this material. responses along with vibration behavior of laminated composite plates have been extensively investigated. Pandya and Kant (6) has proposed a C0 continuous displacement finite element formulation of laminated composite plates under transverse loading using Higher order Shear Deformation theory. The static behavior of laminated composite and sandwich plates is studied analytically using Navier's technique based on higher order refined theory by Kant and Swaminathan (7). Soltani et al. (8) has analyzed nonlinear tensile behavior of glass fibre reinforced aluminium laminates using finite element modeling. Attallah et al. (9) presented 3- D solutions of laminated composite plates by using a combined finite strip and state space approach. Bharet al. (10) analyzed the significance of the HSDT over the FSDT for laminated composite stiffened plates by using FEM. Grover et al. (11) investigated the static and buckling analysis of laminated composite and sandwich plates in the framework of a new inverse hyperbolic shear deformation theory and this theory is based on shear strain shape function. Kumar et al. (12) studied static analysis of thick skew laminated composite plate with elliptical cutout based on 3-D elasticity theories. Riyah and Ahmed (13 investigated stresses of composite plates with different types of cutouts. Rezaeepazhand and Jafari (14) analytically analyzed the stresses of composite plates with a quasi-square cutout subjected to uniaxial tension based on Lekhnitskii's theory. Static analysis of isotropic rectangular plate with various support conditions and loads are studied by Vanamet al. (15) using finite element analysis (FEA). A recent study detailed the effect of cutout orientation on SCF (16).

The method of fundamental solutions for acoustic wave scattering by a single and a periodic array of poroelastic scatterers

The method of fundamental solutions (MFS) is now a well-established technique that has proved to be reliable for a specific range of wave problems such as the scattering of acoustic and elastic waves by obstacles and inclusions of regular shapes. The goal of this study is to show that the technique can be extended to solve transmission problems whereby an incident acoustic pressure wave impinges on a poroelastic material of finite dimension. For homogeneous and isotropic materials, the wave equations for the fluid phase and solid phase displacements can be decoupled thanks to the Helmholtz decomposition. This allows for a simple and systematic way to construct fundamental solutions for describing the wave displacement field in the material. The efficiency of the technique relies on choosing an appropriate set of fundamental solutions as well as properly imposing the transmission conditions at the air–porous interface. In this paper, we address this issue showing results involving bidimensional scatterers of various shapes. In particular, it is shown that reliable error indicators can be used to assess the quality of the results. Comparisons with results computed using a mixed pressure–displacement finite element formulation illustrate the great advantages of the MFS both in terms of computational resources and mesh preparation. The extension of the method for dealing with the scattering by an infinite array of periodic scatterers is also presented.

Numerical solution of crack problems in gradient elasticity

Gradient elasticity is a constitutive framework that takes into account the microstructure of an elastic material. It considers that, in addition to the strains, second-order derivatives of the displacement also affect the energy stored in the medium. Three different yet equivalent forms of gradient elasticity can be found in the literature, reflecting the different ways in which the second-order derivatives can be grouped to form other physically meaningful quantities. This paper presents a general discretisation of gradient elasticity that can be applied to all three forms, based on the finite-element displacement formulation. The presence of higher order terms requires C1-continuous interpolation, and some appropriate two- and three-dimensional elements are presented. Numerical results for the displacement, stress and strain fields around cracks are shown and compared with available solutions, demonstrating the robustness and accuracy of the numerical scheme and investigating the effect of microstructure in the context of fracture mechanics.

Enhanced Biot's Finite Element Displacement Formulation for Porous Materials and Original Resolution Methods Based on Normal Modes

The use of finite element modeling for porous sound absorbing materials is often limited by the numerical cost of the resolution scheme. To overcome this limitation, an alternative finite element formulation for poroelastic materials modelled with the Biot-Allard theory is first presented. This formulation is based on the solid and total displacement fields of the porous medium. Three resolution methods (one semi-analytical and two numerical) based on normal modes are proposed secondly. These methods take benefit from the decoupling properties of normal modes. The semi-analytical method is associated with problems in which the shear wave can be neglected. The numerical methods are a direct and an iterative scheme. The direct method allows a reduction by 2 of the number of degrees without making any approximation. The iterative method provides an approximation corresponding to a controlled tolerance. The finite element formulation is validated by comparison with an analytical model in two mono-dimensional configurations corresponding to a single and a multilayered problem. The efficiency of the two numerical resolution methods is also illustrated in term of computation time in comparison with classical formulations, such as the mixed displacement-pressure formulation.

SUT-DAM: An integrated software environment for multi-disciplinary geotechnical engineering

As computer simulation increasingly supports engineering design, the requirement for a computer software environment providing an integration platform for computational engineering software increases. A key component of an integrated environment is the use of computational engineering to assist and support solutions for complex design. In the present paper, an integrated software environment is demonstrated for multi-disciplinary computational modeling of structural and geotechnical problems. The SUT-DAM is designed in both popularity and functionality with the development of user-friendly pre- and post-processing software. Pre-processing software is used to create the model, generate an appropriate finite element grid, apply the appropriate boundary conditions, and view the total model. Post-processing provides visualization of the computed results. In SUT-DAM, a numerical model is developed based on a Lagrangian finite element formulation for large deformation dynamic analysis of saturated and unsaturated soils. An adaptive FEM strategy is used into the large displacement finite element formulation by employing an error estimator, adaptive mesh refinement, and data transfer operator. This consists in defining new appropriate finite element mesh within the updated, deformed geometry and interpolating (mapping) the pertinent variables from one mesh to another in order to continue the analysis. The SUT-DAM supports different yield criteria, including classical and advanced constitutive models, such as the Pastor–Zienkiewicz and cap plasticity models. The paper presents details of the environment and includes several examples of the integration of application software.

Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity

We model the progressive localisation of deformation which causes failure around thick-walled cylinders under external radial pressure. The study is based on a second-gradient elastoplastic model developed to regularise the ill-posedness caused by material strain-softening behaviour. The stress increment is related to both the strain increment and its Laplacian. The gradient terms introduce an internal length scale to the material allowing robust modelling of its post-peak behaviour. The numerical implementation is based on a C 1 finite element displacement formulation. Mesh insensitivity in terms of load–displacement and failure mechanism is demonstrated. The internal length in the constitutive equations enables modelling of the scale effect in thick-walled cylinders, according to which the load required to induce failure appears to be much larger for small holes than for large holes.

Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and the point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a "fixed basis function" finite element shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite element matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C0 problem; the extension to three-dimensional problems is straight-forward.

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain-softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C1 continuity element. The required higher-order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill-posedness caused by strain-softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post-peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear-band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values. Copyright © 2001 John Wiley & Sons, Ltd.

A modified first-order-warping theory for shear deformation analysis

A modified first-order-warping theory which satisfies zero transverse shear stress conditions on the bottom and the top of the plate is developed. This theory takes into account the effects of non-linear distribution of the displacements through the thickness and the theory requires no shear correction coefficients. Using only one warping function with satisfaction of the transverse shear stress conditions leads to a high-order theory using only five mid-plane displacement parameters, as the Mindlin theory. A displacement finite element formulation is presented. The results from the present theory are compared to the low-order theory of Mindlin and exact solutions available in the literature.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Numerical modelling of large deformation in metal powder forming

In this paper, the transient dynamic analysis of metal powder during the cold compaction process is simulated by the finite element method based on a ‘Total’ and ‘Updated’ Lagrangian formulation. Since the compaction process involves a very large reduction in volume, the behaviour of the powders is assumed to be that of a rate-independent elasto-plastic material. The process is therefore described by a large displacement finite element formulation for the spatial discretization. A generalized Newmark scheme is used for the time domain discretization and then the final nonlinear equations are solved by a Newton-Raphson procedure. A combination of a Mohr-Coulomb and elliptical cap yield model is utilised as a constitutive model to describe the nonlinear behaviour of powder materials. An incremental elasto-plastic material model is used to simulate the compaction process and a plasticity theory for friction is employed in the treatment of the powder-tooling interface. The interfacial behaviour between the die and powder is modelled by using an ‘interface’ element mesh. Finally, the powder behaviour during the compaction of a plane bush, a cutting tool and a rotational flanged component is analysed numerically. The predictive compaction forces at different displacements, the variation with time of the displacement, relative density and stress contours are obtained. It is shown that the proposed large displacement elasto-plastic finite element approach is capable of simulating the metal powder during compaction.

Collocation in finite element analysis of constrained problems

The concept of three-field displacement finite element formulations of constrained problems is explored in this paper. The approach studied uses collocation within the element domain to enforce the constraints of the problem and resembles mixed formulations because in addition to the approximation of the displacement field, all other auxiliary deformation variables that enter the constraint conditions are interpolated independently. Plate-bending as described by the Reissner–Mindlin theory is used as a model problem for the development of the approach. Two triangular plate elements are formulated based on this methodology, and their performance is investigated using several patch tests. Numerical examples are also considered to study the element behavior under locking conditions (which occur with the classical displacement approach at the thin plate limit). The ability of the elements to overcome locking is established using mathematical arguments and practical examples. The influence of the position of collocation points on the computed results is evaluated through sensitivity studies, with the aim to identify the optimal set. Results are compared with those obtained from the exact solutions and the associated classical displacement model with selective reduced integration of the constrained energy terms. Key words: finite elements, collocation, discrete constraints, mixed methods, thick plates, bending of plates, shear deformations.

A novel isoparametric finite element displacement formulation for axisymmetric analysis of nearly incompressible materials

AbstractIt is well accepted that severe numerical difficulties arise when using the conventional displacement method to analyse incompressible or nearly incompressible solids. These effects are caused by the kinematic constraints imposed on the nodal velocities by the constant volume condition. In elastic‐plastic analysis, these effects are due to a conflict between the plastic flow rule and the finite element discretization. Although several methods have been proposed to cope with this problem, none has been based on the appropriate choice of displacement interpolation functions to minimize the constraints. The theoretical formulation of a new six‐noded isoparametric displacement finite element, which is well suited for elastic‐plastic analysis of axisymmetric constrained solids by using a rational displacement interpolation function, is presented in this paper. The proposed displacement interpolation function implies that the displacement in the axial direction and the product of the displacement in the radial direction and the radius should be treated as two independent basic variables. Alternatively, the proposed displacement interpolation function can also be implemented in a conventional displacement formulation simply by using a modified shape function matrix. The suitability of the proposed formulations is first studied theoretically by assessing the number of degrees of freedom per constraint and then verified by performing numerical experiments on typical boundary value problems which involve incompressible behaviour.

Galerkin Finite Element Displacement Formulation of Coupled Thermoelasticity Spherical Problems

When the time rate of change of the thermomechanical forces in a continuum is high enough to produce stress wave, the solution must be sought through the simultaneous consideration of the first law of thermodynamics and the equations of thermoelasticity. The general finite element formulation of coupled thermoelasticity along with a general discussion for inclusion of mechanical and thermal boundary conditions is presented. The case is then considered for a spherical symmetry and the governing coupled thermoelastic equations are reduced for a thick sphere. Based on the Galerkin method, a Kantrovich approximation is applied to the displacement and temperature field and the finite element formulation of the problem is obtained.

Partial hybrid stress method applied to the higher-order laminated plates theory

A partial hybrid stress method (PHSM) applied to the higher-order plate theory is established for orthotropic composite plates. The midplane stress resultants and midplane generalized strains are included in the flexural part and assumed transverse shear stresses are included in the transverse shear part. Therefore, the governing equations of the plate can be derived variationally from a modified Hellinger-Reissner principle, and this variational principle is illustrated to be consistent in itself in plate sense. This new PHSM plate element is demonstrated to be more accurate than finite element displacement formulation in analysis of thin, moderately thick and thick laminated plates, and a rational through thickness distribution of transverse shear stresses is obtained by present method.

Nonlinear analysis of reinforced concrete thin-walled beams and frames

A method of analysis for three-dimensional reinforced concrete thin-walled beams and frames based upon the finite element displacement formulation is presented. A stiffness matrix for the analysis ofthin-walled beams with an arbitrary open cross-section consisting of longitudinal filaments to represent the concrete and reinforcing steel is derived. The description of the element is based on the assumptions introduced by Vlasov, and a method of describing the warping of a cross-section, which is valid for both straight and curved members with either open or closed cross-sections. An updated Lagrangian formulation is used to take into account the material nonlinearities and the effects of changing structural geometry.

A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C∘ continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.

Finite element analysis of laminated composite plates using a higher-order displacement model

A C° continuous displacement finite element formulation of a higher-order theory for flexure of thick arbitrary laminated composite plates under transverse loads is presented. The displacement model accounts for non-linear and constant variation of in-plane and transverse displacement model eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with nine degrees-of-freedom per node. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form, the theory of elasticity and the finite element solutions with another higher-order displacement model by the same authors. A computer program has been developed which incorporates the realistic prediction of interlaminar stresses from equilibrium equations.