Element Displacement Approximation Research Articles

Evaluating Fracture Toughness of Rolled Zircaloy-2 at Different Temperatures Using XFEM

Fracture toughness and mechanical properties of the zircaloy-2 processed by rolling at different temperatures have been investigated, and simulations have been performed using extended finite element method (XFEM). The solutionized alloy was rolled at different temperatures for different thickness reductions (25–85%). Fracture toughness has been investigated by compact tension test. The improved fracture toughness of the rolled zircaloy-2 samples is due to high dislocation density. SEM image of the fractured surface shows the reduction in dimple sizes with the increase in dislocation density due to the formation of microvoids as a result of severe strain induced during rolling. Compact tension test, edge crack, center crack and three-point bend specimen simulations have been performed by XFEM. In XFEM, the cracks are not a part of finite element mesh and are modeled by adding enrichment function in the standard finite element displacement approximation. The XFEM results obtained for compact tension test have been found to be in good agreement with the experiment.

ESL Based Cylindrical Shell Elements with Hierarchical Shape Functions for Laminated Composite Shells

We introduce higher-order cylindrical shell element based on ESL (equivalent single-layer) theory for the analysis of laminated composite shells. The proposed elements are formulated by the dimensional reduction technique from three-dimensional solid to two-dimensional cylindrical surface with plane stress assumption. It allows the first-order shear deformation and considers anisotropic materials due to fiber orientation. The element displacement approximation is established by the integrals of Legendre polynomials with hierarchical concept to ensure theC0-continuity at the interface between adjacent elements as well asC1-continuity at the interface between adjacent layers. For geometry mapping, cylindrical coordinate is adopted to implement the exact mapping of curved shell configuration with a constant curvature with respect to any direction in the plane. The verification and characteristics of the proposed element are investigated through the analyses of three cylindrical shell problems with different shapes, loadings, and boundary conditions.

Higher‐order extended finite elements with harmonic enrichment functions for complex crack problems

AbstractIn this paper, we analyze complex crack problems in elastic media using harmonic enrichment functions in a higher‐order extended finite element implementation. The numerically computed enrichment function of a crack is the solution of the Laplace equation with discontinuous Dirichlet boundary condition along the crack, and its interaction with branches or other cracks is realized by imposing vanishing Neumann boundary conditions along those cracks. The classical finite element displacement approximation is enriched by adding the enrichment function of a crack through the framework of partition of unity. A nested subgrid mesh is used in the Laplace solve with a rasterized approximation of a crack, which simplifies the numerical integration—no partitioning of finite elements is required. Harmonic enrichment functions readily permit the extension to handle multiple interacting and branched cracks without any special treatment around the junction points. Several numerical examples are presented that affirm the accuracy and effectiveness of the method when applied to complex crack configurations under mixed‐mode loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.

P-Version two-dimensional beam element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear formulation (GNL) based on the total Lagrangian approach for a three-node two-dimensional curved beam element. The hierarchical element approximation functions and the corresponding nodal variables are derived directly from the Lagrange family of interpolation functions. The resulting element displacement approximation is hierarchical and can be of arbitrary and different polynomial orders in the longitudinal and the transverse directions of the beam element and ensures C 0 continuity. The element geometry is described by the coordinates of the nodes located on the axis of the beam (middle surface) and the nodal vectors describing the top and bottom surfaces of the element. The element properties are established using the principle of virtual work and the hierarchical element displacement approximation. In formulating the properties of the element complete two-dimensional stresses and strains are considered hence the element is equally effective for very slender as well as extremely deep beams. Incremental equations of equilibrium are derived and solved using the standard Newton-Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those reported in the literature. The formulation presented here removes virtually all of the drawbacks present in existing GNL beam finite element formulations and has many additional benefits. First, the currently available GNL beam formulations are based on fixed order of approximation for the displacements and thus are not hierarchical and have no provision for changing the order of approximation for the displacements u and v. Secondly, the element displacement approximations in the existing formulations are either based on a linearized displacement field for which a true Lagrangian formulation is not possible and the incremental load step size is severely limited or are based on nonlinear nodal rotation function approach in which case even though the description of the displacement field for the element is exact but additional complications arise due to the noncummutative nature of the nonlinear nodal rotation functions. The p-version displacement approximation used here does not involve the traditional nodal rotations that have been used in the existing beam formulations, thus the difficulties associated with their use are not present in this formulation.

The incremental variational principle and finite element displacement approximation for the frictional contact problem of linear elasticity

The paper presents an incremental variational formulation of the frictional contact problem of two thermoelastic bodies moving together. The proper variational inequality corresponds to a local formulation based on the linear elasticity equations in the non-incremental form and the incremental contact mechanics equations and takes the Hamilton principle as a basis. A question of equivalence of the formulations as well as the existence and uniqueness of the solution are considered. The variational principle is then applied to introduce an appropriate finite element displacement approximation. A basis of the solution algorithm is described. The numerical example is presented.

P‐Version plate and curved shell element for geometrically non‐linear analysis

AbstractThis paper presents a p‐version geometrically non‐linear formulation based on the total Lagrangian approach for a nine node three dimensional curved shell element. The element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction. The element approximation functions and the corresponding nodal variables are derived from the Lagrange family of interpolation functions. The resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures C° continuity.The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete three dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells and plates.Incremental equations of equilibrium are derived and solved using the standard Newton–Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those available in the literature.

P‐version hierarchical three dimensional curved shell element for elastostatics

AbstractThis paper presents a hierarchical three dimensional curved shell finite element formulation based on the p‐approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders pξ, pη and pξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders pξ, pη and pξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders pξ, pη and pξ are a subset of those corresponding to the orders (pξ + 1), (pη +1) and (pξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries.The geometry of the element is described by the co‐ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = −1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.

Three-dimensional curved beam element based on p-Version for dynamics

This paper presents a three-node curved three-dimensional beam element for linear dynamic analysis, where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders, p ξ , p η , and p ζ . This is accomplished by first constructing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in ξ, η and ζ directions using Lagrange interpolating polynomials, and then taking the products of these one-dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three-dimensional beam element are hierarchical. The formulation guarantees C 0 continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element, all six components of the stress and strain tensors are retained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors, representing the nodal cross-sections. The results obtained from the present formulation are compared with available analytical solutions and the h-models using isoparametric three-dimensional solid elements. The formulation is equally effective for very slender as well as deep beams, since no assumptions are made regarding such conditions during the formulation. To demonstrate the effectiveness of this hierarchical formulation in dynamics, numerical results are presented for eigenvalue analysis. The frequencies and the mode shapes are presented for beams with various boundary conditions to demonstrate the accuracy, efficiency, modelling convenience and overall superiority of the present formulation. Numerical results obtained from the present formulation are also compared with those given by the Timoshenko beam theory and other available analytical solutions.

P-version hierarchical axisymmetric shell element

A hierarchical formulation for a three-node axisymmetric shell element based on p-version for linear elastic axisymmetric stress analysis is presented. The element displacement field can be of arbitrary polynomial orders p ξ and p η in the longitudinal (ξ) and the transverse (η) directions of the element. The element approximation functions and the corresponding nodal variables are derived by first constructing the one-dimensional hierarchical approximation functions of order p ξ and p η and the corresponding nodal variables operators in the ξ and η directions and then taking their products (also known as tensor products). The element displacement approximation is hierarchical, i.e. the approximation functions and the nodal variables corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). Since the displacement approximation is hierarchical, the resulting element stiffness matrix and the equivalent nodal load vectors are hierarchical also. The formulation ensures C 0 continuity. The element properties are derived using the principal of virtual work and the hierarchical element approximation. In developing the element properties all four components of the stress ( σ θ , σ r , τ rz , σ z ) and strain ( ε θ , ε r , γ rz , ε z ) are retained, i.e. the stress normal to the middle surface of the element is not neglected (an assumption commonly used in developing axisymmetric shell elements) in the present formulation. The element formulation also permits non-differentiable geometries (sharp corners). The stress concentrations at such locations can be calculated accurately due to the presence of a complete state of stress (and strain) and a non-differentiable geometry feature. Numerical examples are presented to demonstrate the accuracy, efficiency, modeling convenience and overall superiority of the present formulation as well as its applications to pressure vessel stress analysis. The results obtained from the present formulation are also compared with those available in the literature as well as with the analytical solutions. The numerical results from the h- models using isoparametric axisymmetric solid elements are also presented for comparison purposes

Completely hierarchical two-dimensional curved beam element for dynamics

This paper presents a completely hierarchical two-dimensional curved beam element formulation for linear dynamic analysis based on p-version. The element displacement field can be of arbitrary polynomial orders p ξ and p η in the axial and the transverse directions of the element. Since the element displacement approximation is hierarchical, the resulting element stiffness matrix, the element mass matrix and the equivalent nodal load vectors are also hierarchical. The formulation ensures C° continuity. The element properties are derived using the principles of virtual work and the hierarchical element approximation. In the development of the element properties complete two-dimensional state of stress ( δ x, δ y, τ xy) and strain ( ϵ x, ϵ y, γ xy) is utilized, i.e. the stress normal to axis of t neglected (an assumption commonly used in the two-dimensional beam element formulations) in the present formulation. The formulation permits variable p-level control in axial and transverse directions of the element. Thus higher order transverse deformation coupled with higher-order longitudinal effects in deep and short beams can be easily and accurately simulated for dynamic applications. The formulation also permits nondifferentiable geometry (sharp corners) and the dynamic stress concentrations at such locations can be calculated accurately. To demonstrate the effectiveness of this hierarchical formulation in dynamics, numerical results are presented for eigenvalue analysis. The frequencies and the mode shapes are presented for a variety of slender and deep beams with various boundary conditions to demonstrate the accuracy, efficiency, modelling convenience and overall superiority of the present formulation. Numerical results obtained from the present formulation are also compared with those given by Timoshenko beam theory and other available analytical solutions.

Completely hierarchical p-version curved shell element for laminated composite plates and shells

This paper presents a new completely hierarchical three dimensional curved shell finite element formulation for linear static analysis of laminated composite plates and shells. The element displacement approximation can be of arbitrary polynomial orders pξ, pη and pξ in the ξ, η, and ξ directions thereby permitting strains of at least (pξ−1), (pη−1) and (pξ−1) order. The element approximation functions as well as the nodal variables are hierarchical. The element formulation ensures C0 continuity. The lamina properties are incorporated by numerically integrating the element stiffness matrix for each lamina. The formulation has no restriction on either the number of laminas or the layup pattern of the laminas. The geometry of the laminated shell element is described by the coordinates of the nodes lying on the middle surface of the element and the lamina thicknesses at each node. The element formulation is equally effective for very thin as well as very thick laminated plates and curved shells. The results obtained from the present formulation are compared with those available in the literature as well as available analytical solutions.

Hierarchical three dimensional curved beam element based on p-version

This paper presents a three node curved three dimensional beam element for linear static analysis where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders pξ, pη and pζ. This is accomplished by, first constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators in ξ, η and ζ directions using Lagrange interpolating polynomials and then taking the products (also called tensor product) of these hierarchical one dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three dimensional beam element were hierarchical. The formulation guarantees C0 continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element all six components of the stress and strain tensor are ratained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors representing the nodal cross-sections. The results obtained from the present formulation are compared with analytical solutions (when available) and the h-models using isoparametric three dimensional solid elements. The formulation is equally effective for very slender as well as deep beams since no assumptions are made regarding such conditions during the formulation.

Curved shell elements based on hierarchical p‐approximation in the thickness direction for linear static analysis of laminated composites

AbstractThis paper presents a new curved shell finite element formulation for linear static analysis of laminated composite plates and shells, where the displacement approximation in the direction of the shell thickness can be of an arbitrary polynomial order p, thereby permitting strains of at least (p ‐ 1) order. This is accomplished by introducing additional nodal variables in the element displacement approximation corresponding to the Lagrange interpolating polynomials in the element thickness direction. The resulting element displacement approximation has an important hierarchical property i.e. the approximation functions and the generalized nodal variables corresponding to an approximation order p are a subset of those corresponding to an approximation order (p + 1). The element formulation ensures C0 continuity or smoothness of displacements across the interelement boundaries.The element properties (stiffness matrix and equivalent load vectors) are derived using the principle of virtual work and the hierarchical element approximation. The formulation is extended for generally orthotropic material behaviour where the material directions are not necessarily parallel to the global axes. Further extension of this formulation for laminated composites is accomplished by incorporating the material properties of each lamina by numerically integrating the element stiffness matrix for each lamina. The formulation has no restriction on either the number of laminas or the layup pattern of the laminas. Each lamina can be generally orthotropic, and the material directions and the lamina thicknesses may vary from point to point within each lamina. The geometry of the laminated shell element is described by the co‐ordinates of the nodes lying on the middle surface of the element and the lamina thicknesses at each node. The formulation permits any desired order displacement or strain approximation in the shell thickness direction without remodelling.Numerical examples are presented to demonstrate the accuracy, efficiency, and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as the analytical solutions.

Curved shell elements for elastostatics with p-version in the thickness direction

A curved shell finite element formulation for elastostatics is presented in which the element displacement field approximation in the shell thickness direction can be of an arbitrary desired polynomial order p, thereby permitting strain of at least p−1 order. This is accomplished by introducing additional degrees of freedom in the element displacement field approximation corresponding to the Lagrange interpolating polynomials in the direction of the element thickness. These elements have the important hierarchical property, i.e. the element approximation functions, nodal degrees of freedom, stiffness matrix and equivalent nodal load vectors corresponding to an approximation order p are a subset of those corresponding to the approximation order ( p + 1). The formulation ensures C 0 continuity across the interelement boundaries. The element geometry is defined using the coordinates of the nodes lying on the middle surface of the element and the nodal point normals to the middle surface. The element displacement approximation consists of hierarchical approximation functions, nodal displacements and the first and the higher derivatives of the displacements at the nodes corresponding to the Lagrange interpolating polynomials (which are complete) in the direction of the shell thickness. The quadratic functional or the total potential energy functional is constructed using the energy approach. The element properties are derived by substituting the hierarchical element displacement approximation into the quadratic functional and then minimizing it with respect to nodal variables or the nodal degrees of freedom. The resulting element stiffness matrix and the equivalent generalized nodal force vectors due to body forces, distributed pressure loads and thermal loadings are all hierarchical. This formulation permits any desired order of displacement approximation in the direction of the shell thickness without remodeling. Numerical examples are presented to demonstrate the accuracy, efficiency, modeling convenience, faster rate of convergence and overall superiority of the present formulation over the existing shell and plate elements based on conventional theories as well as various higher order deformation theories. The results obtained from the present p-approximation shell finite element models are compared with analytical solutions. The h-approximation results (using three thick shells and three-dimensional solids) are also presented for comparison purposes.

Two-dimensional curved beam element with higher-order hierarchical transverse approximation for laminated composites

A new two-dimensional curved beam element formulation with higher-order transverse shear deformation for linear static analysis of laminated composites is presented. The displacement approximation in the transverse direction can be of arbitrary desired polynomial order p, thereby permitting strains of at least order ( p − 1). This is accomplished by introducing additional nodal degrees of freedom in the element displacement approximation that correspond to Lagrange interpolating polynomials in the transverse direction. The element has an important hierarchical property; the element stiffness matrix, generalized nodal displacement vector and the equivalent nodal load vector corresponding to a polynomial order p are a subset of those corresponding to the approximation order p + 1. The element formulation provides displacement continuity across the interelement boundaries, i.e. C 0 continuity is ensured automatically at the mating element boundaries. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is presented for generally orthotropic material behavior, where the material directions may not be parallel to the global axes, as well as for the laminated composites. The laminated composite material behavior is incorporated in the element formulation through numerical integration for each lamina. For each lamina, the material behavior may be generally orthotropic, and the material direction and lamina thicknesses may vary from point to point within the lamina. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation for laminated composite material behavior. Results obtained from the present formulation are also compared with the available analytical solutions.

P-Version hierarchical two dimensional curved beam element for elastostatics

This paper presents a completely hierarchical two dimensional curved beam element formulation where the element displacement field can be of arbitrary polynomial orders p ξ and p η in the axial and the transverse directions of the element. The approximation functions and the corresponding nodal variables for the beam element are derived by first constructing the hierarchical one dimensional approximation functions of orders p ξ and p η , and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product. This procedure yields approximation functions and nodal variables for the curved beam element that correspond to polynomial orders p ξ and p η in ξ and η directions. The element approximation is hierarchical, i.e. the approximation functions and the nodal variables are both hierarchical. Thus, the element matrix and the load vectors corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The element formulation ensures C 0 continuity. The element properties are derived using the principle of virtual work and the hierarchical element displacement approximation. The element geometry is constructed using the coordinates of the nodes located on the elastic axis of the element and the node point vectors indicating nodal depths and the element width at the nodes. The orders of approximation along the length of the element as well as in the transverse direction can be chosen independently to obtain optimum (maximum) rate of convergence. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation. Results obtained from the present formulation are also compared with h-approximation models and available analytical solutions.