Constant Strain Triangle Research Articles

Computational modeling of cracking in cortical bone microstructure using the mesh fragmentation technique

The cortical bone is a hierarchical composite material that, at the microscale, is segmented in an interstitial matrix, cement line, osteons, and Haversian canals. The cracking of the structure at this scale directly influences the macro behavior, and, in this context, the cement line has a protagonist role. In this sense, this work aims to simulate the crack initiation and propagation processes via cortical bone microstructure modeling with a two-dimensional mesh fragmentation technique that captures the mechanical relevance of its constituents. In this approach, high aspect ratio elements are inserted between the regular constant strain triangle finite elements to define potential crack paths a priori. The crack behavior is described using a composed damage model with two scalar damage variables, which is integrated by an implicit-explicit (Impl-Ex) scheme to avoid convergence problems usually found in numerical simulations involving multiple cracks. The approach’s capability of modeling the failure process in cortical bone microstructure is investigated by simulating four conceptual problems and one example based on a digital image of an experimental test. The results obtained in terms of crack pattern and failure mechanisms agree with those described in the literature, demonstrating that the numerical tool is promising to simulate the complex failure mechanisms in cortical bone, considering the properties of its distinct phases.

Damage identification of thin plates using multi-stage PSOGSA and incomplete modal data

ABSTRACT The present work proposes a multi-stage optimization technique for damage detection of the thin plate-like structure using sparse modal data collected from the limited sensors. The response of the thin plate structures is obtained from a finite element model, which is developed with constant strain triangle elements. The iterated improved reduction system (IIRS) is applied to simulate the state of sparse sensors numerically. Next, the damage identification problem is formulated as a constrained optimization problem. A weighted linear combination of the natural frequencies and modes shapes of the structure is used to construct the objective function. The damage extent of each element is considered as the design variable. The formulated objective function is minimized by the proposed multi-stage strategy employing a hybrid metaheuristic algorithm, which is based on particle swarm optimization and gravitational search algorithm (PSOGSA). The algorithm identifies the healthy elements in each stage and eliminates them from the search space of the next stage. Thus, the search space dimension of the optimization problem reduces in each stage, and actual damaged elements are identified effectively. The efficiency of the proposed method is demonstrated through three numerical examples with different damage scenarios.

Fracture Toughness Determination on an SCB Specimen by Meshless Methods

This work investigates fracture characteristics of a marble semi-circular bend (SCB) specimen with a pre-defined crack under a compressive loading condition. It aims at evaluating how the fracture toughness can be affected by the crack and span length variation. Numerically, the model is solved using meshless methods, extended to the linear elastic fracture mechanics (LEFM), resorting to radial point interpolation method (RPIM) and its natural neighbor versions (NNRPIMv1 and NNRPIMv2). Alternatively, to validate the meshless method results, the problem is resolved following the finite element method (FEM) model based on the standard 2D constant strain triangle elements. As a result, fracture toughness and the critical strain energy release rate are characterized following the testing method on the cracked straight through semi-circular bend specimen (CSTSCB). A comparison is drawn amongst the theoretical, meshless methods and FEM results to evaluate the capability of advanced numerical methods. Encouraging results have been accomplished leading to validate the supporting numerical methodologies.

Robust flat‐facet triangular shell finite element

AbstractA flat facet finite element with three nodes and six degrees of freedom per node for linear static and dynamic analysis of thick and thin shells is presented. The membrane response is modeled with constant strain triangles. The bending response treats the transverse shear with an improved version of the discrete‐shear‐gap technique. The main novelty of the present approach is the robust treatment of the drilling rotations: the element internally responds to five degrees of freedom per node, but externally it naturally possesses six degrees of freedom per node. The drilling degree of freedom is decoupled from the bending and twisting, and the undesirable user‐selectable arbitrary stabilization is avoided. The element is shown to be robust for extremely thin shells, and passes tests such as the Raasch hook problem. Performance is illustrated with static and dynamic examples including branched and folded shells.

Mesh-independent damage model for trabecular bone fracture simulation and experimental validation.

We propose in this study a two-dimensional constitutive model for trabecular bone combining continuum damage with embedded strong discontinuity. The model is capable of describing the three failure phases of trabecular bone tissue which is considered herein as a quasi-brittle material with strains and rotations assumed to be small and without viscous, thermal or other non-mechanical effects. The finite element implementation of the present model uses constant strain triangle (CST) elements. The displacement jump vector is implicitly solved through a return mapping algorithm at the local (finite element) level, while the global equilibrium equations are dealt with by Newton-Raphson method. The performance, accuracy and applicability of the proposed model for trabecular bone fracture are evaluated and validated against experimental measurements. These comparisons include both global and local aspects through numerical simulations of three-point bending tests performed on 10 single bovine trabeculae in the quasi-static regime.

A constant strain triangle element oriented multi-material topology optimization with a moved and regularized Heaviside function

This study presents an optimal topology material distribution method in the framework of minimum compliance with a constraint on the total amount of multi-material using constant strain triangle (CST) elements and Moved and Regularized Heaviside Function (MRHF) filters. The sensitivity formulations for objective function and sensitivity for structures are derived in terms of multiphase design variables through triangle elements. Mathematical formulations of topology optimization problem solving the minimum compliance by using multiple materials are an alternating active-phase algorithm with a Gauss-Seidel version as an optimization model of optimality criteria. Moreover, MRHF that has the role of a filter in multiple materials is considered to produce obvious material distributions and improve the convergence of objective values. Some optimal topology results under the influence of rmin and filter are also investigated and verify the CST element-based multi-material topology optimization is appropriate to the use of MRHF and produces reasonable optimal results.

ANALYSIS OF THE STATE OF STRESSES AND STRAINS IN THE CURVED STILT, USING TRIANGULAR FINITE ELEMENTS (CST)

The paper presents a study on the analysis of the distribution of stresses and strains in the curved stilt of the plough-body, using the Finite Element Method. Modeling was performed with the help of triangular finite elements "CST" (Constant Strain Triangle), using a special program for this type of applications “CSTPL.EXE”, developed at University Politehnica of Bucharest, Department of Materials Strength.

Form-finding of shell structures generated from physical models

Vector form intrinsic finite element is a recently developed and promising numerical method for the analysis of complicated structural behavior. Taking the cable-link element as example, the framework of the vector form intrinsic finite element is explained first. Based on this, a constant strain triangle element is introduced, and relevant required equations are deduced. Subsequently, the vector form intrinsic finite element is successfully applied to carry out form-finding of shells generated from physical models, such as hanging models, tension models, and pneumatic models. In addition, the resulting geometries are analyzed with finite element method, thus demonstrating that a dominant membrane stress distribution arises when the shell is subjected to gravitational loading.

Consistent multiscale analysis of heterogeneous thin plates with smoothed quadratic Hermite triangular elements

A consistent multiscale formulation is presented for the bending analysis of heterogeneous thin plate structures containing three dimensional reinforcements with in-plane periodicity. A multiscale asymptotic expansion of the displacement field is proposed to represent the in-plane periodicity, in which the microscopic and macroscopic thickness coordinates are set to be identical. This multiscale displacement expansion yields a local three dimensional unit cell problem and a global homogenized thin plate problem. The local unit cell problem is discretized with the tri-linear hexahedral elements to extract the homogenized material properties. The characteristic macroscopic deformation modes corresponding to the in-plane membrane deformations and out of plane bending deformations are discussed in detail. Thereafter the homogenized material properties are employed for the analysis of global homogenized thin plate with a smoothed quadratic Hermite triangular element formulation. The quadratic Hermite triangular element provides a complete C1 approximation that is very desirable for thin plate modeling. Meanwhile, it corresponds to the constant strain triangle element and is able to reproduce a simple piecewise constant curvature field. Thus a unified numerical implementation for thin plate analysis can be conveniently realized using the triangular elements with discretization flexibility. The curvature smoothing operation is further introduced to improve the accuracy of the quadratic Hermite triangular element. The effectiveness of the proposed methodology is demonstrated through numerical examples.

Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석

The finite element analysis (FEA) is a numerical technique to find solutions of field problems. A field problem is approximated by differential equations or integral expressions. In a finite element, the field quantity is allowed to have a simple spatial variation in terms of linear or polynomial functions. This paper represents a review and an accuracy-study of the finite element method comparing the FEA results with the exact solution. The exact solutions were calculated by solid mechanics and FEA using matrix stiffness method. For this study, simple bar and cantilever models were considered to evaluate four types of basic elements - constant strain triangle (CST), linear strain triangle (LST), bi-linear-rectangle(Q4),and quadratic-rectangle(Q8). The bar model was subjected to uniaxial loading whereas in case of the cantilever model moment loading was used. In the uniaxial loading case, all basic element results of the displacement and stress in x-direction agreed well with the exact solutions. In the moment loading case, the displacement in y-direction using LST and Q8 elements were acceptable compared to the exact solution, but CST and Q4 elements had to be improved by the mesh refinement.

An embedded crack in a constant strain triangle utilizing extended finite element concepts

This paper revisits the formulation of the CST element with an embedded discrete crack taking advantage of the direct formulations developed within the framework of the extended finite element method, XFEM. The result is a simple element for modeling cohesive fracture processes in quasi-brittle materials. The element is easily fitted a standard FEM code, and as such it is an alternative to more cumbersome XFEM elements which require special d.o.f.’s and extra administration. The crack description is embedded, in the sense that extra d.o.f.’s controlling the crack opening are eliminated at the element level. The cracked element is stress-compatible in the sense that stresses are continuous across the crack. A special shape function is introduced to allow for the discontinuous displacements without eradicating the stress compatibility. The simplicity of the element comes at the cost of inter-element discontinuity of displacements. The formulation is based on a variational principle of virtual work involving only the interpolation of displacements. The good performance of the element is demonstrated through the comparison with three benchmark tests in which a single crack is propagated: The center cracked sheet in uni-axial tension, the three-point bending test and the four-point shear beam test.

A method for planar development of free-form surfaces made of anisotropic materials

This paper presents a method for planar development of free-form surfaces especially made of anisotropic materials. A free-form surface is first tessellated into a set of triangular elements (i.e., facet model). If necessary, a facet refinement technique using Euler operators is applied to create a more reliable model that approximates the original surface as closely as possible. The facet model is then roughly flattened on a 2D plane. Next, the roughly flattened model is used as an initial guess for subsequent optimization during which the strain energy method using constant strain triangles (CST) is applied to yield an optimal 2D contour. In fact, various resulting 2D contours can be obtained while varying elasticity ratios between two material axes or development angles. To compare these resulting 2D contours, a contour alignment technique for error estimation is also presented. In this paper, anisotropic materials are considered to be orthotropic as a first approximation. A specific example (lateral surface of a shoe last) is shown to validate the proposed method.

Node-Failure Technique-Based Representative Unit-Cell Lifetime-Dependent Ductile Damage Model Between Elastoviscoplastic Matrix and Rigid Fiber

Numerical fracture mechanics analysis of the composite material presents many difficulties both for complexity of structure and various behaviors of its components. In recent years, there is an increasing trend of study about the initial crack location in internal microstructure of these materials. Numerical iterative computation procedure with the full mesh of composite structure, taking into account the elastoviscoplastic behavior and the failure criterion based on a locally accumulated damage to identify the path of the crack propagation, requires an important execution time, consequently an expensive cost. The aim of this article is the presentation of an original technique expressed as node-failure-based method developed in order to determine the initiation and progressive failure path using a nonlinear damage evolution law. In order to avoid the disadvantage of the full mesh, a representative unit-cell is adopted with adequate periodic boundary conditions to predict debonding initiation at fiber/matrix interface. The analysis shows finite element method–based representative unit-cell can reduce the computing power consumption and node-failure-based method act as a viable tool for prediction fracture pattern, while the well-known element-failure-based method, which requires more refined mesh of constant strain triangle elements is less accurate.

Cellular Automata in Topology Optimization of Continuum Structures

In this paper, an optimization algorithm based on cellular automata (CA) is developed for topology optimization of continuum structures with shear and flexural behavior. The design domain is divided into small triangle elements and each cell is considered as a finite element. The stress analysis is performed by the Constant Strain Triangles (CST) finite elements method. The thicknesses of the individual cells are taken as the design variables, while the weight of the structure and the ratio of the Von Mises equivalent stress to the yield stress in each cell are considered as the two objective functions to minimize. Using the weighted sum method, the multi-objective optimization problem (MOOP) is converted to the single-objective optimization problem (SOOP) and then the optimization problem is solved by the developed method. The paper reports the results of several design experiments, comparing with the existing reported results.

Cellular Automata in Topology Optimization of Continuum Structures

In this paper, an optimization algorithm based on cellular automata (CA) is developed for topology optimization of continuum structures with shear and flexural behavior. The design domain is divided into small triangle elements and each cell is considered as a finite element. The stress analysis is performed by the Constant Strain Triangles (CST) finite elements method. The thicknesses of the individual cells are taken as the design variables, while the weight of the structure and the ratio of the Von Mises equivalent stress to the yield stress in each cell are considered as the two objective functions to minimize. Using the weighted sum method, the multi-objective optimization problem (MOOP) is converted to the single-objective optimization problem (SOOP) and then the optimization problem is solved by the developed method. The paper reports the results of several design experiments, comparing with the existing reported results.

Numerical simulation of residual stress and deformation in glass pressed parts

As an important glass manufacturing process, glass pressing has been an effective process with the advantages of mass‐production, net shape manufacturing, etc., in which the residual stress and deformation in glass pressed parts are key issues. Based on the geometric characteristics of glass pressed parts, the shear plate assumption is employed for residual stress analysis, together with an applicable linear viscoelastic material model. Based on the results of residual stress, the finite element method using a flat triangular shell element is introduced to compute the deformation. The shell element is constructed by combining a constant strain triangle (CST) membrane element and a discrete Kirchhoff triangle (DKT) plate bending element. The comparison shows that the predicted results agree well with the experimental data.

Finite element analysis for the shear strength appearing in in situ rock shear tests

The mechanisms of the strength which appeared in in situ rock shear tests were examined with a finite element analysis in this paper. A set of plaster model laboratory tests simulating the in situ rock shear tests was analyzed. The plaster models were expressed initially with constant strain triangles (CSTs). Then, the displacement was imposed gradually, and each CST was replaced with a triangular element containing an embedded interface at the point when the stress in each CST reached the failure criterion of the material. The cracking patterns and the deformation obtained from the computation resembled those in the laboratory tests. The cracking pattern under each normal stress deviated from that under different conditions; and therefore, the stress path and the shear strength appeared differently under the various normal stress conditions. The relationship between the apparent shear strength and the failure criterion of the material also changed depending on the normal stress. The shear strength measured in the model tests appeared to be lower than the material strength in the lower and in the higher normal stress ranges. Such differences were thought to occur due to the influence of the stress distributions, which were not assumed, but were caused by several different cracking patterns.

A macro plane triangle element from the individual element test

A simple way to make the closed forms stiffness matrix of macro triangle elements is proposed through the individual element test and the computational algebra system in this paper. The newly developed element consists of three constant strain triangles, which only satisfy the convergence conditions according to the individual element test. Computational algebra system MAPLE and the free formulation framework are used to combine the stiffness matrix with the free parameters in terms of the algebraic symbols. The assembly of three constant strain triangles produces the linear strain modes in the resulted macro elements. The rigid body modes and constant strain modes of elements and the newly produced linear strain modes of the macro element are systematically assembled in the basic stiffness and the higher order stiffness in the free formulation procedure. The parameter of the decomposed higher order stiffness was tuned through the bending test. The results of numerical examples show the advantages of the proposed macro element in the distorted mesh and elements with the high aspect ratio.

A direct XFEM formulation for modeling of cohesive crack growth in concrete

Applying a direct formulation for the enrichment of the displacement field an extended finite element (XFEM) scheme for modeling of cohesive crack growth is developed. Only elements cut by the crack is enriched and the scheme fits within the framework of standard FEM code. The scheme is implemented for the 3-node constant strain triangle (CST) and the 6-node linear strain triangle (LST). Modeling of standard concrete test cases such as fracture in the notched three point beam bending test (TPBT) and in the four point shear beam test (FPSB) illustrates the performance. The XFEM results show good agreement with results obtained by applying standard interface elements in FEM and with experimental results. In conjunction with criteria for crack growth local versus nonlocal computation of the crack growth direction is discussed.

A consistent partly cracked XFEM element for cohesive crack growth

AbstractPresent extended finite element method (XFEM) elements for cohesive crack growth may often not be able to model equal stresses on both sides of the discontinuity when acting as a crack‐tip element. The authors have developed a new partly cracked XFEM element for cohesive crack growth with extra enrichments to the cracked elements. The extra enrichments are element side local and were developed by superposition of the standard nodal shape functions for the element and standard nodal shape functions for a sub‐triangle of the cracked element. With the extra enrichments, the crack‐tip element becomes capable of modelling variations in the discontinuous displacement field on both sides of the crack and hence also capable of modelling the case where equal stresses are present on each side of the crack. The enrichment was implemented for the 3‐node constant strain triangle (CST) and a standard algorithm was used to solve the non‐linear equations. The performance of the element is illustrated by modelling fracture mechanical benchmark tests. Investigations were carried out on the performance of the element for different crack lengths within one element. The results are compared with previously obtained XFEM results applying fully cracked XFEM elements, with computational results achieved using standard cohesive interface elements in a commercial code, and with experimental results. The suggested element performed well in the tests. Copyright © 2007 John Wiley & Sons, Ltd.