Voronoi Cell Finite Element Method Research Articles

The numerical simulation of particulate reinforced composites by using a two-dimensional VCFEM formulated with plastic, thermal, and creep strain

The working environment of industrial fields where the Particulate Reinforced Composites (PRCs) are commonly used is complex. In these harsh environments, PRCs produce some nonlinear strains, such as plastic, thermal, and creep strains, which affect the performance of PRCs to some extent. In addition, the actual structure of PRCs contains a massive amount of inclusions, and the number of inclusions could reach the magnitude of millions or even tens of millions. For the large-scale numerical simulation of PRCs, ordinary displacement finite element requires a large number of meshes, which increases their computational complexity and unsolvability. For real PRCs with random particle distribution, homogenisation theory does not accurately capture the true state of stress concentration in PRCs. The Voronoi cell finite element method (VCFEM) can be a good solution to the drawbacks of the above two numerical simulation methods. The subject of the present work explores a modified complementary energy generalized function that takes the constitutive relationship of nonlinear strain into account, proposing a two-dimensional VCFEM, formulated with plastic, thermal, and creep strain. The simulation results are compared with the computed results of a commercial finite element software Marc to verify the validity and accuracy of the VCFEM. Results of the comparison show that the VCFEM has the advantages of simple meshing and faster computation under the same precision. The VCFEM, considering plastic, thermal, and creep strains proposed in this paper, lays the foundation for future large-scale numerical simulations of PRCs containing a large quantity of inclusion structures and can realistically capture the stress concentration state of PRCs. At the end of the article, the impact of the order of stress function on the calculation results is discussed.

Study on hybrid stress element of three-dimensional arbitrary polyhedron

In three-dimensional problems, regular tetrahedron, pentahedron, and hexahedron elements are used in standard finite element software. In practice, however, the three-dimensional model is irregular. Traditional regular elements cannot accurately describe model structure. The field function integral accuracy of traditional displacement finite elements is low, and the number of elements considerably affects the integral results. As the number of elements increases, the computational efficiency decreases as well. Therefore, based on the hybrid stress element method, this paper proposes a new numerical calculation element for three-dimensional problems–a three-dimensional arbitrary polyhedron element. The corresponding calculation scheme is the 3-Dimensional Voronoi Cell Finite Element Method (3DVCFEM). This paper first verifies the effectiveness of the scheme. The stress field model with a singular solution is further simulated, and the change rule of the number of stress terms is obtained. On this basis, 3DVCFEM is introduced into calculating octree elements to solve complex classification problems and difficult calculation of octree elements. Moreover, the efficiency of the octree elements is investigated.

Automatic simulation of the initiation and propagation of interface cracks and matrix cracks with VCFEM

Based on the transformation mechanism of interfacial cracks propagating into a matrix, this paper proposes a model containing the particle phase, interphase, matrix phase, interface cracks, and matrix cracks. The model derives the functional, and we develop a new Voronoi cell finite element method (VCFEM) containing the interphase to describe interface crack and matrix crack initiation and propagation. A new mesh-subdivision method reflecting crack propagation from the interface to the matrix is constructed, which can automatically simulate the initiation and propagation of interface cracks and the transformation and penetration of interface cracks into matrix cracks. With crack propagation, various combined elements models are proposed. For these combined elements, their shared external nodes change as the cracks propagate through the original element, while the external node numbers of other elements remain unchanged. The effectiveness of the proposed method is verified by comparing the results of the new VCFEM with those obtained with Abaqus and analytical solution through several numerical examples. Since the proposed method assumes a high-order stress field within an element and can capture the stress concentration with fewer elements, it is suitable for large-scale calculation.

A multiphase hybrid-stress finite element method for macroscopic and microscopic modeling of composites: An element with multiple interfaces

A multiphase hybrid-stress finite element method (MHSFEM) for microscopic modeling of randomly dispersed closer microstructures and macroscopic modeling of layered structures include adhesive layer of laminated composites is proposed in the work. The stress functions for each phase within the element are constructed taking into account the shape of phase interfaces. The traction reciprocity conditions on the phase-phase interfaces within the element are introduced into the element functional, which is derived based on the minimum complimentary energy principle, by the Lagrange multiplier method. By several numerical examples, the effectiveness and accuracy of the proposed MHSFEM is verified by comparing its results with those of conventional finite element method and those of Voronoi cell finite element method. The proposed method is certainly an effective way for the microscopic and macroscopic mechanical analysis of composite materials with lower computational costs.

Two-scale Modeling of Composites damage with Voronoi cell finite element method for microscale computation

This paper develops an adaptive two-level computational model that includes macro and microscope analysis. Traditional displacement based finite element is used for macroscopic analysis, while Voronoi cell finite element model is used for computations of arbitrary heterogeneous domains. For an adaptive analysis, when local area may be destructive, VCFEM mesh is used instead of corresponding finite element for damage analysis. Because VCFEM model and finite element are all taken the nodal displacements as field variables, therefore, based on the idea of multigrid by the adaptive mesh adjustment, let finite element edge length and VCFEM edge length are basically the same, two kinds of elements were connected together through the use of shared nodes and use the order of the edge displacement interpolation. This formed a two-scale model of the coexistence of macro structure and micro structure. For microscale analysis, three Voronoi cell finite models (VCFEM) are build including perfect interface bonding model, interface debonding model and multiple matrix cracks model. Numerical example is executed for validating the effectiveness of the two-scale model method.

Extended Voronoi cell finite element methods for fracture analysis of composites

The extended Voronoi cell finite element model used to solve the fracture problems of isotropic materials and particle reinforced composites has been proven to be accurate and efficient. In this paper, the extended Voronoi cell finite element method is developed to analyse cracked orthotropic composite structures. The X-VCFEM is enhanced by orthotropic crack-tip enrichment stress functions to accurately capture the crack tip singularity. In addition, the stress functions also include polynomial stress functions. For elliptical internal cracks, reciprocal stress functions associated with polynomial functions is adopted, which are obtained by conformal mapping technique. The traction-free at crack face is introduced into functionals by the Lagrange multiplier method. The proposed method is applied for numerical investigation of fracture mechanics parameters for orthotropic composites, in which the stress intensity factors (SIFs) are calculated using the least square method. The effectiveness of the proposed method is validated by comparing with the results from ABAQUS or reference solutions. This method yields accurate results with fewer elements, which is the advantage of the proposed method.

Extraction of Stress Intensity Factors by Using the P-Version Finite Element Method and Contour Integral Method

The stress intensity factors (SIFs) for two-dimensional cracks are extracted using the p-version finite element method (P-FEM) and the contour integral method. Several numerical experiments, e.g., crack initiating from the edge of a circular hole under an unidirectional uniform tension and two equal-length, unequal-length hole–edge cracks, respectively, at a rectangular plate, an inclined centered crack under uniaxial tension at a square plate and a pipeline crack model, are used to demonstrate the accuracy and effectiveness of the approaches. SIFs are presented for the effects of various crack lengths and length–width ratio. Numerical results are analyzed and compared with reference solutions and results obtained by the Voronoi cell finite element method, boundary element method, high-order extended finite element method (high-order XFEM) and commercial finite element software ABAQUS in the available literature. Numerical results are in good agreement with the benchmark problems and show faster convergence rate, higher accuracy and better numerical stability.

Modelling interfacial cracking and matrix cracking in particle reinforced composites using the extended Voronoi cell finite element method

In this paper, a new extended Voronoi cell finite element method (X-VCFEM), in which two new elements in the crack propagation process are proposed and a remeshing algorithm is constructed, is proposed to model the initiation and propagation of the interface crack and the matrix crack in particulate reinforced composites. In order to accurately capture crack-tip stress concentrations, some singular stress terms of Williams expansion in the vicinity of crack tips are introduced in the stress functions in X-VCFEM. The direction of the crack propagation in matrix is determined by the maximum energy release rate and the interface crack propagation along the interface or into the matrix is assessed on the base of a series of criteria relating to the critical normal stress and the critical hoop stress. Then several numerical examples are used to demonstrate the effectiveness of X-VCFEM by comparing with results obtained by the commercial software ABAQUS. At last, a numerical example is given for a complex structure with 20 randomly distributing inclusions to model the interfacial cracking and matrix cracking. It is certain that this method is a powerful way to model the interfacial and matrix damage of composites containing a great deal of inclusions and cracks.

Extended Voronoi cell finite element method for multiple crack propagation in brittle materials

This paper introduces a new extended Voronoi cell finite-element method (X-VCFEM) for modelling the propagation of multiple cracks in brittle materials. The direction of the crack propagation is adaptively determined in terms of the maximum energy release rate near the crack tip, and we apply a remeshing strategy that a node at a last incremental crack tip in the crack advance process is replaced by a node pairs to realize the gradual crack propagation. In order to accurately capture crack-tip stress concentrations, some singular stress terms of Williams expansion in the vicinity of crack tips are introduced in the assumed stress hybrid formulation which also includes the polynomial terms and the reciprocal terms. Then several numerical examples are used to demonstrate the effectiveness and accuracy of X-VCFEM enriched by some singular stress terms of Williams expansion by comparing X-VCFEM results with those computed by the commercial finite element package ABAQUS or established results in the literature. At last, two numerical examples are given to simulate multiple crack propagation in brittle media. The effects of morphological distributions such as length, orientation and dispersion on the crack propagation are studied. It is obvious that the X-VCFEM has great advantage of analyzing large regions of the microstructure with multiple growing and interacting cracks.

Prediction of debonding growth in two-dimensional RVEs using an extended interface element based on continuum damage mechanics concept

In the present study, a new interface model based on continuum damage mechanics (CDM) is developed to investigate the fiber-matrix interfacial debonding in the composite material. This model differs from Voronoi cell finite element and cohesive zone element method. To this purpose, a 2D classical elasticity formulation with non-interactive strength-based criteria and complex variable method with non-interactive energy-based criteria are applied for the analysis of the debond onset and propagation in an RVE with a single fiber, respectively. Then, the CDM approach is used to bridge between analytical and numerical methods. The developed model based on CDM is implanted into ANSYS commercial software to compare the accuracy of the proposed method with the obtained results by the cohesive zone model in RVEs containing single and multiple fibers. The admissible matching between the obtained stress-strain responses reveals the capability of the proposed approach to predict the fiber-matrix debonding as a primary damage mode in composite materials.

Calculation of stress intensity factors of matrix crack tip in particle reinforced composites using the singular Voronoi cell finite element method

In this paper, a new singular Voronoi cell finite element method (S-VCFEM) is proposed to study the initiation and propagation of interfacial debonding and matrix cracking in particulate reinforced composite. Composites with large-scale inclusions and cracks can be easily simulated with this model. A new assumed stress hybrid variational functional is derived from the element energy function to accommodate the interface traction reciprocity on bonded interface and the interface traction-free on debonded interface and matrix crack surface. S-VCFEM is achieved through enhancements in stress functions in the assumed stress hybrid formulation. In addition to polynomial terms and interaction terms, singular stress terms of Williams expansion are added for stress functions, which are adaptively enriched to accurately capture the crack-tip stress concentrations. After obtaining the stress field, calculation of stress intensity factors is performed, using the least square method based on stress value obtained numerically in the S-VCFEM. The efficiency of the method is verified against numerical results obtained by ABAQUS with XFEM module for the same problem considering a matrix-crack and a debonded interface.

Effect of the Interfacial Transition Zone on Basic Mechanical Properties of a Solid Composite Propellant

The evaluation of basic mechanical properties are shown to be of importance for the solid composite propellant development and application. The numerical analysis approach termed the three-phase Voronoi cell finite element method, was proposed to evaluate of the interfacial transition zone effect. Numerical results showed that the performance of the effective modulus could be essentially enhanced by increasing the zone thickness. The module of the composite with the inhomogeneous zone was higher than that of the composite with the homogeneous one. The effect of different zones on volume fractions and matrix moduli was also calculated. The Voronoi method can also be helpful for analyzing the characteristic properties of other three-phase composites.

Effective Mechanical Property Estimation of Composite Solid Propellants Based on VCFEM

A solid rocket motor is one of the critical components of solid missiles, and its life and reliability mostly depend on the mechanical behavior of a composite solid propellant (CSP). Effective mechanical properties are critical material constants to analyze the structural integrity of propellant grain. They are estimated by a numerical method that combines the Voronoi cell finite element method (VCFEM) and the homogenization method in the present paper. The correctness of this combined method has been validated by comparing with a standard finite element method and conventional theoretical models. The effective modulus and the effective Poisson’s ratio of a CSP varying with volume fraction and component material properties are estimated. The result indicates that the variations of the volume fraction of inclusions and the properties of the matrix have obvious influences on the effective mechanical properties of a CSP. The microscopic numerical analysis method proposed in this paper can also be used to provide references for the design and the analysis of other large volume fraction composite materials.

A Voronoi cell finite element method for estimating effective mechanical properties of composite solid propellants

The mechanical properties of Composite solid propellant (CSP) are the critical material parameters to analyze the structural integrity of propellant grains, and have a significant influence on the life and reliability of solid rocket motors. A Voronoi cell finite element method using an adaptive algorithm in the time domain is proposed for investigating the linear viscoelasticity of CSP in the present paper. A process is brought forward to produce a Representative volume element (RVE) model, which reflects the microstructural features of CSP. Numerical viscoelastic examples are used for varying the accuracy of this method. In addition, finite element simulations are conducted to understand the effect of microstructural morphology and material properties of inclusion on the effective mechanical properties of CSP using the new method. When this method is applied to the design process of CSP, it can reduce the design cost and shorten the design cycle effectively. The current microscopic numerical analysis method can be used to provide guidance for designing and analyzing the mechanical properties of composite viscoelastic materials.

Voronoi Cell Finite Element Method for Fluid-Filled Materials

A modified Voronoi cell finite element method is proposed to allow for fluid pressure within porous materials. Traction on the hole boundary can equal the force generated by the pressure, and multiplying the Lagrange multiplier by the constraints, we derive a new function. The stiffness matrix and load vector were derived from the modified element energy functional. Numerical examples with various boundary problems were evaluated using the proposed model and compared with a conventional displacement-based finite element model. The results show that the proposed method can measure the irregular local stress on porous materials that contain randomly distributed and sized holes with specifics pressures. The proposed method significantly reduces the number of elements and nodes of the calculated porous structure.

The Analysis of Interfacial Debonding Using Voronoi Cell Finite Element Method

This paper bases on the principle of the stress hybrid element, using voronoi cell finite element method to analysis the interfacial debonding phenomenon of a particle reinforced composite materials, then it contrasts by the commercial finite element software MARC in the same conditions of numerical simulation. Research results show that: In the interfacial debonding, especially at the crack tip stress, Stress is the biggest. Particles and matrix interface delamination is the important cause of material damage, at the same time, it has a great impact on the service life of components.

Molecular Dynamics Approach and its Application in the Analysis of Multi-Scale

Numerical simulation of the behavior of materials can be used as a versatile, efficient and low cost tool for developing an understanding of material behavior [. The numerical simulation methods include quantum mechanics, molecular dynamics, Voronoi cell finite element method and finite element method et al. These methods themselves are not sufficient for many fundamental problems in computational mechanics, and the deficiencies lead to the thrust of multiple-scale methods. The multi-scale method to model micro-scale systems by coupled continuum mechanics and molecular dynamics was introduced. This paper describes the basic methods of multi-scale and general simulation process of molecular dynamics was reviewed.

Simulation of Particle Reinforced Composite Materials in Macro- and Meso-Scales

A simulation method of macro-and meso-scales is developed for particle reinforce composite materials. The two-scale modeling based on homogenization theory enables to formulate the macro scale problem with Finite Element Method (FEM), while the meso-scale one with Voronoi Cell Finite Element Method (VCFEM). Dangerous regions are identified in macro scale computing period, which lately be meshed into Voronoi Cells in meso-scale period to get a more accurate solution. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method of particulate reinforce composite materials.

Molecular Dynamics Approach and its Application in the Analysis of Multi-Scale

Numerical simulation of the behavior of materials can be used as a versatile, efficient and low cost tool for developing an understanding of material behavior [. The numerical simulation methods include quantum mechanics, molecular dynamics, Voronoi cell finite element method and finite element method et al. These methods themselves are not sufficient for many fundamental problems in computational mechanics, and the deficiencies lead to the thrust of multiple-scale methods. The multi-scale method to model micro-scale systems by coupled continuum mechanics and molecular dynamics was introduced. This paper describes the basic methods of multi-scale and general simulation process of molecular dynamics was reviewed.

Simulation of Particle Reinforced Composite Materials in Macro- and Meso-Scales

A simulation method of macro-and meso-scales is developed for particle reinforce composite materials. The two-scale modeling based on homogenization theory enables to formulate the macro scale problem with Finite Element Method (FEM), while the meso-scale one with Voronoi Cell Finite Element Method (VCFEM). Dangerous regions are identified in macro scale computing period, which lately be meshed into Voronoi Cells in meso-scale period to get a more accurate solution. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method of particulate reinforce composite materials.