Augmented Finite Element Method Research Articles

Fracture modeling of curved composite shell structures using augmented finite element method

This paper presents an augmented finite element method for the fracture modeling of curved composite shell structures considering geometric nonlinear effects. We first derive the composite shell AFEM (CS-AFEM) formulation using a dynamic implicit algorithm, which enhances the numerical convergence when dealing with unstable crack propagations. Besides, the cohesive zone model featuring with shell kinematics is incorporated into the CS-AFEM to describe the quasi-brittle fracture behaviors of composite laminates. Furthermore, a coordinate transformation scheme is proposed to describe both the local material orientation and the crack evolving path in the curved shell structures. Finally, several benchmark examples are numerically investigated, and the capability of the proposed method in modeling the arbitrary evolving cracks in curved composite shell structures has been thoroughly demonstrated.

Rate-dependent augmented finite element method for fast arbitrary crack growth

Experimental observations indicate that fast crack growth often exhibits a rate dependency, which is caused by the viscosity of the material and the dissipation of energy resulting from the branching of microcracks. To address this phenomenon in simulation modeling, this study introduces a rate-dependent augmented finite element method designed for analyzing fast arbitrary crack growth. The method utilizes the maximum principal stress criterion to simulate arbitrary crack growth and predict the path of crack growth. The elements where crack growth is expected are divided into subelements, and they are re-assembled by incorporating rate-dependent cohesive elements between them to construct augmented finite elements. The cohesive elements are based on a rate-dependent trapezoid traction–separation relationship, where the critical traction and fracture toughness are modeled as functions of the separation rate and crack growth speed. The augmented finite elements only contain external nodal degrees of freedom (DoFs), while internal nodal DoFs are condensed. They can be utilized as conventional elements. A numerical algorithm based on the proposed method was developed in this article, and it was implemented in ABAQUS via user element subroutine. The effectiveness of this method and algorithm is validated through an example of dynamic crack growth in a double cantilever beam. The results demonstrate agreement with experimental and computational findings reported in the literature. Additionally, the method was applied to other structures, yielding reasonable outcomes. Analysis of these examples confirms the method’s utility in predicting fast arbitrary crack growth under dynamic load conditions.

On the Use of Drilling Degrees of Freedom to Stabilise the Augmented Finite Element Method

The augmented finite element method (AFEM) embeds cracks within solid elements. These cracks are modelled without additional degrees of freedom thanks to a dedicated static condensation process. However, static condensation can induce a lack of constraint problem, resulting in singular stiffness matrices. To address this issue, we propose a new method called the stabilised augmented finite element method (SAFEM), which produces non-singular stiffness matrices. We conducted 2D experiments involving stationary traction-free cracks and propagating cohesive discontinuities to compare the performance of the SAFEM with the AFEM. The SAFEM outperforms the AFEM in modelling traction-free cracks.

Progressive Failure Analysis of Composite Laminates Subjected to Transverse Loading with Augmented Finite Element Method

Progressive Failure Analysis of Composite Laminates Subjected to Transverse Loading with Augmented Finite Element Method

Progressive Failure Analysis of Composite Laminates Subjected to Transverse Loading with Augmented Finite Element Method

In this paper, two-dimensional (2D) orthotropic augmented finite element method (A-FEM) is applied to account for progressive failure of composite laminates under transverse loading, which considers all major cracking modes (delamination, fiber kinking/rupture matrix cracking). High-fidelity simulations of different stacking composite laminates under transverse loading are implemented. Both predicted load−deflection curves and damage evolution are in good agreement with that of experimental results, which demonstrates the numerical capability of A-FEM. In addition, the influence of stacking sequence on the failure mechanism is also studied by predicted damage evolution of laminates with different stacking sequence [Formula: see text], [Formula: see text] and [Formula: see text]. Results show that the tensile matrix crack in the bottom laminar is always the first damage mode for the composite laminate, and the subsequent crack propagation is related to ply orientation of adjacent plies which have a blocking effect on crack propagation.

Finite element modelling of traction-free cracks: Benchmarking the augmented finite element method (AFEM)

This paper investigates the accuracy and the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model strong discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns to represent cracks. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the element deletion method (EDM), the phantom node method (PNM), the finite element method (FEM) and the embedded finite element method (EFEM). The h-convergence in the energy norm of the AFEM is studied for the first time and it is shown to outperform the aforementioned numerical methods when cracks are loaded in Mode I.

Numerical simulations of arbitrary evolving cracks in geotechnical structures using the nonlinear augmented finite element method (N-AFEM)

This paper presents an integrated numerical algorithm based on the nonlinear augmented finite element method (N-AFEM) to accurately simulate the arbitrary evolving strong discontinuities (cracks/slip lines) and failure behaviors in geotechnical structures. A novel nonlinear elemental augmentation and condensation scheme was first proposed within the N-AFEM framework, which allows the consideration of nonlinear coupled intra-element cracks without the need of additional nodes or nodal DoFs. Then a modified exponential cohesive zone model (CZM) that considers the coupling effects between tension/compression and shear at the fracture surface was proposed to describe the fracture process of geomaterials. Besides, a nonlinear yielding function based on the Mohr-Coulomb strength theory was introduced into the framework and serves as the fracture initiation criteria for geomaterials. Finally, several benchmark examples were simulated and the predicted results were compared with existing numerical or experimental data to demonstrate the validity of the proposed method.

Progressive fracture analysis of the open-hole composite laminates: experiment and simulation

A conforming augmented finite element method (C-AFEM) is extended to predict the failure process in inter-ply of fiber-reinforced composite laminates. Combining with cohesive zone model (CZM) element, the delamination between inter-plies can also be captured. Matrix fracture, fiber fracture and delamination in inter-plies will occur when the composite laminate fracture, which is a complicated process, and the current method appears capable of modelling failure mechanism of composite laminates, which is demonstrated mesh-independent, high-fidelity and robust. Different stacking open-hole composite laminates subject to tension load are tested by electrical measurement and Digital Image Correlation (DIC) system. The proposed C-AFEM is applied to model progressive failure of open-hole composite laminates and results are found to agree with experiments very well.

Performance assessment of the augmented finite element method for the modeling of weak discontinuities

AbstractThis article investigates the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model weak discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the finite element method (FEM) and the nonconforming FEM. It is shown that the AFEM converges with an error of in the energy norm. The nonconforming FEM shares the same property while the FEM converges at . Yet, the AFEM is on par with the FEM for certain homogenization problems.

3D geometrically nonlinear augmented finite element method for arbitrary cracking in composite laminates

This paper, for the first time in literature, formulated and validated a three-dimensional, nonlinear augmented finite element method (3D A-FEM) that can account for multiple crack evolution in laminated composites under large deformation. The 3D A-FEM accounts for all major cracking events (intra-ply matrix cracking, fiber rupture/kinking, and inter-ply delamination) with explicit cohesive cracks. The computational scheme is achieved by coupling the 3D A-FEs for intra-ply cracks with 3D cohesive interface elements for inter-ply delamination. The strong discontinuities of both intra- and inter-ply cracks are explicitly represented by the geometrically nonlinear cohesive zone models (CZMs). The numerical capability is demonstrated by several benchmark tests with both in-plane and out-of-plane loadings. Results show that the A-FEM predicted progressive damage processes, including the arbitrary initiation of multiple cracks and their nonlinearly coupled progression with delamination all the way up to the final catastrophic failure, are all in good agreement with experimental results.

Nonlinear augmented finite element method for arbitrary cracking in large deformation plates and shells

AbstractThis article presents a nonlinear augmented finite element method (N‐AFEM) for the analysis of arbitrary crack initiation and propagation in large deformation plates and shells. The FE formulations for plate/shell elements and a shell‐like cohesive zone element, both with explicit consideration of geometric nonlinearity, have been derived in detail. The geometrically nonlinear shell‐like cohesive element has the essential feature of 3D but with crack displacements directly extracted from midplane shell element nodes, which enables an accurate description of crack propagation in shells and plates under large deformation. Furthermore, a novel augmentation process that can explicitly account for the discontinuous displacement fields of cracked elements without the need of extra nodes or nodal DoFs has been develop based on a nonlinear Newton‐Raphson method. The numerical performance of the N‐AFEM in modeling a number of benchmark shell/plate fracture problems demonstrates that the method is efficient, accurate, and robust.

Refined analysis and construction parameter calculation for full-span erection of the continuous steel box girder bridge with long cantilevers

To accurately control the full-span erection of continuous steel box girder bridges with complex cross-sections and long cantilevers, both the augmented finite element method (A-FEM) and the degenerated plate elements are adopted in this paper. The entire construction process is simulated by the A-FEM with the mesh-separation-based approximation technique, while the degenerated plate elements are constructed based on 3D isoparametric elements, making it suitable for analysis of a thin-walled structure. This method significantly improves computational efficiency by avoiding numerous degrees of freedom (DoFs) when analyzing complex structures. With characteristics of the full-span erection technology, the end-face angle of adjacent girder segments, the preset distance of girder segments from the design position, and the temperature difference are selected as control parameters, and they are calculated through the structural response of each construction stage. Engineering practice shows that the calculation accuracy of A-FEM is verified by field-measured results. It can be applied rapidly and effectively to evaluate the matching state of girder segments and the stress state of bearings as well as the thermal effect during full-span erection.

A numerical and experimental study on the multiple fracture progression of CFRP T-joints under pull-off load

This paper reports a combined numerical and experimental study on the multiple fracture evolution in carbon fiber reinforced polymer (CFRP) T-joints under pull-off load. The experimental study successfully captured the sequential fracture initiation and evolution involving a complex interplay among noodle cracking, noodle-skin and skin-stiffener delamination, and their correlation with the load-displacement curves. Numerically, two independent numerical methods, (a) the extend finite element method (X-FEM) available in ABAQUS, and (b) the augmented finite element (A-FEM) method, are used to cross-check their predictive capability in modeling progressive fracture evolution in the T-joints. It is found that, although the nonlinear load-displacement curves predicted by both methods are consistent with experimental data, their predictions on fracture progression responsible for final failure are different. The A-FEM prediction is in good agreement with experimental record while the X-FEM's prediction is inaccurate. Finally, the validated A-FEM model is coupled with the Binary Model to quantify the improvement in fracture resistance using the Z-pin reinforcement technique. The simulation results, which are validated by previously reported experimental data, show that properly arranged Z-pin reinforcement can improve the fracture tolerance and significantly delay the final failure of T-joints.

An augmented fully-mixed finite element method for a coupled flow-transport problem

In this paper we analyze the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem, where the viscosity of the fluid and the diffusion coefficient depend on the solution to the transport problem and its gradient, respectively. An augmented mixed variational formulation for both the fluid flow and the transport model is proposed. As a consequence, no discrete inf-sup conditions are required for the stability of the associated Galerkin scheme, and therefore arbitrary finite element subspaces can be used, which constitutes one of the main advantages of the present approach. In particular, the resulting fully-mixed finite element method can employ Raviart–Thomas spaces of order k for the Cauchy stress, continuous piecewise polynomials of degree $$k + 1$$ for the velocity and for the scalar field, and discontinuous piecewise polynomial approximations for the gradient of the concentration. In turn, the Lax–Milgram lemma, monotone operators theory, and the classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations. In addition, suitable estimates, arising from the combined use of a regularity assumption with the Sobolev embedding and Rellich–Kondrachov compactness theorems, are also required for the continuous analysis. Then, sufficiently small data allow us to prove uniqueness of solution and to derive optimal a priori error estimates. Finally, several numerical tests, illustrating the performance of our method and confirming the predicted rates of convergence, are reported.

Micromechanical analysis of damage mechanisms under tension of 0°–90° thin-ply composite laminates

A micromechanical model is used to investigate ply thickness effect on damage evolution of thin-ply carbon fiber reinforced laminate under transverse tensile load. Representative volume element (RVE) for 90° lamina are constructed and sandwiched between two homogenized zero degree plies. Four different thicknesses for 90° RVEs including 30, 60, 90, and 120 μm are considered for analysis. The three dimensional (3D) computational micromechanics are combined with augmented finite element method (AFEM) to provide high-fidelity results of damage evolution. Random arrangement for fibers and normal distribution for interface toughness and strength are considered within RVEs. Damage evolution in different RVEs under tensile loading are discussed and compared. The results show that decreasing 90° lamina thickness alters damage progression mechanism and suppresses cracking within matrix loading. A detailed comparative discussion on the influence and importance of material parameters as well as voids/defects on the process of cracking are given.

Micromechanisms of Failure and Damage Evolution in Low-Thickness Composite Laminates Under Tensile Loading

In this work, failure mechanisms and damage evolution in low-thickness composite laminates are numerically simulated. The effect of thinning ply on the strength and damage evolution of composite laminates is investigated. Micromechanical modeling is performed to study damage initiation and propagation mechanisms and ply thickness or in situ effect in thin-ply carbon fiber-reinforced laminate under transverse tension loading. Two different sets of models are examined. The first set of the thin-ply laminate models contains a representative volume element (RVE) of 90° lamina constrained between two homogenized 0° plies. Thicknesses of models with embedded 90° lamina were 30, 60, 90 and 120 μm. In the second set, two models with 30 and 90 μm of 90° thin ply as the outer layers with embedded homogenized 0° layer are considered. Micromechanical analysis is combined with augmented finite element method to provide high-fidelity results of damage evolution. Random fibers’ arrangement and interface toughness and strength normal distribution are considered to capture the composite stochastic behavior. It is shown that the damage initiation and propagation locations are affected by the thickness of 90° lamina and distributions of fracture properties within the RVE.

An orthotropic augmented finite element method (A-FEM) for high-fidelity progressive damage analyses of laminated composites

In this paper, we extended a recently developed augmented finite element method (A-FEM) to account for the complicated progressive damage processes in laminated composites, which are of orthotropic nature and typically develop multiple types of cracking systems including intra-ply matrix/fiber splitting, fiber rupture in tension and/or kinking in compression, and inter-ply delamination. The orthotropic A-FEM represents all of these major damage modes with improved nonlinear cohesive zone models (CZMs) that explicitly consider the asymmetric tension- and compression-responses. A rigorous verification and validation process demonstrates that the developed orthotropic A-FEM can adequately account for the initiation and propagation of various types of cracks and their coupled evolution under complex stress environments. A-FEM predictions to progressive damage processes in several multidirectional notched and un-notched laminates, including the initiation of multiple cracks and their nonlinearly coupled progression with delaminations all the way up to the final, catastrophic failure, are all in excellent agreement with experimental measurements and observations.

Numerical Modelling of FRCM Materials Using Augmented-FEM

The use of externally-bonded composite materials for strengthening and rehabilitation of existing structures is among the most popular reinforcement techniques. Technologies, such as Fabric Reinforced Cementitious Matrix (FRCM) have been recently developed to address some of the issues of Fiber Reinforced Polymers (FRP), such as sensitivity to elevated temperatures and UV, impermeability, restricted application in presence of moisture or uneven substrate. For a detailed strengthening design with FRCM composites, the mechanical properties of the materials are required. Analytical models in literature discuss the interaction between the FRCM matrix and fabric using a fracture mechanics approach. These analytical laws were simplified using a trilinear curve in which a constant branch correlated to the friction is added. In the United States, “Acceptance Criteria AC434” includes the test methods to evaluate the mechanical properties of the FRCM through a direct tensile test which uses clevis grips. The material characterization per AC434 is in harmony with ACI 549.4R design guidelines. This study deals with the analysis of FRCM materials using 2D Augmented-Finite Element Method (A-FEM) approach. Constitutive material behaviors were used to implement on A-FE model, which can predict the failure modes of the composite material. The damage of the mortar was described by a trilinear curve, and the number and position of the cracks were fixed preliminarily. The fabric was modelled as a continuum layer attached to the mortar with no-thickness cohesive elements. The cohesive law between fabric and mortar was taken from the literature. The tensile test on the FRCM coupon with one layer of fabric was numerically modeled and compared to the experimental stress-strain curves. Results show that the numerical curves matched the experimental ones and capture the three branches of the FRCM constitutive law as well as the failure mode. This modelling tool will allow researchers to predict the constitutive law of an FRCM mater

Sliding frictional contact analysis of a monoclinic coating/isotropic substrate system

Owing to innovative developments and the need to satisfy the requirements of engineering applications, new and improved materials are investigated by engineers to compensate for the drawbacks of conventional materials. There is an urgent need for the tribology community to properly investigate the behavior of these materials before utilizing them in real life. In this study, the sliding frictional contact between a monoclinic coating/isotropic substrate system and a rigid cylindrical punch is provided using an analytical formulation and finite element method (FEM). The analytical expressions for the stress and displacement fields are presented, and the resulting singular integral equations (SIEs) are solved numerically. Details of the analytical formulation and the discretization used in the solution method for the SIEs are also presented. For obtaining the numerical solution, an augmented finite element method is used with plane strain conditions and 6-node higher order finite elements. According to the results, the analytical formulation and FEM results are consistent. Using different lamina orientations, coefficient of friction, and geometric/material parameters, provides the tribology community with a highly qualified parametric study that informs surface wear and surface fatigue crack initiation in contact mechanics problems

A Conforming Augmented Finite Element Method for Modeling Arbitrary Cracking in Solids

Abstract This paper presents a conforming augmented finite element method (C-AFEM) that can account for arbitrary cracking in solids with similar accuracy of other conforming methods, but with a significantly improved numerical efficiency of about ten times. We show that the numerical gains are mainly due to our proposed new solving procedure, which involves solving a local problem for crack propagation and a global problem for structural equilibrium, through a tightly coupled two-step process. Through several numerical benchmarking examples, we further demonstrate that the C-AFEM is more accurate and mesh insensitive when compared with the original A-FEM, and both C-AFEM and A-FEM are much more robust and efficient than other parallel methods including the extended finite element method (XFEM)/generalized finite element (GFEM) and the conforming embedded discontinuity method.