Finite Element Implementation Research Articles

Modeling yield stress scaling and cyclic response using a size-dependent theory with two plasticity rate fields

This work considers a recently developed finite-deformation elastoplasticity theory that assumes distinct tensorial fields describing macro-plasticity and micro-plasticity, where the latter is determined by a higher-order balance equation with associated boundary conditions and evolves according to a contribution to the Helmholtz free-energy density that depends on a Nye-Kröner-like dislocation density tensor and is referred to as the defect energy. The theory is meant to set the onset of micro-plasticity at a stress level lower than that activating macro-plasticity, such as micro-plasticity aims at explaining and characterizing the increase in yield stress with diminishing size. Additionally, the formulation relies on smooth elastic–plastic transitions for both plasticity fields, even if focusing on rate-independent response. This investigation demonstrates the capability of the proposed theory to predict size-effects of interest in small-scale metal plasticity by focusing on multiple loading cycles and, prominently, on the scaling of the apparent yield stress with sample size, the latter being a crucial open issue in the recent literature on modeling size-dependent plasticity. To this end, this work considers the specialization of the theory to small deformations and proposes a finite element implementation for the constrained simple shear problem. Importantly, it is shown that the simplest treatment of plastic strain gradients, which consists of adopting a quadratic defect energy, can be conveniently used to predict reliable size-effects, although in the literature on strain gradient plasticity quadratic defect energies have always been associated with a relatively poor description of size-effects. In fact, in the present theory the limits of the quadratic defect energy are overcome by leveraging on the complex interplay between micro- and macro-plasticity fields. The capability of the proposed theory is quantitatively demonstrated by predicting results from the literature that are obtained from discrete dislocation dynamics simulations on planar polycrystals of grains with variable size subjected to macroscopic pure shear.

Electric circuit element boundary conditions for electromagneto-quasistatic and full wave models in A, φ potentials and their finite element implementation

Electric Circuit Element (ECE) boundary conditions (BC) defined for full-wave (FW) electromagnetic field allow a natural coupling between field devices and electric circuits. The novelty of this paper is that it shows how ECE BC can be implemented into a 3D-finite element method (FEM), when using A, a magnetic vector potential and φ, a scalar potential. Weak formulations are described and implemented in the free environment Open Numerical Engineering LABoratory(onelab). The validation is carried out on 3D examples solved both in frequency (FD) and time domain (TD), for FW formulations in potentials, as well as for corresponding Darwin approximations of Electromagneto-Quasistatic (EMQS) models. Results are compared with those obtained with a formulation in EV, where E is the electric field inside the domain and V is a scalar potential defined solely on the boundary. The results show that: 1) the use of potentials has some advantages over the EV formulation in TD only; 2) excitation type matters, the voltage excitation, here essential in FEM, proved to be the most robust one for the considered examples: a coplanar waveguide and a spiral inductor.

Mathematical modeling of the operation of plate elements when working together with a soil base in conditions of flat deformation

Assessment of the forces arising in the structures of underground parts of structures, such as piles, foundation slabs and walls, as well as in the structures of the pit fence made of metal sheet piling and using the “wall in the ground” technology is a normative mandatory check for the first group of limit conditions. The work of such structures should be evaluated in conjunction with the soil base, the characteristics of which directly affect the final effort. The article provides a detailed mathematical description of the implementation of a three-node finite element describing a beam (plate in the conditions of a flat problem). At the end, a comparison of the calculation results of the test tasks with another computing software package is provided. High convergence of the results was noted. The results obtained can be used in the development of proprietary finite elements or as alternative calculation methods within the framework of scientific and technical support.

Multiscale Modelling of the Two-Dimensional Problem Using the Boundary Element Method

A full coupled multi-scale modelling using the Boundary Element Method for analysing the 2D problem of stretched plates composed of heterogeneous materials, where dissipative phenomena can be considered, is presented. Both the macro-scale and the micro-scale are modelled by BEM formulations where the consistent tangent operator (CTO) is used to achieve the equilibrium of the iterative procedures (see [1]). The equilibrium equation of the plate (macro-continuum) is written in terms of in-plane strains while the equilibrium problem of the microstructure, which is defined by the RVE (Representative Volume Element), is solved in terms of displacements fluctuations (see [2]-[5]). In this kind of modelling, the mechanical behaviour of the material is governed by the homogenized response of the RVE, obtained after solving its equilibrium problem. As this kind of modelling is expensive computationally, it is important to investigate other numerical methods to have faster formulations, but which are still accurate. To validate the presented model, the numerical results are compared to the ones where the material microstructure (RVE) is modelled by the FEM.REFERENCES[1] G. R. Fernandes and E. A. de Souza Neto, “Self-consistent linearization of non-linear BEM formulations with quadratic convergence,” Computational Mechanics, vol. 52, pp. 1125-1139, 2013.[2] G. R. Fernandes, L. H. R. Crozariol, A. S. Furtado and M. C. Santos, “A 2D boundary element method to model the constitutive behavior of heterogeneous microstructures considering dissipative phenomena” Engineering Analysis with Boundary Elements, vol. 99, pp. 1-22, 2019.[3] G. R. Fernandes, J. J. C. Pituba and E. A. de Souza Neto, “FEM/BEM formulation for multi-scale analysis of stretched plates,” Engineering Analysis with Boundary Elements, vol. 54, pp. 47-59, 2015.[4] D. Peric, E. A. de Souza Neto, R. A. Feijóo, M. Partovi and A. J. C. Molina, “On micro to macro transitions for multi scale analysis of non linear heterogeneous materials: unified variational basis and finite element implementation,” Numerical Methods in Engineering, vol. 87, pp. 149-170, 2011.[5] Somer D.D., de Souza Neto E.A., Dettmer W.G., Peric D., A sub-stepping scheme for multi-scale analysis of solids, Comput. Methods Appl. Mech. Engrg., v. 198, p. 1006–1016, 2009.

Specialization of a new flux-based approach to thermal problems. Numerical study of glass façades of buildings with cast shadows

Uneven temperature distributions in the glass panes of building façades can induce thermal stresses exceeding material strength, a critical condition usually referred to as “thermal shock”. Here, we specialize in architectural glazing a recently-proposed flux-based variational approach to thermal problems, by considering heat exchanges resulting from climatic actions, in particular the variability of environmental temperatures and solar radiation, influenced by the shielding of cast shadows. The finite element implementation is facilitated because the unknown field is the heat displacement, whose time derivative is the heat flux, which is much smoother than the temperature field, on which the classical approach via Fourier’s law is based. A custom in-house-made FEM code is used to determine the effect of size and shape of cast shadows on temperature distribution in monolithic and laminated glass panes, also coupled in insulating glass units. We find that the temperature difference between irradiated and shadowed regions is only mildly influenced by the shape and size of the shadows, but the glass thickness determines the width of the transition zone and, hence, the temperature gradient. Practical rules for the design of glass façades are tentatively proposed.

Phase field fracture in elastoplastic solids: a stress-state, strain-rate, and orientation dependent model in explicit dynamics and its applications to additively manufactured metals

Phase field models have gained growing popularity in analysing fracture behaviour of materials. However, few have been explored to simulate dynamic ductile fracture to date. This study aims to develop a phase field framework considering strain rate, stress state, and orientation dependent ductile fracture under dynamic loading. Firstly, governing equations of displacement and phase fields are formulated in an explicit finite element framework. Secondly, constitutive relations are established with a hypoelastic-plasticity framework, encompassing the influence of material orientation and strain rate on both plasticity and fracture initiation. Stress states dependent fracture initiation is also considered. Thirdly, finite element implementation and corotational formulation for constitutive equations are derived. To validate the proposed model, finally, additively manufactured samples, involving material-level and crack propagation specimens are tested under dynamic loading conditions. Overall, the proposed phase field model can properly reproduce the experimental force-displacement curves and crack paths. Uniaxial tension tests reveal that a higher strain rate can lead to a higher hardening curve and lower ductility. Other material specimens further demonstrate the capability to predict stress state and orientation dependent dynamic fracture. To simulate dynamic crack paths accurately, it is necessary to consider anisotropic fracture initiation. Lastly, the phase field model was applied for the first time to predict the dynamic response of triply periodic minimal surface (TPMS) structures. Dynamic crack patterns were effectively captured, and the fracture mechanisms were thoroughly analysed. This study provides an explicit phase field framework for dynamic ductile fracture, with applications to additively manufactured materials and structures.

Free vibration of sandwich beams with graphene nanoplatelets reinforced piezoelectric face sheets

Owing to its exceptional physical properties, graphene is considered as an excellent reinforcement for composite materials. However, the existing higher-order models encounter significant hurdles when attempting to accurately predict the natural frequencies of sandwich beams with graphene nanoplatelets reinforced composite (GNPRC) piezoelectric face sheets. If transverse shear deformations are not accurately modeled, the dynamic behavior of piezoelectric sandwich beams will be noticeably influenced by the disparities of material properties at the interfaces of adjacent layers, as well as the inherent electromechanical coupling characteristic. To overcome these challenges, an advanced electro-mechanical coupling theory will be introduced for the vibration analysis of sandwich beams featuring GNPRC piezoelectric face sheets. Compared to previous higher-order models, the proposed beam model introduces an enhanced interlaminar shear stress field incorporating electromechanical properties. This improved stress field can be involved in the equations of motion based on the Hamilton principle, significantly enhancing the precision in analyzing the free vibration behavior of piezoelectric sandwich beams. Furthermore, a simplification of finite element implementation can be achieved by removing the second-order derivatives of in-plane displacements from the transverse shear stress field. Consequently, a C0-type three-node beam element is constructed for the dynamic analysis of sandwich beams with GNPRC piezoelectric face sheets. To validate the performance of the proposed theory, the 3D elasticity solutions and results obtained from alternative theoretical frameworks are used. Numerical results demonstrate that the present model offers superior accuracy over the existing higher-order theories. Additionally, a comprehensive parametric investigation is conducted to explore the influence of key parameters on the free vibration characteristics of piezoelectric sandwich beams.

Magnetostriction of soft-magnetorheological elastomers

Soft-magnetorheological elastomers (s-MREs) are particulate composites made of a non-magnetic elastomeric matrix dispersed with micron-sized particles of a “soft-magnetic” material. The phenomenon of magnetostriction in specimens made from s-MREs is the change in their shape when they are subjected to an external magnetic field. Experiments in the literature show that for circular cylindrical specimens subjected to an axially applied magnetic field the magnetostriction is strongly dependent on their aspect-ratio, with specimens with a low ratio of the length to the diameter exhibiting a larger tensile magnetostrictive strain than specimens with a large aspect-ratio — the “shape-effect.” This response is also hysteretic because of the underlying viscoelasticity of the matrix material. In this paper we report on a large deformation magneto-viscoelasticity theory for s-MREs and its finite element implementation. Using our theory we show that we can model this non-intuitive geometry-dependent magnetostrictive response of cylindrical s-MRE specimens. We show that the effect of the magnetization m of the cylinder is to decrease the magnetic field h within the cylinder relative to the applied magnetic field happ outside the cylinder, the well-known demagnetization effect, and that this demagnetization is diminished in more slender cylinders due to magnetic fringing at the boundaries of the cylinder. This is the physical reason behind the macroscopically-observed “shape-effect.”Our magneto-viscoelasticity theory is quite broad, and it has many potential applications beyond modeling the magnetostriction of cylindrical specimens. As an example we apply the theory to study the bending actuation response of beams of s-MREs when they are subjected to magnetic fields at different incidence angles to their longitudinal axis.

Web shear buckling of steel-concrete composite girders – advanced numerical analysis

Load-bearing capacity of plate girders, often used in design of bridges and high-rise buildings, is limited by the shear capacity of connected slender plate elements subjected to shear buckling. To quantify this, experimental investigations on five large-scale steel and steel-concrete composite plate girders loaded solely in shear, with a minimal influence of bending moments, have been conducted and evaluated. In this paper, the phenomenon of web shear buckling is investigated within the numerical analysis using the ABAQUS Software. An advanced numerical model has been developed and results validated against existing experimental findings. One of the focal points of this study represents the methodology of developing such a comprehensive numerical model, implementation of suitable analysis procedures, material models, boundary conditions, finite elements and interactions, in order to correctly replicate the observed response in the tests. In addition, case studies tackling the influence of web slenderness, aspect ratio, initial imperfections, shear connection and concrete classes on the structural-mechanical behavior of steel-concrete composite girders in shear as well as the applicability and suitability of the existing analytical model are also presented and analyzed.

A micropolar phase-field model for size-dependent electro-mechanical fracture

This article proposes a micropolar phase-field model for size-dependent brittle fracture in solids under electro-mechanical loading conditions. Considering displacement, micro-rotation, electric potential, and phase-field variable as the kinematic descriptors and employing the virtual power principle, we derive a set of coupled governing partial differential equations (PDEs) for size-dependent solids. Invoking the first and second laws of thermodynamics, we determine the constitutive relations for the thermodynamic fluxes. Carrying out the finite element implementation of the derived governing PDEs using the open-source Gridap package in Julia, we demonstrate the efficacy of the proposed phase-field model through a few representative numerical examples. Especially the importance of the proposed model in incorporating the effect of relative rotation, i.e., the difference between macro- and micro-rotation, on the response of solids under electro-mechanical loading is shown that may not be possible with the existing non-local models such as strain-gradient or couple-stress approaches. To capture the experimentally observed size effects in solids under electro-mechanical loading, the proposed model does not demand higher-order continuity of the field variables, unlike a typically used strain gradient model. To demonstrate the efficacy of the proposed model, we have compared our results against demanding experimental and numerical benchmark results available in the literature. We provide a parametric study to unravel the effect of different micropolar material parameters on the electro-mechanical response of a brittle solid. Interestingly, the proposed micropolar model is less sensitive to the phase-field length scale than the conventional non-polar phase-field models.

Bridging length and time scales in predictive simulations of thermo-mechanical processes

Abstract This work introduces a theoretical formulation and develops numerical methods for finite element implementation of the formulation so as to extend the concurrent atomistic-continuum (CAC) method for modeling and simulation of finite-temperature materials processes. With significantly reduced degrees of freedom, the CAC simulations are shown to reproduce the results of atomically resolved molecular dynamics simulations for phonon density of states, velocity distributions, equilibrium temperature field of the underlying atomistic model, and also the density, type, and structure of dislocations formed during the kinetic processes of heteroepitaxy. This work also demonstrates the need of a mesoscale tool for simulations of heteroepitaxy, as well as the unique advantage of the CAC method in simulation of the defect formation processes during heteroepitaxy.

Anisotropy cyclic plasticity constitutive modelling for Ni-based single-crystal superalloys based on Kelvin decomposition

Nickel-based single-crystal (SC) superalloys exhibited excellent exceptional mechanical properties at high temperatures due to the elimination of internal grain boundaries, contributed a strong orientation-dependent material response. The anisotropy of SC superalloys was modeled viscoplastically from a macroscopic viewpoint based on the Kelvin decomposition theory [1] which was a decomposition of the stress space according to the elastic matrix eigen-directions to control the viscoplastic flow by Kelvin stress decoupled from each other. Compared to the classical phenomenological macro model, the proposed model effectively captures the slip deformation mechanism of SC superalloys with the inherent ability to simulate anisotropic because of the two criterions framework controlled by Kelvin stress. Compared with others, the proposed model was able to simulate time-dependent inelastic deformation and cyclic deformation behavior under complex loading. The kinematic hardening and isotropic hardening models incorporated microscopic quantities, such as dislocation density and channel phase width, connecting the macroscopic mechanical response with the microscopic state to achieve multiscale constitutive modelling. The parameter identification and finite element implementation were conducted on a SC superalloy [2]. Simulation results demonstrated the accuracy of the proposed model in predicting deformation behavior under various orientations, rate-dependent effects, isothermal and non-isothermal cyclic deformation. Comparison with the classical anisotropic matrix macroscopic phenomenological approaches highlights the superior capability of the proposed model to simulate the orientation-dependent mechanical properties of single-crystal alloys.

Parallel two‐scale finite element implementation of a system with varying microstructure

AbstractWe propose a two‐scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying system of partial differential equations with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed‐ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two‐scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.

Calculating shear lag in steel-concrete composite beams under combined compression and bending

Long and complex composite steel-concrete structures are becoming common, requiring a deep understanding of the effects induced by the simultaneous action of axial forces and bending. In fact, the axial force generated, for instance, by cable inclination in cable-supported structures can modify the stress distribution within the elements compared to bending scenarios, thereby necessitating a revision of the effective width to be utilized. Nonetheless, current design codes, including Eurocode specifications and others, lack provisions for addressing the combined effects of axial force and bending, as they are exclusively tailored for bending. This limitation can introduce design complexities, necessitating the implementation of intricate Finite Element (FE) models, which impose substantial computational loads and design efforts. The methodology proposed in this paper overcomes these challenges allowing to assess the stress distribution and resistance of composite deck at Serviceability Limit State (SLS) and Ultimate Limit States (ULS) by leveraging results obtained from standard beam models typically used by structural designers or practitioners. A comprehensive parametric analysis using nonlinear finite element models is performed to validate the developed methodology. A comparison with the Eurocode 4 formulations highlights that the proposed method provides superior accuracy in estimating peak stress in concrete slabs under combined compression and bending. Additionally, it facilitates straightforward verification at the ULS in compliance with Eurocode requirements.

Implementation of a non-conforming finite element in abaqus for dynamic thin plate analyses

Implementation of a non-conforming finite element in abaqus for dynamic thin plate analyses

Mechanism based four-linear cohesive zone model for mode I fracture of different stacking sequence CFRP laminates

Delamination, a prevalent failure mode observed in laminated composites, exerts a significant impact on structural integrity and performance. The occurrence of fiber bridging during the fracture process adds complexity and elevates the research challenges associated with this phenomenon. Existing models exhibit limitations in accurately capturing bridging behavior and discerning its underlying mechanical mechanisms. This study addresses these limitations by analyzing experimental results, employing the J-integral, and analyzing R-curve behavior, proposing a mechanism-based four-linear cohesive zone model along with a new finite element implementation method. Comprising three overlapping bi-linear CZMs, this model effectively simulates mode I fracture behavior in laminates with different stacking sequences. Moreover, it intuitively illustrates the mechanical mechanisms during crack propagation and offers simplicity in implementation. This research contributes to a deeper understanding of composite fracture mechanics and provides a practical model for predicting delamination behavior in laminated structures.

Sprain energy consequences for damage localization and fracture mechanics

The 2023 smooth Lagrangian Crack-Band Model (slCBM), inspired by the 2020 invention of the gap test, prevented spurious damage localization during fracture growth by introducing the second gradient of the displacement field vector, named the "sprain," as the localization limiter. The key idea was that, in the finite element implementation, the displacement vector and its gradient should be treated as independent fields with the lowest ([Formula: see text]) continuity, constrained by a second-order Lagrange multiplier tensor. Coupled with a realistic constitutive law for triaxial softening damage, such as microplane model M7, the known limitations of the classical Crack Band Model were eliminated. Here, we show that the slCBM closely reproduces the size effect revealed by the gap test at various crack-parallel stresses. To describe it, we present an approximate corrective formula, although a strong loading-path dependence limits its applicability. Except for the rare case of zero crack-parallel stresses, the fracture predictions of the line crack models (linear elastic fracture mechanics, phase-field, extended finite element method (XFEM), cohesive crack models) can be as much as 100% in error. We argue that the localization limiter concept must be extended by including the resistance to material rotation gradients. We also show that, without this resistance, the existing strain-gradient damage theories may predict a wrong fracture pattern and have, for Mode II and III fractures, a load capacity error as much as 55%. Finally, we argue that the crack-parallel stress effect must occur in all materials, ranging from concrete to atomistically sharp cracks in crystals.

Phenomenological constitutive laws for the dissipative behaviour of highly compressible elastomers and their finite element implementation

This paper deals with the development of constitutive equations to model the mechanical behaviour of compressible elastomers. These materials are naturally incompressible but can be made compressible by the addition of hollow microspheres, for example. Such a material is referred to as syntactic foam. The CEA (French Commission for Atomic and Alternative Energies) employs such compressible materials as seals in complex structures to reduce the internal stresses in vulnerable components and prevent their failure. The behaviour of these structures is predicted by finite element simulations. It is important to know and model the mechanical behaviour of the seals. Like elastomers, they can undergo large deformations. The microspheres enable the material to undergo large volume change unlike pure elastomer that is nearly incompressible. This compressibility also intensifies dissipative phenomena encountered in elastomers such as viscosity or plasticity. Furthermore, the Mullins stress softening effect is also intensified even for loadings that only bring about volumetric changes. To model these behaviours, a phenomenological approach was developed based on the isochoric/volumetric decomposition of the deformation gradient. The method of intermediate dissipative configurations was employed to introduce multiple phenomena, including viscosity (with several characteristic times) and viscoplasticity, for these two parts of the deformation. The constitutive equations and their flow rules were implemented in Abaqus through a UMAT subroutine and using a numerical approach to define the tangent operator. The parameters of the behaviour law were identified using a model reduction technique known as shape manifold approach. The resulting model can be compared with experimental data.

Mixed variational formulation and finite-element implementation of second-order poro-elasticity

We present a variational formulation of second-order poro-elasticity that can be readily implemented into finite-element codes by using standard Lagrangian interpolation functions. Point of departure is a two-field minimization principle in terms of the displacement and the fluid flux as independent variables. That principle is taken as a basis for the derivation of continuous and incremental saddle-point formulations in terms of an extended set of independent variables. By static condensation this formulation is then reduced to a minimization principle in terms of the displacement and fluid flux as well as associated higher-order fields. Once implemented into a finite-element code, the resulting formulation can be applied to the numerical simulation of porous media in consideration of second-order effects. Here, we analyze the model response by means of several example problems including two standard tests in poro-elasticity, namely the consolidation problems of Terzaghi and Mandel, and compare the results with those of a corresponding first-order model. As becomes clear, the second-order formulation can unleash its full potential when applied to the study of porous media having spatial dimensions comparable to the size of their microstructure. In particular, it is capable to regularize steep field gradients at external as well as internal surfaces and to describe material dilatation effects known from experiments.

Mixed finite element implementation of plane crack problems within strain gradient elasticity

AbstractAn approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method is used, where stresses, strains, and displacements are considered as independent variables. This significantly simplifies the pre‐requirement for approximating functions and allows one to use the simplest triangular finite elements with a linear approximation of displacements. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses in nodes are treated as local unknowns. This discrete problem is solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. The modeling plane crack problem (tension of a square plate with a central crack) is solved. The obtained solution agrees with the results available in the literature. Good convergence is achieved by refining the mesh for all scale parameters.