Incremental Formulation Research Articles

A cluster-based incremental potential approach for reduced order homogenization of bones.

We develop a cluster-based model order reduction (called C-pRBMOR) approach for efficient homogenization of bones, compatible with a large variety of generalized standard material (GSM) models. To this end, the pRBMOR approach based on a mixed incremental potential formulation is extended to a clustered version for a significantly improved computational efficiency. The microscopic modeling of bones falls into a mixed incremental class of the GSM framework, originating from two potentials. An offline phase of the C-pRBMOR approach includes both a clustering analysis spatially decomposing the micro-domain within an RVE and a space-time decomposition of the microscopic plastic strain fields. A comparative study on two different clustering approaches and two algorithms for mode identification is additionally conducted. For an online analysis, a cluster-enhanced version of evolution equations for the reduced variables is derived from an effective incremental variational formulation, rendering a very small set of nonlinear equations to be numerically solved. Several numerical examples show the effectiveness of the C-pRBMOR approach. A striking acceleration rate beyond 104 against conventional FE computations and that beyond 103 against the original pRBMOR approach are observed.

Snap-through eversion of axisymmetric shells under contact indentation

In this paper, we systematically investigate the stability of an axisymmetric shell and the snap-through eversion induced by indentation through a discrete numerical approach. To capture the intricate interplay between the geometric and boundary nonlinearities during contact actuation, we employ the discrete axisymmetric shell model accompanied by the incremental potential formulation for our analysis. Our results reveal that the indentation response of a spherical shell can be classified into three groups, i.e. monotonous, monostable and bistable behaviours, whose boundaries can be characterized by a simple scaling law. We further discover that, for bistable shells, the snap-through eversion happens at a critical state where the configurations are universal, which is independent of the indenter size and can be captured by a simple geometric model. One interesting prediction of our model is that, with increasing indenter size, the contact state between the shell and indenter changes from conformal contact to partial separation, which is validated by a finite element method simulation. Our findings may provide explanations for some biophysical phenomena (e.g. cell fusion) and can also guide optimal designs of intelligent structures (e.g. soft actuators and soft robots).

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.

Command-filtered incremental backstepping attitude control of spacecraft with predefined-time stability

This paper investigates the predefined-time attitude tracking control problem within the theoretical framework of incremental dynamic inversion. The utilization of the Taylor series facilitates the transformation of the attitude control system into a discrete-time plant, with the control input expressed in an incremental form. Additionally, a novel predefined-time stable dynamics system is presented and incorporated into the disturbance observer designing process, serving the purpose of estimating lumped disturbance. It is also employed in a command filter design to approximate the derivatives of the virtual control law within a predefined time. Consequently, an incremental backstepping attitude tracking control scheme is further developed, integrating the proposed predefined-time disturbance observer to ensure the system's robustness and the predefined-time filter to address challenges related to “term explosion” and singularity problem. Rigorous Lyapunov analysis affirms that the attitude control system, when using the incremental backstepping controller, remains predefined-time stable. The effectiveness of the proposed control scheme is subsequently validated through numerical simulations.

Elastoplastic limit analysis of functionally graded materials pipe elbows to predict the damage under in-plane bending moment using XFEM-Technic

Functionally Graded Materials (FGMs) have become a vast field of research, due to their wide range of applications and numerous advantages. The use of FGMs has recently expanded, particularly in the reinforcement of tubular structures. Pipe elbows are among the most important elements in a piping system, responsible for reducing stresses and balancing pressure inside pipes. However, due to significant deformations, most frequently caused by bending moment loads, elbows are more prone to exceeding their stress limits. It is for this reason that the analysis of these tubular structures has attracted the interest of many researchers. As a result, the study of their reinforcement can only be carried out through the study of their damage. In other words, to unlock the difficulties of numerically predicting their responses. The aim of our study is to use the finite element method with the proposed meshing technique in the ABAQUS calculation code, to analyze the behavior of a 90° elbow tubular structure, attached to two straight sections, made of functionally graded FGM Al/SiC (Aluminum/Silicon Carbide) material, with a continuous gradual variation of properties in the direction of wall thickness, per row of finished elements along the entire geometry of the tubular structure, in order to simulate a real pipe system installation. The structure is loaded by two modes of in-plane bending moment (Closing and Opening), with an imposed rotational displacement. The aim of this analysis is to evaluate, firstly, the effects of different grading concepts for the FGM Al/SiC material couple in the model geometry (Simple and Symmetrical Concepts), and secondly, the effect of different power exponent indices (n) of the volume fraction, on the elasto-plastic response up to the damage of the tubular structure. The constitutive behavioral law of our model is based on Von Mises’ equivalent stress flow theory with a hardening variable in incremental form. In addition, the TTO (Tamura-Tomota-Ozawa) model was used to describe the elastic-plastic behavior of the FGM, and the XFEM technique was also applied for crack initiation and propagation. The results obtained under moment-rotation conditions show a significant effect on the response as well as on the level of damage, the effect is also remarkable on the ovalization at the bend cross-section. The approach and reliability of the results obtained were based on a comparison with experimental results, which show good agreement, compared with our numerical model.

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

Deformation twinning as a displacive transformation: computational aspects of the phase-field model coupled with crystal plasticity

AbstractSpatially-resolved modeling of deformation twinning and its interaction with plastic slip is achieved by coupling the phase-field method and crystal plasticity theory. The intricate constitutive relations arising from this coupling render the resulting computational model prone to inefficiencies and lack of robustness. Accordingly, together with the inherent limitations of the phase-field method, these factors may impede the broad applicability of the model. In this paper, our recent phase-field model of coupled twinning and crystal plasticity is the subject of study. We delve into the incremental formulation and computational treatment of the model and run a thorough investigation into its computational performance. We focus specifically on evaluating the efficiency of the finite-element discretization employing various element types, and we examine the impact of mesh density. Since the micromorphic regularization is an important part of the finite-element implementation, the effect of the micromorphic regularization parameter is also studied.

Micromechanical analysis of magnetoelectric coupling of composites containing nonlinear magnetostrictive and piezoelectric constituents

This paper is a reformulation of the simplified unit-cell and Mori-Tanaka micromechanics theories in their applications to nonlinear magnetoelectric analysis of two-phase composite materials with 0–3, 1–3, and 2–2 connectivity types, respectively. We elucidate the similarity between the two models insofar as concentration-factor matrices are concerned. The representations of the bulk magnetostrictive and piezoelectric phases are based on nonlinear constitutive equations. Due to material nonlinearity, the mathematical frameworks are accomplished via incremental formulation that provides a system of linear algebraic equations at each increment which is obviously a great advantage over nonlinear one. The derived nonlinear composite constitutive relations that govern the hysteresis behavior of composites are implemented to study the composite composed of Terfenol-D and PZT constituents. The responses of this composite to a complete cycle of magnetic field are shown and the micromechanics predictions are compared in light of existing experimental data.

Experimental study on the dynamic response of sintered NdFeB and secondary development verification of its Zhu-Wang-Tang damage constitutive model

Currently, there is relatively limited research on the dynamic response of sintered neodymium iron boron (sintered NdFeB). In order to reveal the dynamic response of sintered NdFeB, a Hopkinson pressure bar experimental platform is constructed to study its dynamic response. The results of the study indicate the necessity of establishing a dynamic constitutive model which can reflect its the strain rate sensitivity. Therefore, combining the Weibull distribution with the Zhu-Wang-Tang constitutive model, a constitutive model with damage for sintered NdFeB is created and this model is extended to three-dimensional form, expressing its incremental formulation. Finally, to verify the accuracy of this model in practical applications, the VUMAT subroutine is utilized to compile the damage constitutive relationship of sintered NdFeB. A Hopkinson pressure bar computational model is established, and the results show good consistency between the simulation and the experiment. This study provides a reference for the practical application of the damage constitutive model in actual scenarios.

An algorithm based on B-differentiable equations method for solving problems involving frictional contact coupled with concrete plastic damage

In this paper, an algorithm is proposed to solve the frictional contact problems while considering elasto-plastic material properties. The static and dynamic solution procedures are derived in incremental form, respectively. The proposed method employs a two-layer iteration strategy. In the outer iteration layer, the equilibrium equations are solved, with the stress and stiffness matrix updated at each iteration. Within the inner iteration layer, the contact force is calculated, where the contact equations, expressed as B-differentiable equations, incorporate only contact conditions in normal and tangential directions, and the contact flexibility matrix, obtained by applying unit force pair to all contact node pairs in turn, is introduced. Consequently, the solution for nonlinear equilibrium equations and contact equations is decoupled, requiring only one decomposition of the stiffness matrix for each calculation of the contact flexibility matrix, as it remains constant within the contact iteration layer. In addition, the constitutive model of concrete developed by Lee and Fenves is utilized. This model adopts two independent variables to describe tensile and compressive damage, together considering the strength recovery of the concrete. Numerical examples demonstrate the accuracy of the proposed method by comparing the results from Abaqus. Furthermore, the present algorithm is applied to static and dynamic analyses of a concrete arch dam with several transverse joints, where the effects of gravity, water level, the number of contact surfaces, and seismic load are considered, and some conclusions are obtained.

Precise analysis of geometrical nonlinear mechanical behavior of shear deformable laminated curved composite beams in local coordinates via mixed type finite elements

In the present article, geometrically nonlinear analyses of curved laminated composite beams are carried out using Mixed Finite Element Method (MFEM). Timoshenko beam theory is used to consider shear influence which significantly effects the structural behavior of composite layers. Energy-based functional is generally needed to perform geometric nonlinear analyzes of layered composites. However, obtaining energy-based functional in geometric nonlinear theory is quite challenging due to the difficulty of solving mathematically nonlinear differential equations. In this study, this situation is overcome and a functional suitable for performing geometric nonlinear analyzes of layered composite materials is obtained. Incremental formulation procedure is utilized as a solution method for nonlinear finite element analyzes. This is an innovative application of the non-linear mixed finite element method and has the ability to perform geometric nonlinear analysis of curved laminated composite beams with perfect precision. By obtaining the functional in local coordinates, it is possible to solve special problems in various fields such as construction, machinery, aerospace and biomechanics.

Dynamic modeling of a sliding ring on an elastic rod with incremental potential formulation

Abstract Mechanical interactions between rigid rings and flexible cables find broad application in both daily life (hanging clothes) and engineering system (closing a tether-net). A reduced-order method for the dynamic analysis of sliding rings on a deformable one-dimensional (1D) rod-like object is proposed. In contrast to the conventional approach of discretising joint rings into multiple nodes and edges for contact detection and numerical simulation, a single point is used to reduce the order of the model. To ensure that the sliding ring and flexible rod do not deviate from their desired positions, a new barrier function is formulated using the incremental potential theory. Subsequently, the interaction between tangent frictional forces is obtained through a delayed dissipative approach. The proposed barrier functional and the associated frictional functional are C2 continuous, hence the nonlinear elastodynamic system can be solved variationally by an implicit time-stepping scheme. The numerical framework is initially applied to simple examples where the analytical solutions are available for validation. Then, multiple complex practical engineering examples are considered to showcase the effectiveness of the proposed method. The simplified ring-to-rod interaction model has the capacity to enhance the realism of visual effects in image animations, while simultaneously facilitating the optimisation of designs for space debris removal systems.

Low-cycle fatigue simulation of ductile materials using elasto-plastic gradient damage approach

This work presents a novel computational strategy based on gradient enhanced damage and cyclic yielding with nonlinear mixed hardening behaviour for simulating low-cycle fatigue of ductile materials. A new yield function is introduced which accommodates the cyclically evolving state variables during mixed hardening and the damage induced during the application of repetitive loads. The resulting constitutive relations are derived in an incremental form. The non-local strains facilitate the computation of incremental damage in the material. A finite element scheme is developed from the governing physics through full coupling between damage and cyclic-elastoplastic deformations. In order to obtain an accurate computational response, a stress-return mapping scheme is formulated for the damage-dependent yield function. The developed numerical strategy is investigated for specimens made of various ductile materials exhibiting elasto-plastic behaviour under low-cycle fatigue loads. The stress-strain hysteresis and cyclic softening computed by the present numerical framework agree well with experimental data available in literature. The low-cycle fatigue crack-growth results are also accurately captured for various fatigue loading conditions.

A Nonlinear Mixed Finite Element Method for the Analysis of Flexoelectric Semiconductors

Abstract In this paper, we develop a nonlinear mixed finite element method for flexoelectric semiconductors and analyze the mechanically tuned redistributions of free carriers and electric currents through flexoelectric polarization in typical structures. We first present a macroscopic theory for flexoelectric semiconductors by combining flexoelectricity and nonlinear drift-diffusion theory. To use C0 continuous elements, we derive an incremental constrained weak form by introducing Langrage multipliers, in which the kinematic constraints between the displacement and its gradient are guaranteed. Based on the weak form, we established a mixed C0 continuous nine-node quadrilateral finite element as well as an iterative process for solving nonlinear boundary-value problems. The accuracy and convergence of the proposed element are validated by comparing linear finite element method results against analytical solutions for the bending of a beam. Finally, the nonlinear element method is applied to more complex problems, such as a circular ring, a plate with a hole, and an isosceles trapezoid. Results indicate that mechanical loads and doping levels have distinct influences on electric properties.

A Constitutive Law Modeling the Mixed Hardening Behavior of Particle-Reinforced Composites

Abstract A homogenized elasto-plastic constitutive law (including both constitutive equations and inversed constitutive equations) is proposed in an incremental form for the particle-reinforced composites based on the flow theory of plasticity and the asymptotic homogenization method. The constitutive law can be used to predict the mixed hardening behavior of particle-reinforced composites under arbitrary loading conditions if the uniaxial tension test curve of matrix materials is known. It is found that the constitutive law of particle-reinforced composites is similar in form to the law of matrix materials. There is a simple proportional relationship between the yield stress, the plastic modulus, and the deviatoric back stress of particle-reinforced composites and the corresponding parameters of matrix materials, which is equal to the ratio of the shear modulus of composites to the shear modulus of matrix materials. The tangent modulus of particle-reinforced composites can be calculated using a simple arithmetic formula according to the tangent modulus of matrix materials. A numerical algorithm is suggested.

Investigation on A Viscoelastic Creep Damage Constitutive Model for HTPB Propellant and Its Application in Finite Element Analysis

Solid rocket motor (SRM) is typically kept in storage for a significant portion of its lifespan. During storage, the propellant grain creeps under the action of gravity. To investigate the creep properties of hydroyl-terminated polybutadiene (HTPB) propellant under long-term loading, a creep damage constitutive model is developed based on the generalized Kelvin model and continuum damage mechanics theory. Parameters for this model are determined through creep tests conducted on HTPB propellant under various stress levels. Furthermore, to enable the application of the developed propellant constitutive model in the finite element analysis of SRM, the derived constitutive equation is transformed into an incremental form and integrated into the user subroutine UMAT of the finite element analysis platform ABAQUS. Subsequent to this, a three-dimensional SRM model containing the case, the insulator, and the propellant grain is established. The mechanical response of the grain under thermal and gravitational loads is calculated and analyzed. The results indicate that the stress concentration region in the grain is located at the stress release boot in the forward dome of the motor, where damage occurs. The methods employed in this paper and the conclusions drawn can serve as a reference for evaluating the structural integrity and storage lifespan of the SRM.

Co-rotational 3D beam element using quaternion algebra to account for large rotations: Formulation theory and static applications

This paper presents a co-rotational beam element based on quaternion algebra as a means of parameterizing large rotations. This co-rotational framework is based on a decomposition of beam kinematics into a rigid element frame, which follows the element and its pure deformation. Once the decomposition between rigid body motion and deformation is obtained, the principle of virtual work allows calculating the element response projected onto large displacements and rotations. This 3D co-rotational element lies within the framework of incremental formulations. The special feature of this formulation pertains to the decomposition of kinematics to extract the rigid body motion and pure deformations through use of quaternion algebra by solving an internal nonlinear kinematic equation. Thus, from the 2 quaternions defining global rotations and the 2 displacement vectors at the ends of the beam, this method is able to extract: 1 quaternion and 1 displacement vector parameterizing the rigid body motion, plus 2 other quaternions defining the internal rotations and beam elongation. The proposed formulation based on quaternion algebra therefore constitutes an alternative to the literature’s treatment of large-rotation kinematics using a quaternion algebra. The operators are also linearized in order to obtain the algorithmic stiffness operator. Eleven distinct static numerical applications are presented, along with comparisons from the literature in the aim of demonstrating the efficiency of this proposed element.

Persymmetric design of jointly detection and bearing estimation for a 2D array radar in training demanding scenarios

The problem of joint detection and bearing estimation for a two-dimensional (2D) array radar in training demanding scenarios is addressed. Regardless of the cause of the angle deviation, we model the 2D steering vector as a fully incremental form. Thus, the 2D array requires the optimization of two target cosine offsets. To relax the requirement of sufficient training data, we incorporate persymmetric structure in the design of the receiver. Unlike traditional joint optimization, the persymmetric structure brings a more complex multidimensional matrix form, making detection optimization more difficult. Then, we transform the optimization problem into a fractional-order programming problem and solve it by Dinkelbach algorithm. Finally, numerical results verify the superiority of the proposed method over conventional methods in insufficient training scenarios. Also, the proposed method is more robust.

A 3D viscoelastic–viscoplastic behavior of carbon nanotube‐reinforced polymers: Constitutive model and experimental characterization

AbstractDue to the widespread use of carbon nanotube (CNT)‐reinforced polymers in various industries, investigating the dynamic mechanical response of these materials under impact loading is of significant importance. A three‐dimensional viscoelastic (VE)–viscoplastic (VP) constitutive model for CNT/epoxy nanocomposites is developed. A proper rheological model is established using the generalized Maxwell model to represent the VE behavior and propose an exponential function to express the elasto‐VP response. Employing an empirical logarithmic relationship to estimate material properties at different strain rates, the VP behavior is modeled based on the overstress concept. The 3D numerical discretization of the proposed rate‐dependent material model is also carried out to develop a user‐defined material model (UMAT) for the commercial finite element (FE) software of Abaqus. The performed FE simulations based on the proposed rate‐dependent constitutive material can properly capture stress relaxation and hysteresis behaviors of the nanocomposites. Comparing the estimated tensile and shear stress–strain curves through developed material models with experimental measurements, an excellent agreement is observed.Highlights Tensile/shear properties of CNT/epoxy are measured at three strain rates. An empirical constitutive model is proposed for any arbitrary strain rates Viscoplastic behavior is captured based on over‐stress concept A 3D incremental form of a viscoelastic–viscoplastic material model is developed.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.