Nonlinear Constitutive Relations Research Articles

Multiple solutions of nonlinear coupled constitutive relation model and its rectification in non-equilibrium flow computation

In this study, the multiple solutions of Nonlinear Coupled Constitutive Relation (NCCR) model are observed and a way for identifying the physical solution is proposed. The NCCR model proposed by Myong is constructed from the generalized hydrodynamic equations of Eu, and aims to describe rarefied flows. In the non-equilibrium regime, the NCCR equations are more reliable than the Navier-Stokes equations. And the NCCR equations have an advantage of efficiency over the discrete velocity methods and stochastic particle methods. However, the NCCR model is a complicated nonlinear system. Many assumptions have been used in the schemes for solving the NCCR equations. The corresponding numerical methods may be associated with unphysical solution and instability. At the same time, it is hard to analyze the physical accuracy and stability of NCCR model due to the uncertainties in the numerical discretization. In this study, a new numerical method for solving NCCR equations is proposed and used to analyze the properties of NCCR equations. More specifically, the nonlinear equations are converted into the solutions of an objective function of a single variable. Under this formulation, the multiple solutions of the NCCR system are identified and the criteria for picking up the physical solution are proposed. Therefore, a numerical scheme for solving NCCR equations is constructed. A series of flow problems in the near continuum and low transition regimes with a large variation of Mach numbers are conducted to validate the numerical performance of proposed method and the physical accuracy of NCCR model.

A mathematical justification for nonlinear constitutive relations between stress and linearized strain

A mathematical justification for nonlinear constitutive relations between stress and linearized strain

Anisotropic odd elasticity with Hamiltonian curl forces

Abstract A host of elastic systems consisting of active components exhibit path-dependent elastic behaviors not found in classical elasticity, which is known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd) elastic modulus tensor. Here, from the perspective of geometry, we construct the Hamiltonian formalism to show the origin of the antisymmetry of the elastic modulus that is intrinsically anisotropic. Furthermore, both non-conservative stress and the associated nonlinear constitutive relation naturally arise. This work also opens the promising possibility of exploring the physics of odd elasticity in dynamical regime by Hamiltonian formalism.

Deformation and enclosed volume change of axisymmetric non-linear dielectric elastomer membrane pump: a theoretical study

Abstract This work investigates the change in the enclosed volume of an axisymmetric non-linear hyper-elastic membrane pump subjected to dielectric actuation. The equilibrium equations in cylindrical coordinates, coupled with non-linear constitutive relations, are solved numerically to obtain the deformed membrane geometry under pressure loading (using Variational Calculus). The effect of dielectric actuation is then incorporated by considering the change in material properties and deformation induced by the applied electric field. The deformed geometry under dielectric actuation is determined, and the change in enclosed volume is calculated by comparing the deformed states with and without the applied electric field. The proposed approach enables quantifying the volumetric change in non-linear hyper-elastic membrane pumps due to dielectric actuation for potential applications in soft robotics, adaptive optics, and microfluidics. By non-dimensionalizing variables, key dimensionless parameters governing the problem are identified for broader applicability. Furthermore, this methodology can estimate volume changes for different electric actuations when a specific material is chosen with experimentally derived parameter trends, facilitating material selection or synthesis to meet targeted volumetric requirements.

Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach–Mayergoyz space model

SUMMARY The existence of pores, cracks and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress–strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach–Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress–strain and modulus–stress relationships under different stress protocols. Additionally, we analyse the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach–Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain–displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wave front discontinuities and uneven energy distribution in the 2D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.

Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.

A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines

Abstract In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in a space-time domain. Based on the Babuška–Nečas theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability and accuracy of the proposed approach.

Characterizing nonlinear constitutive behaviors of fiber metal laminates

Abstract This study characterized the nonlinear tensile behavior of fiber metal laminates (FMLs). FMLs comprise layers of thin metallic sheets and fiber-reinforced composite layers, and a constitutive FML model includes the constitutive relationships of the FML’s constituent materials; however, nonlinear behavior is typically only considered for the metal components of an FML. In this study, a nonlinear constitutive relationship for the unidirectional fiber composites was modeled using a one-parameter plastic model. The nonlinear constitutive law for the metal was formulated using the J2 flow rule. These relationships were summed for each layer in accordance with laminated plate theory to obtain a constitutive FML model, which was then used for numerical predictions of nonlinear stress–strain curves. The model was validated by comparing its predictions with experimental results from the literature. Moreover, the effect of the inclusion of nonlinear fiber composite behavior on the model predictions was investigated. Results revealed that the difference between the model predictions and the experimental results was less than 4%. These predictions with nonlinear fiber composite behavior were substantially more accurate than those of the model without this behavior for FMLs with angle-ply fiber composites.

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.

Numerical simulation of high-frequency induction welding in longitudinal welded tubes

In the present paper the high-frequency induction welding process is studied numerically. The mathematical model comprises a harmonic vector potential formulation of the Maxwell equations and a quasi-static, convection dominated heat equation coupled through the Joule heat term and nonlinear constitutive relations. Its main novelties are a new analytic approach which permits to compute a spatially varying feed velocity depending on the angle of the Vee-opening and additional spring-back effects. Moreover, a numerical stabilization approach for the finite element discretization allows to consider realistic weld-line speeds and thus a fairly comprehensive three-dimensional simulation of the tube welding process.

Nonlinear analysis on electro-elastic coupling properties in bended piezoelectric semiconductor beams with variable cross section

In this paper, the effects of nonlinear constitutive relation on the electro-elastic coupling behaviors in non-uniform piezoelectric semiconductor beams are investigated. On the basis of three-dimensional theory for piezoelectric semiconductor and double power series, one-dimensional bending model is developed. In order to deal with the nonlinear problem, a differential quadrature method based iteration procedure is proposed. Comparing with the results from finite element method, the proposed method has good convergence and high accuracy. In numerical analysis, the effects of nonlinearity and non-uniformity in piezoelectric semiconductor are discussed thoroughly. It is found the introduction of nonlinearity induces the loss of symmetric for all electrical field quantities but little changes the mechanical quantities. While the reduction of width along the length direction produces larger mechanical and electrical responses. Importantly, the effects of nonlinearity on the electrical field quantities are more evident in a piezoelectric semiconductor beam with rapidly varied cross section. Besides, we designed the piezoelectric semiconductor beams subject to double shear forces and a beam with locally varied cross section. In the designed structures, the potential barriers and wells are realized, which provide new approaches adjusting the electro-elastic behaviors in piezoelectric semiconductors. The study in this paper could be the guidance designing high performance electronic devices.

Bayesian model class selection of nonlinear constitutive relationships for piezoelectric energy harvesters with small set of observations

The demand for efficient energy harvesting from vibratory environments has notably increased with the proliferation of piezoelectric devices. However, mismatches between model predictions and experimental observations underpin a critical challenge: accurately modeling piezoelectric energy harvesters (PEHs) while accounting for the inherent uncertainties in their electromechanical properties. A comprehensive Bayesian inference framework is posited for identifying electromechanical properties and facilitating model class selection of nonlinear PEHs. This approach addresses uncertainties by integrating prior knowledge and experimental data to update model parameters, offering the potential to balance model complexity and predictive precision systematically. The influence of different excitation amplitudes and the number of observations employed in the framework is studied. The results demonstrated the ability to capture both linear and nonlinear behaviors, while the model class selection effectively determined the simplest constitutive models sufficient for accurate predictions, validated against varying excitation intensities. By elucidating the trade-off between model simplicity and accuracy, this method not only refines the understanding of PEH behavior under diverse operating conditions but also equips designers with a robust predictive tool. Consequently, this methodology serves to optimize the deployment of PEHs in real-world applications, such as structural health monitoring, ensuring reliable energy supply with quantifiable confidence levels.

Machine learning as alternative strategy for the numerical prediction of the shear response in reinforced and prestressed concrete beams

Some materials, such as reinforced and prestressed concrete, involve non-linear constitutive relationships in elasticity problems defined on them. In particular, the shear strength of a reinforced concrete beam may be calculated by considering a diagonal struts field in the context of the so-called “Compression Field Theories” (CFTs). This work presents an efficient Machine Learning method alternative to numerical methods for obtaining the full shear response of reinforced and prestressed concrete beams based on CFT regarding stresses, strains, and crack angles. For that, a regression task is developed using state-of-the-art Machine Learning (ML) models. A ML model per output variable is trained with the existing Newton-Raphson solutions database. The solvability region of the embedded steel constitutive model is also considered, demonstrating the comprehensive character of the proposed method. The model is validated on two real beam responses, where the results obtained demonstrate that this alternative method based on ML algorithms effectively addresses the problem of prediction of the shear response in reinforced and prestressed concrete beams. The proposed ML approximation performs reasonably well without requiring any initial approximations. Moreover, the ML regressor here developed shows a low dependence on the concrete tension stiffening area surrounding the steel reinforcement, which significantly improves the performance of this methodology in a higher number of design cases. Thus, in the practical context of structural engineering, this last approach establishes an efficient alternative procedure for obtaining numerical solutions of mechanical models based on the CFTs framework.

A reactive electrochemomechanical theory for growth and remodeling of polyelectrolyte hydrogels and application to dynamic polymerization of DNA hydrogels

This study develops a framework for growth and remodeling of active polyelectrolyte hydrogels that accounts for effects of compositional changes on the mechanical response. By developing a reactive electrochemomechanical theory, thermodynamical constraints upon reactive and remodeling processes are elucidated within a general framework that allows any number of chemical reactions to evolve the response of the gel and transfer mass and charge between constituents. Fully coupled, nonlinear constitutive relations are adopted for molar fluxes, allowing exploration of effects including cross-diffusion, electrophoresis, and electro-osmosis. A robust finite element implementation is developed in the open source FEBio software (febio.org) by exploiting an equivalence between electrochemomechanics and mixture theory. The implementation is verified against analytical solutions for free swelling, and a proper reduction to a prior chemomechanical theory is demonstrated for neutral gels swollen only by a solvent with no solutes. The theory and implementation are then applied to model the tunable large swelling achieved through dynamic polymerization of DNA crosslinkers seen in our recently developed experimental hydrogel system (Cangialosi et al., 2017). A novel constitutive model for reaction-driven evolution of the locking stretch λL in a non-Gaussian mechanical free energy was developed, where the increasing concentration of DNA crosslinkers makes further swelling energetically favorable. With a single free parameter, excellent agreement was found between measured and predicted equilibrium swelling ratios. This study demonstrated the ability to extend the electrochemomechanical framework to include chemical reactions and composition-aware constitutive models, and showed that development of reactive models allows simulation of complex dynamic polymerization phenomena not treated before. The theoretical frame here can be further expanded in scope to incorporate additional non-ideal and nonlinear phenomena.

A phase field and machining-learning approach for rapid and accurate prediction of composites failure

A new approach is proposed for rapid and accurate prediction for composite failure in combination of the phase field and machine-learning methods. First, using experimentally-fitted tangent modulus instead of elastic modulus as constitutive relationship, a modified phase field method (MPFM) is established for the crack propagation and mechanical response, which can be effectively applied for composites with a nonlinear constitutive relationship. Interestingly, both the crack propagation path and mechanical responses of two typical examples of composites using MPFM are well consistent with previously available experimental and calculated ones. Furthermore, the data-driven back propagation neural network (BPNN) is constructed to greatly accelerate the prediction on a database generated by MPFM, emphasizing several critical parameters, for example, fiber orientations, external load, maximum failure strain, and critical strain energy release rate. Of much importance, the well-trained BPNN builds a bridge between the traditional computational fracture mechanics and machine learning algorithms, enabling non-specialists to accurately calculate the mechanical response of composites, moreover, saving over 99% of computing time in comparison with MPFM.

Modeling and performance analysis of resonant self-biased magnetoelectric transducers

Compared with single-phase multiferroic materials, magnetoelectric (ME) composites composed of piezoelectric and magnetostrictive materials have great ME coupling, and have received widespread attention in various application fields. The application of ME devices in wireless power transfer (WPT) is attractive due to their compactness and ability to operate at lower frequencies than conventional coils. However, traditional ME composites rely on permanent magnets or electromagnets to provide biased magnetic fields, thus leading to problems such as high noise, large size, and high cost, which significantly hinder the advancement of miniaturized and high-performance ME devices. To solve this problem, a self-biased ME laminated structure based on the magnetization grading effect is proposed in this work. Using the equivalent magnetization and nonlinear magnetostrictive constitutive relationship, a finite element simulation model for a self-biased ME transducer operating in L-T mode is constructed. The ME coupling performances without DC bias in bending vibration mode and stretching vibration mode are studied. Based on the model, the corresponding experimental samples are prepared for measurement. The measurement results are in agreement with the simulation data, thereby validating the accuracy and effectiveness of the model. The measured results show that the Metglas/Galfenol/PZT-5A structure can exhibit more significant self-biased ME effect under the stretching resonance mode than under bending resonance mode. Its ME coefficient attains a notable value of 10.7 V·cm<sup>–1</sup>·Oe<sup>–1</sup> at 99.4 kHz, while ME power coefficient reaches 5.01 μW·Oe<sup>–2</sup> at 97.9 kHz. Its on-load ME power coefficient can reach up to 4.62 μW·Oe<sup>–2</sup> at 99.3 kHz without impedance matching. When an external bias magnetic field of 25 Oe is applied, these performance indexes increase significantly to 47.06 V·cm<sup>–1</sup>·Oe<sup>–1</sup> at 99.4 kHz and 82.13 μW·Oe<sup>–2</sup> at 99.0 kHz, respectively. The simulation results further show that the performance of the self-biased ME transducer can be significantly improved by increasing the thickness of the high permeability layer. For example, by increasing the Metglas layer thickness from 30 μm to 90 μm, both the ME coefficient and ME power coefficient increase rapidly by 2.47 times and 6.96 times the original values, respectively. Self-biased ME transducers effectively reduce the dependence on external bias magnetic field, thereby providing a good approach for applying and developing ME composites in low-frequency WPT systems.

Modeling the thermo-mechanical response and phase changes in metallic additive manufacturing (MAM) processes using a dissipative phase-field model

Additive manufacturing (AM) has emerged as a highly promising manufacturing technique, offering unprecedented possibilities for creating complex geometries and functional structures. However, harnessing the full potential of AM requires the development of a robust computational framework capable of capturing the intricate multi-scale and multi-physics nature of the process. The constitutive and structural responses encountered in AM are particularly challenging to reproduce due to the complex behavior of the material involved. This research aims to address these challenges by presenting a comprehensive computational approach that incorporates a material model capable of accurately representing the behavior of different phases occurring during AM. To achieve this, the finite element method, using the Lagrangian framework in the implicit time scheme, is employed through the widely adopted ABAQUS software. Computational implementation is facilitated using the FORTRAN programming language. By employing weakly coupled thermal and mechanical constitutive equations, the framework enables the analysis of thermal stresses, strains, and displacements during realistic solidification processes, which inherently involve highly nonlinear constitutive relations. Through a series of numerical examples, the capabilities of the proposed model are demonstrated across various computational scales, particularly during the rapid melting and solidification phases. These simulations reveal the formation of residual stresses, which can lead to part distortion and have detrimental effects on the mechanical properties of the manufactured components. This research contributes to the advancement of additive manufacturing by providing a reliable computational tool that integrates the complex interplay between thermal and mechanical phenomena. The developed framework enhances our understanding of the AM process, offering valuable insights into the factors influencing the structural integrity and performance of additively manufactured parts.

Modeling of induced currents in a type-II superconducting cylinder of finite length in a transverse magnetic field

Induced currents in the bulk superconducting cylinder of finite length placed in a transverse magnetic field are calculated by the partial and complete separation of variables. The Fourier transform over the angular and axial coordinates reduces the problem to the solution of one-dimensional integral equations. The harmonic balance method is applied to account for the highly nonlinear constitutive relation between the voltage and current. The current and field distribution is compared with the results of FEA. The approximation of the current distribution by a set of cylindrical and disc layers is presented. The magnetic moment of induced currents is analyzed in the increasing applied field when the cylinder is magnetized up to the state of saturation.

Progressive failure analysis of perforated composite laminates considering nonlinear shear effect

Composites are widely used in high performance structures such as aerospace structures due to their excellent properties. The analysis of failure evolution of composite perforated structures by finite element simulation is of great significance for practical work as engineering composite structures often contain notches and voids. In this paper, the numerical simulation of failure evolution and failure modes of carbon fiber reinforced resin composite laminates with large openings was carried out. A UMAT subroutine was written based on the 3D Hashin-Ye failure criterion and progressive damage model theory. The characteristic length and viscosity coefficient were introduced into the model to reduce mesh dependency and improve computational convergence. The nonlinear shear constitutive relationship defined by the Ramberg–Osgood equation was introduced into the continuous damage degradation model. The effect of nonlinear shear on the failure evolution of laminates with different stacking sequence was studied.

APPLICABILITY INDICATORS FOR THE NONLINEAR MAXWELL-TYPE ELASTO-VISCOPLASTIC MODEL WITH POWER MATERIAL FUNCTIONS AND TECHNIQUES TO CALIBRATE THEM

A physically non-linear Maxwell-type constitutive relation with two material functions for nonaging elasto-viscoplastic materials is studied analytically in order to examine the set of basic rheological phenomena that it simulates, to enclose its application field, to obtain necessary phenomenological restrictions which should be imposed on its material functions and to develop identification and validation techniques. Characteristic features of loading-unloading-recovery curves family produced by the model with two power material functions (with four parameters) under loading and unloading at constant stress rates and subsequent rest are analyzed in uni-axial case and compared to general properties of stress-strain-recovery curves produced by the constitutive relation with two arbitrary (increasing) material functions (theorems 1 and 2). Their dependences on loading rate, maximal stress and material functions exponents are examined. Power functions are the most popular in creep models, elastoviscoplasticity, polymer rheology, hydrodinamics of non-newtonian fluids and simulation of superplastic flow. The analysis reveals several specific properties of theoretic loading-unloading-recovery curves produced by power model with four parameters that can be employed as the model applicability indicators which are convenient for check using test data of a material. They should be checked in addition to general applicability indicators for the Maxwell-type constitutive relation with two arbitrary material functions. A number of effective calibration procedures for the model in the class of power material functions are developed. They are more rapid and effective than general identification techniques for two arbitrary material functions developed previously. The first procedure employs a pair of stress-strain curves at different stress rates, the second one is based on a pair of loadingunloading- recovery curves with various maximal stress values and loading rates and the third one needs only one loading-unloading-recovery curve. The explicit expressions are derived for four material parameters via test data. They enable separate and direct evaluation of the material parameters without error accumulation. Identification techniques versions are considered and their advantages and shortcomings are discussed. The ways to minimize the error using additional tests are proposed.