Nonlinear Constitutive Relations Research Articles

Inverse design of mechanical metamaterial achieving a prescribed constitutive curve

Besides exhibiting excellent capabilities such as energy absorption, phase-transforming metamaterials offer a vast design space for achieving nonlinear constitutive relations. This is facilitated by switching between different patterns under deformation. However, the related inverse design problem is quite challenging, due to the lack of appropriate mathematical formulation and the convergence issue in the post-buckling analysis of intermediate designs. In this work, periodic unit cells are explicitly described by the moving morphable voids method and effectively analyzed by eliminating the degrees of freedom in void regions. Furthermore, by exploring the Pareto frontiers between error and cost, an inverse design formulation is proposed for unit cells. This formulation aims to achieve a prescribed constitutive curve and is validated through numerical examples and experimental results. The design approach presented here can be extended to the inverse design of other types of mechanical metamaterials with prescribed nonlinear effective properties.

A Time-Domain Volume Integral Equation Solver to Analyze Electromagnetic Scattering From Nonlinear Dielectric Objects

A time domain electric field volume integral equation (TD-EFVIE) solver is proposed for analyzing electromagnetic scattering from dielectric objects with Kerr nonlinearity. The nonlinear constitutive relation that relates electric flux and electric field induced in the scatterer is used as an auxiliary equation that complements TD-EFVIE. The ordinary differential equation system that arises from TD-EFVIE’s Schaubert-Wilton-Glisson (SWG)-based discretization is integrated in time using a predictor-corrector method for the unknown expansion coefficients of the electric field. Matrix systems that arise from the SWG-based discretization of the nonlinear constitutive relation and its inverse obtained using the Padé approximant are used to carry out explicit updates of the electric field and the electric flux expansion coefficients at the predictor and the corrector stages of the time integration method. The resulting explicit marching-on-in-time (MOT) scheme does not call for any Newton-like nonlinear solver and only requires solution of sparse and well-conditioned Gram matrix systems at every step. Numerical results show that the proposed explicit MOT-based TD-EFVIE solver is more accurate than the finite-difference time-domain method that is traditionally used for analyzing transient electromagnetic scattering from nonlinear objects.

Nonlinear analysis on electrical properties in a bended composite piezoelectric semiconductor beam

In this paper, the interactions between the transverse loads and the electrical field quantities are investigated based on the nonlinear constitutive relation. By considering a composite beam consisting of a piezoelectric semiconductor and elastic layers, the nonlinear model is established based on the phenomenological theory and Euler’s beam theory. Furthermore, an iteration procedure based on the differential quadrature method (DQM) is developed to solve the nonlinear governing equations. Before analysis, the convergence and correctness are surveyed. It is found that the convergence of the proposed iteration is fast. Then, the transverse pressure induced electrical field quantities are investigated in detail. From the calculated results, it can be found that the consideration of nonlinear constitutive relation is necessary for a beam undergoing a large load. Compared with the linear results, the consideration of the nonlinear constitutive relation breaks the symmetry for the electric potential, the electric field, and the perturbation carrier density, and has little influence on the electric displacement. Furthermore, the non-uniform pressures are considered. The results show that the distributions of the electric field quantities are sensitively altered. It indicates that the electrical properties can be manipulated with the design of different transverse loads. The conclusions in this paper could be the guidance on designing and manufacturing electronic devices accurately.

NccrFOAM suite: Nonlinear coupled constitutive relation solver in the OpenFOAM framework for rarefied and microscale gas flows with vibrational non-equilibrium

The nccrFOAM suite is a collection of nonlinear coupled constitutive relation (NCCR) solvers for rarefied and microscale gas flows with vibrational non-equilibrium, in conjunction with the conservation laws implemented in the foam-extend framework which is an open-source solver with a General Public License (GPL 3). nccrFOAM solvers are developed as an extension to the dbnsTurbFoam solver by implementing additional algebraic constitutive relations for the non-conserved quantities of the stress tensor and heat flux vector. In contrast to Navier-Stokes-Fourier (NSF) solvers that employ first-order constitutive relations to calculate non-conserved quantities and consequently suffer from obvious shortcomings when simulating gas flows in high non-equilibrium, the second-order NCCR framework presents a novel alternative to simulate rarefied and microscale gas flows in a better and a more intuitive manner. In addition to the solver for monoatomic gases, the solvers are for the first time implemented for diatomic and polyatomic gases with translational-rotational and vibrational degrees of freedom based on second-order constitutive models in a three-dimensional framework. Towards this, a new foam-extend library has been written based on the two-temperature formulation to handle the translational-rotational and vibrational modes. The new foam-extend solver was validated for several representative problems, and an exhaustive list of tutorials is documented. The solver will certainly benefit the rarefied and microscale gas dynamics and hypersonics communities at large interested in flows involving high degrees of rarefaction, speed, and temperature variations.

Asymptotically correct 3D displacement of the Mooney–Rivlin model using VAM

The current work focuses on the nonlinear analytical analysis (dimensional reduction and recovery relations) of a hyperelastic plate governed by the compressible Mooney–Rivlin model using the Variational Asymptotic Method (VAM). The geometric nonlinearity is accommodated through finite deformations, and generalized 3D warping functions, while material nonlinearity through the hyperelastic material model. VAM mathematically splits the 3D nonlinear elastic problem into the 1D through the thickness analysis and 2D nonlinear plate analysis, using the inherent small parameters. These are the geometric small parameter (ratio of thickness to the characteristic dimension), and the physical small parameter (moderate strains). This work results include the derivation of closed-form analytical expressions of 3D warping functions, 2D nonlinear constitutive relation, and recovery relation to express the 3D displacement field for a plate. The 2D nonlinear constitutive relation is given as an input to the in-house developed 2D nonlinear finite element analysis of the reference surface to determine the 2D displacements and 2D strains. In order to validate the current theory, standard test cases are solved and compared with 3D nonlinear Finite Element Analysis (FEA).

Modeling polymeric lubricants with non-linear stress constitutive relations

Modeling polymeric lubricants with non-linear stress constitutive relations

Magneto-thermo-elastic coupling vibration and bifurcation characteristics of functionally graded rotating cylindrical shell

The magneto-thermo-elastic coupling vibration of rotating ferromagnetic functionally graded cylindrical shell is investigated in this paper. Based on the physical neutral surface theory and Donnell's theory, the expressions of kinetic energy and strain energy are obtained by introducing geometric nonlinearity. The magneto-thermo-elastic vibration equations of the rotating ferromagnetic functionally graded cylindrical shell in multi-physical field are derived via the Hamilton's principle. Considering the magnetization effect of ferromagnetic shell in magnetic field, the expression of magnetization force is obtained according to the nonlinear magnetic constitutive relation. For simply supported boundary conditions, Galerkin's method is used to obtain the discretized ordinary differential equations, and the static deflection of the shell under static load is solved. The steady-state response of the system during primary resonance and the stability of solutions are given by the multi-scale method and the Lyapunov stability theory. Through numerical examples, the characteristic curves and surfaces of amplitude are plotted, and the parameter ranges corresponding to regions of multi-valued solutions and stable solutions are determined. The system dynamics laws describing the coupling effect between multiple parameters such as force, magnetism, heat, and rotational speed are evaluated. The Global bifurcation diagrams and the maximum Lyapunov exponent spectrums under different control parameters are plotted to analyze the effect of parameter changes on the dynamic response of system. The results show that the vibration characteristics of the system are significantly influenced by the external physical fields, in addition to the shell size and material volume fraction. Slight changes in parameters can lead to violent changes in the dynamic response of the system, exhibiting intermittent, jumping, and chaotic characteristics.

Constitutive relationships for osteonal microcracking in human cortical bone using statistical mechanics

The post-elastic mechanical behavior of cortical bone, which is represented by extensive microcracking once the elastic regime is exceeded, has been characterized by a nonlinear constitutive relationship for osteonal microcracking. The relationship/model is based on the formalism of Statistical Mechanics, allowing the degree of irreversibility to be calculated using the increase in entropy associated with the progression of microcracking. Specific tensile and bending tests were conducted to compare theoretical predictions of constitutive relationships to empirical curves. In addition, the tests were utilized to determine the model’s parameters, whose values were used to explicitly calculate the entropy increase. A large sample was used: 51 cortical bone coupons (dog-bone-shaped specimens) were extracted from the 4th ribs of numerous individuals and subjected to uniaxial tensile testing. Additionally, fifteen complete 4th ribs were used for bending tests. Displacement and strain fields were measured for both types of tests using digital image correlation or video recordings of the tests. All experimental specimen data were successfully fitted to the model, and all constitutive parameter values were found to be correlated with anthropometric variables. Explicit entropy calculations indicate that microcracking is minimal for low strain and, initially, stress is nearly proportional to strain. After a certain point, significant microcracking occurs, and the relationship between stress and strain becomes invalid. Several significant associations between constitutive parameters and age have also been identified.

Damage constitutive model of uniaxially compressed coal material considering energy dissipation

Coal plays a dominant role in China's energy production and consumption, which also serves as the cornerstone for energy security. The constitutive relationship of coal material is of great significance for the quantitative characterization and evaluation of mechanical properties of isolated coal pillars. From an energy perspective, coal failure is driven by energy, and the energy-based damage characterization provides a basis for the establishment of constitutive model. In this study, since the constitutive model based on the classical Lemaitre's hypothesis shows a significant error in presenting the nonlinear constitutive relationship of coal material, a correction coefficient considering the compaction effect of coal material is first introduced. Then, a novel damage constitutive model that can accurately describe the overall compression state of coal material is constructed. Finally, the theoretical model is compared with exiting models using the experimental data of five types of coal material from different mines. The results show that the evolution of energy dissipation damage variable is highly consistent with that of stress–strain relations. The established constitutive model considering energy dissipation and compaction effect can more accurately describe the nonlinear strength and deformation characteristics of coal material than the classical and existing models. Notably, this new model can be effectively extended to the single cyclic compression conditions to reproduce the loading-unloading-reloading process of coal material.

Effect of Support Stiffness Nonlinearity on the Low-Frequency Vibro-Acoustic Characteristics for a Mechanical Equipment—Floating Raft—Underwater Cylindrical Shell Coupled System

The low-frequency vibro-acoustic characteristics of a mechanical equipment—floating raft—cylindrical shell—underwater acoustic field coupled system with nonlinear supports are studied in this paper. Firstly, the state space equations were established by a modal superposition theory for the coupled system, and a modal parameter identification method was deduced and verified for the cylindrical shell—underwater acoustic field coupled subsystem. On this basis, the formulas were derived for transmitted power flow in the coupled system, and the nonlinear stiffness constitutive relation of the vibration isolation supports was expressed by softening and hardening characteristics. Finally, dynamic simulations were carried out by the Runge—Kutta method to analyze the effect of nonlinear stiffness characteristic parameters on the low-frequency vibration modes and vibro-acoustic transfer characteristics in the coupled system. The research shows that a superharmonic phenomenon is common in the steady vibration mode of the coupled system with a nonlinear softening (or hardening) stiffness characteristic under harmonic excitation. The stronger the softening (or hardening) stiffness characteristic is, the more complex the vibration form is, and the smaller (or larger) the low-frequency vibro-acoustic transfer level in resonance regions is.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Nonlinear constitutive calculation method of rarefied flow based on deep convolution neural networks

In the field of rarefied gas dynamics, the presence of non-equilibrium flow characteristics poses significant challenges for achieving efficient and accurate numerical simulation methods. These challenges arise from the complex coexistence of these phenomena at multiple scales. The recent advent of intelligent fluid mechanics has introduced the data-driven nonlinear constitutive relation (DNCR) method as a promising approach for expeditious physical modeling of non-equilibrium rarefied flows. To enhance the generalization capabilities of the DNCR method, this study proposes a deep convolutional neural network model (DNCR-CNN) based on data-driven nonlinear constitutive relations, integrated with free-form deformation (FFD). Employing FFD technology, a series of hypersonic geometric shapes are generated for model training, and a multi-task learning-based deep convolutional neural network model is subsequently trained. The prediction of the hypersonic geometric shapes test set is carried out, and the results of the model prediction are substituted in the conservation equation for the iterative solution, thereby enhancing the DNCR method's generalization performance for varying geometric shapes. Upon conducting a comparative analysis of the outcomes obtained from DNCR, Navier–Stokes (NS), and unified gas kinetic scheme (UGKS), it is revealed that the DNCR method can maintain computational resource levels equivalent to those of the NS equation while achieving a level of accuracy comparable to UGKS under diverse geometric shapes and grid resolutions. The enhancements in usability render the DNCR method a potent tool for addressing the challenges posed by rarefied gas, thereby expanding its applicability within the field.

Nonlinear dynamics of pipes composed of Mooney-Rivlin hyperelastic materials conveying unsteady fluid flow

In the present study, a mathematical model for the nonlinear dynamics of a Mooney-Rivlin type hyperelastic pipe conveying unsteady fluid flow is established. The internal fluid velocity is assumed to comprise an average component superimposed on a harmonically varying component. Subsequently, various numerical methods are employed to investigate the nonlinear responses of the pipe through the forms of amplitude-response, disturbance-response, global bifurcation diagrams, phase trajectories, Poincaré maps, and power spectral densities (PSD). Additionally, the contributions of the Coriolis force and geometric nonlinearity are examined. The results reveal that, in comparison to linearelastic pipes, the combined effect of nonlinear geometric relation and nonlinear constitutive relation introduces specific nonlinear terms, leading to distinct dynamical phenomena. With small disturbed coefficient, the asymmetric period-1 and symmetric period-2 oscillations are respectively observed in the primary and sub-critical regimes. Furthermore, the hyperelasticity and fluid-structure interactions (FSI) play significant, unequal, and competitive roles in influencing the dynamic responses of the pipe by affecting either linear or nonlinear stiffness. Additionally, abundant dynamical phenomena are captured within the system versus system parameters with large disturbed coefficient. The study also demonstrates that the combined effect of nonlinear geometric relation and nonlinear constitutive relation ultimately results in weaker hardening behaviors for Mooney-Rivlin type pipes compared to linearelastic ones.

An extended two-scale random field model for stochastic response analysis and its application to RC Short-leg shear wall structure

For the stochastic response analysis of concrete structures, it is necessary to model the heterogeneous concrete with significant spatial variability by random fields. In practical applications, multiple material parameters are typically required to characterize the nonlinear constitutive relationship of concrete. As a result, the material parameters of concrete will be represented by a multivariate random field in the nonlinear stochastic response analysis. Nevertheless, a vast number of random variables are frequently involved in representing the multivariate random field, and it is difficult to quantify the probabilistic dependence among the material parameters. To address the preceding issues, this paper proposes an extended two-scale random field model (ETSRF) to quantify the spatial variability of the constitutive relationship of concrete. The model can accurately depict the fluctuation of material properties at the micro-meso scale and the spatial correlation at the macro scale. Besides, it is proved that there is a linear relationship between the mesoscopic random field and ETSRF. Furthermore, a new framework of stochastic response analysis is proposed, where the probability density evolution method (PDEM) is applied to obtain the probabilistic information of the structural responses, and ETSRF is adopted to describe the spatial variability of the material properties. Finally, the proposed model is validated by two illustrative examples, one of which simulates the random fields of compressive and tensile strength of a plain concrete slab, and the other of which deals with the stochastic response analysis of a RC short-leg shear wall structure. The results indicate that the proposed model can depict the spatial variability of the nonlinear constitutive relationship of concrete effectively.

Fatigue damage assessment using nonlinear critically refracted longitudinal (LCR) wave in pure iron: Experiments and FEM

To achieve early-stage microstructure evolution assessment and damage monitoring of fatigue, a new thickness-independent and nondestructive method based on nonlinear critically refracted longitudinal (LCR) wave is proposed. The nonlinear ultrasonic (NLU) parameter is sensitive to the accumulated fatigue damage, showing two peaks around the 5th and 1000th cycles (0.03% and 6% of fatigue life), respectively. This phenomenon is comprehensively discussed in terms of dislocation structure evolution based on the dislocation string model. A correction considering the weakening of nonlinearity induced by the rough surface is applied to the experiment measurements. The corresponding results are closer to the finite element modeling (FEM) quantitation based on the dislocation-dependent nonlinear constitutive relation. Compared to linear LCR wave, nonlinear LCR wave has weaker sensitivity to roughness effect and higher sensitivity to damage such as dislocation, which offers a new method for monitoring fatigue in engineering components.

A nonlinear substructure method based on a computational plasticity framework for efficient analysis of structural assemblies

Substructure analysis reduces the computational order of a discretized structural domain from the full set of degrees of freedom needed to solve a boundary value problem (e.g., the displacements of all nodes in an FEA mesh) to a predefined and much smaller set of retained degrees of freedom. Given only one initial analysis considering all degrees of freedom, this technique reduces the computational cost associated with subsequent analyses of the same domain by eliminating degrees of freedom, usually internal to the domain, which are not essential for interfacing the domain with a larger system/assembly. For large multiscale applications, such as aircraft and automotive assemblies, substructure analysis enables efficient computation of both static and dynamic responses of larger assemblies consisting of one or more of these previously analyzed and dimensionally reduced domains. However, while existing methods are exact for linear problems (e.g., small-deformation linear elasticity), they are insufficient when considering large deformations or material nonlinearities. In this work, we develop a new nonlinear substructure method to consider general nonlinear responses by leveraging the mathematical framework developed for computational plasticity, including a decomposition of deformations, criteria for nonlinearity initiation, and evolution equations. While computational plasticity provides nonlinear constitutive relationships between six independent stress and strain components, we show that the same mathematical formulation can capture similar relations between an arbitrary number of forces and displacements (i.e., the retained degrees of freedom), and thus, can be implemented with the same numerical solution algorithms. As a notional example to emphasize both the generality of the aforementioned method and the application to design frameworks, we model an infilled lattice structure exhibiting local plasticity and internal nonlinear geometric effects.

Nonlinear modelling for structural damage assessments of reinforced and coated longitudinally coupled slab tracks

Interface debonding and slab end arching of the anchor-reinforced and coated longitudinally continuous slab tracks have been unprecedentedly investigated in this study. A novel finite element model of the longitudinally continuous slab track has been established by incorporating a tailored cohesive zone model to mimic nonlinear constitutive relationships. The new model has been validated by over 100 field measurement data. This paper is the first to assess the influences of the dual applications of anchors and coatings on the damages of the longitudinally continuous slab tracks, and to identify the effectiveness of different track maintenance methods in mitigating track damages. This study exhibits new findings: (1) In comparison with traditional tracks, the tracks constructed with anchors, coatings, and the combined use of both can reduce the maximum vertical relative displacement between the concrete slab and the mortar layer by 75%, 40%, and 85% respectively. (2) In comparison with 4 anchors, 6 or more anchors in each slab can help to prevent inner interface defects. However, an increase in anchor quantity above 6 anchors does not have a significant impact on the interface damage mitigation. (3) For the anchor-reinforced tracks, the organic coating outweighs the inorganic coating to improve interface damage mitigation.

Machine-learned constitutive relations for multi-scale simulations of well-entangled polymer melts

We investigate the use of a machine-learning-based simulation scheme to predict flows of complex fluids with a non-linear constitutive relation. This scheme, originally proposed by Seryo et al. for general complex fluids [N. Seryo et al., “Learning the constitutive relation of polymeric flows with memory,” Phys. Rev. Res. 2, 033107 (2020)], is used to learn the time derivative of the stress as a function of the stress and velocity gradient. However, previous assessments were limited to fluids with linear constitutive relations. In this study, we employ the dual sliplink model to express the dynamics of well-entangled polymers as a fluid exhibiting a non-linear stress response. We test the simulation scheme by using it to predict the flow of a viscoelastic fluid between two parallel plates with only shear deformations and compare the results with those of a multi-scale simulation using microscopic simulators. Overall, our machine-learning method possesses good predictive capabilities, for both the transient response and the non-linear behavior at steady-state, i.e., the shear-thinning. We are able to accurately track the evolution of the stress, for both the weak and strong elastic cases, although the velocity predictions for the latter show a decreased accuracy in the transient regime. Furthermore, our simulation scheme is more computationally efficient than the conventional multi-scale simulation approach, which uses microscopic simulators, containing a system of coarse-grained polymers, to evaluate the macroscopic stress. We discuss possible extensions and improvements for enhancing the predictive capabilities and generality of the method.

Numerical Study of Shock Wave Boundary Layer Interaction Based on an Anisotropic Turbulence Model

This paper presents numerical simulations of incident oblique shock wave impinging on a flat plate with a turbulent boundary layer using a nonlinear anisotropic turbulence model. Three cases with different incident shock angles under the same free-stream Mach number are studied, including attachment flow, incipient separation flow and outstanding separation flows. Convection terms and diffusion terms are calculated using the third-order Roe’s flux difference splitting scheme and the second-order central difference scheme in the code, respectively. The Runge-Kutta time marching method is employed to solve the governing equations for steady state solutions. Comparisons between the simulated results and experimental data are carried out, which include density contours, pressure distribution, skin friction coefficients distribution and streamline vectors. To capture the essential behavior of this complex anisotropic turbulence separation flow caused by the shock-wave-boundary-layer-interaction (SWBLI), an anisotropic eddy viscosity tensor and a nonlinear Reynolds stress-strain constitutive relationship are used in this model. The anisotropic eddy viscosity tensor is composed of turbulent fluctuation velocity vector and characteristic length vector which are obtained by statistical partial average scheme. With proper modeling of anisotropic eddy viscosity tensor, the numerical results are in good agreement with the experimental data. It is worthy to point it out that the computation accuracy decreases with increasing interaction intensity.

Design, fabrication, nonlinear analysis, and experimental validation for an active sandwich beam in strong electric field and thermal environment

In the present work, an efficient one-dimensional finite element (1D-FE) framework considering layerwise mechanics with full electro-mechanical coupling has been developed for the laminated smart sandwich beams submitted to active damping. Nonlinear constitutive relations of piezo-thermo-elasticity at large electric field with temperature-dependent material properties have been considered. The efficient layerwise theory (ELWT) considers the normal deformation of the piezoelectric sensing/actuating patches/layers due to the more significant value of piezoelectric strain coefficient d33. The electric potential variation across the thickness of the PZT sensor/actuator has been assumed quadratic. The state space format of the dynamical system has been constructed based on a reduced order model in modal space. The feedback linearization approach has been applied to the nonlinear system of equations to design an equivalent linear optimal control strategy. An experimental facility has been developed to test the accuracy of the present nonlinear FE model. This facility consists of a thermal chamber with a heating system, air gun, actuation & sensing units with band-pass filters, a smart sandwich beam equipped with PZT-SP5H patches, a data acquisition system, and a computer with LabVIEW software. Open- and closed-loop responses from the present 1D-FE model have been compared with those obtained through experimentation at different temperatures. It has been observed that the current nonlinear FE model correctly captures the dynamic behavior of the vibrating smart sandwich beam at different operating temperatures and strong electric field. The study shows a strong dependence of the PZT materials on thermal chamber temperature and actuation field. New results for a smart sandwich beam are also presented to study the performance of PZT-SP5H patches as sensors/actuators for nonlinear active shape and vibration control in a thermal environment. The effect of piezoelectric actuator/sensor thickness, patch size, position, and applied mechanical load on the active shape and vibration control has been discussed.