Nonlinear Constitutive Equations Research Articles

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

Kenneth Walters. 14 September 1934 — 28 March 2022

Kenneth (Ken) Walters was a leading international figure in the field of rheology. He applied his mathematical training to the analysis of many common test procedures, and contributed to the formulation of nonlinear constitutive equations for viscoelastic fluids. He also pioneered computational studies in rheology, and engaged in fruitful collaborations with industry—notably the oil and china clay industries. He was born in Swansea, South Wales, and he remained a devoted Welshman throughout his life. He studied under James Oldroyd at Swansea and was awarded the PhD degree in 1959. In 1960, after a stint in the USA, he was appointed as a lecturer at Aberystwyth, and he remained there until his death in 2022. He built a very strong research team at Aberystwyth, which became a dominant factor in the rheological scene in the United Kingdom. He was a keen promoter of rheology and assisted in the organization of many conferences and meetings. He was the founding editor of the Journal of Non-Newtonian Fluid Mechanics , and authored or co-authored four well-known books on rheology plus several tracts of a religious nature—Ken was always a devoted Christian. He was also a very able sportsman, especially at cricket and golf. He is survived by his wife Mary, three children and seven grandchildren.

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.

On more insightful dimensionless numbers for computational viscoelastic rheology

Abrupt contraction flows involving viscoelastic fluids represent a longstanding computational challenge within the field of non-Newtonian fluid mechanics. Despite the apparent simplicity of the geometry, these flows have given rise to intricate discussions in the study of viscoelastic phenomena. This study aims to re-examine the numerical solutions for flows through abrupt contractions, offering a fresh interpretation through the lens of reformulated dimensionless numbers. These numbers are designed to consider the characteristic shear rate of the problem, providing a more comprehensive understanding of the underlying dynamics.When investigating models with intermediate levels of complexity, such as the Giesekus and Phan-Thien-Tanner constitutive equations, the usual comparison with the corresponding Oldroyd-B model becomes inadequate because it tends to rely on the nominal relaxation time (λ) and the nominal total viscosity (η) instead of their effective counterparts when defining the Reynolds number (Re), the Weissenberg number (Wi) and the ratio of solvent to total viscosities (β) (β plays a role only in rheological models involving a solvent contribution). If these dimensionless numbers are tailored to account for the characteristic shear rate specific to the problem under investigation, the choice of the corresponding Oldroyd-B flow, at the adequate values of Re, Wi, and β allows for significantly better quantification of the correct effects of nonlinear viscoelasticity of the original model.We show the conventional approach tends to overemphasize the role of the nonlinear parameter in nonlinear constitutive equations, like the Giesekus and PTT models, when examining standard abrupt contraction flow outputs such as the Couette correction and vortex size. This overestimation occurs because the conventional method does not allow the Reynolds and Weissenberg numbers (and possibly β) to carry the portion of the nonlinear effect that can potentially be captured by the linear Oldroyd-B model through the use of characteristic shear rate-based values. We believe the present approach provides a better perspective of the role played by the nonlinear parameter and its extension to more general flows is also discussed.

Temperature Effects on GaN/AlN Heterojunction in a Piezomagnetic and Piezoelectric Semiconductor Composite Fiber

In this article, considering the thermal effects, a PN GaN/AlN heterojunction in a piezomagnetic and piezoelectric semiconductor composite fiber driven by dynamic magnetic loads is studied. The true nonlinear constitutive equations of the electric current density for the composite heterojunction are employed. Thus, a nonlinear finite‐element method (FEM) is adopted to obtain dynamic responses of the composite heterojunction, including the mechanical displacement, electric potential, and electron and hole concentrations. By comparing dynamic responses with those derived by COMSOL Multiphysics, the nonlinear FEM is verified. Based on numerical results, regulation effects of the temperature on distributions of the mechanical displacement, electron and hole concentrations, and temperature–current–voltage characteristics are discussed.

Vibration control of membrane structures by piezoelectric actuators considering piezoelectric nonlinearity under strong electric fields

Membrane structures, which are widely used in aerospace and civil engineering, are susceptible to large amplitude vibrations due to their low structural damping and high flexibility, leading to the degradation of their working performance. Strong electric fields are commonly needed for piezoelectric actuators to control larger amplitude vibration of membrane structures. However, the effect of piezoelectric nonlinearity due to strong electric fields on the control performance of membrane structures is still underexplored. Given the importance of piezoelectric nonlinearity due to strong electric fields, the present study aims to investigate the vibration control performance of piezoelectric actuators for membrane structures. First, based on the piezoelectric nonlinear constitutive equation, the actuation equation of the piezoelectric actuator is generated. Then, following D′Alembert's principle, a nonlinear dynamic control model of membrane structure containing actuation force is derived. Finally, active vibration control based on velocity feedback algorithms is realized. The results show that the predicted strain without considering the piezoelectric nonlinearity is 80 % smaller than the finite element structure at 500 V. Meanwhile, the errors of Root Mean Squared (RMS) control displacement without considering the piezoelectric nonlinearity are more than 24 % when vibration amplitudes achieve 10 % of the membrane thickness. The developed model provides theoretical implications for active vibration control of flexible structures with piezoelectric actuators in strong electric field environments.

An iteration method to study nonlinear wave propagation for a non-Green elastic 1D bar

Abstract The problem of propagation of nonlinear waves in a 1D bar is studied, wherein the linearized strain tensor is considered as a function of the Cauchy stress tensor. Specifically, two constitutive equations for non-Green elastic solids are investigated, introducing a novel numerical iterative method capable of obtaining approximate solutions of one nonlinear constitutive equation for rock, and one constitutive equation that shows a strain-limiting behaviour. The numerical results are compared with exact solutions for the case of a linearized elastic solid.

Microstructure size and packing of fracture surface on the nonlinear breakage mechanical response of different marbles under cyclic loading

The microstructural properties of fracture surfaces are a remarkable reflection of the nonlinear mechanical properties of rocks, which have not been thoroughly evaluated to date. In order to understand the contributions of microstructure size and packing to the nonlinear mechanical response, both micro- (field emission scanning electron microscopy (SEM) experiments on failure specimen section) and macro-scale (unequal amplitude cycle experiment) tests were carried out on different marbles. The size and arrangement properties of microstructures on the fracture surface were determined using digital image recognition technology based on micro-scale experiments. Remarkable differences were observed among different types of marble in terms of breakage evolution, fracture combining form, and nonlinear evolution of energy. The micro-deformation of marbles is characterized by the deformation distribution and microstructure inhomogeneity in strong deformation zones of failure. The microstructures of brittle-ruptured marbles exhibit mechanical fragmentation and fracture of mineral grains. The ductile deformation of marble is characterized by the presence of solid flow traces within the mineral crystals, as well as the loose arrangement of mineral crystal grains in the strongly deformed zones. The constitutive model for marble samples with fine mineral crystals can be derived by nonlinearly separating the Helmholtz free energy during the breakage process and defining the relationship between breakage force and dissipated energy in the absence of residual strength after fracture. The obtained nonlinear breakage constitutive equation is in good agreement with the experimental results. These findings may provide basis and inspiration for engineering construction and geological disaster prevention.

Symmetric Equivalent Circuit Theory and Numerical Simulation Analysis of Nonlinear Magnetoelectric Laminated Composite

In order to further study the nonlinear characteristics of the resonance magnetoelectric coefficient and vibration mode at the resonance frequency, three-layer magnetoelectric composite with length direction magnetization and thickness direction polarization is investigated in the article. Firstly, based on the Z-L model and the numerical solution characteristics of magnetization intensity, the magnetization intensity function was fitted, and the dynamic parameters of the giant magnetostrictive material, including dynamic piezomagnetic coefficient, dynamic elastic compliance coefficient, and dynamic magnetic permeability, were further derived. The effects of bias magnetic field and prestress on the corresponding composite were analyzed; Secondly, based on the nonlinear magnetostrictive constitutive equation, a symmetric magneto-elastic-electric equivalent circuit model of magnetoelectric laminate composite was established, and the expression of magnetoelectric coefficient was derived. The variation curve with bias magnetic field and prestress was analyzed, which is consistent with the conclusions of existing literature [8] and [9]; Finally, in order to compare with the theoretical results, the same parameters were set using COMSOL software, and the corresponding magnetoelectric coefficient frequency curve was plotted. The two results were in good agreement, and the maximum peak modal vibration shape was extracted, which can conveniently observe the vibration of the magneto electric laminate composite in the length direction. The results indicate that the theoretical model of this symmetric magneto-elastic-electric equivalent circuit and the numerical simulation method using COMSOL software are feasible, laying the foundation for further nonlinear analysis of magnetoelectric laminate composite and making it possible to design high-precision magnetoelectric micro devices.

HYBRIDIZATION OF THE LINEAR VISCOELASTIC CONSTITUTIVE EQUATION AND A NONLINEAR MAXWELL-TYPE VISCOELASTOPLASTIC MODEL, AND ANALYSIS OF EVOLUTION SCENARIOS OF THE TRANSVERSE STRAIN COEFFICIENT UNDER CREEP

In this work, a generalization of a physically nonlinear Maxwell-type viscoelastoplastic constitutive equation with four material functions is formulated, whose general properties and range of applicability were discussed in a series of our previous studies. In order to expand the range of rheological effects and materials that can be described by the equation, it is proposed to add a third strain component expressed by a linear integral Boltzmann-Volterra operator with arbitrary functions of shear and volumetric creep. For generality and for the convenience of using the model, as well as for fitting the model to various materials and simulated effects, a weight factor (degree of nonlinearity) is introduced into the constitutive equation, which allows combining the original nonlinear equation and the linear viscoelastic operator in arbitrary proportions and controlling the degree of different effects simulated. Equations are derived for families of creep curves (volumetric, shear, longitudinal and transverse) induced by the proposed constitutive equation with six arbitrary material functions, and an expression is obtained for the transverse strain coefficient as a function of time. Their general properties and dependence on loading parameters and characteristics of all material functions are studied analytically and compared with the properties of similar relations generated by two combined constitutive equations separately. New qualitative effects are identified which can be described by the new constitutive equation in comparison with the original ones, and it is verified that the generalization eliminates some shortcomings of the Maxwell-type viscoelastoplastic constitutive equation, but retains its valuable features. It is confirmed that the proposed constitutive equation is capable of simulating sign alternation, monotonic and nonmonotonic changes in transverse deformation and transverse strain coefficient under constant stress, and their stabilization over time. Generally accurate estimates are obtained for the variation range, monotonicity and nonmonotonicity conditions of Poisson’s ratio, and its negativity criterion over a certain time interval. It is proven that neglecting volumetric creep (the postulate of volumetric elasticity), which simplifies the constitutive equation, greatly limits the range of possible evolution scenarios of the transverse strain coefficient in time: it increases and cannot have extremum and inflection points. The analysis shows that the proposed constitutive equation provides ample opportunities for describing various properties of creep and recovery curves of materials and various evolution scenarios of the transverse strain coefficient during creep. It can significantly expand the range of described rheological effects, the range of applicability of the Maxwell-type viscoelastoplastic equation, and deserves further research and application in modeling.

A coupled thermo-hydro-mechanical framework for simulating the failure response of asphalt concrete under freezing conditions

Freezing-induced damage occurs in extremely cold regions in the form of sudden and simultaneous formation of cracks in dry and wet asphalt concrete. This work investigates the mechanical response, up to and including final failure, of dry and partially saturated asphalt concrete at freezing temperatures as low as −30° (−22 °F). A coupled thermo-hydro-mechanical constitutive framework is proposed within the context of finite element simulation. The framework contains nonlinear viscoelastic-viscodamage constitutive equations, the swelling strain effect associated with the freezing of water content, and the moisture damage constitutive equation. The stress induced in dry and partially saturated asphalt concrete samples due to reducing temperatures is simulated and compared with those of a studied experiment, and an excellent agreement is achieved in terms of the critical features, including the final brittle failure and the swelling effect. A parametric study is then performed to discover the effects of the microstructural constituents of asphalt concrete on its mechanical response under freezing conditions. This work provides a reliable computational framework that can predict deterioration and damage in asphalt concrete structures in extremely cold weather and further leads to the development of novel construction materials that can resist the negative effects of cold temperatures, e.g., in the Arctic.

Analytical solutions for resonant radiation performance of bending-mode magnetoelectric antennas

This paper proposes analytical solutions for the resonant radiation performance of bending-mode magnetoelectric (ME) antennas. The strain-mediated Converse ME (CME) coupling model of bending-mode ME antennas is first established by solving nonlinear constitutive equations and bending governing equations using the elastic mechanics method. Then, the calculated magnetic flux and electric displacement are employed to propose a resonant radiation field model based on the dipole method. The numerical results for the CME coefficient show a good agreement with the experimental data. It can be observed that the volume fraction ratio of the piezoelectric layer can control the CME coefficient and radiation efficiency with the same variation trend since it can determine the bending strain via changing the location of the neutral layer of the ME antennas, which also demonstrates the strain-mediated essence of the ME antennas. In addition, the volume fraction ratio can tune the resonant frequency within a wide range. The gain of the ME antenna is stable and higher than −168 dB with the volume fraction ratio ranging from 0.2 to 0.7. The tensile stress and compressive stress have the opposite effect on the resonant frequency at low and high bias magnetic fields. Meanwhile, the tensile (compressive) stress is beneficial for both the radiation and gain in the low (high) bias field region. This model may facilitate the understanding of the bending-mode radiation mechanism of ME antennas and provide a basis for designing asymmetric ME antennas.

Arbitrary mesh‐moving velocity‐based space‐time finite element method for large deformation analysis of solids

SummaryThis paper presents a new velocity‐based space‐time finite element method for computing the large deformation of solids over arbitrary Lagrange/Eulerian moving meshes. The proposed method is oriented toward quasi‐static deformation problems with hypoelastic and neo‐Hookean hyperelastic constitutive models. The Cauchy stress in the weak form of the linear momentum equation is explicitly described in terms of deformation velocity through consistent time integration of objective stress rates transferred in the arbitrary moving mesh. Consequently, the resultant system equation only contains the space‐time nodal velocity as the primary unknown. An iterative algorithm for solving the discretized system equation is implemented, whereby the nonlinearities induced by geometrical change in the computational domain as well as nonlinear constitutive equations are computed in an iterative manner updating the stress and the strain over the moving mesh of the deforming domain. Three benchmark problems, such as uniform beam compression, thick‐cylinder compression with superimposed rigid body rotation and extrusion of a column involving material loss, have been solved in order to examine the accuracy of the proposed numerical scheme. The numerical results have revealed the proper computation of the stress and the displacement of the solids undergoing the large deformation and rotation over the arbitrary Lagrange/Eulerian moving meshes, and have shown the validity and applicability of the proposed method with linear convergence of the iterative algorithm.

Cohesive crack growth in polyethylene considering Schapery equation using XFEM

A nonlinear rate-dependent cohesive rule based on a novel incorporation by Schapery nonlinear coefficients is suggested to model crack growth in the case of polyethylene which is well-defined through the Schapery nonlinear constitutive equation. As the matter between the crack faces in a polymeric medium like polyethylene may tolerate large strain ranges during crazing and before complete separation, it may involve stress-dependent factors as same as bulk material. A useful damage-creeping cohesive rule is selected and its creeping part is extended in a way to incorporate the nonlinear Shapery parameters. The augmented relation is fitted to both high and low constraint results and applied to an experimented medium using Extended Finite Element Method (XFEM) to evaluate its ability to properly predict cohesive crack propagation. The results are in tight agreement with the existing experimental outputs. The suggested framework may illuminate a way of expressing the damage zone in polymers using nonlinear specifications of the bulk material.

Nonlinear multivariate constitutive equations for modeling hot deformation behavior

Nonlinear constitutive equations are proposed to model variations in flow stress as a function of strain rate and temperature during hot deformation. Modified constitutive are applied to seventy data sets about hot deformation of alloys. Two modifications to conventional constitutive models are introduced, viz. (1) nonlinear and (2) multivariate models with the fitting of flow stress simultaneously with two variables. The predictive accuracy of constitutive equations was evaluated using three statistical parameters and compared with a conventional Arrhenius-like model. It is shown that nonlinear constitutive equations have improved predictive accuracy for variations in flow stress during hot deformation. The advantages of multivariate models include less computation and material parameters that are constants independent of temperature or strain rate. In another type of multivariate model, flow stress is expressed as linear and nonlinear polynomial functions of the Zener-Holloman parameter. This approach gives a single value of the activation energy of hot deformation. The results have indicated that a generalized second-order multivariate constitutive equation can be used to better predict flow stress, as a function strain rate and temperature, during hot deformation.

A self-consistent magnetoelectric coupling model for GaN-based /Terfenol-D composites

A magnetoelectric composites is designed and investigated, which combines GaN-based semiconductor and magnetostrictive Terfenol-D. In this work, we study the polarization charge and strain of the multilayer composites under an applied magnetic field and prestress. In order to quantitatively describe the response of composites under different loading environments, a nonlinear magneto-electric self-consistent coupling model is established based on nonlinear constitutive equation, Schrodinger equation, and Poisson equation. The validity of this model is well-proven by comparing experimental data and numerical results under different magnetic fields. The results demonstrate that the giant magnetostriction effect of Terfenol-D offered large tensile strain to decrease the polarization charge and strain in GaN-based semiconductor layer. These conclusions will be of great help to the basic research of piezomagnetic composite materials as well as guidance for the future device design.

The mathematical study of compression behaviours of silicone rubber composites reinforced by warp-knitted spacer fabrics

To synthetically investigate the compressive mechanism of warp knitted spacer fabric reinforced silicone rubber matrix composite (SRWSF) under 20% deformation, a compression meso-mechanics theoretical model was established in this paper. The silicone rubber was processed as isotropic material according to the nonlinear constitutive equation, the spacer yarn was assumed as a continuous uniform element. The compression meso-mechanics theoretical model consists of the mechanical model of silicone rubber and the compression model of warp knitted spacer fabric (WSF), based on the Euler-Bernoulli beam theory and the Spence Invariant constitutive model theory. To verify the feasibility of the compression meso-mechanics theoretical model of SRWSF, the compression test of SRWSF was carried out. The simulated compression stress-strain curve was compared with the results of the experiment. The result shows that the compression theoretical model has a high agreement with the experimental result under 20% strain, as compared with the Polynomial fitting curve. This phenomenon indicates that the compression theoretical model established in this study can reasonably simulate the actual compression behaviours within 20% deformation for SRWSF. Moreover, the theoretical model can help understand the compression mechanism of SRWSF and optimise the designing of silicone- rubber-matrix composite reinforced by WSF for cushioning applications.

Study on Nonlinear Viscoelastic Constitutive Equations for Rock

Numerical simulation is one of the effective tools for the long-term stability assessment of underground structures. The reliability improvement of the numerical simulation requires precise constitutive equations to reproduce time-dependent behavior of rock such as loading-rate dependence, creep, and relaxation. These time-dependent behaviors of rock have nonlinearity between stress and strain rate and are influenced by surrounding environment. A number of constitutive equations have been proposed in previous studies; however, the equations based on the theory of linear viscoelasticity have difficulty reproducing accurately the time-dependent behavior of rock, and most of the equations based on the theory of viscoplasticity have been applied to the limited loading condition such as creep. In this article, nonlinear viscoelastic constitutive equations for rock proposed by the authors were introduced, and their exact solutions and application to experimental results were summarized. These equations can reproduce nonlinear time-dependent behavior and not only brittle but also ductile failures of rock and can be applied to the behavior under various loading conditions and experiment environment. In the equations, compliance or inelastic strain is assumed to be increased with time, but the functions and their combinations are of great variety. Hence, the differences of the solutions and applicability of the equations were detailed, and in addition future subjects were presented.

Implicit integration of an anisotropic egg-shaped elastoplastic model for saturated soft clay

PurposeThe purpose of this paper is to suggest an implicit integration method for updating the constitutive relationships in the newly proposed anisotropic egg-shaped elastoplastic (AESE) model and to apply it in ABAQUS.Design/methodology/approachThe implicit integration algorithm based on the Newton–Raphson method and the closest point projection scheme containing an elastic predictor and plastic corrector are implemented in the AESE model. Then, the integration code for this model is incorporated into the commercial finite element software ABAQUS through the user material subroutine (UMAT) interface to simulate undrained monotonic triaxial tests for various saturated soft clays under different consolidation conditions.FindingsThe comparison between the simulated results from ABAQUS and the experimental results demonstrates the satisfactory performance of this implicit integration algorithm in terms of effectiveness and robustness and the ability of the proposed model to predict the characteristics of soft clay.Research limitations/implicationsThe rotational hardening rule in the AESE model together with the implicit integration algorithm cannot be considered.Originality/valueThe singularity problem existing in most elastoplastic models is eliminated by the closed, smooth and flexible anisotropic egg-shaped yield surface form in the AESE model. In addition, this notion leads to an efficient implicit integration algorithm for updating the highly nonlinear constitutive equations for unsaturated soft clay.

A novel analysis method for magnetically actuated soft origami mechanisms

The magnetically actuated soft origami mechanisms (MASOMs), which integrates the advantages of magnetic actuation and origami mechanisms, possesses promising prospective applications on soft untethered machines such as soft robots and compliant actuators. Generally, the deformation behavior of these mechanisms are analyzed through finite element analysis (FEA) considering elusive nonlinear constitutive equations of magnetically actuated elastomer that captures magnetic body torque and force induced by a magnetic field. However, it is time-consuming and troublesome to model and analyze the case with heterogeneous magnetization distribution. We propose an equivalent magnetic bar and hinge model to investigate the folding behavior of planar MASOMs under a uniform magnetic field. Both FEA and experiments are conducted to demonstrate the effectiveness of the proposed model in analyzing MASOMs with different system geometries and magnetization profiles. Two case studies with parallel and serial origami creases and different magnetization profiles are provided to show the practicability. The proposed model provides an effective method to investigate and design MASOMs.