Nonlinear Constitutive Behavior Research Articles

Nonlinear electro-elastic dynamics of a hub–cantilever bimorph rotor structure

Dynamics of a rotating mini hub–bimorph structure excited by periodic torque supplied to the hub is studied in this paper. The mathematical model of the system is formulated by means of the extended Hamilton’s principle. In the analysis the nonlinear constitutive behaviour of the piezoceramic material is adopted based on experimental results published in literature. Moreover, nonlinear inertia effects related to the system rotation are taken into account. The derived system of three integro-partial differential equations represents the electro-mechanical behaviour of the beam (transverse displacement and transducer output voltage) and the hub rotation coordinate. The state equations are discretized and solved numerically around the first resonance zone for the system excited by a periodic torque supplied to the hub. The obtained response curves reveal strong softening behaviour with additional unstable solutions identified out of the resonance zones. Within the framework of the performed numerical simulations the impact of torque excitation amplitude, torque mean value, hub inertia and electrical load resistance on system behaviour is discussed. Finally, the responses of the system with parallel vs series connected transducers are compared.

A micromechanical damage model for oxide/oxide ceramic matrix composites with hierarchical porosity under thermomechanical loading

This work presents a micromechanical damage model to describe the microstructural damage behaviors of ceramic matrix composites with hierarchical porosity during thermomechanical loading. The microstructure evolution may cause the nonlinear constitutive behavior, and a hierarchical porosity-based elasto-plastic constitutive model was developed. Damage mechanisms of matrix-crack, hierarchical pore nucleation and fiber-breaking are incorporated into the formulation of the damage model to describe various micromechanical damage modes of ceramic matrix composites accurately. Two damage variables are proposed for the damage evolution of matrix and fiber bundles. The main damage mechanisms in the matrix are matrix-cracking, and fibers breaking in the fiber bundles. The performance of the proposed damage model is verified by comparing with the existing experimental data. The proposed damage model outperforms the existing counterparts by capturing the microstructural damage mechanism and integrated into the damage model, and the contribution of different damage mechanisms can be quantified. The present work will provide a robust tool for describing the damage behaviors of matrix and fiber bundles in the ceramic matrix composites under thermomechanical loading, as well as allow a more accurate characterization of microstructural damage for a large extent of ceramic matrix composites.

Peridynamic stress is the static first Piola–Kirchhoff Virial stress

The peridynamic stress formula proposed by Lehoucq and Silling (2008) is cumbersome to be implemented in numerical computations. In this paper, we show that the peridynamic stress tensor has the exact mathematical expression as that of the first Piola–Kirchhoff static Virial stress originated from the Irving–Kirkwood–Noll formalism (Irving and Kirkwood, 1905; Noll, 1955) through the Hardy–Murdoch procedure (Hardy, 1982; Murdoch, 1983), which offers a simple and clear expression for numerical calculations of peridynamic stress.Several numerical verifications have been carried out to validate the accuracy of the proposed peridynamic stress formula in predicting the stress states in the vicinity of the crack tip and other sources of stress concentration. By evaluating the peridynamic stress in the prototype microelastic brittle (PMB) material model of bond-based peridynamics, it is found that the PMB material model may exhibit nonlinear constitutive behaviors at large deformations. The stress fields calculated through the proposed peridynamic stress formula show good agreements with these calculated by using finite element methods, analytical solutions, as well as experimental data, demonstrating the merit of the peridynamic stress formula in predicting stress states for material failure problems.

Model-Free and Uncalibrated Eye-in-Hand Visual Servoing Approach for Concentric-Tube Robots

This article proposes a model-free and uncalibrated eye-in-hand visual servoing (EiH-VS) approach for controlling concentric-tube robots (CTRs) in minimally invasive surgery (MIS). Traditionally, a closed-loop EiH-VS controller requires an accurate robot kinematic model and hand–eye calibration. However, it is difficult to model CTR accurately when considering torsion, shear, friction, interactive force, and nonlinear constitutive behavior. In this article, we map image deviations to robot actuation variables with numerically calculated image Jacobian. The estimation of dynamic image Jacobian is based on a modified adaptive square-root unscented Kalman filter (MASR-UKF). A customized measurement matrix is constructed to describe the transformation between state and observation vectors. Moreover, an adaptive gain controller is designed to accelerate convergence. As a result, no prior knowledge of the CTR kinematic model and hand–eye calibration is needed in visual servoing tasks. Simulations and experiments have been conducted. The results validated the efficacy and efficiency of the proposed methods.

Contact-Resistance-Free Stretchable Strain Sensors with High Repeatability and Linearity.

Most of the existing stretchable strain sensors are based on the contact-resistance mechanism, where the stretchability and resistance variation depend on the change of the contact relationship of the conductive microstructures. These sensors usually exhibit large sensing ranges and gauge factors but unsatisfactory repeatability and linearity of the electrical responses because the contact is unstable. Here, we report a completely different design for stretchable strain sensors based on a contact-resistance-free structure, i.e., the off-axis serpentine sandwich structure (OASSS), with the mechanism of the stretch-bending-stretch transformation (SBST). Neither unstable contact resistance nor nonlinear constitutive and geometric behaviors occur for the OASSS while the sensor undergoes a large applied strain (50%), which guarantees high repeatability (repeatability error = 1.58%) and linearity (goodness-of-fit >0.999). Owing to such performances, the present sensors are not only applied to monitoring human activities and medical surgery but also to the ground tests of Tianwen-1, China's first Mars exploration mission.

Adiabatic kinetics of phase transformation in shape memory TiNi alloy subjected to shock loading

ABSTRACT In this paper, we study the adiabatic kinetics of martensitic phase transition induced by shock loading. Our main goals are to reveal the effect of thermal–mechanical coupling on phase interface propagation by demonstrating the wave profiles to the impact loading. First, a kinetics and constitutive model for shape memory TiNi alloy with thermo-mechanical coupling effect was preliminarily established based on the laws of thermodynamics. Then the numerical results show that phase transformation shock waves during shock loading and unloading would be aroused due to the nonlinear hardening since adiabatic temperature rise. The shock front and the temperature interface were coupled to each other. Moreover, the adiabatic temperature rise across the transformation shock front effectively reduces the driving force of phase transition and increases the propagation velocity of phase transition wave. Finally, the theoretical results are compared with the experimental results. The comparison results reflect the intrinsic thermo-mechanical coupling characteristic of shape memory TiNi alloy with strong nonlinear constitutive behaviour and kinetics, which will be helpful for temperature controlling for TiNi alloy material and structures subjected to blast or impact loading.

The role of non-linear material constitutive behaviour in modeling of fatigue crack growth of laminated plates

In this paper the effect of material constitutive model non-linearity on the fatigue crack growth rate was revealed and discussed. Article presents also a novel analytical model for the prediction of stress redistribution in the partially cracked fiber metal laminates. The model is based on the isostrain condition and Classical Lamination Theory, where the laminated plate is divided into a coherent and cracked regions. The model application allowed for the analysis and prediction of fatigue crack growth rate of fiber metal laminates. A significant differences in fatigue life was modeled for glass, carbon and hybrid glass/carbon fiber reinforced polymers, while the highest fatigue life was predicted for the laminate containing the stiffest internal composite layer. The model equations are open for further development to potentially include the fiber bridging effect as well as to analyze fatigue crack growth rate in bending and non-symmetrical problems.

Simple analytical models for restrained steel columns in fire

In structural engineering, two distinct types of computational strategies can be used to analyse a frame under the action of forces. The first type applies to any general structure and incorporates all involved complexities in the mathematical model. It aims to represent as accurately as possible all involved phenomena and leads usually to very complex numerical models, such as the Finite Elements ones. In contradistinction to this, the alternative strategies adopt simple but representative mathematical models that focus on a specific phenomenon. Despite their limited range of application, simple mathematical models enable an easier analysis of the key phenomena without forfeiting accuracy. They also provide a good depiction of the true physical response. Within this latter context, the present work presents two simplified structural models for the analysis of restrained elasto-plastic steel columns under fire conditions: i) an adapted Shanley model and ii) a two rigid rods frame joined by a plastic hinge. Only uniform heating is considered, as reference, and all nonlinear properties are taken into account, namely the degradation of the steel’s properties with heating, the nonlinear constitutive behaviour and the instability effects arising from the restraining of the thermal elogation. These two models enable the computation of the critical temperature, which defines the column’s strength in case of fire, with minimum effort and within an acceptable accuracy.

A novel iterative method based on fixed stress rates for hydromechanical problems with nonlinear constitutive relationships

This paper focuses on the development of alternative coupling strategies for hydromechanical applications considering highly nonlinear constitutive models to represent the behavior of porous media. We present a novel consistent iterative coupling formulation based on fixed stress rates to define the coupling terms between mass balance and equilibrium equations. We show that controlling stress rates allows the application of an adaptive iterative coupling scheme with single or multi-rate solutions. The proposed methodology has been implemented in a framework that manages sequential simulations, exchanges of information between models, and checks the convergence of the iterative process. We present several numerical tests in which we evaluate the stability, accuracy, and performance of the proposed coupling technique. From the results, we confirm that the rate form of the fixed stress split guarantees solution stability and yields results as accurate as those obtained through fully implicit solutions, even for highly nonlinear constitutive behavior.

Slope stability analysis using deterministic and probabilistic approaches for poorly defined stratigraphies

The stability of slopes directly affects human lives, the environment, and the economy. Inaccurate geological profiles within numerical slope stability models can lead to potentially catastrophic consequences when model conditions do not appropriately reflect real-life stratigraphy. In cases where localised deposits are prevalent, probabilistic methods are often necessary to accommodate for unknown or poorly defined stratigraphies. Currently, there are no commercial geotechnical software packages that simulate probabilistic constitutive behaviour of materials within finite element methods for large-scale stratigraphic analysis. Instead, commercially available probabilistic methods such as the random limit equilibrium method are incapable of incorporating non-linear constitutive soil behaviour. For this reason, advanced constitutive models are seldom coupled with probabilistically varying soil layers or spatially variable soil parameters. The objective of this research is the implementation of a simplified method for probabilistic stratigraphic analysis within a commercially available FE environment, providing a technique to assess the effects of stratigraphic uncertainty on slope stability. The proposed method is presented, highlighting the impact of localised thin layers of soft material as well as their frequency and location on the slope of an operational open-pit mine. The significance of these stratigraphic effects is presented through a case study of Australia’s second-largest open-pit mine, at which the stability of a collapsed coal slope is analysed. To improve the reliability of the finite element method for slope stability assessment, the Monte Carlo approach has been incorporated to consider varying shear strength distributions for models incorporating advanced constitutive behaviour. Thicker probabilistically generated deposits of silty material resulted in increased slope Factors of Safety. Similarly, greater proportions of silty deposits within a predominantly clayey interseam produced larger safety factors than slopes without localised thin silty layers. Stratigraphic analysis indicated that the Factor of Safety was most sensitive to localised silt layers at depths greater than 83 m below ground level.

A combined theoretical and numerical modeling study of cyclic nonlinear response of sandy seabed

Factors affecting the cyclic response of seabed around offshore foundations are associated with soil structure, wave loading and drainage conditions. Constitutive theories proposed to account for these factors in the design of such soil-structure systems must be able to capture the essentials of seabed dynamics. In this paper, a combined theoretical-numerical study on dynamic response of sandy seabed focusing on nonlinear soil constitutive behavior is presented. The manuscript is of threefold. First part presents a rigorous theoretical study to evaluate the cyclic behavior of seabed using the framework of bounding surface plasticity. Focus here is on more effectively modeling the elemental response, particularly liquefaction. A new hardening law is proposed to incorporate into the modified bounding surface model which is found to be the effective formulation addressing the wave-induced nonlinear seabed behavior. Second part presents a number of cyclic triaxial tests simulated to verify the new constitutive formulation. In the third part, a poro-elasto-plastic numerical model in terms of the extended nonlinear finite element form of the discretized coupled flow-deformation equations is developed. Then, free-field wave-induced dynamic response of sandy seabed is evaluated on various problems presenting the capability of the proposed constitutive formulation in capturing residual liquefaction.

Elastic Properties and Nonlinear Elasticity of the Noncarbon Hexagonal Lattice Nanomaterials Based on the Multiscale Modeling

Abstract This study presents the elastic properties and nonlinear elasticity of the two-dimensional noncarbon nanomaterials of hexagonal lattice structures having molecular structure XY. Four nitride-based and two phosphide-based two-dimensional nanomaterials, having graphene-like hexagonal lattice structure, are considered in the present study. The four empirical parameters associated with the attractive and repulsive terms of the Tersoff–Brenner potential are calibrated for noncarbon nanomaterials and tested for elastic properties, nonlinear constitutive behavior, bending modulus, bending and torsional energy. The mathematical identities for the tangent constitutive matrix in terms of the interatomic potential function are derived through an atomistic–continuum coupled multiscale framework of the extended version of Cauchy–Born rule. The results obtained using newly calibrated empirical parameters for cohesive energy, bond length, elastic properties, and bending rigidity are compared with those reported in the literature through experimental investigations and quantum mechanical calculations. The continuum approximation is attained through the finite element method. Multiscale evaluations for elastic properties and nonlinear stretching of the nanosheets under in-plane loads are also compared with those obtained from atomistic simulations.

Nonlinear isogeometric multiscale simulation for design and fabrication of functionally graded knitted textiles

We present a nonlinear multiscale modeling and simulation framework for the mechanical design of machine-knitted textiles with functionally graded microstructures. The framework operates on the mesoscale (stitch level), where yarns intermesh into stitch patterns, and the macroscale (fabric level), where these repetitive stitch patterns are composed into a fabric. On the mesoscale, representative unit cells consisting of single interlocked yarn loops, modeled as geometrically exact, nonlinear elastic 3D beams, are homogenized to compute their effective mechanical properties. From this data, a B-Spline response surface model is generated to represent the nonlinear orthotropic constitutive behavior on the macroscale, where the fabric is modeled by a nonlinear Kirchhoff–Love shell formulation and discretized using isogeometric finite elements. These functionally graded textiles with locally varying properties can be designed and analyzed by parameterizing the stitch value, i.e., the loop length of a single jersey stitch, and the knitting direction as mesoscopic design variables of the macroscopic response surface constitutive model. To validate the multiscale simulation and design approach, numerical results are compared against physical experiments of different tensile loading cases for various grading scenarios. Furthermore, the versatility of the method for the design of functionally graded textiles is demonstrated.

Physically sound, self-learning digital twins for sloshing fluids.

In this paper, a novel self-learning digital twin strategy is developed for fluid sloshing phenomena. This class of problems is of utmost importance for robotic manipulation of fluids, for instance, or, in general, in simulation-assisted decision making. The proposed method infers the (linear or non-linear) constitutive behavior of the fluid from video sequences of the sloshing phenomena. Real-time prediction of the fluid response is obtained from a reduced order model (ROM) constructed by means of thermodynamics-informed data-driven learning. From these data, we aim to predict the future response of a twin fluid reacting to the movement of the real container. The constructed system is able to perform accurate forecasts of its future reactions to the movements of the containers. The system is completed with augmented reality techniques, so as to enable comparisons among the predicted result with the actual response of the same liquid and to provide the user with insightful information about the physics taking place.

An Improved Lagrangian-Inverse Method for Evaluating the Dynamic Constitutive Parameters of Concrete.

The dynamic constitutive behaviors of concrete-like materials are of vital importance for structure designing under impact loading conditions. This study proposes a new method to evaluate the constitutive behaviors of ordinary concrete at high strain rates. The proposed method combines the Lagrangian-inverse analysis method with optical techniques (ultra-high-speed camera and digital image correlation techniques). The proposed method is validated against finite-element simulation. Spalling tests were conducted on concretes where optical techniques were employed to obtain the high-frequency spatial and temporal displacement data. We then obtained stress–strain curves of concrete by applying the proposed method on the results of spalling tests. The results show non-linear constitutive behaviors in these stress–strain curves. These non-linear constitutive behaviors can be possibly explained by local heterogeneity of concrete. The proposed method provides an alternative mean to access the dynamic constitutive behaviors which can help future structure designing of concrete-like materials.

Out-of-plane response of arching masonry walls to static loads

This paper aims at characterizing the response of arching masonry walls to out-of-plane static loading. An analytical multi-degree-of-freedom model is presented aiming at modeling the nonlinear out-of-plane behavior of arching masonry walls. The model combines physical modeling tools with a formulation that applies variational principles and equilibrium conditions, boundary and continuity conditions, geometrically nonlinear kinematics, and nonlinear constitutive laws. The latter take into account the nonlinear response in compression and in tension, including cracking and inelastic response of the joints. The model refers to one-way flexural action of unreinforced masonry walls. The masonry units are modeled as rigid bodies, whereas the mortar joints are represented by arrays of springs with nonlinear kinematics and nonlinear constitutive behavior. A series of numerical examples that explore the capabilities of the model, reveal various aspects of its physical behavior and compare the analysis with experimental results reported in the literature are presented. The examined aspects include the cracking process, the development of the arching mechanism, and the effect of inelastic behavior of the mortar material on the response. The combination of the above provides a deeper look into the unique and complex physical behavior of arching walls and sets a modeling approach for its quantitative assessment.

DBFFT: A displacement based FFT approach for non-linear homogenization of the mechanical behavior

Most of the FFT methods available for homogenization of the mechanical response use the strain/deformation gradient as unknown, imposing their compatibility using Green’s functions or projection operators. This implies the allocation of redundant information and, when the method is based in solving a linear equation, the rank-deficiency of the resulting system. In this work we propose a fast, robust and memory-efficient FFT homogenization framework in which the displacement field on the Fourier space is the unknown: the displacement based FFT (DBFFT) algorithm. The framework allows any general non-linear constitutive behavior for the phases and direct strain, stress and mixed control of the macroscopic load. In the linear case, the method results in a linear system defined in terms of linear operators in the Fourier space and that does not require a reference medium. The system has an associated full rank Hermitian matrix and can be solved using iterative Krylov solvers and allows the use of preconditioners. A preconditioner is proposed to improve the efficiency of the system resolution. Finally, some numerical examples including elastic, hyperelastic and viscoplastic materials are solved to check the accuracy and efficiency of the method. The computational cost reduction respect the Galerkin-FFT was around 30%.

Mixed beam formulation with cross-section warping for dynamic analysis of thin-walled structures

This paper presents the formulation of a three-dimensional beam finite element (FE) that accounts for cross-section warping and dynamic inertia effects. The model is the extension of an existing mixed formulation, originally developed for the static analysis of thin-walled beams, to the case of dynamic loading conditions.Four independent fields are considered to derive the element governing equations, i.e. material rigid displacements, strains and stresses and an additional displacement field, describing the out-of-plane warping displacement of the beam cross-sections. The latter is independently interpolated in the element volume by including additional degrees of freedom (DOF) to the nodal translations and rotations classically considered in beam formulations.To obtain a consistent form of the element mass matrix, the cross-section displacement shape functions are computed, relating the generalized cross-section displacement fields to the element nodal variables. In mixed FE formulations, these are not assigned a priori, as in displacement-based approaches, but are derived on the basis of material stiffness and element geometry, together with compatibility conditions. Thus, the Unit Load method is applied to deduce the expressions of the shape functions consistent with the force-based approach, assuming the simply-supported beam as reference element configuration.As opposed to the original FE model, the additional warping DOFs are not condensed-out with the definition of the element quantities but are treated as additional global unknowns. This permits a correct description of the inertia effects and ensures continuity of the warping displacement fields between adjacent FEs.Correlation studies are presented to validate the proposed model and investigate the effects of cross-section warping on the dynamic behavior of thin-walled structures. For selected specimens, the studies compare solutions obtained adopting the proposed beam element with those resulting from shell or brick FE models. Modal decompositions and time-history analyses are conducted, assuming both linear elastic and nonlinear constitutive behavior for the latter.

Numerical modeling of crack growth in polymer-bonded explosive with cavity subject to compression

Polymer-bonded explosives are solid high-explosive particles that exhibit brittle behavior in uniaxial tension, quasi-brittle in uniaxial compression, and ductile when subjected to high confining pressure. Tension cracking is the primary failure mode of polymer-bonded explosives quasi-brittle solid, which will lead to overall failure of structural integrity. One characteristic of brittle or quasi-brittle solids, such as polymer-bonded explosives, is that when subjected to overall compressive loading, the tensile cracks will initiate inside the material due to existence of imperfection within the materials. In this study, extended finite element method is applied to analyze the cracking failure mechanism in the PBX 9502 plate-like specimen with cavity subjected to overall compression. The nonlinear constitutive behaviors and failure of polymer-bonded explosives under complex stress states were described by means of stress state–dependent strength surface, non-associated flow rule, and cohesive failure model. Analysis indicates that the tensile stress around the cavity arises in the specimen under overall compression, and this local tensile stress will lead to cracking initiation. The comparison between simulation results and the experimental data reported by Liu and Thompson shows that they are in agreement with each other on some aspects of crack behaviors, including overall development of crack history and inflection, crack initiation moment, and crack initial speed.

Free radical polymerization simulation and molecular entanglement effect on large deformation behavior

In this paper, we proposed a novel molecular dynamics (MD) free radical polymerization algorithm that can dynamically simulate chain growth of the thermoplastic network and produce unit cells having experimentally observed molecular weight distributions. In the proposed algorithm, a coarse-grained force field was implemented by combining bonded terms from the PCFF and nonbonded terms from the TraPPE-UA to alleviate computational times. We controlled molecular weight distribution by the number of initiator radicals. Effects of the molecular weight on density, and glass transition temperature were investigated. A linear relationship of the conversion ratio with specific volume change was discovered. More importantly, the PMMA system by the proposed MD polymerization algorithm could explain physical causes of stress-softening behavior often observed in the large deformation tensile tests in terms of molecular weight and unfolding of chain entanglement. The proposed polymerization algorithm will facilitate studies on large deformation nonlinear constitutive behavior and modelings of thermoplastic polymers.