Nonlinear Elastic Material Research Articles

Floating Wind Turbine Model Test to Verify a moordyn Modification for Nonlinear Elastic Materials

Abstract As the floating offshore wind industry matures, it has become increasingly important for researchers to determine the next-generation materials and processes that will allow platforms to be deployed in intermediate (50–85 m) water depths, which challenge the efficiency of traditional catenary chain mooring systems and fixed-bottom jacket structures. One such technology, synthetic ropes, has in recent years come to the forefront of this effort. A significant challenge of designing synthetic rope moorings is capturing the complex physics of the materials, which exhibit viscoelastic and nonlinear elastic properties. Currently, numerical tools for modeling the dynamic behavior of floating offshore wind turbines (FOWTs) are limited to mooring materials that lack these strain rate-dependent properties and have a linear tension–strain response. In this article, a mooring modeling module, moordyn, which operates within the popular FOWT design and analysis program, openfast, was modified to allow for nonlinear elastic mooring materials to address one of these shortcomings in the numerical tools. Simulations from the modified openfast tool were then compared with 1:52-scale test data for a 6 MW FOWT semi-submersible platform in 55 m of water subjected to representative design load cases. A strong correlation between the simulations and test data was observed.

From force chains to nonclassical nonlinear dynamics in cemented granular materials.

In this letter, we present evidence for a mechanism responsible for the nonclassical nonlinear dynamics observed in many cemented granular materials that are generally classified as mesoscopic nonlinear elastic materials. We demonstrate numerically that force chains are created within the complex grain-pore network of these materials when subjected to dynamic loading. The interface properties between grains along with the sharp and localized increase of the stress occurring at the grain-grain contacts leads to a reversible decrease of the elastic properties at macroscopic scale and peculiar effects on the propagation of elastic waves when grain boundary properties are appropriately considered. These effects are observed for relatively small amplitudes of the elastic waves, i.e., within tens of microstrain, and relatively large wavelengths, i.e., orders of magnitude larger than the material constituents. The mechanics are investigated numerically using the hybrid finite-discrete-element method and match those observed experimentally using nonlinear resonant ultrasound spectroscopy.

Ice-shedding-induced vibration of conductors with active vibration control

A method is proposed for attenuating the vibration of suspended conductors. The idea involves the application of vibration control that contributes to reducing substantially the conductor rebound height following ice shedding. The theoretical background of the method is developed, and its application to vibration following ice shedding is tested by numerical simulations. A simplified model of a conductor with vibration damper is constructed, which simulates the dynamic behaviour of the conductor with damper at the location where the damper is attached. The model takes into account the nonlinear elastic material behaviour of the conductor together with the effects of a vibration damper with active control. Such vibration control may successfully attenuate even high-amplitude vibration that follows ice shedding. Simulation results provide to what extent the vibration amplitude is reduced due to the application of active control. After further improving the method, it may also be adaptable to decrease the amplitude of forced vibrations which are induced by wind and characterized by high amplitude.

Optical Investigation of the Limits of Modeling the Nonlinear Elastic Behavior of PA6 Using Linear Elastic Material Models

One of the most critical issues during polymer finite element simulations is the selection of the proper material models. The widely used and accepted multilinear material models require load case-specific material tests, which are time and cost demanding. Data for these characteristics must be acquired by standardized measurements. On the other hand, the parameters required to create a linear elastic material model in most cases are easy to obtain, and the establishment of the model is a shorter process. This research is aimed to provide information to engineers about the possibility of modeling the nonlinear elastic materials by using linear elastic material models and about the limits of such models. To create the most accurate material models, laboratory measurements were performed on polyamide (PA6) material, which is a widely used raw material in the industry. Test specimens were manufactured to obtain material constants according to the ISO 527-2 standard, and for validating the effectiveness of the applied material models, three different tensile specimens were created, which were tested under quasi-static loading in the elastic region. A comprehensive finite element investigation was performed, and the numerical results were then compared to laboratory measurements using the GOM Aramis digital image correlation (DIC) system. By comparing the optically measured strain data to the numerical results, it was determined that the nonlinear elastic materials can be modeled using linear elastic models in a well identifiable strain range with sufficient accuracy.

Tearing energy calculation in hyperelastic fracture mechanics using the local and global complex-variable finite element method

The “local” and “global” complex-variable finite element methods were extended for computing the energy release rate (ERR) of materials undergoing nonlinear elastic deformation at both small and large strains. These methods compute the derivative of the strain energy with respect to crack length by applying a virtual crack extension through the imaginary domain of complex variables. The “global” complex finite element method (G-ZFEM) is a general sensitivity method that provides accurate ERR estimates regardless of the material and loading conditions in the cracked body. However, as G-ZFEM duplicates all the degrees of freedom of a finite element model to store information pertinent to the ERR, this analysis adds significant computational overhead. Local-based methods, analogous to the J-integral, can be employed to improve the computational efficiency of the fracture analysis. The “local” complex-variable finite element method (L-ZFEM) is a fracture-specific approach where complex-valued calculations are only conducted at a small group of elements surrounding the crack tip, adding minimum overhead to the analysis. The G-ZFEM and L-ZFEM methods were incorporated within the commercial finite element software ABAQUS through a two-dimensional user element that was linked to several material models, including Ramberg–Osgood, Neo-Hookean, and Mooney–Rivlin. The ERR, known as tearing energy (TE) in hyperelastic fracture problems, was obtained by the proposed methods in benchmark problems and found to be in excellent agreement with analytical formulations and the J-integral results. Unlike L-ZFEM and the J-integral, the G-ZFEM method does not exhibit variations in TE estimates with respect to the size or direction of the virtual crack extension in nonhomogeneous cracked bodies; thus, G-ZFEM estimates can be regarded as the reference tearing energy solutions. In addition, by using multidual or multicomplex algebra, the complex finite element methods allow the computation of arbitrary, higher-order derivatives of the strain energy with respect to the model’s material, loading, or geometric properties, providing additional information to design structural components with maximum resistance to fracture. The complex-variable methods are versatile and can be easily adapted to any nonlinear elastic material formulation and extended to 3D in a straightforward manner.

Investigation of two-way fluid-structure interaction of blood flow in a patient-specific left coronary artery.

The blood flow in the human artery has been a subject of sincere interest due to its prime importance linked with human health. The hemodynamic study has revealed an essential aspect of blood flow that eventually proved to be paramount to make a correct decision to treat patients suffering from cardiac disease. The current study aims to elucidate the two-way fluid-structure interaction (FSI) analysis of the blood flow and the effect of stenosis on hemodynamic parameters. A patient-specific 3D model of the left coronary artery was constructed based on computed tomography (CT) images. The blood is assumed to be incompressible, homogenous, and behaves as Non-Newtonian, while the artery is considered as a nonlinear elastic, anisotropic, and incompressible material. Pulsatile flow conditions were applied at the boundary. Two-way coupled FSI modeling approach was used between fluid and solid domain. The hemodynamic parameters such as the pressure, velocity streamline, and wall shear stress were analyzed in the fluid domain and the solid domain deformation. The simulated results reveal that pressure drop exists in the vicinity of stenosis and a recirculation region after the stenosis. It was noted that stenosis leads to high wall stress. The results also demonstrate an overestimation of wall shear stress and velocity in the rigid wall CFD model compared to the FSI model.

Multiscale plasticity of geomaterials predicted via constrained optimization‐based granular micromechanics

AbstractA general framework to derive nonlinear elastic and elastoplastic material models from granular micromechanics is proposed, where a constraint‐based variational structure is introduced to classical grain contact‐based homogenization methods of hyperelasticity. Like the classical hyperelastic methods, reference solutions for closed‐form hyperelastic material models are analytically derived from the grain‐scale contact mechanics. However, unlike prior methods, the proposed homogenization framework defines closed‐form hyperelastoplastic material models that extend multiscale variational methods to granular plasticity. The proposed framework is used to develop novel granular micromechanics‐based macroscopic models for a Mises type solid, Drucker–Prager type plasticity, and grain‐contact cohesive‐debonding with a deviatorically and volumetrically coupled nonlinearly elastic response. Macroscopic plastic parameters and yield criteria are explicitly related to their microscale counterparts, for example, the friction coefficient governing intergranular slip. Numerical examples and comparison to measurements from the literature, including triaxial compaction of concrete, are provided to investigate model predictions and demonstrate calibration to experimental data.

Effect of material inhomogeneity under creep and plastic to creep transition of cracks

The driving force of a crack in a non-linear elastic material is described by the conventional J-integral, J (Rice 1968). For elastic−plastic materials, J does not describe the crack driving force. However, it describes the intensity of the crack tip field and can be, therefore, used to quantify the fracture toughness (Rice 1968). For the high-temperature behaviour of materials where creep deformation dominated, analogous parameter to J-integral were deduced, the C*-integral and a similar parameter, the Ct-integral (Landes and Begley 1976). C* has been proven to have a good correlation with the creep crack growth rate for many materials, mostly under extensive creep conditions. However, it does not describe the crack driving force.Based on the concept of configurational forces, a J -integral for elastic–plastic materials, Jep, was derived, which is able to quantify the true crack driving force in accordance with incremental theory of plasticity [4]. Hereby, the plastic strain is treated as an eigen-strain. Jep can be applied also in cases of non-proportional loading, e.g. for a growing crack (Simha et al. 2008) or for cyclic loading conditions (Ochensberger and Kolednik 2015). In this work, the concept of configurational force is applied to evaluate the crack driving force under small, medium and extensive creep conditions. Hereby, the creep strain is treated as a time dependent eigenstrain. A case study is performed where the path dependences of the elastic–plastic J-integral, Jep, and the conventional parameters Ct and C* are studied for materials which undergo elastic+creep deformation, with material inhomogeneity.It has been confirmed in the recent studies (Kolednik et al. 2014, Tiwari et al. 2020) that the material inhomogeneity term affects the crack depending on the nature of material inhomogeneity. This effect is unknown for creep deformation and the same has been studies on idealized fracture specimen under creep deformation using configurational forces and finite element method simulating real situations of dissimilar metals used at high temperatures in nuclear power plants and jet engines.

G-MAP123: A mechanistic-based data-driven approach for 3D nonlinear elastic modeling — Via both uniaxial and equibiaxial tension experimental data

This work proposes a data-driven approach, G-MAP123, using discrete data directly for nonlinear elastic materials to solve boundary value problems, avoiding analytic-function based constitutive models. G-MAP123 is formulated in the current configuration in which the Cauchy stress and the left Cauchy–Green strain are adopted as the stress–strain measures of the data. Data generated under both uniaxial tension and equibiaxial tension experiments is used. A data search employing stress triaxiality as the index is here proposed for the stress update. Furthermore, including additional data from other loading paths is also rendered possible. Comparison with reference analytic-function based models such as Arruda–Boyce, Yeoh, Mooney–Rivlin and Van der Waals is carried out. Results show that the predictions from G-MAP123 are in agreement with all those of the reference models. Moreover, the classic experimental data of rubber from Treloar is here used to demonstrate the capability of the proposed G-MAP123 in the practical setting. This approach opens a new avenue to modeling soft materials accurately and conveniently at large deformation, directly from the data.

Mechanical-Geometrical Modeling Of The Hyperelastic Materials At Uniaxial Stretching

Hyperelastic materials, such as rubber, occupy an important place in the design and operation of various technological equipment and machines. The article analyzed the deformation behavior of hyperelastic materials using a mechanical-geometric model. The method of mechanical-geometric modeling is a new method for obtaining constitutive relations and strain energy density functions for nonlinear elastic solids. It is based on physically and geometrically consistent prerequisites. The resulting models can describe broad classes of nonlinear elastic materials (both isotropic and anisotropic) depending on the mechanical and geometric properties “embedded” in them at the first stages of design. This paper discusses two basic types of models based on different initial geometry. The mechanical parameters of the models are constants, and the models themselves are considered in a statement corresponding to isotropic hyperelastic materials. The article presents the most common diagrams of deformation of artificial and natural rubbers, as well as steel. Hyperelastic materials, depending on the task, can be described in the nonlinear theory of elasticity as ideal incompressible, or as weakly compressible. Parameters of expressions of strain energy density functions of mechanical-geometric models obtained for cases of incompressible and weakly compressible continuous solids were identified. Stretch diagrams and diagrams of the transverse deformation function of the obtained mechanical-geometric models for the two cases mentioned above are plotted. The extension diagram for the model with parameters corresponding to the classic structural material of the steel type is also shown. Comments are given on the possibility of further paths of developing the method of mechanical-geometric modeling to obtain results not only in the field of nonlinear theory of elasticity, but also viscoelasticity.

A Nonlinear Elastic Model for Compressible Aluminum Alloys with Finite Element Implementation.

In this paper, a three-dimensional model of nonlinear elastic material is proposed. The model is formulated in the framework of Green elasticity, which is based on the specific elastic energy potential. Equivalently, this model can be associated to the deformation theory of plasticity. The constitutive relationship, derived from the assumed specific energy, divides the material’s behavior into two stages: the first one starts with an initial almost linear stress–strain relation which, for higher strain, smoothly turns into the second stage of hardening. The proposed relation mimics the experimentally observed response of ductile metals, aluminum alloys in particular. In contrast to the classic deformation theory of plasticity or the plastic flow theory, the presented model can describe metal compressibility in both stages of behavior. The constitutive relationship is non-reversible expressing stress as a function of strain. Special attention is given to the calibration process, in which a one-dimensional analog of the three-dimensional model is used. Various options of calibration based on uniaxial stress test are extensively discussed. A finite element code is written and verified in order to validate the model. Solutions of selected problems, obtained via ABAQUS, confirm the correctness of the model and its usefulness in numerical simulations, especially for buckling.

Thermal stress effects on size-dependent nonlinear axial vibrations of nanorods exposed to magnetic fields surrounded by nonlinear elastic medium

Taking von Kármán geometric nonlinearity into account, this research proposes a nonlinear model of nanorod, based on the nonlocal elasticity and axial beam theories. Imperfect nanorod is surrounded by nonlinear elastic medium and is subjected to the axial compression or various fields in terms of thermal and transvers magnetic loads. Pinned–pinned boundary conditions are immovable and free for the nanorod under axial compression considering thermal and magnetic loads. The governing equilibrium equations of the nanorod are derived by means of Hamilton's principle. The coupled nonlinear dimensionless differential equations are solved employing He's variational method. To evaluate the accuracy of the results, the results of this method are compared with the values obtained from the finite element method. Numerical results are provided to explore the influences of the low and high temperatures, nonlocal parameter, magnetic force of transverse field, amplitude of vibrations, and linear and nonlinear elastic materials for nanorods.

A high-fidelity human cervical muscle finite element model for motion and injury studies

Abstract Active muscle response is a key factor in the motion and injury of the human head and neck. Due to the limitations of experimentation and the shortcomings of previous finite element models, the influence of material parameters of cervical muscle on motions of the head and neck during a car crash have not been comprehensively investigated. In the present work, a model of the cervical muscle in a 50th-percentile adult male was constructed. The muscles were modelled using solid finite elements, with a nonlinear-elastic and viscoelastic material and a Hill material modelling the passive and active parts of each muscle, respectively. The head dynamic responses of the model were validated using results obtained from volunteer sled tests. The influence of the material parameters of a muscle on head and neck motions were determined. Our key finding was that the greater the stiffness and the contraction strength of the neck muscles, the smaller the rotation angle of the head and the neck, and, hence, the lower the risk of head and neck injury to occupants in a car crash.

Theory of uniformity applied to elastic dielectric materials and piezoelectricity

Building on the celebrated works by Toupin (1963) on non-linear elastic dielectric materials and by Noll (1967) on the theory of material uniformity, we use the method of linear perturbations to obtain a generalised form of the equations of piezoelectricity in the linear regime, i.e., below the ferroelectric threshold. More specifically, in their most general form, the piezoelectric constitutive equations that we obtain include an anelastic strain and an anelastic polarisation, both arising from the intrinsic inhomogeneity of the body and according to the theory of material uniformity. These are distinct from the irreversible strain and polarisation that occur at values of stress and electric field above the ferroelectric threshold. Some of the known standard forms of the piezoelectricity equations are retrieved as particular cases.

Bending failure analysis and modeling of thin fiber reinforced shells

In this work, the impact of high stress gradients, found in bending of thin unidirectional fiber reinforced shells (~0.1 to 1 mm), on compressive micro-buckling failure, was analyzed. Such thin shells show increased resistance to compressive failure under high curvatures, which may even allow tensile fiber damage to drive ultimate failure for very low thickness (e.g. <0.5 mm). The main scope of this work is to analyze this increased resistance to compressive failure and propose a robust modeling scheme. The mechanical failure response was captured by a shell-buckling experimental campaign. The origins of the increased compressive failure resistance were initially attributed to the reduction of shear stresses acting on the most susceptible domain of a representative wavy fiber. This effect was effectively described by an analytically derived, stress-gradient-dependent parameter. The hypothesis for the establishment of this parameter was corroborated by a numerical micromechanical model adopting the embedded cell approach. This model also revealed important micromechanical interactions which were incorporated by simple stress and strain factors. The derived failure prediction scheme was further extended to include the non-negligible, non-linear elastic material behavior of carbon fibers by means of a numerical algorithm. The validity of the failure prediction model was demonstrated by the successful comparison with results acquired from the shell-buckling experiments on a unidirectional carbon-fiber reinforced epoxy system. To this end, the validity of the initial hypothesis of stress-gradient-dependence on compressive failure was corroborated. Major effect on the overall behavior modeling has carbon fiber's material non-linearity, as well as micromechanical interactions.

Viscoelastic surface electrode arrays to interface with viscoelastic tissues.

Living tissues are non-linearly elastic materials that exhibit viscoelasticity and plasticity. Man-made, implantable bioelectronic arrays mainly rely on rigid or elastic encapsulation materials and stiff films of ductile metals that can be manipulated with microscopic precision to offer reliable electrical properties. In this study, we have engineered a surface microelectrode array that replaces the traditional encapsulation and conductive components with viscoelastic materials. Our array overcomes previous limitations in matching the stiffness and relaxation behaviour of soft biological tissues by using hydrogels as the outer layers. We have introduced a hydrogel-based conductor made from an ionically conductive alginate matrix enhanced with carbon nanomaterials, which provide electrical percolation even at low loading fractions. Our combination of conducting and insulating viscoelastic materials, with top-down manufacturing, allows for the fabrication of electrode arrays compatible with standard electrophysiology platforms. Our arrays intimately conform to the convoluted surface of the heart or brain cortex and offer promising bioengineering applications for recording and stimulation.

An exact solution to the Lame problem for a hollow sphere for new types of nonlinear elastic materials in the case of large deformations

Constitutive relations of two classes are proposed for nonlinear elastic isotropic materials, which, in case of purely volumetric deformation, are reduced to the Murnaghan’s equation of state. Exact analytical solution of the Lame problem of the radially symmetric deformation of a hollow sphere is obtained for one of these material classes. Nonlinear effects are studied. The non-uniqueness of solution is obtained for the case in which the sphere radii are specified in the initial configuration. It is shown for this case that there is a limiting pressure, above which the problem has no solution. The strong ellipticity conditions are tested. The obtained results can be used in geomechanics for modeling the recrystallization of metamorphic rocks.

Constitutive modeling of magneto-viscoelastic polymers, demagnetization correction, and field-induced Poynting effect

Magneto-active polymers are polymer-based composites consisting of magnetizable micro-particles embedded in an elastomeric matrix material. The presence of magnetic particles provides strong tunability to the stiffness and damping properties of the polymeric composites under the magnetic field. We propose here a novel free energy-based phenomenological modeling for magneto-active polymers. Coupled mechanical and magnetization constitutive equations are derived by considering magneto-viscoelasticity in a finite deformation framework. The model calibration takes into account the demagnetization effect, which depends on the specimen size, shape, and its own magnetization. The model prediction is verified with the available experimental data. Finally, a coupled simple shear problem is solved to demonstrate the influence of magnetic field on the Poynting effect. Classical nonlinear soft elastic material shows positive Poynting effect, i.e., shearing planes try to expand. We found the possibilities of field-induced reverse or negative Poynting effect in shear for magneto-active polymers. Such a negative Poynting effect could potentially affect a very diverse range of applications from the performances of miniature sensors and actuators to controlled drug delivery.

The remarkable bending properties of perforated plates

Motivated by recent experiments on the bending of porous strips towed through a fluid bath, a combined theoretical and experimental study is made of the bending response of perforated plates. The focus is on the practically relevant class of thin plates (with thickness h) made of a homogeneous isotropic material that is perforated with periodic distributions (with unit-cell size ε) of monodisperse holes spanning a large range of porosities, from the dilute limit to nearly the percolation threshold. From the theoretical point of view, with the objective of quantifying the roles that the various constitutive and geometric inputs play on the their bending response, the perforated plates are modeled by means of three different approaches: (i) as 3D structures made of a perforated nonlinear elastic material and as 2D structures made of homogeneous linear elastic materials whose effective properties result from (ii) first taking the limit of dimension reduction (h↘0) and then that of homogenization (ε↘0) and, vice versa, (iii) first taking the limit of homogenization (ε↘0) and then that of dimension reduction (h↘0). From the experimental point of view, laser-engraved molds are utilized to fabricate perforated elastomeric plates with hexagonal distributions of elliptical holes. The resulting perforated plates are then subject to cantilever-bending due to their self weight and their pointwise deformation measured by means of X-ray tomography. Remarkably, counter to the general expectation from available mathematical results for heterogeneous plates at large, the results indicate that the bending response of perforated plates is fairly insensitive to whether the holes are smaller or larger than the plate thickness. Instead, it is dominated by their porosity, while the spatial distribution and shape of the underlying holes only have relatively marginal effects.

Mechanics of hyperelastic composites reinforced with nonlinear elastic fibrous materials in finite plane elastostatics

A model for the mechanics of a hyperelastic material reinforced with unidirectional and bidirectional fibers is presented in finite plane elastostatics. This includes the refinement/development of a series of continuum-based prediction models to accommodate the nonlinear responses of both the matrix material and the reinforcing fibers. The kinematics of the embedded fibers, including the torsional kinematics between two adjoining fibers, are formulated via the first and second gradient of continuum deformations. Within the framework of variational principles and a virtual work statement, the Euler equation and the admissible boundary conditions are derived. To this end, a set of inhouse experiments are performed for the purpose of cross-examination and model implementation. The obtained models successfully predict the strain-stiffening responses of the elastomer – polyester fiber composites together with other key design considerations such as, deformation profiles, shear strain distributions, and the deformed configurations of a local unit fiber mesh. The Euler – Almansi strain integrate model is also proposed through which the strain-softening behaviors of a certain type of polyurethane fiber composites are predicted, yet further implementation of the obtained Euler – Almansi model remains to be determined due to the paucity of available data. The practical utility of the proposed models may be expected in the design and analysis of hyperelastic composites exhibiting strain-stiffening/softening responses by providing instant estimations of the resultant properties of intended composites.