Finite Deformation Model Research Articles

A novel rotor internal bend actuator for the wireless compensation of inherent eccentricities

A novel internal rotor bend actuator is presented that enables compensating strains to be applied about the rotor centre. Axial reaction tubes are used to maintain the structural stability of the bend actuator and to provide feedback on the level of actuation. The system is powered by an onboard battery and the actuation is derived from the operation of lead screw motors. The actuation is commanded wirelessly, thus providing the functionality for an autonomous rotor. The integrated rotor is demonstrated in isolation via laser tracker measurements and for the rotating prototype system. Actuated bending gave rise to radial displacements at the central point relative to shaft ends of approximately 50µm peak-to-peak. These results validate a finite element deformation model. The rotor was levitated on active magnetic bearings, then rolling elements bearings, to demonstrate the synchronous vibration control achievable by static internal bending. The potential now exists for the commanded actuation to compensate for rotor eccentricities arising from manufacturing errors and dynamically induced unbalance for the cancellation of underlying synchronous vibration.

Incompatible Deformations of Elastic Plates

This article considers the methods for mathematical modeling of incompatible finite deformations of elastic plates by using the principles of the differential geometry theory underlying continuously distributed defects. Equilibrium equations were derived by asymptotic expansions of the finite strain measures with respect to two small parameters. One parameter defines the order of smallness of displacements from the reference shape (self-stressed state), while the other specifies the thickness. Asymptotic orders were different for the deflections and displacements in the plate plane, as well as for their derivatives. They were selected in such a way that, with additional assumptions on the possibility of ignoring certain terms in the resulting expressions and the compatibility of deformations, the equations could be reduced to the system of F¨oppl–von Ka´rm´an equations.

Incompatible Deformations in Hyperelastic Plates

The design of thin-walled structures is commonly based on the solutions of linear boundary-value problems, formulated within well-developed theories for elastic plates and shells. However, in modern appliances, especially in MEMS design, it is necessary to take into account non-linear mechanical effects that become decisive for flexible elements. Among the substantial non-linear effects that significantly change the deformation properties of thin plates are the effects of residual stresses caused by the incompatibility of deformations, which inevitably arise during the manufacture of ultrathin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of this paper. To this end, the local unloading hypothesis is used. This makes it possible to define smooth fields of local deformations (inverse implant field) for the mathematical formalization of incompatibility. The main outcomes are field equations, natural boundary conditions and conservation laws, derived from the least action principle and variational symmetries taking account of the implant field. The derivations are carried out in the framework of elasticity theory for simple materials and, in addition, within Cosserat’s theory of a two-dimensional continuum. As illustrative examples, the distributions of incompatible deformations in a circular plate are considered.

Research on reconstruction algorithm of finite element deformation model based on digital twin

Research on reconstruction algorithm of finite element deformation model based on digital twin

Research on the reconstruction algorithm of a finite element deformation model based on digital twins

Research on the reconstruction algorithm of a finite element deformation model based on digital twins

Design and Characterization of Deformable Superstructures Based on Amine-Acrylate Liquid Crystal Elastomers.

Deformable superstructures are man-made materials with large deformation properties that surpass those of natural materials. However, traditional deformable superstructures generally use conventional materials as substrates, limiting their applications in multi-mode reconfigurable robots and space-expandable morphing structures. In this work, amine-acrylate-based liquid crystal elastomers (LCEs) are used as deformable superstructures substrate to provide high driving stress and strain. By changing the molar ratio of amine to acrylate, the thermal and mechanical properties of the LCEs are modified. The LCE with a ratio of 0.9 exhibited improved polymerization degree, elongation at break, and toughness. Besides an anisotropic finite deformation model based on hyperelastic theory is developed for the LCEs to capture the configuration variation under temperature activation. Built upon these findings, an LCE-based paper-cutting structure with negative Poisson's ratio and a 2D lattice superstructure model are combined, processed, and molded by laser cutting. The developed superstructure is pre-programmed to the configuration required for service conditions, and the deformation processes are analyzed using both experimental and finite element methods. This study is expected to advance the application of deformable superstructures and LCEs in the fields of defense and military, aerospace, and bionic robotics.

A strain energy function for large deformations of compressible elastomers

Elastomers are typically considered incompressible or slightly compressible. However, we present simple tension and bulk tests showing that, under large deformations, these materials can undergo significant volume changes. A review of the literature reveals the lack of an accurate hyperelastic model for finite volumetric deformations of elastomers. Therefore, we propose a new volumetric strain energy density (SED) that overcomes the limitations of the current models. The main advantages of the proposed SED are: (1) accurate description of the response of rubbers for both small and large volumetric deformations; (2) ability to reproduce diverse behaviors during volume shrinkage and expansion; (3) adaptability to other compressible materials, such as soft tissues, foams and hydrogels. Using the deviatoric–volumetric split of the strain energy, the proposed volumetric SED is combined with a suitable deviatoric part selected from the literature. The parameters of the combined SED are calibrated by fitting the model to the experimental data from simple tension and bulk tests. As a result, an accurate description of the response of elastomers under both shape and volume deformations is provided. The proposed SED can be implemented in numerical codes to capture the effects of volumetric deformations on the equilibrium solutions for various stress states.

Predicting non-axisymmetric growth and facet evolution during lithiation of crystalline silicon anode particles through orientation-dependent interface reaction

A finite deformation modeling framework is developed to capture the experimentally observed two-phase, non-axisymmetric growth during the first lithiation of crystalline silicon cylinder in an improved fashion. The key feature of this model is an orientation-dependent interface reaction constant. Furthermore, the migration of lithium through the lithium-rich, growing amorphous zone is modeled through two additional features. First, an alloying–dealloying reaction that distinguishes between “movable” and “immovable” lithium parts. Second, a strong two-way coupling between diffusion and stress in contrast to existing mathematical models of non-axisymmetric growth. These features enable the prediction of a non-uniform distribution of concentration and stresses. The most important ability of this framework is that it can capture the evolution of facets at the crystalline–amorphous interface shown in experimental studies. The ensuing orientation-dependent interface velocity breaks the axisymmetric shape of the nanowire despite the initial circular geometry and the maintenance of an axisymmetric lithium influx. A very interesting finding is that the deposition of lithium and induced stresses are higher at the sites with lower interface velocity, which occur at the junction of two facets. The resulting stress concentration, in turn, pushes towards the further accumulation of lithium at those sites. It is expected that the improved modeling capabilities of this framework will contribute a step in moving beyond the limitations of simplistic structural design of lithium-ion battery particles.

A thermoviscoelastic finite deformation constitutive model based on dual relaxation mechanisms for amorphous shape memory polymers

ABSTRACT This paper proposes a new thermoviscoelastic finite deformation model incorporating dual relaxation mechanisms to predict the complete thermo-mechanical response of amorphous shape memory polymers. The model is underpinned by the detailed microscopic molecular mechanism and effectively reflects the current understanding of the glass transition phenomenon. Novel evolution rules are obtained from the model to characterize the viscous flow, and a new theory named an internal stress model is introduced and combined with the dual relaxation mechanisms to capture the stress recovery. The rationality of the constitutive model is verified as the theoretical results agree well with the experimental data. Moreover, the constitutive model is further simplified to facilitate engineering applications, and it can roughly capture the characteristics of shape memory polymers.

Reference shape of bodies with finite incompatible deformations

The work develops differential-geometric methods for modeling of finite incompatible deformations of hyperelastic solids. Deformation incompatibility can be caused, for example, by inhomogeneous temperature fields and distributed defects. As a result, residual stresses and distortion of geometric shape of the body occur. These factors determine the critical parameters of modern high-precision technologies, in particular, in additive manufacturing technologies. In this regard, the development of methods for their quantitative description is an urgent problem of modern solid mechanics.
 The application of methods of differential geometry is based on the representation of a body as a smooth manifold equipped with a metric and a non-Euclidean connection. This approach allows one to interpret the body as a global stress-free shape and to formulate the physical response and material balance equations with respect to this shape. Within the framework of the geometric method, deformations are characterized by embeddings of non-Euclidean shape into physical space, which is still considered to be Euclidean. Measures of incompatibility are identified with the invariants of the affine connection, namely, torsion, curvature, and nonmetricity, and the connection itself is determined by the type of physical process.

A Phase Field Model for the Damage and Fracture of Multiple Network Elastomers

Abstract This work develops a continuum phase field model for predicting the damage initiation and crack propagation in multiple network elastomers. Previous researches have revealed that failure of multiple network elastomers involves microscopic damage initiation by the chain scission of filler network and macroscopic fracture by penetrating crack of matrix network. However, most existing models for multiple network elastomers only deal with its finite deformation and strain softening process, which are unable to capture the initiation and propagation of cracks. In this work, to bridge the microscopic damage and the macroscopic fracture of multiple network elastomers in the finite deformation model, we incorporate the phase field variable of crack surface density to model the crack propagation and the internal damage variable to model the chain scission. By forming a multi-field variational framework, the developed model can be used to simulate the macroscopic deformation and fracture of multiple network elastomers. Through a finite element implementation of the phase field model, previous experiment results obtained from uniaxial tension and unilateral fracture can be well predicted. Moreover, experimentally observed damage zone formed by sacrificing filler network to achieve toughening effect is also numerically illustrated in simulation, giving much clearer pictures for the contributions of different energy dissipation mechanisms.

In Situ Strain Measurements Within Helmet Padding During Linear Impact Testing.

Drop and Impact testing of helmets are used extensively in the design process and eventual certification of helmets. These techniques have traditionally relied heavily on the measurement of the kinematic response to impact, which provides an indirect measurement of the liner response that is subject to interpretation during the design process. In the present work, we introduce an in situ experimental technique that provides a time-resolved measurement of the deformation of the helmet and its components during an impact event. The data collected from a high-speed X-ray imaging system can provide a full description of the deformation at the component level, which provides a helmet designer further insight into the performance of their helmet, while also returning the traditional kinematic metrics. The data presented focuses on the deformation of a commercial hockey helmet subjected to a series of linear impacts with three different impactor caps at speeds ranging from 2.4 to 4.5m/s. Deformation of the liner was monitored in the midsagittal and a parasagittal plane of the helmet. The results show that there is a clear dependence on the maximum strain achieved in the foam that is dependent on the impact type, the impactor shape, and the resulting strain rate of deformation in the foam liner. These techniques can provide the first data for a direct validation and calibration of finite element helmet deformation models, while also providing a new tool-set to improve the efficacy of helmet design.

A unified physically-based finite deformation model for damage growth in composites

Two 3D homogenized models for damage growth in a unidirectional (UD) composite ply are simplified and merged into a unified model. The fibre kinking behaviour is based on fibre kinking theory handled in a finite deformation framework. The nonlinear shear behaviour is pressure dependent and is modelled by combining damage and friction on the fracture plane. Fibre kinking growth and transverse behaviour are modelled with a single damage variable. This allows both modes to occur simultaneously and mutually influence each other in an efficient and physically-based way.For validation the model is tested against micro-mechanical Finite Element (FE) simulations under pure longitudinal compression and influenced by shear. The results show nearly perfect agreement for stiffness, strength and crushing stress. The model validation is performed against two different components under three-point bending and a quasi-static crash scenario. Both simulation show good correlation with experiments, validating thus the present unified model.

Theoretical and Experimental Investigation of Shape Memory Polymers Programmed below Glass Transition Temperature.

An epoxy-based shape memory polymer (SMP) is synthesized and examined for its deterioration in shape fixity due to springback and isothermal viscoelastic recovery at different ambient temperatures. Shape fixity depends not only on material properties but also on programming conditions. A constitutive finite deformation model is incorporated to predict the behavior of the proposed SMP and find maximum shape fixity. A programming approach is followed in which, in contrast to hot programming, the SMPs are neither heated before deformation nor cooled afterward but are deformed at ambient temperature and then stress-relaxed. The proximity of the programming temperature to the glass transition temperature plays a crucial role in determining the shape fixity of SMP. It has been found that the SMP with a glass transition temperature of 42.9 °C can achieve maximum shape fixity of 92.25% when programmed at 23 °C with 100 min stress relaxation time. Thermal contraction and dynamic tests are performed in the Dynamic Mechanical Analyzer (DMA) to determine structural relaxation properties and distinguish the programming temperature in the cold, warm or hot temperature zone. The shape memory tests are carried out using temperature-controlled UTM to determine the shape fixity and shape recovery of SMP. The SMPs are subjected to a full thermomechanical cycle with different stress relaxation times and programming temperatures.

Implementation of a Finite Element Deformation Model Within an Elasto-Hydrodynamic Lubrication Numerical Solver for a Ball in Socket Tribopair

In the framework of the elasto-hydrodynamic lubrication simulation algorithms of lubricated tribopairs, a key role is played by the chosen deformation model, since it affects the surfaces’ separation, which guarantees the existence of a thin lubricant film thickness, even when the tribo-system is subjected to high loads. The aim of this article is to merge a finite element deformation model based on linear tetrahedra, previously developed by the same authors, within the Reynolds equation solver in the elasto-hydrodynamic mode, with reference to a generic ball in socket lubricated tribo-system. The main novelty of this research is the implementation of the finite element deformation model, allowing the authors to relate the deformation vector to the pressure one through an influence matrix which takes into account the spherical motion of the ball with respect to the socket. The computer code for the problem–solution was developed in a MATLAB environment and simulated a planar motion condition in terms of eccentricity and angular velocity vectors, in order to calculate the meatus fluid pressure field, surfaces’ separation, shear stress, deformation, and wear depth. The integration over time of the output fields led to the time evolution of the load vector, friction torque vector, and wear volume. Moreover, the lubrication algorithm takes into account the fluid non-Newtonian behavior and the surfaces’ progressive geometrical modification over time due to cumulated wear. The obtained results reproduced the classical elasto-hydrodynamic shapes of the involved quantities, following the meatus minimum thickness predicted by the Hamrock–Dowson model; furthermore, it provided information about the mechanical behavior of the whole bodies belonging to the spherical joint thanks to the finite element deformation model.

Modeling and Design of Periodic Polygonal Lattices Constructed from Microstructures with Varying Curvatures

Lattice structures can exhibit unusual mechanical properties by appropriate design and arrangement of the microstructures, and have already found applications in areas such as tissue engineering, stretchable electronics, and soft robotics. However, the designed microstructures generally follow simple geometric patterns with constant curvatures and theoretical models are developed for each geometry particularly. It lacks a universal method to model and design the periodic lattice constructed from microstructures with varying curvatures. In this paper, we develop a universal approach to design polygonal lattices with a wide range of microstructures with varying curvatures using instant curvature and parametric functions. A finite deformation model is proposed for the microstructures and the corresponding lattices, which considers the wavy or curled microstructure, hierarchical geometry, and large deformation. Experiments and finite-element simulations are conducted to validate the theoretical model. All of Young's modulus, stretchability, and Poisson's ratio can be inversely designed using the developed theoretical model. Results show that the stretchability of the highly stretchable lattice can be further increased by more than 4 times using densely curled microstructure. The signs of Poisson's ratio can also change with the axial strain by rationally designing the microstructures. By combining curve fitting and curvature and parametric functions, the mechanical behaviors of lattices constructed by bioinspired microstructures can also be predicted. The lattice structures are then demonstrated to reinforce a hydrogel with modulus increased by around 2 orders and act as a biomimetic soft-strain sensor. This work could aid the design of a hierarchical lattice and have potential applications in tissue engineering, stretchable electronics, and soft robotics.

Study of bending of plate steel with a through-the-thickness gradient of strength properties

The paper considers the studies of bending of a plate made of A32 ship steel with a through-the-thickness gradient of strength properties. The grading was produced by accelerated one-sided cooling of the plate from the austenitic area. As a result, a spectrum of microstructures was formed over the thickness of the plate: from ferrite-bainite on the cooled surface to ferrite-perlite on the other. During elastic-plastic bending of a steel plate with a homogeneous microstructure, the neutral surface shifts towards the compressed fibers, which is explained by the greater resistance of the material to compression than to tension. The purpose of this work was to develop a finite plastic deformation model of bending of a steel plate with tension/ compression (T/C) asymmetry and a strength gradient to confirm the expediency of one-sided thermal reinforcement of rolled sheets. It is confirmed that the displacement of the neutral surface caused by T/C asymmetry depends on the asymmetry ratio and does not depend on the steel microstructure, and is directed towards the compressed fibers. The displacement caused by the strength gradient depends on the absolute value of this gradient and is directed towards it. Calculations revealed that the critical bending moment for a plate made of A32 steel with a strength gradient is not less than that for the normalized and thermally hardened (by quenching and tempering ) states, at any direction of the strength gradient with respect to the bending direction. It is concluded that the proposed technology of thermal reinforcement of heavy-plate rolled products made of carbon and low-alloy steels using accelerated one-sided cooling provides mechanical properties not worse than for the thermally hardened state. This saves up to 40 % of cooling water.

A TRANSITION METHOD FROM DISCRETE SIMULATION TO ELASTIC FEA OF CONTINUOUS MEDIA

A transition method from discrete molecular dynamics (MD) simulation to continuum elastic finite element analysis (FEA) is proposed. Firstly, the moving position and displacement of crystal material atoms is obtained by MD calculation, and then the finite element deformation model under the assumption of continuous medium is constructed according to the characteristics of crystal structure. Further, the strain and stress fields are obtained combined with the constitutive relationship of material mechanical behavior. In order to test the effectiveness of the MD-FEA method, this method is applied to analyze the tensile deformation of Al-Ni soft-hard composite nano cylinder and the nano indentation of substrate Al with spherical diamond indenter. The stress and the strain fields of the above two problems are obtained by MD-FEA method, and the calculated results are compared with the atomic strain calculated by discrete deformation gradient and the atomic potential stress in traditional MD method. The difference between the stress and strain field calculated by MD-FEA method and the traditional MD atomic strain and potential stress is discussed in detail, and the effectiveness of MD-FEA method and its advantages over traditional MD method are discussed. The result shows that the MD-FEA method and the traditional MD method are consistent when the stress and strain change softly in the volume, and in the area where stress and strain change rapidly and in the surface/interface area, the MD-FEA method can calculate the result more precise. Meanwhile, the MD-FEA method avoid the selection of the cutoff radius and weighting functions, which is necessary in traditional MD method and can lead to human error in some circumstances. When the strain is large, there are obvious difference between the small strain calculated by MD-FEA method and the Green strain calculated by traditional MD method. Thus the MD-FEA method is more suitable for the situation that the stress and strain is small.

An Auxetic System Based on Interconnected Y-Elements Inspired by Islamic Geometric Patterns

A 2D mechanical metamaterial exhibiting perfectly auxetic behavior, i.e., Poisson’s ratio of −1, is proposed in this paper drawing upon inspiration from an Islamic star formed by circumferential arrangement of eight squares, such as the one found at the exterior of the Ghiyathiyya Madrasa in Khargird, Iran (built 1438–1444 AD). Each unit of the metamaterial consists of eight pairs of pin-jointed Y-shaped rigid elements, whereby every pair of Y-elements is elastically restrained by a spiral spring. Upon intermediate stretching, each metamaterial unit resembles the north dome of Jameh Mosque, Iran (built 1087–1088 AD), until the attainment of the fully opened configuration, which resembles a structure in Agra, India, near the Taj Mahal. Both infinitesimal and finite deformation models of the effective Young’s modulus for the metamaterial structure were established using strain energy approach in terms of the spiral spring stiffness and geometrical parameters, with assumptions to preserve the eight-fold symmetricity of every metamaterial unit. Results indicate that the prescription of strain raises the effective Young’s modulus in an exponential manner until full extension is attained. This metamaterial is useful for applications where the overall shape of the structure must be conserved in spite of uniaxial application of load, and where deformation is permitted under limited range, which is quickly arrested as the deformation progresses.

Nonlinear compressive deformations of buckled 3D ribbon mesostructures

Three-dimensional (3D) mesostructures with complex ribbon geometries have widespread applications in a diversified range of areas. The mechanically-guided 3D assembly approach based on precisely controlled buckling provides a versatile route to complex 3D ribbon-shaped mesostructures in a broad set of high-performance materials, thereby allowing the development of 3D flexible electronic devices with new functionalities. In many scenarios of practical operations, especially the biointegrated applications, the 3D flexible electronics would inevitably experience the out-of-plane compression, due to loadings of surrounding bones/organs or clothes, which could affect device functions significantly. A clear understanding of the deformation mode of 3D ribbon-shaped mesostructures is thereby essential to the design of 3D flexible electronics. This paper presents a systematic investigation of nonlinear compressive deformations and mode transitions of bridge-shaped and helical mesostructures assembled through controlled lateral buckling. Based on the elastica theory, a finite-deformation model is developed for the bridge-shaped mesostructure compressed by a frictional wall. Validated by results of experiments and finite element analyses (FEA), this theoretical model enables the establishment of the phase diagram of all possible deformation modes, indicating the crucial roles of friction coefficient and in-plane compressive strain on the mode transition. Experimental and computational studies further show that five distinct deformation modes could possibly occur during the compression of helical mesostructures. The mode transitions in the compressed helical mesostructures are found to arise mainly from the downward collapse, either at the middle or the side region of the mesostructure. These results can serve as design guidelines for the development of 3D flexible electronic devices, especially those for biointegrated applications.