Large Elastic Deformations Research Articles

Implementation of the Zienkiewicz–Pande Model into a Four‐Dimensional Lattice Spring Model for Plasticity and Fracture

ABSTRACTPlasticity and fracture problems have always been hot topics in numerical methods. In this work, a universal implementation procedure for the elasto‐plastic constitutive model is developed in the four‐dimensional lattice spring model (4D‐LSM), in which the Jaumann stress rate is incorporated to exclude the influence of the rigid rotation in the particle stress, expanding the ability of 4D‐LSM to deal with large elastic deformation problems by its own to large plastic deformation problems. As an example, the Zienkiewicz–Pande (ZP) constitutive model is implemented. Several numerical examples are carried out to check the performance of the implemented model. Through a comparison with analytical solutions, available experimental data, and other numerical results, the stability of the developed plastic framework and the correctness of the stress calculation scheme are verified. Meanwhile, numerical results show that the developed code is capable of solving elasto‐plastic large deformation problems. With the advantage of 4D‐LSM in handling fracture problems, the ability of the embedded model to solve plastic fracture problems is verified with a simple maximum deformation failure criterion.

HLFEMP: A coupled MPM-FEM method under a hybrid updated and total Lagrangian framework

Coupling the Material Point Method (MPM) with the Finite Element Method (FEM) provides a powerful approach for addressing problems characterized by the coexistence of large and small deformations. The majority of existing MPM-FEM methods, which operate within the updated Lagrangian framework, introduce inherent issues, such as cell-crossing noise and numerical fracture, compromising the overall accuracy of the coupled method. This paper presents a coupled MPM-FEM method under a hybrid updated and total Lagrangian framework (HLFEMP). Within the HLFEMP, updated Lagrangian FEM is employed to handle small deformations, while total Lagrangian MPM is adopted to model large deformations. A particle-to-surface contact algorithm with auxiliary thickness layer is utilized to couple both methods. Notably, by imposing the weak form of total Lagrangian MPM in the undeformed configuration, HLFEMP is rendered free from cell-crossing noise and numerical fracture, resulting in better computational stability. The performance of HLFEMP in terms of energy behavior, accuracy, efficiency and stability under large elastic and elasto-plastic deformations are investigated through a series of numerical simulations.

Hyperinelasticity. Part II: A stretch-based formulation

A generalisation of the hyperinelasticity modelling framework devised in Part I of this sequel is formulated here, by presenting a (principal) stretches-based hyperinelastic deformation energy function WF. This generalisation is based on the premise that the (principal) stretches λj may assume any arbitrary real-valued exponents, rather than being restricted to the prescriptive powers 2 and −2, as in principal invariants-based models. The motivation behind this extension is to reduce the overall number of model parameters and thereby increase the versatility of the application of the hyperinelasticity framework, as well as to provide a more universal model. The ensuing hyperinelastic model is then applied to a wide range of extant experimental datasets encompassing foams, glassy and semi-crystalline polymers, hydrogels and liquid crystal elastomers, over both elastic and inelastic deformation ranges including yield, softening and plateau, and hardening behaviours, under tensile and compressive deformations. Upon demonstrating the favourable simulation of the foregoing behaviours by the model, its application is then extended to account for other nuanced aspects of inelasticity such as the effects of rate of deformation, crystallinity volume and angle of printing in 3D printed lattice structures. This augmentation is done via devising a generalised modelling framework which allows for the incorporation of a generic tensorial (including rank zero scalar) field of inelasticity-inducing factors into the core model, resulting in the model parameters to evolve with an appropriate measure of the factor of interest; e.g., deformation rate, crystallinity volume ratio etc. The proposed modelling framework will be shown to capture these effects proficiently. Given the simplicity of this modelling approach, as essentially an extension in the application of hyperelasticity, its versatility of implementation, and the favourable capturing of both elastic and inelastic behaviours, the devised hyperinelasticity framework is presented for application to the large elastic and inelastic deformation of polymers and elastomers.

Finite Elastic Deformations of Initially Flat Thin Rubber-Like Materials

This contribution aims to derive a nonlinear FE formulation for large elastic deformation of hyperelastic thin rubber-like elastomeric materials. The formulation accounts for isotropic as well as transversely isotropic rubber-like materials. Besides, with the developed formulation, deformation of incompressible as well as compressible materials can be investigated. The formulation is applicable for in-plane and out-of-plane deformation of thin membranes. Several problems are inspected including uniaxial and equibiaxial extension of rubber, axisymmetric deformation of annular sheet, stretching a perforated rubber, extension of an anisotropic rubber, and inflation of an initially-flat circular and a square membrane. The results show very good agreement compared with those reported in the literature. Moreover, this model is able to model the inflation of initially flat membranes which is not possible to be done by commercial software.

Long-term creep behavior of glulam member under compression and bending moment

ABSTRACT In this study, 12 glulam members were designed and tested, considering the effects of stress ratio and relative eccentricity. After the 180-day test period, the results showed that increasing the stress ratio and relative eccentricity led to large initial elastic deformation and creep deformation, and the creep coefficients at 180 days for the glulam members under higher stress ratio and relative eccentricity were lower than those for the glulam members under lower stress ratio and relative eccentricity. Typical empirical and mechanical models and a modified model were chosen to fit the creep coefficient curves. The fitting analysis indicated that the values of the R 2 for all creep models were larger than 0.90, reflecting the goodness of fit. Employing the modified model, the prediction of the creep coefficient for the compressive-bending glulam member over 50 years was conducted. The results found that the glulam members under high relative eccentricity and stress ratio exhibited low creep coefficients, of which values approached to 0.7. Compared to creep coefficients specified in current standards and literature, the creep coefficient of 1.5 at a 50-year period for the compressive-bending glulam members was recommended, aiming to provide a reference to the long-term design of the compressive-bending glulam member in practice.

On elastic deformations of cylindrical bodies under the influence of the gravitational field.

Large elastic deformations of gravitating cylindrical bodies play a significant role in everyday life. On the other hand, tiny such deformations are relevant for state-of-the art experiments, as they affect the physical properties of materials under consideration, impacting wave propagation. This is of key importance for a recently planned experiment to explore the influence of the gravitational field on entangled photons propagating in waveguides. The purpose of this work is to determine this influence. We use the methods of linear elasticity, including thermoelasticity, to determine the stresses and strains of the medium. For this, the symmetry of the cylinder allows us to solve the problem by using Mitchell's solutions of the equations satisfied by the Airy functions. The boundary conditions are implemented by an approximation of the Hertz contact method. We calculate the displacements, the stresses and strains for several classes of boundary conditions, and give explicit solutions for a number of physically motivated configurations. The influence of the resulting deformations on the planned GRAVITES experiment is determined. The results are relevant for fiber interferometry experiments sensitive to the effects of the gravitational field on photon propagation. Our calculations give stringent bounds on the environmental variables, which need to be controlled in such experiments.

Reusable energy-absorbers design harnessing snapping-through buckling of tailored multistable architected materials

Abstract Multistable metamaterials are artificially engineered materials that possess microarchitectures capable of maintaining multiple stable configurations. However, the realization of mechanical metamaterials with numerous programmable stable configurations using Double-Curved Beam (DCB) elements remains an ongoing challenge. In this study, we exploit the snapping-through buckling phenomenon exhibited by architected DCB structures to devise a mechanical metamaterial with a unique deformation mode, encompassing Multi-stability, Multi-path, Multi-platform, and Multi-step characteristics, hence referred to as a 4M mechanical metamaterial. By employing DCB as fundamental building block elements, architected metamaterials with two-dimensional (2D) series or parallel lattices are successfully constructed, as well as three-dimensional (3D) tubular geometries, denoted as DCB-n-m-C and DCB-n-m-M metamaterials, respectively. These metamaterials exhibit reversible energy absorption characteristics and the stiffness can be transformed from positive to negative under both small and large elastic deformations. Functional gradient design and tailored deformation capability are given by adjusting the wall thickness of each layer of double-curved beams, thereby demonstrating the multi-path deformation features inherent in 4M metamaterials. DCB-n-m-M metamaterials has multiple energy platforms in the process of snapping-through, which reflects the multi-platform characteristics of 4M metamaterial. Consequently, novel properties such as multistability, programmability, and reusable energy absorption characteristics are achieved. To comprehensively understand the mechanical response of the metamaterials, a thorough investigation into the influence of geometric parameters is conducted, including the number of polygonal edges, the number of the layers, and the aspect ratio Q. This investigation involves a combination of theoretical analyses, numerical simulations, and experimental verifications. The introduced design strategy paves a way for the innovative design of multistable, multi-step, tailored, and reversible deformation metamaterials.

Highly Flexible and Superelastic Graphene Nanofibrous Aerogels for Intelligent Sign Language.

Highly flexible and superelastic aerogels at large deformation have become urgent mechanical demands in practical uses, but both properties are usually exclusive. Here a trans-scale porosity design is proposed in graphene nanofibrous aerogels (GNFAs) to break the trade-off between high flexibility and superelasticity. The resulting GNFAs can completely recover after 1000 fatigue cycles at 60% folding strain, and notably maintain excellent structural integrity after 10000 cycles at 90% compressive strain, outperforming most of the reported aerogels. The mechanical robustness is demonstrated to be derived from the trans-scale porous structure, which is composed of hyperbolic micropores and porous nanofibers to enable the large elastic deformation capability. It is further revealed that flexible and superelastic GNFAs exhibit high sensitivity and ultrastability as an electrical sensors to detect tension and flexion deformation. As proof, The GNFA sensor is implemented onto a human finger and achieves the intelligent recognition of sign language with high accuracy by multi-layer artificial neural network. This study proposes a highly flexible and elastic graphene aerogel for wearable human-machine interfaces in sensor technology.

Geometrically exact 3D beam theory with embedded strong discontinuities for modeling of localized failure in bending

In this work, we propose the elastoplastic model of a three-dimensional (3D) geometrically exact beam, which can capture localized bending deformation leading to a plastic hinge. This is a follow-up work to Tojaga et al. (2023), where we only studied the fracture of fiber-like beam structures that break in mode I or mode II, here extended to handle beam-like behavior with bending failure and the main difficulty of the non-vectorial character of large 3D rotations. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model, which leads to the multiplicative decomposition of the rotation tensor. We shown how to replace this with the corresponding additive decomposition of the rotation vector derivatives, to be performed in the material description in the tangent space of SO(3) manifold in the initial configuration. The plasticity model for which we give a more detailed implementation employs a Rankin-like multi-surface plasticity criteria, where each moment vector component (i.e. torsion or bending) is considered separately. The non-local variational formulation is proposed to deal with the softening phenomena characteristic of localized bending failure, where the fracture energy is introduced as the main parameter for softening. The discrete approximation is built in terms of the embedded-discontinuity finite element method (ED-FEM), which introduces a jump in rotation vector that can be handled at the level of a particular beam element. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best-approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and moments) from local (plastic curvature) variables. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Neural Physical Simulation with Multi-Resolution Hash Grid Encoding

We explore the generalization of the implicit representation in the physical simulation task. Traditional time-dependent partial differential equations (PDEs) solvers for physical simulation often adopt the grid or mesh for spatial discretization, which is memory-consuming for high resolution and lack of adaptivity. Many implicit representations like local extreme machine or Siren are proposed but they are still too compact to suffer from limited accuracy in handling local details and a long time of convergence. We contribute a neural simulation framework based on multi-resolution hash grid representation to introduce hierarchical consideration of global and local information, simultaneously. Furthermore, we propose two key strategies: 1) a numerical gradient method for computing high-order derivatives with boundary conditions; 2) a range analysis sample method for fast neural geometry boundary sampling with dynamic topologies. Our method shows much higher accuracy and strong flexibility for various simulation problems: e.g., large elastic deformations, complex fluid dynamics, and multi-scale phenomena which remain challenging for existing neural physical solvers.

On two results in nonlinear elasticity obtained on the basis of triality theory

A result concerning global extrema in a nonsmooth nonconvex variational problem that appears in applications (e.g. in a large deformation elasticity problem) is investigated in comparison with results of D. Y. Gao and R. W. Ogden. The tools used are elementary and the derived result improve upon and correct two similar results obtained by the aforementioned authors.

Novel polyurethane foam with low released heat for application in bone binder

AbstractBecause of the significant advantages of bone adhesives compared with traditional mechanical fixation, many scholars have studied polyurethane bone adhesives. However, polyurethane will release a lot of heat in the reaction process, producing high temperature. The high temperature does not only easily cause a human discomfort but also will destroy the tissue structure of human. In this paper, a new polyurethane composite based on phase transition materials and hydroxyapatite is prepared to reduce released heat for application in bone binder. The mechanical properties, adhesive strength, and released heat‐induced temperature of polyurethane composite are investigated in detail. It exhibits good viscoelastic performance with a large elastic deformation (>30%) and good adhesive strength (>0.2 MPa). Furthermore, the released heat of polyurethane composite is effectively reduced, in which the temperature drastically decreases from 110.4 to 41.8°C. The work does not only confirm formation of high‐performance bone binder but also provides a new method to reduce released heat of polyurethane for various applications.

On a class of nonlinear isotropic thermo-elastic body that is non-Green elastic: Applications to large elastic deformations

Summary A new class of constitutive equation is proposed for isotropic thermoelastic solids, wherein the Hencky strain tensor is assumed to be a function of the Cauchy stress tensor, via a Gibbs potential. The solid is assumed to be incompressible in the referential state, but the volume can change due to differences in the temperature relative to a reference temperature. The change in volume only depends on temperature. Some restrictions are found for the Gibbs potential, resulting in a constitutive equation for isotropic solids, wherein the volume depends on temperature. Using the resulting constitutive equation, some boundary value problems are studied, considering some relatively simple distributions for the temperature, deformations and stresses.

A numerical model for chemo-thermo-mechanical coupling at large strains with an application to thermoresponsive hydrogels

The aim of this work is the derivation and examination of a material model, accounting for large elastic deformations, coupled with species diffusion and thermal effects. This chemo-thermo-mechanical material model shows three key aspects regarding its numerical formulation. Firstly, a multiplicative split of the deformation gradient into a mechanical, a swelling and a thermal part. Secondly, temperature-scaled gradients for a numerical design comprising symmetric tangents and, thirdly, dissipation potentials for the modelling of dissipative effects. Additionally, the derived general material model is specialised to thermoresponsive hydrogels to study its predictive capabilities for a relevant example material class. An appropriate finite element formulation is established and its implementation discussed. Numerical examples are investigated, including phase transition and stability phenomena, to verify the ability of the derived chemo-thermo-mechanical material model to predict relevant physical effects properly. We compare our results to established models in the literature and discuss emerging deviations.

A novel 3D polygonal double-negative mechanical metamaterial with negative stiffness and negative Poisson’s ratio

Materials with negative stiffness (NS) and materials with negative Poisson’s ratio (NPR) are two fundamental classes of mechanical metamaterials. The previous researches of double-negative metamaterials with simultaneous negative stiffness and negative Poisson’s ratio are either limited to two dimensions or challenging to realize experimentally. Here, a three-dimensional polygonal double-negative (PDN) metamaterial is proposed, which is formed by the auxetic framework and NS dome contact pairs. The main principle is that auxetic behavior is realized by sliding mechanism, whereby the sliding parts convert the applied vertical load into the horizontal compression load of the NS dome units, thus generating the negative stiffness characteristics. The double-negative metamaterial with quadrilateral profile shape (QDN) is studied by theoretical analysis and FEA simulation, and the response of the inversed-stacked QDN structure is verified by experiments. Furthermore, double-negative metamaterials with quadrilateral, hexagonal and cylindrical profile shapes are presented and their displacement conversion efficiency and Poisson’s ratio are analyzed and compared. The 3D polygonal double-negative structure can be applied to energy absorption due to its frictional action and elastic large deformation.

Mathematical modeling and analysis of the tailor rolled blank manufacturing process

A tailor rolled blank (TRB) is a plate of continuously varying thickness with the advantages of lightweight, high strength, and good surface quality. However, it is required that the roll gap can be adjusted online, and the plate and roll produce large elastic deformation in the TRB manufacturing process. These characteristics make the modeling of the force and deformation parameters more difficult. A new mathematical model of roll separating force in the TRB manufacturing process is established taking into account the feature of deformation in this research. The elastic deformation of the strip and the flattening deformation of the roll are considered in order to improve the prediction accuracy of the results. The roll separating forces of the entrance elastic compression region and the exit elastic recovery region are computed using the generalized Hooke's law. A new velocity field is proposed based on the deformation characteristics of the plastic region and velocity boundary conditions, and the power functionals of the internal deformation, shear, friction, and tension are calculated. The analytical models of force and deformation parameters combined the elastic and plastic deformation regions are obtained through iterative operation, which satisfies the convergence condition of the roll separating force and roll flattening. The roll separating force data obtained by the model in this research are compared with the actual measured values in the laboratory, and the deviation value is small, which proves the computational precision of the present model. Additionally, the variations of the entrance velocity and flow volume per second during the TRB manufacturing process are shown, and the influences of the tension and friction factor on the neutral angle and roll separating force are illustrated.

Computational fluid–structure interaction framework for passive feathering and cambering in flapping insect wings

AbstractIn flapping insect wings, veins support flexible wing membranes such that the wings form feathering and cambering motions passively from large elastic deformations. These motions are essentially important in unsteady aerodynamics of insect flapping flight. Hence, the underlying mechanism of this phenomenon is an important issue in studies on insect flight. Systematic parametric studies on strong coupling between a model wing describing these elastic deformations and the surrounding fluid, which is a direct formulation of this phenomenon, will be effective for solving this issue. The purpose of this study is to develop a robust numerical framework for these systematic parametric studies. The proposed framework consists of two novel numerical methods: (1) A fully parallelized solution method using both algebraic splitting and semi‐implicit scheme for monolithic fluid–structure interaction (FSI) equation systems, which is numerically stable for a wide range of properties such as solid‐to‐fluid mass ratios and large body motions, and large elastic deformations. (2) A structural mechanics model for insect flapping wings using pixel modeling (pixel model wing), which is combined with explicit node‐positioning to reduce computational costs significantly in controlling fluid meshes. The validity of the proposed framework is demonstrated for some benchmark problems and a dynamically scaled model incorporating actual insect data. Finally, from a parametric study for the pixel model wing flapped in fluid with a wide range of solid‐to‐fluid mass ratios, we find a FSI mechanism of feathering and cambering motions in flapping insect wings.

A more explosion-resistant bamboo fiber reinforced composite structure through balancing bamboo anisotropy

Explosion-resistance of two kinds of bamboo composite plates with different laminated structures was investigated by in-field explosion experiments. It is revealed that both unidirectional parallel bamboo strand lumber (PBSL) plates and cross-laminated bamboo (CLB) plates can resist large dynamic elastic deformation and have good anti-explosion ability. Through balancing PBSLs’ anisotropy by cross-laminated structure, the CLB plate is far superior to the PBSL in explosion resistance. In this research, the critical scaled distance is between 1.293 m/kg1/3 and 1.086 m/kg1/3 for the PBSL, while it is significantly reduced and between 0.543 m/kg1/3 and 0.407 m/kg1/3 for the CLB. Matrix cracking penetrates the full thickness of the PBSL plate and attenuates its explosion resistance, while this shortcoming is overcome by the CLB whose main damage modes are fiber fracture and spalling. The CLB is a more explosion-resistance bamboo structure.

Dynamic Forced Performance of an O-Rings Sealed Squeeze Film Damper Lubricated With a Low Supply Pressure and a Simple Method to Quantify Air Ingestion

Abstract Contemporary squeeze film dampers (SFDs) in air-breathing engines are short in length to limit weight and part count and lubricated with a low feed pressure to reduce oil storage and pumping power. In SFDs, O-rings (ORs) restrict side leakage and increase the viscous damping while a adding a modest centering stiffness. Continuing a long-term project characterizing SFDs for aircraft engines and extending the original work (San Andrés and Rodríguez, 2021, “On the Experimental Dynamic Force Performance of a Squeeze Film Damper Supplied Through a Check Valve and Sealed With O-Rings,” ASME J. Eng. Gas Turb. Power, 143(11), p. 111011) the paper details measurements of the forced performance of an OR sealed damper (OR-SFD) with diameter D = 127 mm, land length L = 0.2 D, and radial clearance c = 0.0022 D. Lubricant ISO VG 2 supplied at 0.69 bar(g) fills an upstream oil plenum and flows into the middle of the land through a single orifice configured with a check valve. Measurements of applied single-frequency dynamic loads, along with the ensuing damper displacements and accelerations serve to identify the parameters of the test structure, ORs, and SFD. The tests encompass centered whirl motions with amplitude r = 0.05–0.45c, and a range of whirl frequencies, ω = 10–130 Hz. Note the squeeze velocity vs = rω reaches 102 mm/s. The ORs force coefficients are nearly invariant with frequency but do depend on the orbit amplitude. The ORs' stiffness (KOR) decreases by 75% as the motion amplitude increases, r → 0.45c, likely due to the large elastic deformations and slow recovery of the ORs material. For small amplitude motions (r/c = 0.05 and 0.10), the ORs damping coefficient (COR) is ∼10% of the overall coefficient for the lubricated system (CL), while for r/c > 0.25, COR ∼ 0.03CL. For small amplitudes of whirl (r ≤ 0.25c), the SFD experimental viscous damping (CSFD) and added mass (MSFD) coefficients, identified over a shorter frequency range (vs<30 mm/s) equal theoretical magnitudes for a fully sealed damper. As the orbit size grows to r = 0.45c MSFD drops nearly by 75% and CSFD decreases by ∼40% due to the onset of both lubricant cavitation and air ingestion occurring for squeeze velocities vs ≥ 24.5 mm/s, as also seen in the recorded dynamic pressures and video recordings of a bubbly mixture leaving the damper. A comprehensive flow model predicts CSFD and MSFD about 8% and 12% larger than the experimentally identified coefficients. A novel approach enables the estimation of the gas volume fraction (GVF) generated in the damper which rapidly increases as vs grows. The simple procedure draws into a deflated balloon the material contents in the squeeze film, weighs the sample, and identifies its volume to produce an estimate of the GVF. The procedure to quantify the GFV will assist in the validation of predictive tools.

Research on design and milling performance of unequal gradient helix and variable pitch end mill

It is difficult to obtain excellent surface quality and dimensional accuracy of the low-rigidity frame and beam parts made by titanium alloy, due to the obvious vibration and large elastic deformation during side milling. The variable helix and pitch cutting tool is usually used to reduce vibration, which does not consider the unequable stiffness in the milling area. Therefore, the concept of Unequal Gradient Helix and Variable Pitch (UGHVP) is proposed in this research. The helix angles of the UGHVP vary with the stiffness of the milling area position, which will lead to milling forces and vibration decreasing as the stiffness weakens. As a result, the surface quality and dimensional accuracy will be increased, of which the advantages are analyzed. The mathematical models of the UGHVP circumferential cutting edges and groove sections are derived, which provide theoretical support for tool grinding. The models are visualized by numerical simulation. Finally, the comparison experiment with the conventional variable helix and pitch end mill was completed by side-milling the designed feature samples. The results showed that the milling forces perpendicular to the milling surface by the UGHVP were smaller and less fluctuating, which could obtain less elastic deformation theoretically. The milling vibration was smoother, especially when cutting in and out. Besides, the larger the axial depth of cut, the more obvious the superior performance of the UGHVP tool. Better surface quality and dimensional accuracy were obtained, which proved the excellent performance of the UGHVP.