Finite Deformation Framework Research Articles

Predicting buckling regimes during first lithiation of a crystalline silicon cylindrical anode particle: Influence of size and charging rate

A mathematical model for buckling analysis during the first lithiation of cylindrical crystalline silicon (c-Si) anode particles is presented using the finite deformation framework. A reaction–diffusion model that incorporates a reversible alloying–dealloying reaction (ADR) captures the lithiation in the expanding amorphous zone, while an addition reaction models the dynamics of the crystalline-amorphous silicon interface. A “movable” lithium part and an “immovable” lithium part make up the full lithium. In sharp contrast to the case of amorphous silicon (a-Si), the axial force evolution in c-Si is found to show two different peaks during the initial and final stages of lithiation, respectively. These peaks form the basis of two different buckling criteria, which, in turn, are shown to be sensitively influenced by the influx of lithium and a crucial size-dependent parameter. This framework gives an insight towards investigating the buckling phenomena for lithiation with an evolving amorphous silicon zone due to the interface movement, which is still absent in the present literature. Additionally, the present model enables us to obtain a quantitative and mechanistic insight into and capture the previous experimental observation that the mechanical stability of a cylindrical silicon particle may be improved by maintaining a crystalline silicon core inside the amorphous shell through controlled first lithiation. It is shown that such a crystalline-amorphous core–shell structure can be advantageous to more complicated designs involving carbon fiber as the core. It is thus hoped that this model will contribute towards improving the theoretical underpinnings of an improved design of mechanically robust electrodes for next-generation batteries.

A thermodynamically consistent anisotropic damage model for metallic glasses

AbstractThe recent success in manufacturing large‐size, also called bulk metallic glasses (BMGs) using 3D‐printing based on laser powder bed fusion (LPBF) opens an avenue for the broad application of this material class. To explore the great potential of both as‐cast and 3D‐printed BMGs, a comprehensive understanding and an accurate prediction of the plastic deformation and damage behaviour of this material class are indispensable. In this study, we develop a thermodynamically consistent anisotropic damage model incorporating tension‐compression asymmetry (TCA) to describe the unique inelastic deformation and damage behaviours of metallic glasses. The widely observed normal stress sensitivity and plastic dilatancy in metallic glasses are considered by using an extended Mohr‐Coulomb criterion in the constitutive description. Furthermore, a second‐order damage tensor is adopted for describing the anisotropic damage behaviour. By augmenting the Helmholtz free energy function to be dependent on the gradient of the free volume concentration and the gradient of the nonlocal damage parameter, the governing equations for the corresponding internal variables are derived within the framework of finite deformation. The damage localisation and mesh dependency are correspondingly alleviated. The simulation result shows that the shear band patterns are in good agreement with the experimental observations from literature.

Stiffness and toughness of soft/stiff suture joints in biological composites

Biological composites can overcome the conflict between strength and toughness to achieve unprecedented mechanical properties in engineering materials. The suture joint, as a kind of heterogeneous architecture widely existing in biological tissues, is crucial to connect dissimilar components and to attain a tradeoff of all-sided functional performances. Therefore, the suture joints have attracted many researchers to theoretically investigate their mechanical response. However, most of the previous models focus on the sutural interface between two chemically similar stiff phases with (or without) a thin adhesive layer, which are under the framework of linear elasticity and small deformation. Here, a general model based on the finite deformation framework is proposed to explore the stiffness and toughness of chemically dissimilar suture joints connecting soft and stiff phases. Uniaxial tension tests are conducted to investigate the tensile response of the suture joints, and finite element simulations are implemented to explore the underlying mechanisms, considering both material nonlinearity and cohesive properties of the interface. Two failure modes are quantitively captured by our model. The stored elastic energy in the soft phase competes with the energy dissipation due to the interface debonding, which controls the transition among different failure modes. The toughness of the suture joints depends on not only the intrinsic strengths of the constituent materials and their cohesive strength, but also the interfacial geometry. This work provides the structure-property relationships of the soft/stiff suture joints and gives a foundational guidance of mechanical design towards high-performance bioinspired composites.

A plate theory for inflatable panels

An inflatable panel is an airtight membrane structure whose envelope has two parallel flat sides when pressurized. It acquires load-carrying capacity due to its internal pressure. While pressurized tubes have been studied extensively over the past decades, panels were less investigated. In this work, a model of inflatable panels including shear effects is proposed. The nonlinear equations of motion are derived from the principle of virtual power within the framework of finite deformations and exhibit the essential pressure terms that allow for the correct prediction of the panel behavior. The equations are then linearized around the pre-stressed reference configuration and solved for a simply-supported circular panel subjected to a uniform load. The solution is found to be in good agreement with 3D finite element results.

Elastoplastic model for chemo-mechanical behavior of porous electrodes using image-based microstructure

The microstructure of electrodes is intrinsically complex and thus should be determined prior to the analysis of the chemo-mechanical performance during charge/discharge cycles. In this study, a microstructurally resolved, fully coupled chemo-mechanical model was developed to investigate the structure–property relationship of electrodes in the framework of elastoplastic finite deformation. The active particles, binder, and pores of real electrode images were segmented and reconstructed using the region-growing method. The effects of the electrode microstructure on the diffusion-induced stress (DIS) distribution, local electric potential and electrical resistance, interaction between the binder and particles, and mechanical failure of the particle–binder interface (PBI) were elucidated considering the finite concentration effect, concentration-dependent modulus and yield strength. The results indicate that the microstructure of electrodes significantly influences the distribution of DIS, local electric potential, and electrical resistance. The elastoplastic finite deformation theory can accurately describe the stress singularities that occur near sharply concave regions. Additionally, the use of a relatively soft binder was found to accommodate the expansion of active particles and mitigate the effects of particle interaction. Furthermore, being able to ensure that the active particles maintain a simple convex shape can reduce the likelihood of particle pulverization and PBI debonding. These results will enhance the understanding of the evolution of stress generation, the local electric potential and electrical resistance, and the mechanical failure of the PBI within the microstructure of porous electrodes; the results also emphasize the need for an optimized microstructure design for porous electrodes.

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.

Generalized Theory for DISes in a Large Deformed Solid

Analysis of diffusion-induced stress in the electrode materials of metal-ion batteries, especially in silicon-anode lithium-ion battery with large deformation, involves an in-depth understanding of the interaction between atomic migration and matrix volume expansion. Under large migration velocity of solute atoms in a host matrix, such chemo-mechanical coupling can result in the damage or structural degradation of the matrix (electrode), leading to capacity fading. Incorporating the effects of concentration changing rate on strain energy in Helmholtz free energy and the finite deformation framework, a generalized theory has been established to solve stresses in an elasto-plastic electrode induced by lithiation. Numerical calculations of lithiation-induced stresses are performed in an amorphous nano-spherical silicon electrode by using the free-volume-based constitutive relation of amorphous materials and the generalized theory. We analyze the effects of the volumetric expansion coefficient associate with solute atoms and the volumetric expansion coefficient associate with the concentration changing rate of solute atoms on the stress evolution. The results reveal that the concentration changing rate of solute atoms likely accelerates surface cracking of electrode materials during lithiation.

Buckling of an elastic layer based on implicit constitution: Incremental theory and numerical framework

A general class of implicit bodies was proposed to describe elastic response of solids, which contains the Cauchy–Green tensor as a function of Cauchy stress. Here, we consider the buckling of solids described by such a subclass of implicit constitutive relation. We present a general linear incremental theory and carry out bifurcation analysis of a uniaxially compressed rectangular layer described by an implicit constitution. We then provide general governing equations regarding the mixed unknowns, i.e., displacement and stress fields, within the framework of finite strain deformation. Thereby, the combination of Asymptotic Numerical Method (ANM) and spectral collocation method is applied to solve the resulting nonlinear equations. We validate our numerical framework by comparing the computational results with analytical ones, and then explore effects of width-to-length ratio and material nonlinearity on the buckling and post-buckling behavior. The larger the width-to-length ratio is, the larger the critical buckling load is. The material parameter, η, has a significant impact up to 28% on the critical buckling load and influence up to 50% on the post-buckling behavior. Our model based on the implicit constitutive relation well predicts the buckling and post-buckling of Gum metal alloy.

Effect of Degradation and Osteoarthritis on the Viscoelastic Properties of Human Knee Articular Cartilage: An Experimental Study and Constitutive Modeling

Articular cartilage, as a hydrated soft tissue which covers diarthrodial joints, has a pivotal role in the musculoskeletal system. Osteoarthritis is the most common degenerative disease that affects most individuals over the age of 55. This disease affects the elasticity, lubrication mechanism, damping function, and energy absorption capability of articular cartilage. In order to investigate the effect of osteoarthritis on the performance of articular cartilage, the mechanical behavior of human knee articular cartilage was experimentally investigated. Progressive cyclic deformation was applied beyond the physiological range to facilitate degradation of the tissue. The relaxation response of the damaged tissue was modeled by means of a fractional-order nonlinear viscoelastic model in the framework of finite deformations. It is shown that the proposed fractional model well reproduces the tissue’s mechanical behavior using a low number of parameters. Alteration of the model parameters was also investigated throughout the progression of tissue damage. This helps predict the mechanical behavior of the osteoarthritic tissue based on the level of previous damage. It is concluded that, with progression of osteoarthritis, the articular cartilage loses its viscoelastic properties such as damping and energy absorption capacity. This is also accompanied by a loss of stiffness which deteriorates more rapidly than viscosity does throughout the evolution of tissue damage. These results are thought to be significant in better understanding the degradation of articular cartilage and the progression of OA, as well as in the design of artificial articular cartilages.

A finite deformation framework for mechanism-based constitutive models of the dynamic behavior of brittle materials

A finite deformation mechanism-based thermodynamically consistent constitutive framework is presented for describing the dynamic behaviors of brittle materials under impact loading. The framework is developed based upon a multiplicative decomposition of the deformation gradient in terms of multiple mechanisms, including recoverable elasticity, crack-induced damage, and other inelastic mechanisms such as subgrain and granular plasticity. The finite deformation kinematics that captures the multiple mechanisms is structured within a thermodynamically consistent framework, and the consequent coupling of the various mechanisms is articulated. Specific constitutive equations are formulated for a Mie–Grüneisen equation of state, micromechanics-based dynamic-fracture-induced damage growth, subgrain or lattice plasticity for slip and other deformation modes, and granular plasticity for granular flow and pore collapse post-fragmentation. Using hot-pressed silicon carbide (SiC-N) as the model material, this integrative model is calibrated using available experiments that interrogate specific mechanisms. The effects of loading rate, the influence of confinement, and the path-dependent constitutive behaviors of the material predicted by the model are demonstrated. The model performance at the application scale is then evaluated by simulating previously performed sphere-on-cylinder impact experiments.

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.

Granular micromechanics‐based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations

AbstractAlthough the primacy and utility of higher‐gradient theories are being increasingly accepted, values of second gradient elastic parameters are not widely available due to lack of generally applicable methodologies. In this paper, we present such values for a second‐gradient continuum. These values are obtained in the framework of finite deformations using granular micromechanics assumptions for materials that have granular textures at some ‘microscopic’ scale. The presented approach utilizes so‐called Piola's ansatz for discrete‐continuum identification. As a fundamental quantity of this approach, an objective relative displacement between grain‐pairs is obtained and deformation energy of grain‐pair is defined in terms of this measure. Expressions for elastic constants of a macroscopically linear second gradient continuum are obtained in terms of the micro‐scale grain‐pair parameters. Finally, the main result is that the same coefficients, both in the 2D and in the 3D cases, have been identified in terms of Young's modulus, of Poisson's ratio and of a microstructural length.

Improving predictions of amorphous-crystalline silicon interface velocity through alloying-dealloying reactions in lithium-ion battery anode particles

An improved mathematical model for the first lithiation of crystalline silicon is presented in the finite deformation framework. The crystalline-amorphous silicon interface kinetics is modelled through an addition reaction while the lithiation in the growing amorphous zone is captured through a reaction–diffusion model, incorporating a reversible alloying-dealloying reaction (ADR). The entire lithium is divided into two parts: a “movable” lithium part and an “immovable” lithium part. Through a parametric analysis for the different possible combinations of alloying and dealloying reaction constants, appropriate values are obtained that reproduce experimentally observed interface velocity. Thus, a key result of our study is the improvement of previous theoretical predictions through the incorporation of ADR. Comparative results are shown with and without ADR. Importantly, with ADR, a higher state of charge is obtained while the interface moves slower compared to the case without ADR. The non-uniformity in the lithium concentration distribution, which is a major criterion of generating diffusion-induced stress, is represented through the variance of concentraton. With ADR, this variance is initially higher but this trend reverses as the interface moves forward. Corresponding effects are observed for the stresses as well. It is expected that the improved predictions from this model will contribute towards better structural design of next-generation lithium-ion battery electrodes.

Mean‐field homogenization based constitutive modeling of austenitic TRIP‐steels at the single crystal scale

AbstractIn order to accurately describe the effective mechanical behavior of single‐crystalline, austenitic TRIP‐steels, a two‐phase mean‐field homogenization scheme is presented in a finite deformation framework, which also includes transformation‐induced eigendeformations. The constitutive behavior of the two phases, namely austenite and martensite, is described by a rate‐independent single‐crystal plasticity model that captures elastic anisotropy as well as anisotropic slip and hardening behavior. The response of the two‐phase model under homogeneous deformation states is investigated and a particle strengthening effect is predicted in the absence of transformation‐induced eigendeformations and a suitable choice of material constants, while upon the inclusion of eigendeformations a gradual stress softening is observed.

Investigation of the Hall-Petch Relationship Using Strain Gradient Plasticity Model for Finite Deformation Framework

The Hall-Petch relationship in metals is investigated using the strain gradient plasticity theory within the finite deformation framework. For this purpose, the thermodynamically consistent constitutive formulation for the coupled thermomechanical gradient-enhanced plasticity model is developed. The corresponding finite element method is performed to investigate the characteristics of the Hall-Petch relationship in metals. The proposed model is established based on an extra Helmholtz-type partial differential equation, and the nonlocal quantity is calculated in a coupled method based on the equilibrium conditions. An excellent agreement between the simulation results and the test data is resulted in the Hall-Petch graph. Furthermore, it is observed that the Hall-Petch constants do not remain unchanged but vary with the strain level.

A microstructural-based approach to model magneto-viscoelastic materials at finite strains

Magneto-active polymers (MAPs) consist of a polymeric matrix filled with magnetisable particles. MAPs may change their mechanical properties (i.e., stiffness) and/or mechanical deformation upon the application of an external magnetic stimulus. Mechanical responses of MAPs can be understood as the combined contributions of both polymeric matrix and magnetic particles. Moreover, the magnetic response is defined by the interaction between magnetisable particles and the external field. Common approaches to model MAPs are based on phenomenological continuum models, which are able to predict their magneto-mechanical behaviour but sometimes failed to illustrate specific features of the underlying physics. To better understand the magneto-mechanical responses of MAPs and guide their design and manufacturing processes, this contribution presents a novel continuum constitutive model originated from a microstructural basis. The model is formulated within a finite deformation framework and accounts for viscous (rate) dependences and magneto-mechanical coupling. After the formulations, the model is calibrated with a set of experimental data. The model is validated with a wide range of experimental data that show its predictability. Such a microstructurally-motivated finite strain model will help in designing MAPs with complex three-dimensional microstructures.

Rheological model for rock-type materials under large deformations

In this paper the behaviour of rock-type materials is described with respect to the current configuration through rate-type constitutive equations, within the finite deformation framework, using the relative motion. The viscoplastic part of the constitutive equations concerns the long-time effects, i.e. the deformation by creep, while the instantaneous elastic response characterizes the short-time effect. The instantaneous elastic response expresses an appropriate objective time derivative of Cauchy stress tensor via the rate of strain, with the aid of fourth-order elastic stiffness tensor. Using mathematical arguments it follows that our new model extends, to finite strains, Cristescu’s rheological model Cristescu (1989)[2]. To solve boundary value problems for rock-type materials described by the proposed model, the appropriate variational equality for the velocity field, at a time t, is derived starting from the weak formulation of the incremental equilibrium problem. The rate form of the constitutive model is coupled with the variational equality, and serves to update the current state of the material.

Concentration dependent properties and plastic deformation facilitate instability of the solid-electrolyte interphase in Li-ion batteries

Lithium-ion batteries (LIBs) often suffer from capacity fading and poor cyclic performance due to mechanical degradation of the solid-electrolyte interphase (SEI). Here we perform numerical simulations and theoretical analysis to elucidate the role of plasticity in wrinkling and ratcheting behaviors of an SEI/electrode system. A coupled diffusion and finite deformation framework is formulated and numerically implemented as a user-element subroutine (UEL) to describe transient lithium diffusion and accompanying elastic–viscoplastic deformation of the electrode. It is found that concentration dependent properties and plastic deformation facilitate wrinkling in such a system. A wrinkled morphology may further lead to ratcheting and related failure under cycling. A phase diagram of four types of cyclic behaviorsis identified in terms of the charging rate and time. Our analysis suggests several potential strategies to avoid wrinkling and ratcheting instabilities, such as charging/discharging the electrode at a sufficiently slow rate, and/or introducing a thick artificial SEI with a pre-tension.

Mechanism and Model of a Soft Robot for Head Stabilization in Cancer Radiation Therapy.

We present a parallel robot mechanism and the constitutive laws that govern the deformation of its constituent soft actuators. Our ultimate goal is the real-time motion-correction of a patient's head deviation from a target pose where the soft actuators control the position of the patient's cranial region on a treatment machine. We describe the mechanism, derive the stress-strain constitutive laws for the individual actuators and the inverse kinematics that prescribes a given deformation, and then present simulation results that validate our mathematical formulation. Our results demonstrate deformations consistent with our radially symmetric displacement formulation under a finite elastic deformation framework.

Strain gradient finite element model for finite deformation theory: size effects and shear bands

In this work, a thermodynamically consistent constitutive formulation for the coupled thermomechanical strain gradient plasticity theory is developed in the context of the finite deformation framework. A corresponding finite element solution is presented to investigate the microstructural features of metallic volumes. The developed model is established based on an extra Helmholtz-type partial differential equation, and the nonlocal quantity is calculated in a coupled method based on the equilibrium conditions. This approach is well known for its computational strength, however, it is also commonly accepted that it cannot capture the size effect phenomenon observed in the micro-/nanoscale experiments during hardening. In order to resolve this issue, a modified strain gradient approach which can capture the size effects under the finite deformation is constructed in this work. The shear problem is then solved to carry out the feasibility study of the developed model on the size effect phenomenon. Lastly, a plane strain problem under uniaxial tensile loading with shear bands is examined to perform the mesh sensitivity tests of the model during softening.