Finite Deformation Kinematics Research Articles

Dynamic fracture with a continuum‐kinematics‐based peridynamic and a phase‐field approach

AbstractThe notion of dynamic fracture with continuum‐kinematics‐based peridynamics is presented in this work. A geometrically precise version of peridynamics called continuum‐kinematics‐based peridynamics adds surface‐ or volume‐based interactions to the traditional peridynamic bonds, accurately capturing the finite deformation kinematics. The point families produced from the horizon of the material points are used to construct the surfaces and volumes taken into account for these non‐local interactions.In continuum kinematics‐based peridynamics, the traditional bond‐stretch damage technique is insufficient for fracture. Due to the loss of strength in the internal force densities of the material points, it is now extended to the surface‐ and volume‐based interactions by new failure factors. Numerical examples demonstrate that the proposed approach effectively manages crack propagation, impact damage, and spontaneous crack initiation under dynamic loading circumstances with large deformations. When the results are compared to phase‐field calculations, there is a remarkable agreement concerning the damage patterns.

Effect of microstructure on the dynamic behavior of Ultra-High-Molecular-Weight Polyethylene (UHMWPE) composites

We study the effects of composite microstructure on the dynamic response of Ultra-High Molecular Weight Polyethylene (UHMWPE) fiber-reinforced composites when loaded in out-of-plane compression. Using a Kolsky bar, we dynamically load the composites (of differing microstructures) in out-of-plane compression while simultaneously capturing the deformation history with high speed video with μs temporal resolution and approximately 10μm spatial resolution. Image analysis is performed on the resulting high speed video to obtain the finite deformation kinematics of the specimen, including the dynamic lateral deformations, and obtain the complete deformation gradient history during the tests. Using a finite deformation formulation, we measure the evolution of the volumetric deformations and relate this to porosity evolution within the specimen. Specimens are also tested at quasistatic strain rates and their response is compared in the range of strain-rates from 10−3s−1 to 3000s−1. These materials are quite compressible, and their volume can change very rapidly during out-of-plane compression. The volume change is accommodated by a reduction in porosity at low to moderate strains. In the uniaxial stress samples, high strains lead to volume change by ejection of material, and the evolution of material ejection is rate dependent. In both microstructures, deformation at the higher strain rate led to material ejection being initiated at higher stress, ejection evolving more slowly as a function of applied stress, and developing a plateau at a lower stress (than in the quasistatic case). At quasistatic strain-rates, there was little difference between the response of the two microstructures; however at high strain rates of 103s−1, the microstructure with lower porosity displayed a higher strength. The rate sensitivity of the strength was shown to depend on the microstructure, suggesting the defect distribution plays a significant role in the dynamic strength of these materials.

Dynamic fracture with continuum-kinematics-based peridynamics

<abstract><p>This contribution presents a concept to dynamic fracture with continuum-kinematics-based peridynamics. Continuum-kinematics-based peridynamics is a geometrically exact formulation of peridynamics, which adds surface- or volume-based interactions to the classical peridynamic bonds, thus capturing the finite deformation kinematics correctly. The surfaces and volumes considered for these non-local interactions are constructed using the point families derived from the material points' horizon. For fracture, the classical bond-stretch damage approach is not sufficient in continuum-kinematics-based peridynamics. Therefore it is here extended to the surface- and volume-based interactions by additional failure variables considering the loss of strength in the material points' internal force densities. By numerical examples, it is shown that the presented approach can correctly handle crack growth, impact damage, and spontaneous crack initiation under dynamic loading conditions with large deformations.</p></abstract>

A finite-deformation constitutive model of particle-binder composites incorporating yield-surface-free plasticity

Abstract We present a constitutive model for particle-binder composites that accounts for finite-deformation kinematics, nonlinear elasto-plasticity without apparent yield, cyclic hysteresis, and progressive stress-softening before the attainment of stable cyclic response. The model is based on deformation mechanisms experimentally observed during quasi-static monotonic and cyclic compression of mock Plastic-Bonded Explosives (PBX) at large strain. An additive decomposition of strain energy into elastic and inelastic parts is assumed, where the elastic response is modeled using Ogden hyperelasticity while the inelastic response is described using yield-surface-free endochronic plasticity based on the concepts of internal variables and of evolution or rate equations. Stress-softening is modeled using two approaches; a discontinuous isotropic damage model to appropriately describe the softening in the overall loading-unloading response, and a material scale function to describe the progressive cyclic softening until cyclic stabilization. A nonlinear multivariate optimization procedure is developed to estimate the elasto-plastic model parameters from nominal stress-strain experimental compression data. Finally, a correlation between model parameters and the unique stress-strain response of mock PBX specimens with differing concentrations of aluminum is identified, thus establishing a relationship between model parameters and material composition.

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.

A variational phase-field model For ductile fracture with coalescence dissipation

A novel phase-field model for ductile fracture is presented. The model is developed within a consistent variational framework in the context of finite-deformation kinematics. A novel coalescence dissipation introduces a new coupling mechanism between plasticity and fracture by degrading the fracture toughness as the equivalent plastic strain increases. The proposed model is compared with a recent alternative where plasticity and fracture are strongly coupled. Several representative numerical examples motivate specific modeling choices. In particular, a linear crack geometric function provides an “unperturbed” ductile response prior to crack initiation, and Lorentz-type degradation functions ensure that the critical fracture strength remains independent of the phase-field regularization length. In addition, the response of the model is demonstrated to converge with a vanishing phase-field regularization length. The model is then applied to calibrate and simulate a three-point bending experiment of an aluminum alloy specimen with a complex geometry. The effect of the proposed coalescence dissipation coupling on simulations of the experiment is first investigated in a two-dimensional plane strain setting. The calibrated model is then applied to a three-dimensional calculation, where the calculated load-deflection curves and the crack trajectory show excellent agreement with experimental observations. Finally, the model is applied to simulate crack nucleation and growth in a specimen from a recent Sandia Fracture Challenge.

A Mechanism‐Based Model for the Impact Response of Quartz

AbstractThe mechanical response of geomaterials to extreme dynamic conditions is essential to understanding planetary impacts, other geological and seismic events, and underground explosions. In this study, we present a multimechanism constitutive model to describe the dynamic response of polycrystalline quartz (e.g., nonporous quartzite), and implement it using a large scale computational analysis to simulate impact events. The active deformation mechanisms within quartz under dynamic loading consist of amorphization, fracture and granular flow. All of these mechanisms are integrated within the model in terms of their competition, evolution, and interaction during a complex loading history. A generalized amorphization model is introduced within a thermodynamically consistent framework on the basis of earlier work in Zeng et al. (2019a, https://doi.org/10.1016/j.jmps.2019.06.012), and accounts for the kinematics of finite deformation inside amorphization bands and the distinct physical properties of the amorphous phase. Fracture in the form of microcracking emanating from internal defects is also considered. Once the material is fully fragmented, the behavior transitions to granular flow. After determining the model parameters for polycrystalline quartz, we illustrate the model response using a representative volume element, where the competition of mechanisms is revealed under several typical loadings. Plate impact experiments and dynamic compression on a polycrystalline quartz foil are then simulated. The simulation results are compared with experiments to prove the predictive capability of the model in capturing the key features of quartz during impact events.

An extended eight-chain model for hyperelastic and finite viscoelastic response of rubberlike materials: Theory, experiments and numerical aspects

Rubberlike materials exhibit strong rate-dependent mechanical response which manifests itself in creep and relaxation tests as well as in the hysteresis curves under cyclic loading. Unlike linear viscoelasticity, creep and relaxation response of rubber is nonlinear and amplitude-dependent. Within this context, the contribution of this work is three fold. (i) On the experimental side, the characterization of equilibrium and non-equilibrium responses are carried out by means of uniaxial and equibiaxial extension tests. Also performed are the creep and relaxation experiments at various stress and strain levels. (ii) On the theoretical side, we extend the well-known eight-chain model via incorporating a simple yet instrumental tube-constraint term composed of the second invariant into the non-affine network contribution reflecting the ground state equilibrium response. For the non-equilibrium response, we propose a new evolution equation for the creep/relaxation behavior of rubberlike materials based on a relaxation kinetics of a single polymer chain. The geometric non-linearity is incorporated via the finite deformation kinematics based on the multiplicative split of the deformation gradient into elastic and viscous parts, whereas the volumetric and isochoric split of the deformation gradient is entirely discarded. The rheology uses a nonlinear spring responsible for equilibrium elastic response in parallel to n number of Maxwell elements, leading to a generalized Maxwell-Wiechert viscoelastic solid. (iii) The algorithmic implementation of the model features the spectral decomposition of the respective terms and is demonstrated within the context of the finite element method. The developed model is validated by fitting both the elastic and viscoelastic model responses with respect to the experimental data in the sense of uniaxial, (equi)biaxial extensions, and pure shear tests. Relaxation and creep behavior of the model are thoroughly assessed. Also presented in the manuscript is the capability and the performance of the model in the face of a relevant non-homogeneous boundary value problem. The quality of the findings earns the model vast utilization areas from an engineering perspective.

Serial deformation maps and elasto-plastic continua

This paper deals with the kinematics of finite plastic deformation. The starting point is E.H. Lee’s multiplicative decomposition of the deformation gradient into its elastic and plastic parts. We diverge from Lee’s (and others’) derivation of the deformation rate and spin tensors and prove that the elastic and plastic deformation rate and spin tensors are additive. It is then shown that the elastic and plastic strains are also additive and equal to the total strain. A central result, and a major deviation from continuum mechanics, is that not one, but two deformation maps are necessary for the constitutive response in elasto-plasticity. The two-map decomposition is then extended to the n-th order, and additivity is shown there as well. Elastoplastic constitutive equations in large deformation are presented and discussed in the context of path-dependent (endochronic) plasticity.

A continuum mechanics constitutive framework for transverse isotropic soft tissues

In this work, a continuum constitutive framework for the mechanical modelling of soft tissues that incorporates strain rate and temperature dependencies as well as the transverse isotropy arising from fibres embedded into a soft matrix is developed. The constitutive formulation is based on a Helmholtz free energy function decoupled into the contribution of a viscous-hyperelastic matrix and the contribution of fibres introducing dispersion dependent transverse isotropy. The proposed framework considers finite deformation kinematics, is thermodynamically consistent and allows for the particularisation of the energy potentials and flow equations of each constitutive branch. In this regard, the approach developed herein provides the basis on which specific constitutive models can be potentially formulated for a wide variety of soft tissues. To illustrate this versatility, the constitutive framework is particularised here for animal and human white matter and skin, for which constitutive models are provided. In both cases, different energy functions are considered: Neo-Hookean, Gent and Ogden. Finally, the ability of the approach at capturing the experimental behaviour of the two soft tissues is confirmed.

Self-induced parametric amplification in ring resonating gyroscopes

We investigate self-induced parametric amplification that arises from dispersive nonlinear coupling between degenerate modes in systems with circular symmetry that rotate about the axis of symmetry. This phenomenon was first observed in micro-electromechanical ring/disk gyroscopes, where it provided enhanced readout gain using purely passive nonlinear effects [Nitzan et al. [22], 2015]. The goal of this investigation is to provide a fundamental description of this phenomenon, which is an example where nonlinear dynamics can improve the performance of a practical device. To describe this behavior, we consider the in-plane vibrations of a thin ring surrounded by electrodes that rotates about its symmetry axis at a rate Ω much smaller than its vibration frequencies ωn, as is the case in applications. The focus is on the pair of degenerate elliptical modes, one of which is taken as the drive mode and the other as the sense mode for the sensor. These modes are coupled through both inertial (Coriolis) and geometric nonlinear effects, as described by general forms of the kinetic and potential energies that account for finite deformation kinematics, as well as electrostatic effects. We investigate the specific effects of this coupling on the system performance and its sensitivity when used as a sensor for the spin rate. Specifically, we show that drive mode vibrations with sufficiently high amplitude affect the sense mode dynamic behavior in the form of parametric pumping, which leads to a considerable amplification of the sense mode response. As this response amplitude is proportional to Ω, it results in a substantial increase of the gyroscope sensitivity with respect to the external angular rate. We also illustrate that the effects of the sense mode vibrations on the drive mode dynamics can be neglected in the model when Ω/ωn≪1. Finally, we illustrate the applicability of our results by considering the dynamic response of a representative MEMS gyroscope model and quantifying the predicted benefits of these nonlinear effects.

Numerical implementation of a crystal plasticity model with dislocation transport for high strain rate applications

This paper details a numerical implementation of a single crystal plasticity model with dislocation transport for high strain rate applications. Our primary motivation for developing the model is to study the influence of dislocation transport and conservation on the mesoscale response of metallic crystals under extreme thermo-mechanical loading conditions (e.g. shocks). To this end we have developed a single crystal plasticity theory (Luscher et al (2015)) that incorporates finite deformation kinematics, internal stress fields caused by the presence of geometrically necessary dislocation gradients, advection equations to model dislocation density transport and conservation, and constitutive equations appropriate for shock loading (equation of state, drag-limited dislocation velocity, etc). In the following, we outline a coupled finite element–finite volume framework for implementing the model physics, and demonstrate its capabilities in simulating the response of a [1 0 0] copper single crystal during a plate impact test. Additionally, we explore the effect of varying certain model parameters (e.g. mesh density, finite volume update scheme) on the simulation results. Our results demonstrate that the model performs as intended and establishes a baseline of understanding that can be leveraged as we extend the model to incorporate additional and/or refined physics and move toward a multi-dimensional implementation.

Isogeometric collocation for large deformation elasticity and frictional contact problems

Isogeometric collocation methods have been recently proposed as an alternative to standard Galerkin approaches as they provide a significant reduction in computational cost for higher-order discretizations . In this work, we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems. We first derive the non-linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization . Some numerical examples demonstrate the performance of collocation in its basic and enhanced versions, differing by the enforcement of Neumann boundary conditions . For problems with strong singularities , enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to Galerkin, whereas for weaker or no singularities enhanced and basic collocation may give very similar results. A large deformation contact formulation is subsequently developed and tested in the frictional setting, where collocation confirms the excellent performance already obtained for the frictionless case. Finally, it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three-dimensional setting with arbitrarily inclined non-matching discretizations, thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints.

Frequency division using a micromechanical resonance cascade

A coupled micromechanical resonator array demonstrates a mechanical realization of multi-stage frequency division. The mechanical structure consists of a set of N sequentially perpendicular microbeams that are connected by relatively weak elastic elements such that the system vibration modes are localized to individual microbeams and have natural frequencies with ratios close to 1:2:⋯:2N. Conservative (passive) nonlinear inter-modal coupling provides the required energy transfer between modes and is achieved by finite deformation kinematics. When the highest frequency beam is excited, this arrangement promotes a cascade of subharmonic resonances that achieve frequency division of 2j at microbeam j for j = 1, …, N. Results are shown for a capacitively driven three-stage divider in which an input signal of 824 kHz is passively divided through three modal stages, producing signals at 412 kHz, 206 kHz, and 103 kHz. The system modes are characterized and used to delineate the range of AC input voltages and frequencies over which the cascade occurs. This narrow band frequency divider has simple design rules that are scalable to higher frequencies and can be extended to a larger number of modal stages.

Thermomechanical constitutive modelling of shape memory polymer including continuum functional and mechanical damage effects

A multi-mechanism-based phenomenological model is developed within the finite deformation kinematics framework for capturing the thermomechanical behaviour of shape memory polymers (SMPs) both during programming and in service. Particularly, the damage mechanisms in SMPs are studied within the continuum damage mechanics (CDMs) framework in which they are classified into mechanical or physical damage, induced during service condition, e.g. fatigue and functional damage induced during thermomechanical cycles, e.g. shape recovery loss. Statistical mechanics is incorporated to describe the initiation and saturation of these deformation mechanisms. The main advantage of the presented viscoplastic model, comparing to the existing counterparts, is its simplicity by minimizing the need for curve fitting, and capability in simulating the nonlinear stress–strain behaviour of amorphous, crystalline or semicrystalline SMPs. The developed viscoplastic CDM model takes into account several distinctive deformation mechanisms involved in the thermomechanical cycle of SMPs, including glass transition loss events, temperature-dependent material properties, stress relaxation, shape recovery transient events and damage effects. The established model correlates well with the experimental results and its computational capabilities provide material designers with a powerful design tool for future SMP applications.

Formulation of Lucas–Kanade Digital Image Correlation Algorithms for Non‐contact Deformation Measurements: A Review

ABSTRACTDigital image correlation (DIC) metrology has been increasingly used in a wide range of experimental mechanics research and applications. The DIC algorithm used so far is however limited mostly to the classic forward additive Lucas–Kanade type. In this paper, a survey is given about the formulation of other types of Lucas–Kanade DIC algorithms that have been appeared in computer vision, robotics, medical image analysis literature and so on. Concise notations consistent with the finite deformation kinematics analysis in continuum mechanics are used to describe all Lucas–Kanade DIC algorithms. An intermediate image is introduced as a frame of reference to clarify the so‐called compositional algorithms in a two‐frame DIC analysis. Explicit examples about the additive and compositional updating of deformation parameters are given for affine deformation mapping. Extensions of these algorithms to the so‐called consistent or symmetric types are also presented. The equivalency of final numerical solutions using additive, compositional and inverse compositional algorithms is shown analytically for the case of affine deformation mapping. In particular, the inverse compositional algorithm for affine image subset deformation is highlighted for its superior computational efficiency. While computationally less efficient, consistent and symmetric algorithms may be more robust and less biased and their potentials in experimental mechanics applications remain to be explored. The unified formulation of these Lucas–Kanade DIC algorithms collected all together in this paper can serve as a useful guide for researchers in experimental mechanics to further evaluate the merits as well as limitations of these non‐classic algorithms for image‐based precision displacement measurement applications.

Partitioning of elastic energy in open-cell foams under finite deformations

The challenges associated with the computational modeling and simulation of solid foams are threefold—namely, the proper representation of an intricate geometry, the capability to accurately describe large deformations, and the extremely arduous numerical detection and enforcement of self-contact during crushing. The focus of this study is to assess and accurately quantify the effects of geometric nonlinearities (i.e. finite deformations, work produced under buckling-type motions) on the predicted mechanical response of open-cell foams of aluminum and polyurethane prior to the onset of plasticity and contact. Beam elements endowed with three-dimensional finite deformation kinematics are used to represent the foam ligaments. Ligament cross-sections are discretized through a fiber-based formulation that provides accurate information regarding the onset of plasticity, given the uniaxial yield stress–strain data for the bulk material. It is shown that the (hyper-) elastic energy partition within ligaments is significantly influenced by kinematic nonlinearities, which frequently cause strong coupling between the axial, bending, shear and torsional deformation modes. This deformation mode-coupling is uniquely obtained as a result of evaluating equilibrium in the deformed configuration, and is undetectable when small deformations are assumed. The relationship between the foam topology and energy partitioning at various stages of moderate deformation is also investigated. Coupled deformation modes are shown to play an important role, especially in perturbed Kelvin structures where over 70% of the energy is stored in coupled axial-shear and axial-bending modes. The results from this study indicate that it may not always be possible to accurately simulate the onset of plasticity (and the response beyond this regime) if finite deformation kinematics are neglected.

Interface-reaction controlled diffusion in binary solids with applications to lithiation of silicon in lithium-ion batteries

Solid state diffusion in a binary system, such as lithiation into crystalline silicon, often involves two symbiotic processes, namely, interfacial chemical reaction and bulk diffusion. Building upon our earlier work (Cui et al., 2012b, J. Mech. Phys. Solids, 60 (7), 1280–1295), we develop a mathematical framework in this study to investigate the interaction between bulk diffusion and interfacial chemical reaction in binary systems. The new model accounts for finite deformation kinematics and stress–diffusion interaction. It is applicable to arbitrary shape of the phase interface. As an example, the model is used to study the lithiation of a spherical silicon particle. It is found that a dimensionless parameter β=kfeVmBR0/D0 plays a significant role in determining the kinetics of the lithiation process. This parameter, analogous to the Biot number in heat transfer, represents the ratio of the rate of interfacial chemical reaction and the rate of bulk diffusion. Smaller β means slower interfacial reaction, which would result in higher and more uniform concentration of lithium in the lithiated region. Furthermore, for a given β, the lithiation process is always controlled by the interfacial chemical reaction initially, until sufficient silicon has been lithiated so that the diffusion distance for lithium reaches a threshold value, beyond which bulk diffusion becomes the slower process and controls the overall lithiation kinetics.

Viscoplasticity analysis of semicrystalline polymers: A multiscale approach within micromechanics framework

The crystalline segment of semicrystalline polymers can undergo phase separation and creates distinguishable microscale crystalline and amorphous domains. In general severe mechanical and thermal loading conditions may change the amorphous, crystalline and morphological textures of these semicrystalline polymers. Due to the fact that the mechanical responses of these individual components, i.e. crystalline and amorphous polymers, are well-studied in the literature, a multiscale analysis can effectively correlate these micro-constituents properties to the macroscopic mechanical responses of the semicrystalline structures. Although the multiscale Finite Element Analysis (FEA) of the real microstructures encounters with some difficulties, such as computational costs, the micromechanics approach compensate for the FEA limitations. The micromechanics framework correlates the macroscopic mechanical responses to the microscale constitutive behaviors by either analytical or numerical methods. In this study a multiscale theory is developed within the micromechanics framework which links the microscale and macroscale constitutive behaviors of the semicrystalline polymers and also it accounts for the texture updates. While in the developed multiscale approach the crystal plasticity can effectively describe the inelastic responses of the crystalline sub-phase, a novel viscoplastic constitutive relation for the amorphous glassy polymers is developed to enhance the performance of the multiscale approach. The proposed amorphous viscoplastic model minimizes the numbers of material parameters, and it significantly facilitates the calibration process. Also the developed constitutive relations for the amorphous polymers provide the mathematical competency to capture a wide range of experimental results. A reformation of the Transformation Field Analysis (TFA), developed by Dvorak (1990, 1992) and Dvorak and Benveniste (1992), is then presented where the two-phase TFA solution is generalized for capturing local inelastic responses of semicrystalline polymers. Its performance is then examined for the case of finite deformation kinematics. Also the reported TFA deficiency in overestimating the inelastic responses, e.g. Chaboche et al. (2001), is addressed herein in which the elasto-plastic stiffness together with some phenomenological material parameters are incorporated into the TFA formulation to soften its mechanical responses. The multiscale approach and the proposed viscoplastic model provide good correlation with the experimental results.

Effects of high-order surface stress on buckling and resonance behavior of nanowires

A theoretical framework, accounting for high-order surface stress, is implemented in a continuum mechanics model based on the Euler-Bernoulli theory of beams and columns to simulate the buckling and resonance behavior of nanowires (NWs). Closed-form expressions for the critical buckling load of uniaxial compression of NWs are derived for different types of end conditions. Size-dependent overall Young’s modulus is characterized versus the diameter of the NWs and is compared with the experimental data. The resonance frequency of NWs is also studied and compared with the simulation results based on nonlinear, finite deformation kinematics. We demonstrate that the present prediction considering both surface moment and surface stress agrees well with the experimental data, while the pure surface stress model may not be able to capture the general trend when the NW’s diameter is less than a certain size. We conclude that the present continuum mechanics approach, considering both high-order surface effects, could be served as one of the feasible tools to analyze the mechanical behavior of nanostructures.