Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.

An efficient Galerkin meshfree formulation for shear deformable beam under finite deformation

This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam.The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection,large deformation as well as finite rotation.The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation.The Lagrangian meshfree shape function is utilized to discretize the variational equation.Subsequently to resolve the shear and membrane locking issues and accelerate the computation,the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation.Numerical results reveal that the present formulation is very effective.

New finite strain elastoplastic equations for accurately and explicitly simulating pseudoelastic-to-plastic transition effects of shape memory alloys

A new finite strain elatoplastic J2-flow model with coupling effects of both isotropic and anisotropic hardening is proposed with the co-rotational logarithmic rate.In terms of certain single-variable shape functions representing uniaxial loading and unloading curves,explicit multi-axial expressions for the three hardening quantities incorporated in the new model proposed are derived in unified forms for the purpose of automatically and accurately simulating complex pseudoelastic-to-plastic transition effects of shape memory alloys(SMAs)under multiple loading-unloading cycles.Numerical examples show that with only a single parameter of direct physical meaning for each cycle,accurate and explicit simulations may be achieved for extensive data from multiple cycle tests.

On Numerical Modelling of Industrial Powder Compaction Processes for Large Deformation of Endochronic Plasticity at Finite Strains

Compaction processes are one the most important par ts of powder forming technology. The main applications are focused on pieces for a utomotive, aeronautic, electric and electronic industries. The main goals of the compaction processes are to obtain a compact with the geometrical requirements, without cracks, and with a uniform distribution of density. Design of such proc esses consist, essentially, in determine the sequence and relative displacements of die and punches in order to achieve such goals. A.B. Khoei presented a gener al framework for the finite element simulation of powder forming processes based on the following aspects; a large displacement formulation, centred on a total and updated Lagrangian formulation; an adaptive finite element strategy based on error estimates and automatic remeshing techniques; a cap model based on a hard ening rule in modelling of the highly non-linear behaviour of material; and the use of an efficient contact algorithm in the context of an interface element fo rmulation. In these references, the non-linear behaviour of powder was adequately desc ribed by the cap plasticity model. However, it suffers from a serious deficiency when the stress-point reaches a yield surface. In the flow theory of plasticit y, the transition from an elastic state to an elasto-plastic state appears more or less abruptly. For powder material it is very difficult to define the locati on of yield surface, because there is no distinct transition from elastic to ela stic-plastic behaviour. Results of experimental test on some hard met al powder show that the plastic effects were begun immediately upon loading. In such mater ials the domain of the yield surface would collapse to a point, so making the di rection of plastic increment indeterminate, because all directions are normal to a point. Thus, the classical plasticity theory cannot deal with such materials and an advanced constitutive theory is necessary. In the present paper, the constitutive equations of powder materials will be discussed via an endochronic theory of plasticity. This theory provides a unifi ed point of view to describe the elastic-plastic behaviour of material since it places no requirement for a yield surface and a ’loading function’ to disting uish between loading an unloading. Endochronic theory of plasticity has been app lied to a number of metallic materials, concrete and sand, but to the knowledge of authors, no numerical scheme of the model has been applied to powder material . In the present paper, a new approach is developed based on an endochronic rate independent, density-dependent plasticity model for describing the isothermal deformation behavior of metal powder at low homologous temperature. Although the concept of yield surface has not been explicitly assumed in endochronic theory, it is shown that the cone-cap plasticity yield surface (Fig.1), which is the m ost commonly used plasticity models for describing the behavior of powder materi al can be easily derived as a special case of the proposed endochronic theory. Fig.1 Trace of cone-cap yield function on the meridian pl ane for different relative density As large deformation is observed in powder compaction process, a hypoelastic-pl astic formulation is developed in the context of finite deformation plasticity. Constitutive equations are stated in unrotated frame of reference that greatly s implifies endochronic constitutive relation in finite plasticity. Constitutive e quations of the endochronic theory and their numerical integration are establish ed and procedures for determining material parameters of the model are demonstra ted. Finally, the numerical schemes are examined for efficiency in the model ling of a tip shaped component, as shown in Fig.2. Fig.2 A shaped tip component. a) Geometry, boundary conditio n and finite element mesh; b) density distribution at final stage of

A Modified Symmetric and Antisymmetric Decomposition-Based Three-Dimensional Numerical Manifold Method for Finite Elastic-Plastic Deformations

There are relatively few studies on large rotation or deformation by means of the three-dimensional(3D)numerical manifold method(NMM).A new modified symmetric and antisymmetric decomposition(MSAD)theory is developed and implemented into the 3D NMM,eliminating the false-volume expansion and false-rotation strain/stress problems.The Jaumann rate is used to measure the material rotation,and the geometric stiffness built on the Jaumann rate is deduced.The incremental formulas of the MSAD-based 3D NMM and a practical guide on the implementation of the MSAD theory are given in detail and exemplified.The new theory and formulas can be applied to analyze both large rotation and large deformation problems.Based on the hypoelasto-plasticity theory and the unified strength theory,the unified yield criterion with associated flow rule is implemented into the MSAD-based 3D NMM.Several typical examples are studied,showing the advantage and potential of the new MSAD theory and the MSAD-based 3D NMM.

Mechanics of nonbuckling interconnects with prestrain for stretchable electronics

The performance of the flexibility and stretchability of flexible electronics depends on the mechanical structure design,for which a great progress has been made in past years.The use of prestrain in the substrate,causing the compression of the transferred interconnects,can provide high elastic stretchability.Recently,the nonbuckling interconnects have been designed,where thick bar replaces thin ribbon layout to yield scissor-like in-plane deformation instead of in-or out-of-plane buckling modes.The nonbuckling interconnect design achieves significantly enhanced stretchability.However,combined use of prestrain and nonbuckling interconnects has not been explored.This paper aims to study the mechanical behavior of nonbuckling interconnects bonded to the prestrained substrate analytically and numerically.It is found that larger prestrain,longer straight segment,and smaller arc radius yield smaller strain in the interconnects.On the other hand,larger prestrain can also cause larger strain in the interconnects after releasing the prestrain.Therefore,the optimization of the prestrain needs to be found to achieve favorable stretchability.

Electromechanical responses and instability of electro-active polymer cylindrical shells

Based on the theories of finite deformation elasticity,electromechanical responses and instability of an incompressible electro-active polymer(EAP) cylindrical shell,which is subjected to an internal pressure and a static electric field,are studied.Deformation curves and distribution of stresses are obtained.It is found that an internal pressure together with an electric field may cause the unstable non-monotonic deformation of the shell.It is also shown that a critical thickness for the shell exists,and the shell may undergo the unstable deformation if its thickness is less than this critical value.In addition,the effects of the electric field,axial stretch,thickness,and internal pressure on the instability of the shell are discussed.

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 structureproperty relationships of the soft/stiff suture joints and gives a foundational guidance of mechanical design towards high-performance bioinspired composites.

Large deformation plasticity From basic relations to finite deformation

The theory of plasticity as a special field of continuum mechanics deals with the irreversible,i.e.permanent,deformation of solids.Under the action of given loads or deformations,the state of the stresses and strains or the strain rates in these bodies is described.In this way,it complements the theory of elasticity for the reversible behavior of solids.In practice,it has been observed that many materials behave elastically up to a certain load(yield point),beyond that load,however,increasingly plastic or liquid-like.The combination of these two material properties is known as elastoplasticity.The classical elastoplastic material behavior is assumed to be time-independent or rate-independent.In contrast,we call a time-or rate-dependent behavior visco-elastoplastic and visco-plastic—if the elastic part of the deformation is neglected.In plasticity theory,because of the given loads the states of the state variables stress,strain and temperature as well as their changes are described.For this purpose,the observed phenomena are introduced and put into mathematical relationships.The constitutive relations describing the specific material behavior are finally embedded in the fundamental relations of continuum theory and physics.Historically,the theory of plasticity was introduced in order to better estimate the strength of constructions.An analysis based purely on elastic codes is not in a position to do this,and can occasionally even lead to incorrect interpretations.On the other hand,the entire field of forming techniques requires a theory for the description of plastic behavior.Starting from the classical description of plastic behavior with small deformations,the present review is intended to provide an insight into the state of the art when taking into account finite deformations.

Improved Carroll’s hyperelastic model considering compressibility and its finite element implementation

In engineering component design,material models are increasingly used in finite element simulations for an expeditious and less costly analysis of the design prototypes.As such,researchers strive to formulate models that are less complex,robust,and accurate.In the realm of hyperelastic materials,phenomenological-based Carroll’s model is highly promising due to its simplicity and accuracy.This work proposes its further improvement by modifying the strain energy density function to comply with the restriction that it should vanish at reference configuration and adding a compressible term to capture the practical behavior of elastomeric materials and to avoid numerical problems during finite element implementation.The model constants for both the original and the modified versions were obtained by fitting their respective expressions to the classical Treloar’s experimental data using the Levenberg-Marquardt algorithm.The modified model was implemented using Abaqus CAE 2016 via a vectorized user material(VUMAT)subroutine.Comparisons of the model predictions with Treloar’s experimental data demonstrated the superiority of the modified version particularly in the equibiaxial loading mode.Moreover,the simulated and the experimentally observed behavior agreed to a great accuracy thus making the modified model suitable for simulating the loading response of components fabricated of elastomeric materials.

An Elasto-plastic Damage Constitutive Theory Based on Pair Functional Potentials and Slip Mechanism

The deformation work rate can be expressed by the time rate of pair functional potentials which describe the energy of materials in terms of atomic bonds and atom embedding interactions.According to Cauchy-Born rule,the relations between the microscopic deformations of atomic bonds and electron gas and macroscopic deformation are established.Further,atomic bonds are grouped according to their directions,and atomic bonds in the same direction are simplified as a spring-bundle component.Atom embedding interactions in unit reference volume are simplified as a cubage component.Consequently,a material model composed of spring-bundle components and a cubage component is established.Since the essence of damage is the decrease and loss of atomic bonding forces,the damage effect can be reflected by the response functions of these two kinds of components.Formulating the mechanical responses of two kinds of components,the corresponding elasto-damage constitutive equations are derived.Considering that slip is the main plastic deformation mechanism of polycrystalline metals,the slip systems of crystal are extended to polycrystalline,and the slip components are proposed to describe the plastic deformation.Based on the decomposition of deformation gradient and combining the plastic response with the elasto-damage one,the elasto-plastic damage constitutive equations are derived.As a result,a material model formulated with spring-bundle components,a cubage component and slip components is established.Different from phenomenological constitutive theories,the mechanical property of materials depends on the property of components rather than that directly obtained on the representative volume element.The effect of finite deformation is taken into account in this model.Parameter calibration procedure and the basic characteristics of this model are discussed.

A review on material models for isotropic hyperelasticity

Dozens of hyperelastic models have been formulated and have been extremely handy in understanding the complex mechanical behavior of materials that exhibit hyperelastic behavior(characterized by large nonlinear elastic deformations that are completely recoverable)such as elastomers,polymers,and even biological tis-sues.These models are indispensable in the design of complex engineering com-ponents such as engine mounts and structural bearings in the automotive and aerospace industries and vibration isolators and shock absorbers in mechanical systems.Particularly,the problem of vibration control in mechanical system dy-namics is extremely important and,therefore,knowledge of accurate hyperelastic models facilitates optimum designs and the development of three‐dimensional finite element system dynamics for studying the large and nonlinear deformation beha-vior.This review work intends to enhance the knowledge of 15 of the most com-monly used hyperelastic models and consequently help design engineers and scientists make informed decisions on the right ones to use.For each of the models,expressions for the strain‐energy function and the Cauchy stress for both arbitrary loading assuming compressibility and each of the three loading modes(uniaxial tension,equibiaxial tension,and pure shear)assuming incompressibility are pro-vided.Furthermore,the stress–strain or stress–stretch plots of the model's pre-dictions in each of the loading modes are compared with that of the classical experimental data of Treloar and the coefficient of determination is utilized as a measure of the model's predictive ability.Lastly,a ranking scheme is proposed based on the model's ability to predict each of the loading modes with minimum deviations and the overall coefficient of determination.

Mechanics of molecular beacons for mRNA detecting

Decoding genetic information is crucial for gene therapy and cancer diagnosis,which has attracted growing interest in the field of clinical medicine and life science.In this study,we conducted a comprehensive exploration to obtain the detection mechanism of molecular beacons from a mechanics point of view.The potential energy function of molecular beacon/target system is established firstly,based on which the profile of molecular beacons is solved by genetic algorithm optimization.The length of stem and the total energy are further calculated when the target is hybridized with loop and stem.The results show that the hybridization between target and stem is energetically favorable compared with that between target and loop,indicating a new detection strategy.These analyses may cast light on understanding the mechanism of molecular beacons detection,and further help to design novel molecular beacons with high resolution and quantification.

The Non-local Effects Induced by Rapid Transient Mass Diffusion in a Spherical Silicon Electrode of Lithium-ion Batteries

For the development of the fast charging technology of lithium-ion batteries,describing the rapid transient mass diffusion process accurately is the premise to avert the mechanical degradation caused by the diffusion-induced stress.In this paper,we present a diffusion-elasticity model based on the finite deformation theory with the consideration of nonlocal effects of mass transfer.The proposed model is then applied to analyze the evolution and distribution of stress and lithium concentration in a silicon electrode during the rapid charging process under potentiostatic operation.The cases with and without the contribution of non-local diffusion effects are both calculated for comparative analysis.The dependence of the influence of non-local effects on diffusion relaxation time r〇is discussed.The results show that a diffusive wave appears when the value of r〇is sufficiently large,and it moves inside gradually during the rapid charging process,which is qualitatively consistent with the results from literature.