Finite Deformation Theory Research Articles

Chaotic motion of piezoelectric material hyperbolic shell under thermoelastic coupling

Piezoelectric material, which exhibits excellent electro-mechanical conversion properties, is widely used in smart sensors and structures for sonar systems, weather detection and remote sensing. Hyperbolic shell structure made of piezoelectric material is liable to break down when it is used in high temperature environment, which is caused by the unexpected chaotic dynamic motion under the coupling effect of thermal filed and force field. Therefore, the chaotic nonlinear dynamic vibration of simply-supported piezoelectric material hyperbolic shell is studied under the combined action of temperature field and simple harmonic excitation. Based on the theory of finite deformation, the non-linear vibration equation and coordination equation of the hyperbolic shell are established. The non-linear dynamic equation of the structure is obtained by the Bubnov-Galerkin principle. The corresponding undisturbed Hamilton system has a homoclinic orbit. Using Melnikov function, the chaotic motion condition of the dynamic system under the criterion of Smale-horseshoe transformation is obtained. Furthermore, the mathematical model is established by Simulink software and the numerical simulations are performed by the fourth-order Runge-Kutta method. The simulation results accord well with those from the Melnikov method. The bifurcation diagram, the Lyapunov exponent diagram, the phase diagram and Poincaré section diagram are acquired to analyze the influence of temperature field on the non-linear characteristic of piezoelectric material hyperbolic shell system. When the temperature is close to 32℃ and 41℃, the Lyapunov index is less than 0 and the corresponding movement of the system is in the periodic zone, which is the same as that for a temperature range from 36℃ to 37℃. When the Lyapunov index is greater than 0, the corresponding movement of the system is in chaos zone. Therefore, the change of temperature has an additional effect on the stiffness of the system which affects the vibration of the system. The chaos and periodic zones of the system alternate with the increase of temperature and the vibration characteristics of the system can be controlled by changing the temperature field. Therefore, adjusting the temperature field can control the motion state of the system, which helps to improve reliability of the structure.

Mechano-Electrochemical Analysis in Cylindrical Composition-Gradient Electrodes with Varying Young's Modulus of Lithium-Ion Battery

Composition-gradient electrode material (CGEM) is one of the most promising materials in lithium-ion battery, and this paper studies its performance from the stress analysis. The finite deformation theory and the stress-induced diffusion hypothesis are adopted to establish the constitutive equations, and the nonlinear influence of finite deformation is considered. The aim is to investigate diffusion-induced stresses (DISs) generated in a cylindrical composition-gradient electrode with the slope of −0.5 for Young's modulus E(R). Compared with stress distributions in a homogeneous electrode, CGEM is able to make the stress fields smaller and flatter and improve the state of charging (SOC). Then we change the elastic modulus of electrode materials from 10GPa to 150GPa gradually, and those advantages still exist. Finally, diffusion-induced buckling of cylindrical electrodes is analyzed and CGEMs make the cylindrical electrode more difficult to buckle. Therefore, many electrode materials can be made into CGEMs to prevent the electrode from cracking. The results can provide a theoretical guidance for the design of CGEMs.

A finite deformation theory of desolvation and swelling in partially photo-cross-linked polymer networks for 3D/4D printing applications.

Photopolymerization is a process strongly dependent on the light field in the resin. This typically results in a non-uniformly crosslinked network where some parts of the network are fully cross-linked while other parts are partially crosslinked. The partially crosslinked part could exhibit a high volume expansion upon swelling and a high volume shrinkage upon desolvation. Through control over the light field in the photopolymer resin, this feature has been used to create solvent responsive shape changing structures as well as 3D/4D printed smart devices, showing promising application potential. In this paper, we develop a finite deformation theory to consider the nonuniform crosslink density of the network and the interaction between different species inside the network. The mechanical properties of the network are correlated with the reaction process and the existence of residual uncrosslinked monomers is included in the partially crosslinked network. The efficiency of the theory is proved by the finite element simulations of two special applications of the partially crosslinked network.

Acoustic radiation pressure in laterally unconfined plane wave beams

The acoustic radiation pressure in laterally unconfined, plane wave beams in inviscid fluids is derived via the direct application of finite deformation theory for which an analytical accounting is made ab initio that the radiation pressure is established under static, laterally unconstrained conditions, while the acoustic wave that generates the radiation pressure propagates under dynamic (sinusoidal), laterally constrained conditions. The derivation reveals that the acoustic radiation pressure for laterally unconfined, plane waves along the propagation direction is equal to (3/4)〈2K〉, where is the mean kinetic energy density of the wave, and zero in directions normal to the propagation direction. The results hold for both Lagrangian and Eulerian coordinates. The value differs from the value traditionally used in the assessment of acoustic radiation pressure, obtained from the Langevin theory or from the momentum flux density in the Brillouin stress tensor. Errors in traditional derivations leading to the Brillouin stress tensor and the Langevin radiation pressure are pointed out and a long-standing misunderstanding of the relationship between Lagrangian and Eulerian quantities is corrected. The present theory predicts a power output from the transducer that is 4/3 times larger than that predicted from the Langevin theory. Tentative evidence for the validity of the present theory is provided from measurements previously reported in the literature, revealing the need for more accurate and precise measurements for experimental confirmation of the present theory.

Finite deformation theory of crystal plasticity for disclination density based on micropolar model

Finite deformation theory of crystal plasticity for disclination density based on micropolar model

Hydrostatic Pressure Effect on Metallic Glasses: Finite Deformation Theory

Metallic glass does not have well-defined atomic planes and the related dislocation motion during plastic deformation. The atomic displacement during deformation instead contains a large degree of randomness, which necessarily leads to volume change. However, how the volume change contributes to the mechanisms of mechanical deformation, especially the pressure sensitivity of yielding and fracture strength remains unclear. Although volume dilatation has been observed in various mechanical deformation, experimental investigation so far shows negligible pressure effect on yield and fracture strength. Here using finite deformation theory, we give a theoretical analysis of the pressure effect on metallic glasses subject to pure shear in the presence of a large range of applied hydrostatic pressure. We found that shear deformation does couple to applied pressure as manifested through the dependence of shear strength on applied hydrostatic pressure. We argue that the strong tendency of deformation localization and the omnipresent sample imperfections unique for amorphous solids may cause the discrepancy between the theoretical and experimental results.

Modeling of finite deformation of pseudoelastic NiTi shape memory alloy considering various inelasticity mechanisms

A crystal plasticity finite element (CPFE) model is developed for pseudoelastic NiTi shape memory alloy (SMA). The model includes various inelastic mechanisms of deformation such as martensite transformation, dislocation glide in austenite phase and twinning in the martensite phase. This model is based on the finite deformation theory and predicts phase transformation, stress field and residual strain due to slip and twinning. Following the work of Anand and Gurtin (2003), the Helmholtz free energy is derived and further the thermodynamic driving forces for slip, twin and phase transformation are obtained by using Clausius-Duhem inequality. The developed constitutive model has been implemented within a user material subroutine interface VUMAT in ABAQUS™/Explicit. The activation of different inelasticity mechanisms at different temperatures and strains are first verified by performing various simulations of uniaxial tensile deformation. Then the model is calibrated and validated against the experimental stress-strain response of NiTi single crystal as reported in the literature. The single crystal constitutive model is extended for modeling the response of a polycrystal with both initial random and textured crystallographic orientations using Taylor scale transition technique. A series of simulations are performed on polycrystalline representative volume element (RVE) at various temperatures and strains. The effect of temperature and imposed strain on phase transformation and residual strain is investigated systematically. The residual strain increases due to slip in austenite phase and twinning in the martensite phases that are activated as temperature and imposed strain increase respectively. Further the uniaxial loading conditions are extended to multiaxial loading and the results are compared against the experimental data.

Effect of Prestresses on Quasi-Lamb Waves in a System Consisting of a Compressible Viscous Liquid Half-Space and an Elastic Layer

The problem of propagation quasi-Lamb waves in a prestrained elastic layer interacting with a half-space of compressible viscous fluid is considered. The problem is solved using the three-dimensional linearized equations of theory of finite deformations for the elastic layer and the three-dimensional linearized Navier–Stokes equations for the compressible viscous fluid. A problem statement and approach based on the general solutions of linearized equations for the elastic body and fluid are used. The dispersion equations describing the propagation of quasi-Lamb waves in hydroelastic systems over wide range of frequencies are derived. The effect of the initial stresses and the thickness of the elastic layer and compressible viscous liquid half-space on the phase velocities and damping factors of quasi-Lamb modes are analyzed. The approach developed and the results obtained make it possible to establish the limits of applicability of the models for wave processes, based on different versions of the theory of small initial deformations, the classical theory of elasticity, and the model of ideal fluid. The numerical results are presented in the form of graphs and analyzed.

Size-dependent effects sensitively determine buckling of a cylindrical silicon electrode particle in a lithium-ion battery

The influence of size-dependent effects on mechanical behavior of a cylindrical silicon electrode particle within a lithium-ion battery is investigated, and it is shown that these effects sensitively determine critical buckling behavior. The mathematical framework used for this investigation incorporates the bond-order-length-strength (BOLS) theory correlation in a general finite deformation theory model that includes the two-way coupling between diffusion-induced stress and stress-influenced diffusion. Additionally, the possibility of plastic flow is accounted for in the model to allow probing situations with relatively high charging rates. Significant differences, arising due to the consideration of the size-dependent effects and captured through the BOLS theory, are highlighted. In particular, it is found that, in most cases studied, both stresses and plastic stretches are amplified in magnitude. Tensile to compressive stress reversal also takes place faster on considering the BOLS theory. The modeling framework is further applied to estimate critical lengths that are safe against buckling. Thus, it is expected that the presented framework will provide an improved aid for the design of nanowire-based lithium ion battery silicon electrodes.

Three-dimensional nonlinear bending analysis of FG-CNTs reinforced composite plates using the element-free Galerkin method based on the S-R decomposition theorem

A three-dimensional element-free Galerkin method (EFG) based on the Strain-Rotation (S-R) decomposition theorem, named 3D-SR-EFG, is developed to investigate the nonlinear bending behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. Due to its overcoming the deficiencies of classic finite deformation theories, S-R decomposition theorem can provide a reliable theoretical support for the geometrically nonlinear simulation. The incremental variational formulation based on the S-R decomposition theorem for three-dimensional static large deformation problems is derived from the updated co-moving coordinate formulation and principle of potential energy rate. Global weak-form EFG is adopted to obtain the discrete form of the formulation. Convergence and comparison studies are conducted to validate the numerical stability and accuracy of the proposed 3D-SR-EFG. The influences of volume fraction and distributions of CNTs, plate’s aspect ratio and width-to-thickness ratio, boundary conditions on the nonlinear bending response of the CNTRC plates are numerically analyzed and discussed in parametric studies. Results demonstrate that the 3D-SR-EFG can effectively predict the nonlinear bending behavior of the CNTRC plates. This work also further extends the applications of the S-R decomposition theorem.

Effect of Prestresses on Lamb Waves in a System Consisting of an Ideal Liquid Half-Space and an Elastic Layer

A problem is formulated for quasi-Lamb waves propagating in a prestrained elastic layer interacting with a half-space of compressible ideal fluid. The results are obtained using the three-dimensional equations of linearized theory of finite deformations for the elastic layer and the three-dimensional linearized Euler equations for the compressible ideal fluid. The problem statement and the approach are based on the general solutions of the linearized equations for elastic solid and fluid. The dispersion equations that describe the propagation of quasi-Lamb waves in hydroelastic systems over a wide frequency range are obtained. The effect of initial stresses and half-space of compressible ideal fluid and the thickness of the elastic layer on the phase velocities of quasi-Lamb modes is analyzed. The approach developed and the results obtained for wave processes allow establishing the limits of applicability of the models based on different versions of the theory of small initial deformations. The numerical results are presented in the form of graphs and are analyzed.

Buckling analysis and control of layered electrode structure at finite deformation

This work has proposed a coupled mechanochemical model to investigate the buckling performance of electrode and thereby enhance the mechanical integrity of lithium-ion batteries. This method based upon finite deformation theory is capable to simulate the state of charge (SOC) and path dependent buckling phenomenon of the layered structure. It is revealed that the buckling behavior of an electrochemical loaded electrode depends not only on the material properties and the electrode geometry but also on the charging/discharging process, SOC as well as the operation current rate. Moreover, with regarding to the lithiation stiffening active material, the results show that increasing the current density in lithiation while decreasing it upon delithiation can increase the relatively critical buckling load, and contribute the electrode to withstand buckling failure. As for the lithiation softening one, it is disclosed that the bucking resistance would be impaired in lithiation and recovered upon full delithiation. In addition, to eliminate the adverse effect of lithiation soften on the electrode buckling resistance, design insights into the current collector have been provided.

Distortion-gradient plasticity theory for an isotropic body in finite deformation

This study presents a distortion-gradient model for an isotropic plastically deformed solid within the framework of finite deformation theory. The work aims to provide an alternative form of the Gurtin and Anand (Int J Plast 21:2297–2318, 2005) finite deformation strain-gradient model with a view to relaxing the constraint of irrotationality of plastic flow. The kinematic gradient of deformation is assumed to admit the Kroner–Lee decomposition into elastic and plastic parts. The obtained microforce balance, constitutive relations and plastic flow rule are similar to that obtained by Gurtin and Anand but different in that the present theory used a codirectionality hypothesis to obtain thermodynamically consistent constitutive relations for the dissipative microstresses.

An Alternative Material Model Using a Generalized <i>J2</i> Finite-Strain Flow Plasticity Theory with Isotropic Hardening

In this paper an alternative material model using a generalized <i>J2</i> finite-strain flow plasticity theory with isotropic hardening is presented. The model is based on a new nonlinear continuum mechanical theory of finite deformations of elasto-plastic media which allows for the development of objective and thermodynamically consistent material models. As a result, the constitutive equation, the evolution equation and even the ‘normality rule’, characterising the plastic flow in the material during return mapping, can be expressed in various forms, using several instances of the yield surface and corresponding pairs of stress measures and strain rates, respectively, which are conjugate with respect to the internal mechanical power and its arbitrary higher order time derivative. Therefore the results of the material model when used in numerical analyses are not affected by the description and particularities of the material model formulation. Here, we briefly outline the nonlinear continuum theory along with a detailed description of the material model and finally present the model in a numerical example using a cross-shaped specimen in biaxial tension.

Experimental and Numerical Analysis of Titanium Alloy Microtube Tube-End Nosing Forming

This study is mainly based on five sets of mold cone angle and friction coefficient of micro-tube tube end necking forming analysis, and the tool cone angle of 60° experimental verification is carried out to analyze the titanium alloy (Grade 1) micro-tube for different mold cone angle and the different friction coefficient caused by the difference between the shrinkage forming. In this paper, Prandtl-Reuss's plastic flow rule, combined with finite element deformation theory and updated Lagrangian formulation (ULF) concept, establish an incremental elasto-plastic finite element analysis program for simulating the miniature tube end necking. The forming process also uses the generalized rmin algorithm to deal with elasto-plastic state and contact problems. From the simulation data of necking process, deformation history, punch load and punch stroke, stress and strain distribution is obtained. The analysis results show that by increasing the mold cone angle and friction coefficient, the thickness tends to be thicker in the certain area.

A finite deformation theory for the climbing habits and attachment of twining plants

Twining plants, using their special attachment structure to cling to a support and grow upward to capture more sunshine without losing stability, have fascinated many scholars. A mechanics model to interpret this natural selection phenomenon based on the finite deformation theory is developed here. Results show the pitch angle of the twiner, the elongation in the stem and the radius ratio of the twiner to its support play an important role in the twining habit. To get more sunshine overhead, twiners tend to increase the initial pitch angle but to decrease the final twining angle to maintain their positions and stability. By taking into account of the tuning of the interfacial strength with the microstructure, the maximum values of the elongation and the support radius are determined, over which twiners will lose their ability to wind around. The proposed model and the results can not only clarify the mechanism of climbing habits and attachment, but also show the potential applications for the twining-mimic designs.

Architected multi-directional functionally graded cellular plates

Inspired by natural materials like bamboo and the dentin-enamel junction of tooth, the concept of architected multi-directional functionally graded cellular materials (FGCMs) is introduced. These architected porous materials are made by assembling porous unit cells of dissimilar densities and cell topologies. To evaluate the potential of multi-directional FGCMs for improving the performance of lightweight structural elements, the mechanical properties of functionally graded cellular (FGC) plates are analyzed. For the numerical analyses, standard mechanics homogenization is used to predict the mechanical properties of cells with arbitrary superellipse voids. The homogenized effective properties along with finite element method and shear deformation theory are exploited to predict the mechanical responses of plates made of multi-directional FGCMs. Numerical results reveal substantial improvement in structural responses when FGCMs are appropriately used; e.g. 56% improvement in bending stiffness is found in an FGC rectangular plate compared to a cellular plate with the same weight and with uniform distribution of constitutive cells. Numerical results show that cell variation through the thickness of FGC plates is more effective on the structural responses than variation through the length or width. Finally, multi-objective optimization is implemented to show the maximum improvement achievable via architecture variation within FGC structures.

Finite element approximation of the fields of bulk and interfacial line defects

A generalized disclination (g.disclination) theory (Acharya and Fressengeas, 2015) has been recently introduced that goes beyond treating standard translational and rotational Volterra defects in a continuously distributed defects approach; it is capable of treating the kinematics and dynamics of terminating lines of elastic strain and rotation discontinuities. In this work, a numerical method is developed to solve for the stress and distortion fields of g.disclination systems. Problems of small and finite deformation theory are considered. The fields of a single disclination, a single dislocation treated as a disclination dipole, a tilt grain boundary, a misfitting grain boundary with disconnections, a through twin boundary, a terminating twin boundary, a through grain boundary, a star disclination/penta-twin, a disclination loop (with twist and wedge segments), and a plate, a lenticular, and a needle inclusion are approximated. It is demonstrated that while the far-field topological identity of a dislocation of appropriate strength and a disclination-dipole plus a slip dislocation comprising a disconnection are the same, the latter microstructure is energetically favorable. This underscores the complementary importance of all of topology, geometry, and energetics in understanding defect mechanics. It is established that finite element approximations of fields of interfacial and bulk line defects can be achieved in a systematic and routine manner, thus contributing to the study of intricate defect microstructures in the scientific understanding and predictive design of materials. Our work also represents one systematic way of studying the interaction of (g.)disclinations and dislocations as topological defects, a subject of considerable subtlety and conceptual importance (Aharoni et al., 2017; Mermin, 1979).

A theoretical resistance-curve based on nonproportional plastic strain

Using an exact solution for an elliptical hole in a perfectly plastic material, an expression is derived for the resistance of a ductile material undergoing subcritical crack propagation in the plane. This resistance curve is based on an analogy to the J-integral where an energy dissipation rate is determined rather than an energy release rate. The Tresca yield condition under plane stress loading conditions is employed in this derivation as well as finite deformation theory. This resistance curve is applicable to the initial stage of subcritical crack growth for a ductile material subject to crack tip blunting.

Voltage-driven torsion of electroactive thick tubes reinforced with helical fibers

This work studies the voltage-driven torsional deformation of a thick-walled circular cylindrical tube composed of a dielectric elastomer (DE) reinforced with a family of helical fibers. The Helmholtz free energy of the DE composite tube involves separate DE matrix and fiber contributions. Upon the application of an electrical voltage difference on its inner and outer surfaces, the DE matrix contracts through the thickness and expands in cylindrical surfaces due to the Maxwell stress. Since the composite is stiffened along the fiber direction, it preferentially expands in the direction orthogonal to the fiber within the cylindrical surfaces. As a result, voltage-driven torsional deformation will occur in the helical fiber-reinforced DE tube. Although membrane theory was usually adopted to model fiber-reinforced thin DE membranes, the finite deformation theory is utilized in the present work to model relatively thick DE composite tubes. It is found that the bending effect plays an important role in the voltage-driven torsional deformation. The reorientation of the fibers and the distribution of the stress carried by the fibers in the tubes are discussed in detail so as to provide an interpretation to some new voltage-driven torsional behaviors. The theoretical model and physical insights provided in this work will aid the design and optimization of soft electroactive actuators and soft robotics.