Energy Density Function Research Articles

Remarks on bifurcation of an inflated and extended swellable isotropic tube

We consider a tubular membrane whose mechanical behavior is defined by a strain energy density function that combines the neo-Hookean model with the Demiray model. Both, the neo-Hookean and the Demiray models are isotropic material models, which found their application in the modeling of the mechanical behavior of biological soft tissue. This tubular membrane is subjected to an inner pressure, an axial stretch, and a change in the material volume due to swelling or deswelling. The interplay between the cylinder geometry and the loading conditions and different instability modes, namely, bulging bifurcation, bending bifurcation, and prismatic bifurcation, are studied. It is shown that a change in the material volume has a strong effect on the occurrence of these bifurcation modes because a change in the material volume may stabilize the cylinder against a particular bifurcation mode and may trigger another bifurcation mode at the same time. Despite the restriction to isotropic material behavior, this article shows that the material response to pressure, axial stretch, and material volume change is quite complex.

Thermodynamics and Ferroelectric Properties of Pb1-xSrxTiO3 Solid Solutions

Pb1-xSrxTiO3 solid solutions have been utilized to compositionally tune the phase transitions and ferroelectric properties and thus are attractive for potential applications in tunable microwave devices and pyroelectric energy harvesters. However, a reliable thermodynamic description of the Pb1-xSrxTiO3 system is currently lacking. Here, we establish a thermodynamic free energy density function for the Pb1-xSrxTiO3 solid solution system taking into account both the polarization and octahedral tilt, and validated it by reproducing the experimentally measured polarization, lattice parameters, and dielectric constant values. Based on the thermodynamic model, we successfully explain the experimentally observed dielectric anomalies by analyzing the interactions between ferroelectric and octahedral tilt order parameters and constructed a comprehensive temperature-composition Pb1-xSrxTiO3 phase diagram which can be used to guide future experimental synthesis and thin film growth. The thermodynamic model developed in this work can be employed not only to predict the ferroelectric and structural states and their properties for compositions that have not yet been synthesized but also used as input to model the formation of mesoscale domain structures using the phase-field method.

The nonlinear theory of magnetoelasticity and the role of the Maxwell stress: a review

This paper reviews macroscopic aspects of the theory of magnetoelastostatics, starting with a brief summary of the experimental and theoretical contributions leading to the development of the current state-of-the-art. It offers some different perspectives than hitherto, with incompressible materials being the main concern. The use of the so-called total energy (density) functions is highlighted along with their associated total stress tensors and succinct forms of the constitutive equations. The symmetry of the total Cauchy stress tensor, which incorporates the non-symmetric Maxwell stress within the material, is emphasized and it is noted that the use of such a Maxwell stress, often appearing in the literature, is thereby avoided. The theory is illustrated for some simple prototype boundary-value problems, specifically the homogeneous deformation of an infinite slab of magnetoelastic material in the presence of a magnetic field and the non-homogeneous extension and inflation of an infinitely long circular cylindrical tube in the presence of either an axial or a circumferential magnetic field.

New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers

A nonlinear finite element formulation for large deformation analysis of hyperelastic membranes in Total Lagrangian framework is developed. In many cases, it is convenient to utilize the invariant-based strain energy density function to analyze the structures when employing the finite element method. However, in this contribution, both invariant-based and stretch-based formulations are developed. Formulations are completely general and not restricted to axisymmetric loading and geometry of membrane. Furthermore, initially flat and non-flat membrane can be investigated by this method. Incompressible as well as compressible materials can be modeled. A method for imposing the plane stress condition to the formulation for compressible materials is presented. Moreover, isotropic and anisotropic materials can be considered. Some problems are analyzed and results compared to the available ones in the literature. The comparison between solution based on invariant-based method and stretch-based method with exact fourth-order elasticity tensor obtained in this work demonstrates that the stretch-based method derived in this paper works very well. The presented examples demonstrate that the developed formulations have no restriction or simplification with respect to the modeled object geometry and inherent anisotropy.

Wave propagation in generalized thermo-microstretch elastic solid containing voids

A linear theory of generalized thermo-microstretch elastic solid containing voids is formulated. Lord and Shulman [2] theory of thermoelasticity is employed to incorporate thermal effects. Free energy density function is constructed to develop the constitutive relations and field equations for an isotropic homogeneous generalized thermo-microstretch elastic solid containing voids. The possibility of propagation of plane waves is investigated in the medium of infinite extent. It is found that there may exist four sets of coupled longitudinal waves, two sets of coupled transverse waves and an independent longitudinal microrotational wave traveling with distinct speeds. Each set of coupled longitudinal waves is found to be attenuating and dispersive in nature, while an independent longitudinal microrotational wave and the remaining two sets of coupled transverse waves are found to be dispersive but non-attenuating in nature. All the possible waves are influenced by the polar property of the medium; however, all the coupled longitudinal waves are influenced by the stretch, voids and thermal properties of the medium. It is also found that all coupled longitudinal waves exist for all non-negative frequencies, while the independent longitudinal microrotational wave and one of the sets of coupled transverse waves exist only after certain cutoff frequency.

An experimentally validated anisotropic visco-hyperelastic constitutive model for knitted fabric-reinforced elastomer composites

A new anisotropic visco-hyperelastic constitutive model has been developed to predict the time-dependent deformation response of flexible, knitted fabric-reinforced elastomer composites. These composites exhibit a time-dependent response due to the viscoelastic behavior of the elastomer, while the anisotropic reinforcement effect is attributed to the knitted fabrics. The proposed modeling approach features an invariant-based strain energy density function that incorporates the architecture of knitted fabrics and the hyper-viscoelastic behavior of elastomers. In this study, the elastomer is considered as a viscous neo-Hookean material, and the knitted fabrics are modeled as cords with negligible stiffness in bending. The proposed material model is implemented in commercial Finite Element Analysis (FEA) software, Abaqus (version 2017), through a user-defined material subroutine, UMAT. This approach enables the determination of the constitutive behavior of the composite based on the constituent elastomer, fiber properties, and fabric architectures. The proposed model is verified through a discrete 3D FEA model, in which the elastomer and fabric reinforcement are modeled explicitly. The approach is further validated through physical testing including uniaxial tension, torsion, and three-point bending. Since the proposed strain energy-based constitutive model can predict the deformation response of knitted fabric-reinforced elastomer composites based on the fabric architecture, it can be employed as an efficient modeling tool to aid the design of knitting pattern for target mechanical properties.

Probabilistic fatigue evaluation of notched specimens considering size effect under multiaxial loading

Fatigue evaluation of notch components is key to ensuring the structural integrity of grand equipment and implementing fatigue resistant design, while fatigue models that effectively couple multiaxial stress condition, notch and size effects are still lacking. Accordingly, a new probabilistic fatigue model by combining the weakest link theory with critical plane theory is proposed in this work. Firstly, the critical plane location is determined by calculating the stationary point of the relevant strain energy density function. Then, a new effective stress concept is proposed to establish the connection between experimental data of smooth specimens and equivalent stress in the critical damage region of notched specimens. Meanwhile, a critical damage region considering the size effect is defined to determine the effective stress. Finally, the experimental data of EN8 medium carbon steel and Q345 alloy steel are used for model validation and comparison and the results show that the model has a higher accuracy than the other four models. Additionally, the curves P−σeff‾−Nf can effectively describe the fatigue life scatter.

Data-driven hyperelasticity, Part I: A canonical isotropic formulation for rubberlike materials

Data-driven hyperelasticity shows great promise for modeling the mechanical response of rubberlike materials. It enables an automated linkage between experimental data and mechanical response without a priori knowledge of specific analytical expression for the strain energy density function or the stress expression. In this study, we propose a new data-driven approach with three distinct kinematic approaches; (i) invariant-based formulation, (ii) modified invariant-based approach, and (iii) principal stretch-based formulation to model the hyperelastic response of rubberlike materials. Within this context, we replace the partial derivatives of the strain energy density functions with appropriate B-spline interpolations, using a set of control points to implement various multiaxial loading scenarios, such as uniaxial tension, pure shear, and equibiaxial tension deformations. To ensure a polyconvex and stable constitutive response, we enforce the polyconvexity requirement into the B-spline interpolation along with appropriate normalization conditions. In order to obtain the control points of the B-splines, we train the proposed data-driven approach with respect to the Treloar and Kawabata datasets. On the numerical side, the stress and moduli expressions are derived for the finite element implementation. The predictive capabilities of the proposed approach are demonstrated through representative boundary value problems.

Nonlinear 2D-DQ volume-preservative global–local dynamic analysis of composite sandwich plates with soft hyperelastic cores and viscoelastic Winkler-Pasternak foundations

Nonlinear large amplitude vibration and dynamic deformations of sandwich plates with composite face sheets and hyperelastic cores that encounter large thickness changes during the vibration are investigated here for the first time. Moreover, a new global–local zigzag hyperelastic plate theory that considers the incompressibility and large transverse deformability of the core, and employs transverse normal strains whose order is higher than that of the in-plane strains is proposed. The plate is resting on a viscoelastic Winkler-Pasternak substrate, exposed to various spatial and time distributions of loads, and experiencing a diversity of edge conditions. The governing equations of motion are obtained by using Hamilton’s principle, von Kármán’s assumptions, and the series expansion of the volume-preserving strain energy density function. A 2D differential quadrature (2D-DQ) technique is employed for the spatial discretization of a structure with both constitutive and kinematic nonlinearities, for the first time. The time-dependent combination of the nonlinear coupled motion equations and boundary conditions is treated by an iterative/updating Runge-Kutta time-marching technique and the effects of the thickness and aspect ratio of the plate, viscoelastic and shear parameters of the foundation, types of the edge conditions, constitutive parameters of the composite materials, fiber orientation angle, and magnitude and distribution pattern of the loads on the resulting dynamic lateral deflections are studied. Results reveal that while the vibration frequencies and amplitudes of the face sheets are quite different, the damping is more remarkable when stiffer cores are utilized. Moreover, the effects of the higher vibration modes are more apparent in looser edge conditions, smaller fiber angles, and special excitation frequencies.

The electro-mechanical coupling responses of functionally graded piezoelectric nanobeams with flexoelectric effect

The aim of this paper is to study the electro-mechanical coupling responses of functionally graded (FG) piezoelectric cantilever nanobeam under the concentrated load, and the material properties follow an exponential distribution in the thickness direction. The constitutive equations are derived from the electric Gibbs free energy density function. The governing equations and boundary conditions of the Euler–Bernoulli FG beam can be obtained by variational principle, and then, the analytical expressions for its deflection, polarization intensity, and induced potential are obtained. The results show that the flexoelectric effect has a significant impact on the deflection, polarization intensity, and induced potential of the FG cantilever nanobeam.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

Calculation of Strain Energy Density Function Using Ogden Model and Mooney-Rivlin Model Based on Biaxial Elongation Experiments of Silicone Rubber.

Strain energy density functions are used in CAE analysis of hyperelastic materials such as rubber and elastomers. This function can originally be obtained only by experiments using biaxial deformation, but the difficulty of such experiments has made it almost impossible to put the function to practical use. Furthermore, it has been unclear how to introduce the strain energy density function necessary for CAE analysis from the results of biaxial deformation experiments on rubber. In this study, parameters of the Ogden and Mooney-Rivlin approximations of the strain energy density function were derived from the results of biaxial deformation experiments on silicone rubber, and their validity was verified. These results showed that it is best to determine the coefficients of the approximate equations for the strain energy density function after 10 cycles of repeated elongation of rubber in an equal biaxial deformation state, followed by equal biaxial elongation, uniaxial constrained biaxial elongation, and uniaxial elongation to obtain these three stress-strain curves.

Automatedly Distilling Canonical Equations From Random State Data

Abstract Canonical equations play a pivotal role in various sub-fields of physics and mathematics. However, for complex systems and systems without first principles, deriving canonical equations analytically is quite laborious or might even be impossible. This work is devoted to automatedly distilling the canonical equations solely from random state data. The random state data are collected from stochastically excited, dissipative dynamical systems either experimentally or numerically, while other information, such as the system characterization itself and the excitations, is not needed. The identification procedure comes down to a nested optimization problem, and the explicit expressions of the momentum (density) functions and energy (density) functions are identified simultaneously. Three representative examples are investigated to illustrate its high accuracy of identification, the small requirement for data amount, and high robustness to excitations and dissipation. The identification procedure serves as a filter, filtering out nonconservative information while retaining conservative information, which is especially suitable for systems with unobtainable excitations.

A Novel Energy and Momentum Consistent Mixed Formulation for Coupled Nonlinear Electro‐Thermo‐Elastodynamics

AbstractIn the present contribution, a novel mixed formulation aimed at the energy‐momentum consistent simulation of coupled nonlinear electro‐thermo‐elastodynamic systems, in particular dielectric elastomer actuators, is proposed. It is essentially based on a mixed framework for elastodynamics in the case of polyconvex stored energy functions. In accordance with this framework, the properties of the rediscovered tensor cross product are exploited in a first step to derive a mixed formulation via a Hu‐Washizu type extension of the stored energy function. Afterwards, the corresponding strong form is derived and supplemented with the (initial) boundary value problems of thermodynamics and electrostatics. By additionally choosing an appropriate polyconvexity‐inspired energy density function, this procedure leads to a fully coupled electro‐thermo‐elastodynamic formulation that benefits from the properties of the underlying mixed framework.Furthermore, an energy‐momentum consistent time integration scheme is proposed for the novel framework, where discrete derivatives in the sense of Gonzalez are employed. The formulation is second‐order accurate and stable even for large time step sizes. Eventually, the performance of the novel formulation is illustrated in a numerical example.

A flexoelectric actuator model with shear-lag and peel stress effects

Flexoelectric materials are being extensively studied in recent years for potential sensing, actuation, energy harvesting and structural health monitoring applications. However, no analytical model exists in the literature to predict the stress transfer between a flexoelectric transducer and the host structure, considering bonding compliance. In this article, we present an analytical model for the interfacial stresses and deformations of flexoelectric patch actuator(s) bonded to an isotropic host plate considering both shear-lag and peel stress effects at the interface(s). The model is valid for both piezoelectric and flexoelectric effects. The system’s one-dimensional governing equations are derived consistently from the three-dimensional equations for linear dielectric solids obtained using the bulk electric Gibbs energy density function and extended Hamilton’s principle. The transducer and substrate, modelled as classical Kirchhoff plates, undergo both extension and bending deformations. The formulation produces a seventh-order differential equation for the interfacial shear stress, which is solved analytically, satisfying all boundary conditions. The solution is validated by comparing with the available results for piezoelectric actuators with a non-rigid interface and those of flexoelectric actuators based on the pin-force-moment model assuming a rigid interface. Detailed numerical studies are presented to show the effect of the plate, actuator and adhesive parameters on the interfacial stresses and deformations. Finally, the size effect of the actuators at the micro-scale is illustrated.

Damage as a Material Phase Transition

AbstractWe propose paradigmatic examples to show how material damage phenomena can be efficiently described as a solid-solid phase transition. Starting from the pioneering work of J.L. Ericksen (J. Elast. 5(3):191–201, 1975) and the extensions of R.L. Fosdick and other authors to three-dimensional non linear elasticity, we describe the insurgence of damage as a hard → soft transition between two material states (damage and undamaged) characterized by two different energy wells. We consider the two separate constitutive assumptions of a simple Neo-Hookean type damageable material and a more complex microstructure inspired damageable Gent type material with variable limit threshold of the first invariant. In both cases we study two different deformation shear classes, one homogeneous and the other one inhomogeneous and obtain fully analytic description of the system damage response under cyclic loading. The considered constitutive assumptions and deformation classes are aimed at attaining fully analytic descriptions. On the other hand, we remark that the proposed, Griffith type, variational approach of damage, based on two different energy density functions for the damaged and undamaged material phases, and a resulting non (rank-one) convex energy, can be extended to systems with more complex energy functions, possibly with a larger number of wells representing an increasing degree of damage.

Kinetostatic Modeling of Soft Robots: Energy-Minimization Approach and 99-Line MATLAB Implementation.

Soft robots have received a great deal of attention from both academia and industry due to their unprecedented adaptability in unstructured environment and extreme dexterity for complicated operations. Due to the strong coupling between the material nonlinearity due to hyperelasticity and the geometric nonlinearity due to large deflections, modeling of soft robots is highly dependent on commercial finite element software packages. An approach that is accurate and fast, and whose implementation is open to designers, is in great need. Considering that the constitutive relation of the hyperelastic materials is commonly expressed by its energy density function, we present an energy-based kinetostatic modeling approach in which the deflection of a soft robot is formulated as a minimization problem of its total potential energy. A fixed Hessian matrix of strain energy is proposed and adopted in the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, which significantly improves its efficiency for solving the minimization problem of soft robots without sacrificing prediction accuracy. The simplicity of the approach leads to an implementation of MATLAB with only 99-line codes, which provides an easy-to-use tool for designers who are designing and optimizing the structures of soft robots. The efficiency of the proposed approach for predicting kinetostatic behaviors of soft robots is demonstrated by seven pneumatic-driven and cable-driven soft robots. The capability of the approach for capturing buckling behaviors in soft robots is also demonstrated. The energy-minimization approach, as well as the MATLAB implementation, could be easily tailored to fulfill various tasks, including design, optimization, and control of soft robots.

Electroelasticity of copolymer networks

We introduce a novel statistical mechanics-based analysis of electroactive copolymer networks. This enables to express the coupled response of a multi-species polymer network in terms of a Gibbs-like energy-density function. Following the common affine deformation assumption we derive the expressions for the network response in terms of the species’ properties and molar fraction. Next, we specialize the results to the class of monomers with spontaneous electric dipoles and determine the monomers’ orientational distribution in terms of a generalized Langevin function. We reveal that the coupled response of the copolymer depends on the contrast between the intensities of the dipolar moments of the species and their fraction of the chain end-to-end length. In the long chains limit we obtain closed-form expressions for the electrical and mechanical fields. Next, we consider two-species copolymers and find that the long chains approximation is valid for actuation stretch ratios up to 40% of the locking stretch. We also demonstrate that the maximal susceptibility is attained when the contour lengths of the two species are equal. Moreover, the overall electroelastic response is enhanced if the monomers with the weaker dipole are longer than those with the stronger one. Finally, we show that an appropriate choice of actual monomers may lead to copolymers with relative dielectric constants of more than 30.

Parametric optimization of multi-pass electron beam melting for molybdenum alloy containing 47.5 wt% rhenium

In this study, the effects of multi-pass electron beam melting (EBM) processes under vacuum condition on microstructure and mechanical behavior of a molybdenum alloy containing 47.5 wt% rhenium were systematically investigated. Nine experimental designs were performed to improve the surface quality and minimize the defects. Results showed that beam power, scan speed, processing step and penetration depth played a critical role in the specific energy density function, which would further influence the porosity and microhardness. Meanwhile, specific energy density was significantly influenced by beam power at higher scan speed (e.g. 15 mm/s) whereas it was mainly related to scan speed under lower beam power (e.g. 2 kW). With a processing step of 0.5 mm (half of the beam diameter), a continuous melt morphology and less porosity were obtained, indicating that a significant overlap between the scan tracks was beneficial for the optimization of microstructure. Microhardness results showed that mechanical performance can be improved with the reduction of porosity together with the increased specific energy density. Thus, a combination of 10 kW beam power, 2 mm/s scan rate and 0.5 mm processing step can realize surface optimization in melted plates through multi-pass EBM technique.

Numerical Simulation of Assembly Process and Sealing Reliability of T-Rubber Gasket Pipe Joints

Underground pipelines are vital parts to urban water supply, gas supply, and other lifeline systems, affecting the sustainable development of cities to a great extent. The pipeline joint, which is a weak link, may be seriously damaged during natural disasters such as earthquakes. The failure of pipe joints can cause leakage accidents, resulting in system failure and interruption, and even some secondary disasters. Herein, based on uniaxial and plane tensile test results of a T-rubber gasket material, the assembly process and sealing performance of a T-rubber gasket joint of a ductile iron pipe are numerically simulated using the Ogden third-order strain energy density function to fit the material constant. The simulation accounts for severe nonlinearities, including large deformations, hyperelasticity, and complex contacts. The effects of the assembly friction coefficient, assembly depth, and radial clearance deviation of the socket and spigot on the seal contact pressure are analyzed. The results suggest that the entire history of the deformation and stress variations during assembly can be clearly visualized and accurately calculated. For the different friction coefficients, the assembly depth corresponding to the sliding friction condition of the spigot pipe was 74 mm, while the minimum pushing force required to assemble the T-rubber gasket joint of a DN300 ductile iron pipe was 6.8 kN at the ideal situation with a friction coefficient of 0. The effective contact pressure of the rubber gasket seepage surface under various operating conditions is much higher than the normal pressure of municipal pipelines, thus indicating that the rubber gasket joint exhibits the ideal sealing performance. Furthermore, a certain deviation, which is about 20 mm, is allowed for the assembly depth of the rubber gasket joint such that the axial displacement of the pipe joint can be adapted under an earthquake or ground displacement.