Strain Energy Expression Research Articles

Differential growth and shape formation of a flower-shaped structure

Morphogenesis, which is a complex interplay of biological, chemical, and physical processes, facilitates differential growth in various biological systems, particularly in plant organs, such as petals and leaves. Although recent studies have increasingly delved into the mechanical aspects of these biological structures, there remains a lack of a comprehensive quantitative understanding of shape formation induced by differential growth in plant organs. Thus, this study addressed this gap by employing a multifaceted approach encompassing theoretical analyses and comprehensive finite element analysis of the effect of differential growth on the shape of a cylinder into a flower shape. Based on the derivation of the strain energy expressions for the axisymmetric and asymmetric configurations, the shape function of the growing flower-like structures were calculated through mathematical optimization. The findings of this study shed light on the influence of the growth function, geometric characteristics, and material properties. As the wave number increased, the final configuration tended to have smaller waves, whereas a longer cylinder buckled more easily and the thickness had a minimal effect. This study offers insights that can pave the way for innovative geometrical designs, thereby providing inspiration for applications on both micro-and macroscales, such as in the realms of self-assembly of soft robotics and flexible electronics.

Bending of bidirectional functionally graded nonlocal stress-driven beam

This paper provides a comprehensive analysis, using nonlocal stress-driven integral theory, of the static behavior of a nanoscale beam of bidirectionally graded materials. After a brief explanation of the mathematical formulation of BDFGMs, the work done and strain energy expressions derived from the displacement field are discussed. Variational formulations and Hamilton's principle are used to develop the equilibrium equation. An analytical development of the nonlocal kernel for stress-driven integral theory and formulated governing equation which was nondimensionalized later. Explicit equations for displacement and moment are obtained by solving this equation using the Laplace transformation. Three different boundary conditions are examined, and differences in the maximum displacement with respect to the nonlocal parameter and the two material FGM parameters are displayed both visually and in table form. The results exhibit excellent agreement and provide a standard for further research when they are closely compared to the existing numerical data. This work contributes to the knowledge of BDFGMs under nonlocal effects generated by stress-driven integral theory and offers solutions that have been confirmed for further investigation.

Misfit stress and energy in composite nanowire with polygonal core

In this article we present for the first time an analytical solution to the boundary-value problem in the classical theory of elasticity for a core-shell cylinder with an eccentric prismatic core having a trapezoidal cross section and subjected to uniform dilatation eigenstrain. In doing so, we represent the trapezoidal prism as a superposition of long straight dilatational lines located along the cylinder axis. The complex potential approach has been employed to satisfy the boundary conditions on the cylinder free surface. Found in a concise and transparent closed form, the elastic stresses exerted by the dilatational line and dilatational prismatic inclusion (of trapezoidal cross section) in cylindrical body are demonstrated with maps. The mean stress is employed to obtain the analytical expression of strain energy stored by the dilatational inclusion in form of regular prism symmetrically placed in cylindrical body. It is proved that the strain energy is identical to the one of cylindrical inclusion with the same cross section area.

Temperature dependent tensile fracture strength model of rubber materials based on Mooney-Rivlin model

With the increasingly widespread application of rubber in many fields, the demand for quantitative characterization of temperature dependent mechanical properties in service environments over a wide temperature range is increasing. Tensile fracture strength is one of the most important mechanical properties of rubber materials, but its temperature dependent model without fitting parameters is still rarely reported. In this work, based on Mooney-Rivlin hyperelastic constitutive model, a strain energy expression that explicitly includes tensile fracture strength is derived, and a temperature dependent tensile fracture strength model for rubber materials without fitting parameters is further developed in combination with the Force-Heat Equivalence Energy Density Principle. The prediction results of the model in a wide temperature range are in good agreement with the experimental results. Furthermore, a quantitative analysis of the effect of elastic modulus on tensile fracture strength at different temperatures is conducted. Finally, based on the above conclusions, some useful suggestions on the selection of rubber materials for service in a wide temperature range are proposed.

Differential-Growth-Induced Center Wave Buckling

Morphogenesis is a result of complex biological, chemical, and physical processes in which differential growth in biological systems is a common phenomenon, especially notable in plant organs such as petals and leaves. Mechanisms of these biologic structures have been studied in recent years with a growing focus from the mechanics point of view. However, understanding differential-growth-induced shape formation quantitatively in plant organs remains largely unknown. In this study, we conduct quantitative experimental measurement, theoretical analysis, and sufficient finite element analysis of constrained differential growth of a thin membrane-like structure. By deriving the corresponding strain energy expression of a buckled growing sample, we can calculate the shape function of such membrane structures explicitly. The results of this work demonstrate the effect of growth function, geometry characteristics, and material property. Our research points to potential approaches to novel geometrical design and inspirations on the microscale and the macroscale for items such as soft robotics and flexible electronics.

Buckling behavior and damage mechanism analysis of fiber-reinforced shape memory polymer composites

Shape memory polymer composites (SMPCs) are emerging as an attractive material for space deployable structures due to their variable stiffness, programmable deformation, high packing ratio, and controllable deployment. However, the research on the deformation behaviors and damage mechanisms of SMPC was scarce. The buckling behaviors and damage mechanisms of out-of-plane buckling and in-plane buckling of fiber-reinforced SMPC were investigated in this study. The expression and evolution law of strain energy and key parameters of fiber buckling were studied. The effects of material parameters and size parameters on damage mode and critical damage curvature were analyzed quantitatively. It was found that in-plane buckling was more likely to occur without external constraints. The damage modes of in-plane buckling are classified as delamination, matrix cracking and fiber tensile fracture. Out-of-plane buckling is more susceptible to damage and has one more fiber buckling fracture damage mode than in-plane buckling. When the fiber volume content is below a certain value, only fiber tensile fracture damage mode will appear, regardless of the thickness. Delamination is more likely to occur in cases of out-of-plane buckling, and matrix cracking is more frequent in cases of in-plane buckling. Subsequently, the correctness of the theoretical analysis was verified by the bending test. The theoretical analysis in this study provides a theoretical foundation for SMPCs-based structural design.

A modified strain gradient beam constraint model

The objective of the present paper is to develop a modified strain gradient beam constraint model (MSGBCM) to improve the modeling accuracy of small‐scale compliant mechanisms. First, a simple nano/micro flexure beam under the effect of end loads is considered. The virtual work principle is employed to formulate the nonlinear load–displacement behavior of the system, based on the modified strain gradient theory. It is observed that as the size of the structure becomes smaller, the elements of the elastic stiffness and load stiffening matrices severely deviate from their corresponding values in the beam constraint model (BCM). A closed‐form nonlinear strain energy expression is also derived for nano/micro flexure beams in terms of their tip displacements. This energy expression is utilized to model the nonlinear load–displacement relationships of small‐scale parallelogram (P) flexures. Moreover, analytical formulas are derived for the axial, transverse, and rotational stiffness values of P‐flexures. The most important observation is that in the small‐scale P‐flexures, the axial stiffness loss of small‐scale P‐flexures resulted from the movement of the stage in the transverse direction, may be seriously overestimated by BCM. The developed MSGBCM can be easily extended for investigating the static and dynamic behavior of more complicated micron and submicron size flexures.

Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis

Abstract This paper approximately analyzes the dynamic response mode of building materials under impact load according to the Lagrange equation. We use the redundant constraint force of the statically indeterminate structure to list its strain energy expression. At the same time, the paper introduces the Lagrange multiplier and combines the static equilibrium equation to obtain the internal force of the statically indeterminate structure. The calculation results show that the load duration and peak value are important factors determining structural members’ failure mode. This study lays the foundation for the research on the structural building material’s dynamic control and mechanism performance. At the same time, our research results also provide methods and ideas for the rigid body dynamics modeling of other structural redundant parallel mechanisms.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

The coupling free vibration characteristics of a rotating functionally graded shaft-disk system in thermal field

The numerical model of a functionally graded (FG) shaft-disk rotor system with flexible shaft and flexible various-thickness disk as components is established to investigate the coupling free vibration behavior of this rotor system in thermal field. Considering the thermophysical properties of the material, the properties of functionally graded material vary continuously according to the power law with the thickness of the disk. Based on the theory of rotor dynamics, the kinetic energy expression and strain energy expression of rotating shaft-disk system are obtained. In the framework of Ritz method, the global control differential equation of the rotor system assembled by shaft and FG disk is established by using differential quadrature finite element method. On this basis, a set of numerical examples are presented to perform convergence, correctness and effectiveness verification of the current model. Moreover, the influence of temperature difference, rotating speed, geometric and material parameters on the thermoelastic coupling vibration characteristics of light-weight shaft-disk rotor is discussed in detail, which can help to improve the rotor vibration performance in design work and future research.

Vibrations Analysis of Cable-Harnessed Plates: Continuum Modeling and Experimental Validation

Abstract Dynamic modeling of spacecraft structures is imperative to their successful design for flight missions. A large number of these structures’ dry mass consists of signal and power cables, dynamics of which is not well-predicted using ad hoc cable lumped mass models. Hence, accurate modeling techniques are required to understand these cable dynamics effects. In the past, efforts toward developing analytical models for cable-harnessed structures have been primarily focused on beam-like host structures. The presented paper is aimed to fill this gap by obtaining analytical solutions through an energy-equivalent homogenization approach for cable-harnessed plate-like structures and to ultimately help with understanding the dynamic effects of signal and power cables on two-dimensional plate-like structures. As a first step, systems of periodic geometries with parallel cable configurations are considered. The strain and kinetic energy expressions for the fundamental repeated elements are found using linear displacement fields and Green–Lagrange strain tensors. The governing partial differential equation (PDE) for the out-of-plane motion of the cable-harnessed system is then found using Hamilton’s principle. Experimental modal testing is performed for the purpose of validations of the frequency response functions (FRFs) obtained from the model for the cable-harnessed plates under clamped-free-free-free boundary conditions. The results clearly show the dynamic effects of the (CFFF) cables which are also well-predicted by the model. Finally, modal assurance criterion (MAC) analysis has been used for further validations of the mode shapes obtained from the model.

The Mechanical Properties of Origami Structure Determined by the Improved Virtual Crease Method

The mechanical properties and deformation of Origami structures are studied in this paper. Usually, it is a coupling problem of crease rotation and shell deformation. Here, the creases are simplified as torsional springs, whose rotational stiffness [Formula: see text] is obtained by the experiment of compressing a creased shell. While the shells that may have large deformation are simplified as rigid plates connected by virtual creases, whose rotational stiffness is roughly expressed as bending stiffness divides width of the shell. Hence, a coupling factor [Formula: see text] is defined to evaluate the coupling effect of creases and shells. Implementing the obtained rotational stiffnesses of real and virtual creases into the expression of strain energy, an improved Virtual Crease Method (VCM) is proposed. By analyzing the bi-stability of creased shell and Miura-Ori structure, the accuracy and convergence of this improved VCM is proved.

Effect of axially graded constraining layer on the free vibration properties of three layered sandwich beams with magnetorheological fluid core

The free linear vibration of an adaptive sandwich beam consisting of a frequency and field-dependent magnetorheological fluid core and an axially functionally graded constraining layer is investigated. The Euler-Bernoulli and Timoshenko beam theories are utilized for defining the longitudinal and lateral deformation of the sandwich beam. The Rayleigh-Ritz method is used to derive the frequency-dependent eigenvalue problem through the kinetic and strain energy expressions of the sandwich beam. In order to deal with the frequency dependency of the core, the approached complex eigenmodes method is implemented. The validity of the formulation and solution method is confirmed through comparison with the results available in the literature. Finally, the effects of the magnetic field, axially functionally graded material power-law index, constraining, and core layers thickness on the free vibration behavior of the sandwich beam are studied thoroughly for clamped–clamped, simply supported, and clamped-free boundary conditions. It is shown that any change in each of these parameters would have significant effects on the beam's free vibration properties.

Imperfect Bifurcation and Chaos of Slightly Curved Carbon Nanotube Conveying Hot Pressurized Fluid Resting on Foundations

AbstractUnintended slight curvature of a straight pipe and temperature variation in a pipe has been found to create uncertainties in tubes and pipes. Fluttering, divergence, and chaotic instabilities of slightly curved carbon nanotubes (SCCNT) conveying hot pressurized fluid are investigated in this paper. The SCCNT is modeled on the basis of large deformation strains. Their gradients are included in the strain energy expression and the velocity and its gradients in the kinetic energy derivation. In modeling the size effects, both the static and kinetic length scales in the energy equations were considered. Governing equation is derived using Lagrangian approach. The effects of geometric imperfection (which leads to cusp bifurcation), small length scale, and kinetic material length parameter on the static and dynamic instability characteristics of the pipes are studied. Analysis is performed using the eigenfunction expansion method. It is found that the material length scale parameter increase tends to shift instability to the lower fluid velocity while the kinematic material length parameter increase does not change the buckling point but lowers the frequency. In the nonlinear dynamic case, both the parameters lead to chaos of the nanotube beyond the critical fluid velocity. The thermal loading changes the sudden supercritical pitchfork bifurcation to cusp bifurcation. The increasing linear and nonlinear foundation stiffness leads the system to chaotic features after the critical point.

Coupled vibration analysis for equivalent dynamic model of the space antenna truss

A novel equivalent dynamic model is developed for coupled vibration analysis of the space antenna truss to enhance the design capacity of vibration controllers. Based on energy equivalence principle, the space antenna truss with two important features of rigid joints and complicated configuration is equivalent to a spatial anisotropy Timoshenko beam model. According to the kinematic assumptions, strain and kinetic energy expressions of the spatial periodic element can be obtained in accordance with displacement components at its center. Hamilton's principle is carried out to formulate the governing partial differential equations of motion for the equivalent beam model (EBM) which is divided into two sets of PDEs. Each set of PDEs includes three degrees of freedom and describes bending-torsion and bending-extension couplings respectively, due to the asymmetry of the antenna truss. An exact analytical method is developed to solve the two sets of coupled motion equations. The natural characteristics of the EBM are shown to be in excellent agreement with those of the finite element method, which demonstrates that the proposed EBM can provide a satisfactory accuracy for the antenna truss. In addition, since the mode shapes of the EBM are expressed as analytical functions of the spatial coordinate, such an approach may lead the investigations of dynamic property and the design of vibration control law to become convenient for the space antenna truss.

Damping Characteristics of Bending Vibration of Thick-Walled CFRP Hollow Shafts

The damping characteristics of the composite thick-walled hollow shaft directly affect its amplitude of rotational vibration at low speeds and stability at high speeds. In this study, the shaft is made of carbon fiber reinforced plastic (CFRP) material. In order to study the damping characteristics of the composite hollow shaft, a 3D numerical damping model was established based on finite element method and strain energy expression. In this regard, both the contributions of in-plane and out-of-plane strain energy dissipation were studied. Then, the validity of 3D damping model in bending was verified by experiments. On this basis, the flexural damping characteristics of the thick-walled composite shaft were studied, and the effects of orientation angle and stacking sequences on the flexural damping were obtained.

A phenomenological expression of strain energy in large elastic deformations of isotropic materials

A new model is constructed to validate a non-linear elastic response of rubber-like materials. The proposed model is phenomenological and is the combination of Rivlin expression truncated in invariant $$\mathop I\nolimits_{1}$$ with a term in $$\mathop I\nolimits_{2}$$ coming from the Hart-Smith model and which has been modified by Gornet et al. This model has five rheological parameters adjusted with a genetic algorithm. The genetic algorithm method is used in binary code and two sets of experimental data (Treloar data and Nunes and Moreira data) from the literature to compare the proposed model. With Treloar rubber data the simple tension suffices for the determination of the parameter values. Thus, the fit of experimental data is shown to provide an accurate response and predictive description in biaxial tension and pure shear with the same sets of material parameters. Unlike, with Moreira data, the proposed model has given direct material parameters in simple shear with good correlation. On the other hand, a comparison is made between the proposed model and the Beda model (2007), Gornet et al. model and hybrid integral approach (HIA) model. The proposed model is the most efficient among three other classical models because the proposed model emerges from very small relative errors with Treloar's experimental data: 0.42 ± 0.08 in simple tension, 0.44 ± 0.07 in biaxial tension and 0.42 ± 0.05 in pure shear. This model also incorporates all the criteria for model validation described in the literature.

An extended JKR model for adhesion of a rigid sphere on a supported compressible elastic thin layer

This study aimed to develop an explicit JKR-type model for adhesive contact of a rigid sphere (of radius R) on a compressible elastic thin layer (of thickness h, Young’s modulus E, and Poisson ratio $$\nu $$) bonded or sliding on a rigid substrate. Based on a simple Kerr model for a compressible elastic layer, an explicit expression for strain energy of the elastic layer is derived in terms of the two JKR-type variables $$\left( {\delta ,\;a} \right) $$, where a is the radius of contact zone and $$\delta $$ is the indentation depth of the rigid sphere. Thus the equilibrium values of $$\left( {\delta ,\;a} \right) $$ can be determined as the stationary point of the potential energy. The explicit model is justified by detailed comparison of the predicted results (for Poisson ratio $$\nu \le 0.45$$) with known data reported in recent literature. For example, the validity and accuracy of the present model are demonstrated for moderately soft elastic thin layers under the condition $${2\left( {1-v^{2}} \right) WR^{2}} \big /{\left( {Eh^{3}} \right) }\ge 100$$ (where W is adhesion energy per unit contact area, for instance, for typical materials with $$R=500\ \upmu \mathrm{m}$$ $$E=100\ \mathrm{KPa}$$ and $$W=100\ \mathrm{mJ}/\mathrm{m}^{{2}}$$, the condition requests $$\;h<20\ \upmu \mathrm{m}$$). As compared to existing methods that request more substantial numerical calculations, the present model achieves an explicit expression for strain energy of a compressible elastic layer and could offer a simpler analytical method for adhesion mechanics of a rigid sphere on a compressible thin elastic layer bonded or sliding on a rigid substrate.

Hydrothermal effect on Bi-stability of composite cylindrical shell

Bi-stable composite cylindrical shells have attracted many attentions due to their lightness, simple structure, high packaging efficiency, high specific strength and stiffness. However, the anti-symmetric cylindrical shell is susceptible to the effect of ambient temperature and humidity. In this paper, an analytical model of shell structure considering hydrothermal effect was built, and its strain energy expressions were derived. Based on the minimum energy principle, the effects of moist and heat on the bi-stability of cylindrical shell structure were analyzed. In addition, the finite element model of the anti-symmetric cylindrical shell was established to simulate its stable configurations. This study could provide reference for the practical application of bi-stable structure.

Optimized valence force field model for the lattice properties of non-ideal III-nitride wurtzite materials

It has been discussed the optimization of the valence force field model for non-ideal III-nitrides wurtzite materials. To describe the static and dynamical lattice properties of nitrides within the same model, the anisotropy of the wurtzite structure and additional second-nearest-neighbor atomic interaction terms are included in the lattice strain energy expression. The sets of the force constants have been optimized through an iterative scheme in which calculated lattice properties are fitted to experimentally reported data. The model has been applied to the binary wurtzite compounds (GaN, InN) and random InxGa1-xN alloys. The calculated phonon dispersions, elastic constants and internal strain parameters of GaN and InN are in reasonable agreement with the experimental/ab initio values.