Modeling and simulation of the deformation of multi-state hydrogels subjected to electrical stimuli

In this paper, a mathematical model capable of simulating pH-sensitive hydrogels is refined to extend its application to electric-sensitive hydrogels. This is achieved by consideration for large deformation as well as the reformulation of the fixed charge density. The present model, termed the refined multi-effect-coupling electric-stimulus (rMECe) model, consists of nonlinear partial differential governing equations with chemo-electro-mechanical coupling effects and the fixed charge density, which includes the effect of externally applied electric field. A comparison between simulation results and experimental data is carried out so as to validate the rMECe model, and very good agreement is observed. The multi-state rMECe model accurately predicts the responsive deformation of the hydrogels when placed in a bath solution, which is subjected to externally applied electric field. The influences of several physical parameters, such as the applied voltage, initial fixed charge density and ionic concentration of surrounding solution, are examined and discussed in detail.     

Thermostatics of an infinite mixture of two elastic solids and a spherical thermal inclusion problem

This work is concerned with the linear theory of a binary mixture of two elastic solids. With the help of displacement potentials, two differential equations that govern the displacement of each constituent are obtained. Then, more generalised forms of Betti's reciprocal theorem and Maysel's formula for the mixture are found. Finally, a solution of spherical thermal inclusion problem in an infinite mixture of two elastic solids is obtained by using generalised Maysel's formula and by direct integration of the governing differential equations.     

Modeling diffusion of sulfate through concrete using mixture theory

Concrete exposed to sulfates deteriorates. Here, an attempt is made to see whether the framework of mixture theory can be used to model the changes that occur in concrete exposed to sodium sulfate. Toward this, diffusion and reaction of sulfates with concrete is modeled within the framework of mixture theory. Appropriate choices are made for the Helmholtz free energy and interaction momentum so that the process of diffusion of sulfates in concrete can be captured. As expected in mixture theory, diffusion causes deformation of the solid. The parameters in the mixture theory model are determined by comparing the steady-state concentration profile of the diffusing sodium sulfate solution and inlet velocity of the fluid with that of Fick’s solution for the same boundary conditions. An assumption is made for the mass production term to capture the reaction of sulfates with concrete. The rate constant and order of the reaction are estimated using the concentration of gypsum, calcium hydroxide and ettringite reported in the literature, for various duration of exposure of cement pastes to known concentration of sodium sulfate solution. Finally, the governing equations for the combined problem of steady-state diffusion and reaction of sulfates with concrete are presented. A numerical scheme to solve the governing equations is outlined. Long-term concentration profiles of sodium sulfate predicted by the framework agree qualitatively with the experimentally observed profile reported in the literature.     

The Galerkin vector solution for a mixture of two elastic solids and Boussinesq problem

This work is concerned with the linear theory of a binary mixture of two elastic solids. Using the constitutive equations for the mixture which are given by Green and Steel, the displacement equations in the case of isotropic mixture of two elastic solids are derived. By use of the Galerkin vector solution, the displacement vector of each component in the mixture is obtained. Finally, an equilibrium solution for the Boussinesq problem of the mixture of two elastic solids in an infinite half-space is examined.     

Electromagnetoelastic equations for ferroelectrics based on a micromorphic dielectric model

A micromorphic continuum model is exploited to derive the governing equations for the electromagnetoelastic field superposed to a ferroelectric configuration of polarizable dielectrics. The microcontinuum structure accounts for charge microdensity to obtain dipole and quadrupole densities and their evolution equations. These quantities enter as arguments of the constitutive tensors, besides the microstrain measures. A strain-free polarized initial state is considered, and a set of non-linear equations is derived for the superposed field. These equations are linearized to show a simple relation between microstrain and incremental polarization. The case of micropolar continua is analyzed to compare the present model with the formulation of the same problem, obtained in the past on the basis of a continuum theory of electrodynamics.     

A nonlinear Timoshenko beam formulation based on the modified couple stress theory

This paper presents a nonlinear size-dependent Timoshenko beam model based on the modified couple stress theory, a non-classical continuum theory capable of capturing the size effects. The nonlinear behavior of the new model is due to the present of induced mid-plane stretching, a prevalent phenomenon in beams with two immovable supports. The Hamilton principle is employed to determine the governing partial differential equations as well as the boundary conditions. A hinged–hinged beam is chosen as an example to delineate the nonlinear size-dependent static and free-vibration behaviors of the derived formulation. The solution for the static bending is obtained numerically. The solution for the free-vibration is presented analytically utilizing the method of multiple scales, one of the perturbation techniques.     

The refined theory of magnetoelastic rectangular beams

Summary. The problem of deducing a one-dimensional theory from a three-dimensional theory for a soft ferromagnetic elastic isotropic body is investigated. Based on the linear magnetoelasticity, the refined theory of magnetoelastic beams is presented by using the general solution for the soft ferromagnetic elastic solids and the Lure method. Based on the refined theory of magnetoelastic beams, the exact equations and solutions for the homogeneous beams are derived and the equations can be decomposed into three governing differential equations: the fourth-order equation, the transcendental equation and the magnetic equation. Moreover, the approximate equations and solutions for the beam under transverse loadings and magnetic field perturbations are derived directly from the refined beam theory. By omitting higher order terms and coupling effects, the refined beam theory can be degenerated into other well-known elastic and magnetoelastic theoretical models.     

A general laminated plate theory accounting for continuity of displacements and transverse shear stresses at material interfaces

A general two-dimensional theory, suitable for the static and/or the dynamic analysis of transverse shear deformable laminated plates, is presented. This displacement-based theory is capable of satisfying continuity of both displacements and transverse shear stresses at the plate material interfaces. The derivation of its governing differential equations is based on the application of Hamilton's principle in conjunction with the method of Lagrange multipliers. Moreover, this new theory is capable of accounting for unlimited multiple choices of continuous displacement distributions, through the plate thickness, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. With such a double-infinite freedom, it is concluded that for the analysis of the particular laminated plate considered one may start with the solution of the governing equations of the 5-degrees-of-freedom theory derived for relatively simple choices of through-the-thickness displacement distributions. Then, either increasing the number of the degrees of freedom or reforming, suitably, the aforementioned displacement distributions, one may iteratively improve the efficiency of the theory until a sufficient degree of accuracy is achieved for the results obtained.     

Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory

An analytical solution of the static governing equations of exponentially graded plates obtained by using a recently developed higher order shear deformation theory (HSDT) is presented. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of exponentially graded plates and boundary conditions are derived by employing the principle of virtual work. A Navier-type analytical solution is obtained for such plates subjected to transverse bi-sinusoidal loads for simply supported boundary conditions. Results are provided for thick to thin plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and with other well-known trigonometric shear deformation theory. From the obtained results, it can be concluded that the present HSDT theory predict with good accuracy inplane displacements, normal and shear stresses for thick exponentially graded plates.     

Solutions of a twelfth order thick plate theory

Summary A system of equilibrium equations governing a twelfth-order theory for the bending of thick plates is shown to be equivalent to a biharmonic equation together with four Helmholtz equations. These equations are closely related to equations derived by Cheng for an elasticity based thick plate theory. Detailed comparisons between the solutions for the displacements and stresses predicted by the approximate plate theory and an exact theory give some basis for deciding the applicability of the plate theory. As an example of the application of the solution procedure presented here, some earlier results for the decay parameters for the end problem for finite width plates are extended to the present case of twelfth-order plate theory.With 5 Figures     

The exact theory of one-dimensional quasicrystal deep beams

Without employing ad hoc assumptions, various equations and solutions for quasicrystal beams are deduced systematically and directly from the plane problem of one-dimensional quasicrystals. These equations and solutions can be used to construct the exact theory of deep beams for extension or compression and bending deformation forms. A method for the solution of two-dimensional equations is presented, and with the method the exact theory can now be explicitly established from the general solution of quasicrystals and the Lur’e method. The exact governing equations for beams under transverse loadings are derived directly from the exact beam theory. In three illustrative examples of quasicrystal beams it is shown that the exact or accurate solutions can be obtained by use of the exact theory.     

饱和弹性裂隙介质的混合物理论

本文首先用损伤力学的方法,按孔隙的配置及几何结构,分别定义了含各向异性分布裂隙的固体介质的二阶连续法向裂纹张量和切向裂纹张量。然后,在裂隙内充满流体时,对组分速度、组分偏应力等混合物理论的基本变量进行了各向异性修正。并用混合物理论,建立了饱和裂隙介质中各组分的质量和动量平衡方程。最后,在仅考虑裂纹的单一张开度时,针对线弹性骨架材料,得到了由不可压缩材料构成的各组分的动力学控制方程。     

Hygrothermomechanical evaluation of porous media under finite deformation. Part I - finite element formulations

The coupled thermomechanical responses of fluid-saturated porous continua subjected to finite deformation are investigated. Field equations governing the transient response of the media are derived from a continuum thermodynamics mixture theory based on mass balance, momentum balance and energy balance laws as well as the Clausius-Duhem inequality. Finite element procedures for the two-dimensional response, employing updated Lagrangian formulations for the solid skeleton deformation and the weak formulations for fluid and thermal transport equations, are implemented in a fully implicit form. Temperature-dependent mechanical properties for the non-linear solid matrix, characterized by Perzyna's viscoplastic model, are assumed. An iterative scheme based on the full Newton-Raphson method is presented for simultaneously solving the coupled non-linear equations.     

Free vibration analysis of generally laminated beams via state-space-based differential quadrature

A new method of state-space-based differential quadrature is presented for free vibration of generally laminated beams. By discretizing the state space formulations along the axial direction using the technique of differential quadrature, new state equations at discrete points are established. Applying end conditions and using matrix theory, the general solution is derived. Taking account of the boundary conditions at the top and bottom planes, frequency equation governing the free vibration of generally laminated beams is then formulated. The method is validated by comparing numerical results with that available in the literature.     

A Three-Dimensional Asymptotic Theory of Laminated Piezoelectric Shells

An asymptotic theory of doubly curved laminated piezoelectric shells is developed on the basis of three-dimensional (3D) linear piezoelectricity. The twenty-two basic equations of 3D piezoelectricity are firstly reduced to eight differential equations in terms of eight primary variables of elastic and electric fields. By means of nondimensionalization, asymptotic expansion and successive integration, we can obtain recurrent sets of governing equations for various order problems. The two-dimensional equations in the classical laminated piezoelectric shell theory (CST) are derived as a first-order approximation to the 3D piezoelectricity. Higher-order corrections as well as the first-order solution can be determined by treating the CST equations at multiple levels in a systematic and consistent way. Several benchmark solutions for various piezoelectric laminates are given to demonstrate the performance of the theory.     

Bending of a Thin Rectangular Isotropic Micropolar Plate

In this note, deflection of a thin rectangular isotropic micropolar plate is observed under the influence of transverse loading. From the properties of asymptotic solution, a set of hypotheses is considered and the corresponding governing equations of bending are derived. The solutions are validated by comparing the numerical results and with their counterparts reported in literature for classical Timoshenko plate theory and the Kirchhoff’s theory of plate deformation.     

A new higher order shear deformation theory for sandwich and composite laminated plates

A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.     

Refined theory for the analysis of laminated orthotropic structures

A new higher-order theory for the analysis of laminated orthotropic plates and shells subject to both mechanical and thermal loads is developed. Using the variational approach the system of governing differential equations and corresponding boundary conditions are derived. Two refined models of the stress and strain state are considered, their application and accuracy are discussed. The analytical solution is obtained for plates and shells with the Navier boundary conditions on the side surfaces. The results of calculations are given and compared with an exact three-dimensional solution available in the literature. The influence of the laminated structure upon the exactness of results and the characteristics of stress–strain state is studied and discussed.     

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory

This article proposes a higher-order shear deformation beam theory for free vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in a thermal environment. The temperature-dependent material properties of functionally graded carbon nanotube-reinforced composite beams are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. The governing equations and boundary conditions are derived by using Hamilton's principle, and the Navier solution procedure is used to achieve the natural frequencies of the sandwich beam in a thermal environment. A parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio, and core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Numerical results are also presented in order to compare the behavior of sandwich beams including uniformly distributed carbon nanotube-reinforced composite face sheets to those including functionally graded carbon nanotube-reinforced composite face sheets.     

A hybrid finite element formulation of the linear biphasic equations for hydrated soft tissue

A hybrid finite element method has been developed for application to the linear biphasic model of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive governing equations for its mechanical behaviour. These equations are time dependent, involving both fluid and solid velocities and solid displacement, and will be solved by spatial finite element and temporal finite difference approximation. The first step in the derivation of this hybrid method is application of a finite difference rule to the solid phase, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C° continuous interpolations of the solid and fluid phase velocities and discontinuous interpolations of the pore pressure and elastic stress, is then derived. The stress and pressure functions are chosen so that the total momentum equation of the mixture is satisfied; they are jointly referred to as an equilibrated stress and pressure field. The corresponding weighting functions are chosen to satisfy a relationship analogous to this equilibrium relation. The resulting matrix equations are symmetric. As an illustration of the hybrid biphasic formulation, six-noded triangular elements with complete linear, several incomplete quadratic, and complete quadratic stress and pressure fields in element local co-ordinates are developed for two dimensional analysis and tested against analytical solutions and a mixed-penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred elements are identified on the basis of these results.