An explicit formulation of a three-dimensional material damping model with transverse isotropy

A constitutive three-dimensional (3D) damping model is derived for transversely isotropic material symmetry, using the augmented Hooke's law [Intl. J. Solids Struct. 32 (1995) 2835] as a starting point. The proposed material model is tested numerically, via finite-element techniques, on a laminate structure built from stacked aluminium and Plexiglas plates. Effective 3D transversely isotropic material properties are given in terms of homogeneous material damping functions in connection with homogenised elastic laminate properties. Comparisons made between the results from the elastic (undamped) eigenvalue problem of the detailed (layerwise) model of the laminate and the effective 3D elastic model show that the homogenised model is reasonably accurate, in terms of predicted elastic eigenfrequencies for the first 20 modes. The dynamic homogenisation process, with damping included, is evaluated in terms of forced vibration response for the laminate structure, using effective transversely isotropic frequency dependent material properties. The dynamic 3D effective homogeneous material model is found to simulate very closely the detailed model in the studied frequency interval for the numerical test case.     

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.     

Enhanced first-order theory based on mixed formulation and transverse normal effect

An accurate prediction of displacements and stresses for laminated and sandwich plates is presented using an enhanced first-order plate theory based on the mixed variational theorem (EFSDTM) developed in this paper. In the mixed formulation, transverse shear stresses based on an efficient higher-order plate theory (EHOPT) developed by Cho and Parmerter [Cho, M., Parmerter, R.R., 1993. Efficient higher-order composite plate theory for general lamination configurations. AIAA Journal 31, 1299–1306] are utilized and modified to satisfy prescribed lateral conditions, and displacements are assumed to be those of a first-order shear deformation theory (FSDT). Relationships between the modified EHOPT and the FSDT are systematically derived via both the mixed variational theorem and the least-square approximation of difference between in-plane stresses including the transverse normal stress effect. It is shown that the transverse normal stress effect should be considered in predicting the in-plane stresses when the Poisson effect is dominant. The developed EFSDTM preserves the computational advantage of the classical FSDT while allowing for important local through-the-thickness variations of displacements and stresses through the recovery procedure. The accuracy and efficiency of the present theory are assessed by comparing its results with various plate models as well as the three-dimensional exact solutions for thick laminated and sandwich plates.     

Inspection of free energy functions in gradient crystal plasticity

The dislocation density tensor computed as the cud of plastic distortion is regarded as a new constitutive variable in crystal plasticity. The dependence of the free energy function on the dislocation density tensor is explored starting from a quadratic ansatz. Rank one and logarithmic dependencies are then envisaged based on considerations from the statistical theory of dislocations. The rele- vance of the presented free energy potentials is evaluated from the corresponding analytical solutions of the periodic two-phase laminate problem under shear where one layer is a single crystal material undergoing single slip and the second one remains purely elastic.     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

基于余能原理的有限变形问题有限元列式

利用基面力概念,推导了一种基于余能原理的有限变形问题显式有限元列式,可应用于结构的大位移、大转动问题。以基面力为状态变量来表达单元的余能,将有限变形情况下的单元余能分解为变形余能部分和转动余能部分,利用Lagrange乘子法推导出余能原理有限元的控制方程,编制了相应的非线性有限元程序。通过算例分析,说明该列式和程序的可靠性和精确性。     

一种基于路径优化的非平衡自由能预测方法

提出一种基于路径优化的非平衡自由能预测方法。首先,通过建立蛋白质复合物解离的多目标优化模型找出一条能耗小、速度快的拉伸分子动力学SMD(Steered Molecular Dynamics)解离路径,然后,沿此路径计算解离自由能。与Jarzynski按平均力势PMF(Potentials of Mean Force)的外推方法相比,本文方法有较高的预测效率,数值算例也给出了与实验值比较的预测精度。通过拉伸分子动力学模拟还可以揭示小配体与蛋白质之间的解离全过程,为药物设计提供重要的结构信息。     

Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Summary Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.We thank Dipl.-Ing. H. Romanowski for his support and helpful remarks.     

Description of plastic anisotropy effects at large deformations. Part II: the case of transverse isotropy

In a previous paper (see Tsakmakis, 1999) a general thermodynamically consistent (visco-) plasticity theory has been developed, which accounts for anisotropy effects. For simplicity, isotropic hardening has not be regarded, while anisotropy arises from kinematic hardening and orientational evolution of the underlying substructure. In the present paper the capabilities of this theory are discussed for the study case of transverse isotropy. Anisotropy effects are elaborated in the free energy and the yield function by means of structural tensors. Characteristic features of the transversely isotropic model are illustrated for the case of homogeneous simple shear.     

Optimization of energy extraction in transverse galloping

A numerical method to analyse the stability of transverse galloping based on experimental measurements, as an alternative method to polynomial fitting of the transverse force coefficient C_z, is proposed in this paper. The Glauert–Den Hartog criterion is used to determine the region of angles of attack (pitch angles) prone to present galloping. An analytic solution (based on a polynomial curve of C_z) is used to validate the method and to evaluate the discretization errors. Several bodies (of biconvex, D-shape and rhomboidal cross sections) have been tested in a wind tunnel and the stability of the galloping region has been analysed with the new method. An algorithm to determine the pitch angle of the body that allows the maximum value of the kinetic energy of the flow to be extracted is presented.     

Alternative state space formulations for magnetoelectric thermoelasticity with transverse isotropy and the application to bending analysis of nonhomogeneous plates

By introducing two displacement functions and two stress functions, the governing equations of the linear theory of magneto-electro-thermo-elasticity with transverse isotropy are simplified. On selecting certain physical quantities as the basic unknowns, two new state equations are established. Each of them is order reduced when compared with the one reported recently in literature, leading to a higher numerical efficiency. The material inhomogeneity along the axis of symmetry (z-direction) can be taken into account and an approximate laminate model is employed to facilitate deriving analytical solutions. The validity of new formulations is examined by considering a laminated magneto-electro-elastic rectangular plate and good agreement is obtained with existent results. A plate with a functionally graded property is then analyzed. The effect of magnetoelectric coupling in a BaTiO₃–CoFe₂O₄ composite predicted from the micromechanics simulation is studied quantitatively.     

A curvature‐free multiphase flow solver via surface stress‐based formulation

In this note, we show the link between the classical continuous surface stress and continuous surface force approaches together with special finite element method techniques toward a fully implicit level set method. Based on a modified surface stress formulation, neither normals nor curvature has to be explicitly calculated. The method is space‐dimension independent. Prototypical numerical tests of benchmarking character for a rising 2D bubble are provided for validating the accuracy of this new approach. We show additionally that the explicit redistancing can be avoided using a nonlinear PDE so that a fully implicit and even monolithic formulation of the corresponding multiphase problem gets feasible.     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

An enhanced single-layer variational formulation for the effect of transverse shear on laminated orthotropic plates

A novel mixed formulation is derived by means of Reissner's variational approach-based on Castigliano's principle of least work in conjunction with a Lagrange multiplier method for the calculus of variations. The governing equations present an alternative theory for modeling the important three-dimensional structural aspects of plates in a two-dimensional form. By integrating the classical Cauchy's equilibrium equations with respect to the thickness co-ordinate, and enforcing continuity of shear and normal stresses at each ply interface, condenses the effect of the thickness. A reduced system of partial differential equations of sixth-order in one variable, is also proposed, which contains differential correction factors that formally modify the classical constitutive equations for composite laminates. The theory degenerates to classical composite plate analysis for thin configurations. Significant deviations from classical plate theory are observed when the thickness becomes comparable with the in-plane dimensions. A variety of case studies are presented and solutions are compared with other models available in the literature and with finite element analysis.     

An energy formulation of continuum magneto-electro-elasticity with applications

We present an energy formulation of continuum electro-elasticity and magneto-electro-elasticity. Based on the principle of minimum free energy, we propose a form of total free energy of the system in three dimensions, and then systematically derive the theories for a hierarchy of materials including dielectric elastomers, piezoelectric ceramics, ferroelectrics, flexoelectric materials, magnetic elastomers, magnetoelectric materials, piezo-electric–magnetic materials among others. The effects of mechanical, electrical and magnetic boundary devices, external charges, polarizations and magnetization are taken into account in formulating the free energy. The linear and nonlinear boundary value problems governing these materials are explicitly derived as the Euler–Lagrange equations of the principle of minimum free energy. Finally, we illustrate the applications of the formulations by presenting solutions to a few simple problems and give an outlook of potential applications.