Effects of transverse shear on the nonlinear bending of rectangular plates laminated of bimodular composite materials

This paper investigates the application of Dynamic-Relaxation (DR) method to the problems of nonlinear bending of rectangular plates laminated of bimodular composite materials. The classical lamination theory and a shear deformation theory of layered composite plates, taking account of large rotations (in the von Karman sense) are employed separately to analyze the subject. It has been found here that the estimation of the fictitious densities which control the convergence and numerical stability of nonlinear DR solution considering transverse shear effect still needs to be further investigated. In this paper, a procedure to calculate fictitious densities has been presented; hence the numerical stability of this topic has been ensured. In this paper the main steps of solving the nonlinear bending of bimodular composite laminates by means of DR method are outlined. The numerical results are given for simply supported, two-layer cross-ply rectangular plates made of mildly bimodular material (Boron-Epoxy (B-E)) and highly bimodular materials (Aramid-Rubber (A-R) and Polyester-Rubber (P-R)) under sinusoidally distributed and uniformly distributed transverse loads. The results obtained have been compared with linear results and those obtained for laminates fabricated from conventional composite materials, the elastic moduli of which are identical with the tensile moduli of the bimodular materials. In addition, the effect of transverse shear deformation on the nondimensionalized center deflection has been studied.The main contents of this paper were presented at the International Symposium of Composite Materials and Structures (June 1986, Beijing).The authors thank Prof. Zhou Li for his guidance.     

A two parameter elasto-plastic formulation for hardening pressure-dependent materials

A new pressure-dependent yield function is proposed by introducing a plastic Poisson's ratio within the theoretical formulation of the plastic potential. In analogy with other classical models, an equivalent stress and an equivalent plastic strain increment are defined. Then, according to these definitions, the equivalent stress–strain curve is derived and an exponential hardening law is introduced. The advantage of the proposed formulation over alternative approaches relies in explicit closed-form expressions of the flow rules and of the plastic multiplier.     

The applications of generalized variational principles in nonlinear structural analysis

This paper discusses the generalized variational principles founded by thetechnique of Lagrangian multipliers in structural mechanics and analyzes the nonlinearstatically indeterminate structures.It is assumed that the stress-strain relationship ofthe materials of structures has the form ofσ=Bε~(1/m)orτ=Cγ~(1/m),namely,thephysical equations of structures have the shape of exponential functions.Severalexamples are given to illustrate the statically indeterminate structures such as thetrusses,beams,frames and torsional bars.     

A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials

In relation to electroelastic media with thermopiezoelectric coupling, the system of one-dimensional equations is consistently derived so as to accommodate the high-frequency vibrations of a rod with temperature-dependent material. In the first part of the paper, a unified variational principle of differential type is presented which describes the fundamental equations of thermopiezoelectricity with second sound, including the physical and geometrical non-linearities. In the second part, the hierarchic system of rod equations is systematically deduced from the three-dimensional fundamental equations by use of Mindlin's method of reduction. The hierarchic system of equations which is derived in both differential and variational forms is capable of predicting the extensional, thickness-shear, flexural and torsional as well as coupled vibrations of the rod of uniform cross-section. All the higher-order effects are taken into account as deemed pertinent in any particular case. In the third part, attention is confined to certain cases involving special motions, materials and geometry. Besides, the uniqueness is investigated in solutions of the linearized system of rod equations and the sufficient conditions are enumerated for the uniqueness of solutions.     

A nonlinear finite element formulation for axisymmetric torsion of biphasic materials

In this paper, we present a finite element formulation for describing the large deformation torsional response of biphasic materials, with specific application to prediction of nonlinear coupling between torsional deformation and fluid pressurization in articular cartilage. Due to the use of a cylindrical coordinate system, a particular challenge arises in the linearization of the weak form. The torsional axisymmetric case considered gives rise to additional geometric terms, which are important for the robustness of the numerical implementation and that would not be present in a Cartesian formulation. A detailed derivation of this linearization process is given, couched in the context of a variational formulation suitable for finite element implementation. A series of numerical parametric studies are presented and compared to experimental measurements of the time dependent response of cartilage.     

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

各向异性弹性力学问题Hamilton正则方程的一般形式

本文从修正后的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，导出了由２１个弹性常数组成的各向异性材料的混合方程，井证明它们即是Ｈａｍｉｌｔｏｎ正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Ｈａｎｉｌｔｏｎ正则形式。     

THE MATHEMATICAL MODELS AND GENERALIZED VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS FOR PERFORATED THIN PLATES

ＴＨＥＭＡＴＨＥＭＡＴＩＣＡＬＭＯＤＥＬＳＡＮＤＧＥＮＥＲＡＬＩＺＥＤＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＳＯＦＮＯＮＬＩＮＥＡＲＡＮＡＬＹＳＩＳＦＯＲＰＥＲＦＯＲＡＴＥＤＴＨＩＮＰＬＡＴＥＳＣｈｅｎｇＣｈａｎｇｉｕｎ（程昌钧）；ＹａｎｇＸｉａｏ...     

Energy principles in theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], various energy principles in theory of elastic materials with voids can be established systematically. In this paper, an important integral relation is given, which can be considered essentially as the generalized pr. inciple of virtual work. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem of work in theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field generalized variational principles, and the principle of minimum potential and complementary energies. Furthermore, with this appro ach, the intrinsic relationship among various principles can be explained clearly. The project supported by the National Natural Science Foundation of China     

A simplified model for nonlinear dynamic analysis of steel column subjected to impact

This paper presents a new simplified model of the nonlinear dynamic behavior of a steel column subjected to impact loading. In this model, the impacted column, which undergoes large displacement, consists of two rigid bars connected by generalized elastic–plastic hinges where the deformation of the entire steel column as well as the connections is concentrated. The effect of the rest of the structure on the column is modeled by an elastic spring and a point masse both attached to the top end of the column which is also loaded by a compressive force. The plastification of the hinges follows the normality rule with a yield surface that accounts for the interaction between M and N. The latter is described by a super-elliptic yield surface that allows ones to consider a wide range of convex yield criterion by simply varying the roundness factor that affects the shape of the limit surface. By including these features, the model captures both geometry and material nonlinearities. Both the flow rule and the equations of motion are integrated using the midpoint scheme that conserves energy. The non-smooth nature of impact is considered by writing the equations of motion of colliding masses using differential measures. Contact conditions are written in terms of velocity and combined with Newton's law to provide the constitutive law describing interactions between masses during impact. Numerical applications show that the model is able to capture the behavior of a column subjected to impact.     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF HETEROGENEOUS MATERIALS WITH VORONOI FINITE ELEMENT METHOD

The Voronoi cell finite element method (VCFEM) is adopted to overcome the limitations of the classic displacement based finite element method in the numerical simulation of heterogeneous materials. The parametric variational principle and quadratic programming method are developed for elastic-plastic Voronoi finite element analysis of two-dimensional problems. Finite element formulations are derived and a standard quadratic programming model is deduced from the elastic-plastic equations. Influence of microscopic heterogeneities on the overall mechanical response of heterogeneous materials is studied in detail. The overall properties of heterogeneous materials depend mostly on the size, shape and distribution of the material phases of the microstructure. Numerical examples are presented to demonstrate the validity and effectiveness of the method developed.     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center     

三维编织复合材料渐进损伤的非线性模型及强度分析

建立了考虑周期性位移边界条件的细观体胞模型,对三维编织复合材料的渐进损伤过程进行数值模拟。采用Eshelby-Mori—Tanaka方法计算含损伤裂纹的材料的剐度矩阵,并将有限元网格尺寸和单元裂纹尺寸引入损伤演化方程,有效地降低了模拟结果对有限元网格的依赖程度。通过计算得到了材料应力应变的非线性关系和失效时的极限强度,并分析了材料的破坏机理。结果表明,大编织角材料的破坏模式主要是基体失效与纤维横向拉剪破坏,模拟计算结果与文献中的实验值吻合较好。     

圆柱形含能材料热爆炸临界参数计算

根据变分原理,计算了各表面散热条件不同的有限长圆柱含能材料的热爆炸临界参数和临界温度,研究了长径比对热爆炸临界参数和临界温度的影响;计算了各表面散热条件不同的有限长圆柱材料的热爆炸临界参数,得出了Biot数和对热爆炸临界参数和临界温度的作用。计算表明,用变分的方法将求解热爆炸导热方程问题转化为对一个特征值方程求解,求解过程要简单得多,且精度高、速度快。     

A matrix method of displacement analysis of the general spatial 7R mechanism

An input-output equation of the general spatial 7R mechanism is derived in this paper by using the method in [1] and applying the rotation matrices. The result is the same as [2], but the process of derivation is simpler. Applying the character of rotation matrices, it is not difficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors, to calculate the scalar products and triple products of these unit vectors, and to derive the 6th constraint equation. Moreover, an algorithm, which consists of successive applications of row transformation and expansion based on Laplace’s Theorem, is given to evaluate the 16 × 16 determinant according to its characteristic, so that the evaluation is much simplified.     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.     

A finite element formulation for analysis of functionally graded plates and shells

Summary A finite element formulation is derived for the thermoelastic analysis of functionally graded (FG) plates and shells. The power-law distribution model is assumed for the composition of the constitutent materials in the thickness direction. The procedure adopted to derive the finite element formulation contains the analytical through-the-thickness integration inherently. Such formulation accounts for the large gradient of the material properties of FG plates and shells through the thickness without using the Gauss points in the thickness direction. The explicit through-the-thickness integration becomes possible due to the proper decomposition of the material properties into the product of a scalar variable and a constant matrix through the thickness. The nonlinear heat-transfer equation is solved for thermal distribution through the thickness by the Rayleigh-Ritz method. According to the results, the formulation accounts for the nonlinear variation in the stress components through the thickness especially for regions with a variation in martial propperties near the free surfaces.     

A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress–strain curves of human skins. The underlying relations between the J-shaped stress–strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress–strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress–strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress–strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress–strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.     

Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure

Based on the assumption of finite deformation, the Hamilton variational principle is extended to a nonlinear elastic Euler-type beam-column structure located on a nonlinear elastic foundation. The corresponding three-dimensional (3D) mathematical model for anaiyzing the nonlinear mechanical behaviors of structures is established, in which the effects of the rotation inertia and the nonlinearity of material and geometry are considered. As an application, the nonlinear stability and the post-buckling for a linear elastic beam with the equal cross-section located on an elastic foundation are analyzed.One end of the beam is fully fixed, and the other end is partially fixed and subjected to an axial force. A new numerical technique is proposed to calculate the trivial solution,bifurcation points, and bifurcation solutions by the shooting method and the Newton-Raphson iterative method. The first and second bifurcation points and the corresponding bifurcation solutions are calculated successfully. The effects of the foundation resistances and the inertia moments on the bifurcation points are considered.     

An analytical formulation in 3D domain for the nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength within the body. Some of the nonlinear phenomena observed under weak electric fields near resonance frequency excitation are the presence of superharmonics in the response spectra and the jump phenomena etc. In this work, an analytical solution for the nonlinear response of rectangular piezoceramic slabs have been obtained by Rayleigh–Ritz method and perturbation technique in the 3-D domain using a generalized nonlinear electric enthalpy density function. Forced vibration experiments (excitation with electric field) have been conducted on a rectangular piezoceramic slab at varying electric field amplitudes and the analytical solutions have been shown to compare very well with the experimental results.