Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.     

Postbuckling behaviors of open section composite struts with edge delamination using a layerwise theory

We investigate the mechanical stability of L-section and T-section composite struts with single edge delamination. We propose a solution procedure based on a layerwise theory and the first order shear deformation theory by taking into consideration of the von Karman geometrical nonlinearity. We derive the nonlinear equilibrium equations according to the minimum total potential energy principle, and solve them using Rayleigh–Ritz method and Newton–Raphson method. In modeling the delaminated L-section and T-section struts, we divide the structures into regions, and exert continuity conditions between different regions. The proposed model is capable of analyzing both local buckling of the base laminate and sublaminate as well as the global buckling of the whole structure. We present numerical results to provide an insight into effects of size of delamination on buckling mode and post-buckling behaviors of the struts. We perform the three-dimensional finite element analysis using the ABAQUS commercial software. The results show a very good agreement with those obtained by the analytical method. The results indicate that the presence of delaminations not only reduces the load-carrying capacity of open section struts remarkably, but also plays a pivotal role in the critical buckling load and buckling mode shape of the struts.     

Postbuckling analysis of columns resting on an elastic foundation

The postbuckling response of perfect and geometrically imperfect elastic columns resting on an elastic Winkler type foundation is thoroughly discussed. This is established by employing an approximate analytic technique leading to very reliable results in the vicinity of the critical state. It was found that the critical state of perfect columns is a stable symmetric bifurcation point and consequently there is no sensitivity to initial geometrical imperfections. Moreover, a simple but readily analyzed mechanical model is proposed to simulate the salient features of buckling mechanism of the column on elastic foundation with those of the model. The simplicity, reliability and efficiency of the proposed analysis as well as the successful modeling of the buckling mechanism of the column by that of a single mode mechanical model are illustrated with the aid of numerical examples.     

Materially and geometrically non-linear woven composite micro-mechanical model with failure for finite element simulations

A computational micro-mechanical material model of woven fabric composite material is developed to simulate failure. The material model is based on repeated unit cell approach. The fiber reorientation is accounted for in the effective stiffness calculation. Material non-linearity due to the shear stresses in the impregnated yarns and the matrix material is included in the model. Micro-mechanical failure criteria determine the stiffness degradation for the constituent materials. The developed material model with failure is programmed as user-defined sub-routine in the LS-DYNA finite element code with explicit time integration. The code is used to simulate the failure behavior of woven composite structures. The results of finite element simulations are compared with available test results. The model shows good agreement with the experimental results and good computational efficiency required for finite element simulations of woven composite structures.     

A coupled interface-multilayer approach for mixed mode delamination and contact analysis in laminated composites

An accurate laminate model developed by using multilayer shear deformable plate modeling and interface elements, based on fracture mechanics and contact mechanics, is proposed to analyze mixed mode delamination in composite laminates. Perfect adhesion along the undelaminated portion of the delamination plane is simulated by treating interface stiffnesses as penalty parameters, whereas to enforce interface displacement continuity between plate elements constituting each sub-laminate above or below the delamination plane, the Lagrange multiplier method is used. The governing differential equations are derived through a variational procedure by using a modified total potential energy functional. Results are obtained by numerical integration of the non-linear three-point boundary value problem modeling mixed-mode delamination of the laminate plate subjected to end loading, which accounts also for the frictionless contact condition.The coupling of a penalty procedure with the Lagrange multiplier method, results in an accurate and direct energy release rate evaluation. Comparisons with results available from the literature obtained with a local continuum approach, show that mode partition may be performed to the desired accuracy by refining multilayer plate models for each sub-laminate. In addition, original analytical formulas for mode partition are obtained by coupling the interface approach and fracture mechanics concepts, evidencing the effectiveness of the proposed approach and gaining a better insight into the influence of shear effects on mode decomposition. Numerical computations for practical problems, evidence both the relative simplicity and the efficiency of the proposed model to represent mixed mode interlaminar fracturing as well as crack–face interaction.     

Geometrically non-linear analysis of laminated composite structures using a 4-node co-rotational shell element with enhanced strains

To demonstrate the solutions of linear and geometrically non-linear analysis of laminated composite plates and shells, the co-rotational non-linear formulation of the shell element is presented. The combinations of an enhanced assumed strain (EAS) in the membrane strains and assumed natural strains (ANS) in the shear strains improve the behavior of 4-node shell element. To secure computational efficiency in the incremental non-linear analysis, the present element uses the form of the resultant forces pre-integrated through the thickness. The transverse shear stiffness of the laminates is defined by an equilibrium approach instead of the shear correction factor. Numerical examples of this study show very good agreement with the references.     

Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method

The buckling, postbuckling and postbuckled vibration behaviour of composite skew plates subjected to nonuniform inplane loadings are presented here. The skew plate is modelled using first order shear deformation theory accounting for von-Kármán geometric nonlinearity and initial geometric imperfections. The different types of nonuniform loads that have been considered in this study are concentrated load, partial load and parabolic load. The explicit analytical expressions for prebuckling stress distributions within composite skew plate subjected to three different types of nonuniform inplane loadings are developed by solving plane elasticity problem using Airy's stress function approach. It is observed that the inplane normal stress distributions within the skew plate due to above nonuniform loadings do not become uniform even at mid-section. The generalized differential quadrature (GDQ) method is used to solve the nonlinear governing partial differential equations. It is observed that the postbuckled load carrying capacity of skew plate under concentrated loading is the lowest compared to other nonuniform and uniform loadings.     

Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading.This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.     

应用平板壳单元DKQl6分析板壳结构的几何非线性

应用新近开发的四边形十六自由度离Kirchhoff平板壳单元DKQl6，分析了板壳结构的几何非线性问题，采用Total Lagrange格式，在小应交、中等转动的假定下，建立了该单元几何刚度阵和大位移矩阵．非线性方程采用位移引导或弧长引导的牛顿-拉夫森增量迭代法求解．讨论了网格和加载步效对收敛性的影响，通过对典型算例的计算以及与其它单元的比较，说明了DKQl6单元在板壳结构几何非线性分析中也有良好的精度．     

Postbuckling analysis of functionally graded graphene platelet-reinforced polymer composite cylindrical shells using an analytical solution approach

An analytical approach is proposed to study the postbuckling of circular cylindrical shells subject to axial compression and lateral pressure made of functionally graded graphene platelet-reinforced polymer composite(FG-GPL-RPC). The governing equations are obtained in the context of the classical Donnell shell theory by the von K′arm′an nonlinear relations. Then, based on the Ritz energy method, an analytical solution approach is used to trace the nonlinear postbuckling path of the shell. The effects of several parameters such as the weight fraction of the graphene platelet(GPL),the geometrical properties, and distribution patterns of the GPL on the postbuckling characteristics of the FG-GPL-RPC shell are analyzed.     

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.     

An efficient algorithm for elasto-viscoplastic vibrations of multi-layered composite beams using second-order theory

An efficient time-domain algorithm for plane non-linear flexural vibrations of multi-layered composite beams, which are driven into the inelastic range by severe transverse loadings, is presented. The influence of an axial static preload is considered in the sense of the quasi-linear second-order theory of structures. The inelastic parts of strain are treated as additional sources of selfstress in the linear elastic background-structure, driving the elastic response into the inelastic one. The efficiency of this exact formulation lies in the fact that linear solution techniques can be used in their most powerful form: Rubin's useful formulation for the quasi-static second-order transfer-matrix of linear elastic structures is applied in combination with modal analysis. Having in mind multi-metal beams, the classical lamination theory is assumed to be valid. Beams with overhang composed of ideal elastic-plastic and viscoplastic layers are studied as example structures. The fictitious sources of selfstress are calculated from the different material laws of the layers in a numerical time-stepping procedure, where a generalized midpoint-rule in combination with Crisfield's secant-Newton procedure is used.     

Dynamic bifurcation and sensitivity analysis of non-linear non-planar vibrations of geometrically imperfect cantilevered beams

The non-linear non-planar dynamic responses of a near-square cantilevered (a special case of inextensional beams) geometrically imperfect (i.e., slightly curved) and perfect beam under harmonic primary resonant base excitation with a one-to-one internal resonance is investigated. The sensitivity of limit-cycles predicted by the perfect beam model to small geometric imperfections is analyzed and the importance of taking into account the small geometric imperfections is investigated. This was carried out by assuming two different geometric imperfection shapes, fixing the corresponding frequency detuning parameters and continuation of sample limit-cycles versus the imperfection parameter. The branches of periodic responses for perfect and imperfect (i.e. small geometric imperfection) beams are determined and compared. It is shown that branches of periodic solutions associated with similar limit-cycles of the imperfect and perfect beams have a frequency shift with respect to each other and may undergo different bifurcations which results in different dynamic responses. Furthermore, the imperfect beam model predicts more dynamic attractors than the perfect one. Also, it is shown that depending on the magnitude of geometric imperfection, some of the attractors predicted by the perfect beam model may collapse. Ignoring the small geometric imperfections and applying the perfect beam model is shown to contribute to erroneous results.     

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.The project supported by the National Natural Science Foundation of China (10172023)     

复合材料层合板精化高阶理论及其精化三角形板单元

提出一种新的精化高阶理论，该理论满足层间位移、应力连续条件，由此建立了三角形精化板单元。该单元满足单元间Ｃ１类弱连续条件，其收敛性得到保证，且具有简单、高效率的优点。     

Postbuckling analysis of laminated composite plates subjected to the combination of in-plane shear,compression and lateral loading

This study deals with postbuckling behavior of laminated composite plates under the combination of in-plane shear, compression and lateral loading using an Element-based Lagrangian formulation. Natural co-ordinate-based strains, stresses and constitutive equations are used in the present shell element. The Element-based Lagrangian formulation described in this paper, in comparison with the traditional approaches, is more attractive not only because it uses only single mapping but also it converges faster. In addition, the finite element (FE) formulation based on the assumed natural strain method for composite structures shows excellence from the standpoints of computational efficiency as well as its ability to avoid both membrane and shear locking behavior. The numerical results obtained are in good agreement with those reported by other investigators. In particular, new results reported in this paper show the influence of various types of loading, materials and number of layers on postbuckling behavior.     

Stability analysis of internally damped rotating composite shafts using a finite element formulation

This paper deals with the stability analysis of internally damped rotating composite shafts. An Euler–Bernoulli shaft finite element formulation based on Equivalent Single Layer Theory (ESLT), including the hysteretic internal damping of composite material and transverse shear effects, is introduced and then used to evaluate the influence of various parameters: stacking sequences, fiber orientations and bearing properties on natural frequencies, critical speeds, and instability thresholds. The obtained results are compared with those available in the literature using different theories. The agreement in the obtained results show that the developed Euler–Bernoulli finite element based on ESLT including hysteretic internal damping and shear transverse effects can be effectively used for the stability analysis of internally damped rotating composite shafts. Furthermore, the results revealed that rotor stability is sensitive to the laminate parameters and to the properties of the bearings.     

Viscoelastic analysis of single-component and composite PEG and alginate hydrogels简

Hydrogels, three-dimensional hydrophilic polymer networks, are appealing candidate materials for study- ing the cellular microenvironment as their substantial water content helps to better mimic soft tissue. However, hydrogels can lack mechanical stiffness, strength, and tough- ness. Composite hydrogel systems have been shown to improve upon mechanical properties compared to their single- component counterparts. Poly （ethylene glycol） dimethacrylate （PEGDMA） and alginate are polymers that have been used to form hydrogels for biological applications. Single- component and composite PEGDMA and alginate systems were fabricated with a range of total polymer concentrations. Bulk gels were mechanically characterized using spherical indentation testing and a viscoelastic analysis framework. An increase in shear modulus with increasing polymer con- centration was demonstrated for all systems. Alginate hydro- gels were shown to have a smaller viscoelastic ratio than the PEGDMA gels, indicating more extensive relaxation over time. Composite alginate and PEGDMA hydrogels exhib- ited a combination of the mechanical properties of the con- stituents, as well as a qualitative increase in toughness. Additionally, multiple hydrogel systems were produced that had similar shear moduli, but different viscoelastic behaviors. Accurate measurement of the mechanical properties of hydrogels is necessary in order to determine what parameters are key in modeling the cellular microenvironment.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses

The available accurate shell theories satisfy the interlaminar transverse stress continuity conditions based on linear strain-displacement relations. Furthermore, in majority of these theories, either influence of the transverse normal stress and strain or the transverse flexibility of the shell has been ignored. These effects remarkably influence the non-linear behavior of the shells especially in the postbuckling region. Furthermore, majority of the buckling analyses performed so far for the laminated composite and sandwich shells have been restricted to linear, static analysis of the perfect shells. Moreover, almost all the available shell theories have employed the Love-Timoshenko assumption, which may lead to remarkable errors for thick and relatively thick shells. In the present paper, a novel three-dimensional high-order global-local theory that satisfies all the kinematic and the interlaminar stress continuity conditions at the layer interfaces is developed for imperfect cylindrical shells subjected to thermo-mechanical loads.In comparison with the layerwise, mixed, and available global-local theories, the present theory has the advantages of: (1) suitability for non-linear analyses, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions, considering the transverse flexibility, and releasing the Love-Timoshenko assumption, (3) less required computational time due to using the global-local technique and matrix formulations, and (4) capability of investigating the local phenomena. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.     

An evolutionary search technique to determine natural frequencies and mode shapes of composite Timoshenko beams

Natural frequencies and mode shapes of composite Timoshenko beams are determined by a diversity guided evolutionary algorithm (DGEA) with different boundary conditions. After applying boundary conditions, frequency equation is obtained in determinant form. Then, natural frequencies and consequently mode shapes are obtained using DGEA where the absolute value of determinant is the subject of optimization. Advantages of employing DGEA are: first, all natural frequencies are produced in a simple run, second, its simplicity for implementation and third, the procedure is not computationally prohibitive. Results clearly show the applicability of the proposed method for obtaining natural frequencies and mode shapes.