Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

Three-dimensional continuum analysis for a unidirectional composite with a broken fiber

The three-dimensional problem of a periodic unidirectional composite with a penny-shaped crack traversing one of the fibers is analyzed by the continuum equations of elasticity. The solution of the crack problem is represented by a superposition of weighted unit normal displacement jump solutions, everyone of which forms a Green’s function. The Green’s functions for the unbounded periodic composite are obtained by the combined use of the representative cell method and the higher-order theory. The representative cell method, based on the triple discrete Fourier transform, allows the reduction of the problem of an infinite domain to a problem of a finite one in the transform space. This problem is solved by the higher-order theory according to which the transformed displacement vector is expressed by a second order expansion in terms of local coordinates, in conjunction with the equilibrium equations and the relevant boundary conditions. The actual elastic field is obtained by a numerical evaluation of the inverse transform. The accuracy of the suggested approach is verified by a comparison with the exact analytical solution for a penny-shaped crack embedded in a homogeneous medium. Results for a unidirectional composite with a broken fiber are given for various fiber volume fractions and fiber-to-matrix stiffness ratios. It is shown that for certain parameter combinations the use of the average stress in the fiber, as it is employed in the framework of the shear lag approach, for the prediction of composite’s strength, leads to an over estimation. To this end, the concept of “point stress concentration factor” is introduced to characterize the strength of the composite with a broken fiber. Several generalizations of the proposed approach are offered.     

Dynamic stability analysis of finite element modeling of piezoelectric composite plates

The dynamic stability of negative-velocity feedback control of piezoelectric composite plates using a finite element model is investigated. Lyapunov’s energy functional based on the derived general governing equations of motion with active damping is used to carry out the stability analysis, where it is shown that the active damping matrix must be positive semi-definite to guarantee the dynamic stability. Through this formulation, it is found that imperfect collocation of piezoelectric sensor/actuator pairs is not sufficient for dynamic stability in general and that ignoring the in-plane displacements of the midplane of the composite plate with imperfectly collocated piezoelectric sensor/actuator pairs may cause significant numerical errors, leading to incorrect stability conclusions. This can be further confirmed by examining the complex eigenvalues of the transformed linear first-order state space equations of motion. To overcome the drawback of finding all the complex eigenvalues for large systems, a stable state feedback law that satisfies the second Lyapunov’s stability criteria strictly is proposed. Numerical results based on a cantilevered piezoelectric composite plate show that the feedback control system with an imperfectly collocated PZT sensor/actuator pair is unstable, but asymptotic stability can be achieved by either bonding the PZT sensor/actuator pair together or changing the ply stacking sequence of the composite substrate to be symmetric. The performance of the proposed stable controller is also demonstrated. The presented stability analysis is of practical importance for effective design of asymptotically stable control systems as well as for choosing an appropriate finite element model to accurately predict the dynamic response of smart piezoelectric composite plates.     

Three-dimensional modeling of elastic guided waves excited by arbitrary sources in viscoelastic multilayered plates

This paper provides a modal solution for the three-dimensional modeling of Lamb and SH waves excited by sources of arbitrary shape. This solution is applicable to elastic and viscoelastic plates, in the far-field as well as in the near-field regions, under the assumption of transverse isotropy about the thickness direction. The theoretical developments are conducted based on a semi-analytical finite element formulation. This formulation yields a one-dimensional modal problem, fast from a computational point of view, and allows to readily handle heterogeneous materials having depth-varying properties (multilayered, piecewise or continuously varying, functionally graded). The modal solution is shown to be expressed in terms of Hankel functions of multiple order thanks to a proper application of inverse transforms and Cauchy residue calculus. The link between the proposed formulation and a fully analytical approach is discussed. The solution of this paper is then successfully compared to literature results and degenerates to the point source case. Formulas are presented to calculate point source excitabilities from lines sources. These formulas remain valid for non-propagating modes, viscoelastic materials and account for the near-field contribution. Finally, the example of a viscoelastic bilayer waveguide excited by a rectangular source is considered in order to check the theoretical results.     

Strength of a unidirectional composite in tension

International Applied Mechanics -     

Creep of concentrically loaded circular plates

Using a strain-energy approach and the Rayleigh-Ritz procedure, a nonlinear compressible theory was developed to predict the creep deflections of circular plates laterally loaded by a point force acting at the center of the plates. Both clamped and simply supported boundary conditions were considered. The theory applies to transient as well as to steady-state creep. The stress-strain-time relations for the material were represented by a family of isochronous stress-strain curves. For analytical purpose, each curve was approximated by an arc hyperbolic sine function. Experimental data were obtained from eight plates made of high-density polyethylene tested in a controlled-atmosphere room. Material properties were obtained from tension and compression specimens. All test members were subjected to the same load history. An initial load was held constant for a specified time and then increased by 10 percent of its initial value at four equal intervals of time. Good agreement was found between theory and experiment.     

Numerical analysis of delamination growth for stiffened composite laminated plates

A study of postbuckling and delamination propagation behavior in delaminated stiffened composite plates was presented. A methodology was proposed for simulating the multi-failure responses, such as initial and postbuckling, delamination onset and propagation, etc. A finite element analysis was conducted on the basis of the Mindlin first order shear effect theory and the von-K6rm~n nonlinear deformation assumption. The total energy release rate used as the criteria of delamination growth was estimated with virtual crack closure technique (VCCT). A self-adaptive grid moving technology was adopted to model the delamination growth process. Moreover, the contact effect along delamination front was also considered during the numerical simulation process. By some numerical examples, the influence of distribution and location of stiffener, configuration and size of the delamination, boundary condition and contact effect upon the delamination growth behavior of the stiffened composite plates were investigated. The method and numerical conclusion provided should be of great value to engineers dealing with composite structures.     

Elastic-plastic analysis of biaxially loaded center-cracked plates

Center-cracked panels loaded in biaxial tension are examined in this paper. Calibration relations for the J integral and the Q constraint factor are presented for a Ramberg–Osgood power law hardening material under plane stress and plane strain loadings. Two cases are examined: an isolated crack and a periodic array of cracks both under biaxial loading conditions. The latter has previously been studied for plane stress conditions. A number of different J estimation schemes are proposed based on the remote load and displacement and their dependence on geometry, biaxiality, and material properties is discussed. The variation of constraint, as characterised by Q, is also presented for plane stress and plane strain conditions. Simple slip line field solutions are derived for perfectly plastic conditions and the resulting limit load solutions are compared with numerically determined values. Implications for failure of cracked plates under biaxial loading are discussed.     

Birefringent-coating analysis of laterally loaded perforated plates

The purpose of this investigation was to obtain experimental stress and deflection data for thick, circular, simply supported plates, containing circular transverse perforations in square motif, under uniform lateral loading. The stress-concentration factor and the deflection-multiplier factor, the ratio of the maximum principal stress and the maximum deflection of the perforated plate to that of the solid-plate specimen, respectively, were obtained for each perforated specimen. These factors can be conveniently used for the design of tube sheets, perforated heads, or other similar structural components.     

双层纤维压电智能薄板几何非线性建模与分析

纤维压电MFC(Micro-Fiber Composite)的强致动力和高柔性等特点具有广泛的应用前景,但是材料组成结构复杂给建模带来了难处。基于Reissner-Mindlin假设,采用冯卡门非线性、中等转角及大转角几何非线性等理论,建立了MFC压电智能结构的多种几何非线性有限元模型。同时该模型考虑了压电纤维角度变化对结构形变的影响。分别对两种不同结构的MFC进行了建模与仿真,分别是MFC-d31和MFC-d33,前一种主要利用压电d31效应,而后一种主要利用压电d33效应。随后,通过一压电悬臂梁结构实验数据验证了模型的准确性。最后,利用所建模型对一种双层纤维压电智能薄板结构进行了几何非线性的计算与仿真。     

Three-dimensional modeling of tires

Modeling the stress-strain state of pneumatic tires in the conditions of steady-state and transient rolling is of interest for mechanics of composites and computational mechanics and important from the applied point of view. Mechanical models of various levels of complexity can be used for numerical modeling. In quite a few papers, the corresponding models are derived from the theory of orthotropic shells [1]. However, more thorough and accurate studies of the stress-strain state can be carried out on the basis of three-dimensional models based on the elasticity or viscoelasticity equations. As far as Russian authors are concerned, this approach has first been suggested and implemented in [2]. Another, combined approach uses both the shell theory and the three-dimensional equations of elasticity theory [3, 4]. This approach is reasonable, because the tire structure includes both volumes filled with rubber and thin layers of the rubber cord. The rubber cord layers can be considered as a composite whose structural components possess very different properties. Also, it is quite admissible to consider the rubber cord as a structure periodic in the horizontal projection. Note that the mathematical theory of periodic composites has been developed in [5]. Owing to strong anisotropy and inhomogeneity of the material, large shape distortions of the tire, and, in some cases, its large deformations, viscoelastic properties of rubber play an important role, so that the mechanic model of the tire turns out to be quite complex. The large property differences between various structural components make the matrix of the resulting system of linear equations ill-conditioned, which complicates its numerical solution [6].In this paper, theoretical aspects of a three-dimensional tire model and its numerical implementation are considered.     

Experimental-theoretical correlations of impulsively loaded clamped circular plates

Clamped circular plates are impulsively loaded with sheet explosive, and the resulting large-defection response is monitored using a high-speed streak camera, and dynamic-strain measurements. Dynamic and final-plate deflections as well as strain-time histories of various locations on the plates are compared to deflections and strains obtained with the elastic-plastic structural computer program DEPROSS. It is shown that DEPROSS adequately computes the dynamic response of this highly nonlinear biaxial-stress problem.     

Shape forming of shock wave loaded viscoplastic plates

If metal plates are subjected to impulsive loadings a conical shape of the inelastic deformed plates can occur. In the contrary quasi-statically loaded structures, show a more spherical shape of their inelastic deflections. In the present study, an explanation for the development of conical shapes in the case of shock wave loaded plates is proposed based on wave propagation phenomena. The effect of shape forming is studied experimentally with a shock tube and numerically with finite element simulations.     

Crack propagation in rock plates loaded by projectile impact

Photoelastic coatings were bonded to limestone and granite plates which were then struck on edge with a steel projectile. The resulting fracture zone growth was observed with a reflecting polariscope and recorded with an image converter camera. It is shown that the maximum velocity at which cracks propagate under these circumstances is very much higher than predicted by quasi-steady theories. ‘Apparent’ supervelocity cracks can be generated by shock reflection from bounding interfaces, producing tensile waves of sufficient amplitude to activate and join isolated flaw. Photoelastic coatings are also shown to very much enhance surface-crack visualization subsequent to impact.     

Three-dimensional cohesive fracture modeling of non-planar crack growth using adaptive FE technique

In this paper, the three-dimensional adaptive finite element modeling is presented for cohesive fracture analysis of non-planer crack growth. The technique is performed based on the Zienkiewicz–Zhu error estimator by employing the modified superconvergent patch recovery procedure for the stress recovery. The Espinosa–Zavattieri bilinear constitutive equation is used to describe the cohesive tractions and displacement jumps. The 3D cohesive fracture element is employed to simulate the crack growth in a non-planar curved pattern. The crack growth criterion is proposed in terms of the principal stress and its direction. Finally, several numerical examples are analyzed to demonstrate the validity and capability of proposed computational algorithm. The predicted crack growth simulation and corresponding load-displacement curves are compared with the experimental and other numerical results reported in literature.