The dynamic response of composite sandwich beams to transverse impact

The dynamic response of glass fibre–vinylester composite beams is measured by impacting the beams at mid-span with metal foam projectiles. The beams exist in composite monolithic form, and in sandwich configuration with composite face-sheets and a core made from PVC foam or end-grain balsa wood. High-speed photography is used to measure the transient transverse deflection of the beams and to record the dynamic modes of deformation and failure. For both monolithic and sandwich configurations, a flexural wave travels from the impact site towards the supports. Ultimate failure of the monolithic and sandwich beams is by tensile tearing of the faces. The sandwich beams also exhibit cracking of the core, and face-sheet delamination. The dynamic strength of the beams is quantified by the maximum transient transverse deflection at mid-span of the beams as a function of projectile momentum. It is demonstrated that sandwich beams can outperform monolithic beams of equal mass. The trade-off between core strength and core thickness is such that a low density PVC foam core outperforms a higher density PVC foam core. End-grain balsa wood has a superior stiffness and strength to that of PVC foam in compression and in shear. Consequently, sandwich beams with a balsa core outperform beams with a PVC foam core for projectiles of low momentum. The order reverses at high values of projectile momentum: the sandwich beams with a balsa wood core fail prematurely in longitudinal shear by splitting along the grain.     

Determining the contact force during the transverse impact of plates

The contact force during the transverse impact of a plate is determined from dynamic strain-gage measurements made on the plate. Experimental results for the impact of an aluminum plate are presented, and comparisons are made with finite-element predictions and measurements from a force transducer. Paper was presented at the 1986 Spring Conference on Experimental Mechanics held in New Orleans, LA on June 8–13.     

Low-velocity impact of sandwich composite plates

In this paper we investigate impact and compression after impact properties of plain weave carbon fiber sandwich composites. Impact tests were conducted on different sample types to obtain information about absorbed energy and maximum impact force. The different samples consisted of foam-filled and hollow honeycomb cores with four-layer carbon fiber facesheets on one or both sides. The impact and compression after impact data provided valuable information to allow for comparisons between the different sample types. Also, the compression after impact tests were conducted in order to determine the reduction in compressive strength when comparing impacted to non-impacted samples. In conclusion, a two-degrees-of-freedom spring/mass model was compared to experimental results. The comparison helped illustrate the limitations of current impact theory. This paper was presented, in part, at a symposium honoring Dr Christian P. Burger, Novel Applications of Experimental Methods in Mechanics, held at the 2003 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, June 2–4, 2003, Charlotte, North Carolina.     

A BEM solution to transverse shear loading of composite beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of composite beams of arbitrary constant cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross-section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear centre of the cross-section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.     

A simple calculation of the transverse impact on beams and its experimental verification

The tests of several scientific workers have shown that the central maximum loading by transverse impact on beams is independent of the boundary conditions. In this case, the length of the beam is so long that the elastic waves reflected from the supports return to the point of contact after the central peak stress has developed. Now, by these conditions it is possible to simplify the integral equation for the impact force due to the transverse impact. The simplification is realized by calculating separately the series of the eigenfunctions and the eigenfrequencies. Introducing a special reference time and dimensionaless variables, the altered integral equation may be treated in a relative simple way by a computer. The results given in dimensionless form, namely the impact force and the central bending strain as a function of time, allow quickly the calculation of the mechanical loading. The magnitudes of the impact values also depend on a parameter which may be signified as a characteristic value of the transverse impact. This parameter contains the impact velocity, the striking mass, the Hertz constant and a parameter of the beam. This simplified theory of transverse impact was verified by measurements with strain gages, displacement device and PhotoStress method. The contact time was also measured and the data of the tests were changed, i.e., the velocity of the striking mass and the magnitude of the mass. In order to obtain the axial central strain at the lower side of the beam, the added action of the impact force must be taken into account according to the theory of Wilson-Stokes. Using the normalized curves of data of the impact, it is possible for untrained engineers to calculate the mechanical loading.     

An experimental investigation on impact response of laminated composite beams

Experiments based on a new technique were carried out to study the response of glass/epoxy laminated composite beams subjected to impact loading. A number of glass/epoxy composite beams with different fiber orientations, spans, thicknesses, and support conditions were impacted with an instrumented impact hammer and the responses were picked up using an accelerometer in conjunction with a spectrum analyzer. Free vibration test for some sample beams were also carried out to determine their fundamental frequencies.     

Research note on evaluation of the plastic deformation of beams under transverse impact

The permanent deformation of a rigid perfectly plastic beam subjected to a time-dependent line load is calculated. The beam has finite length, and the load is applied transversely in the middle of the span. At the initial time, the load is zero. Thereafter, the load has a triangular time dependence. For sufficiently large values of the peak load, static and moving plastic hinges appear, and the beam is subjected to plastic bending. Previous studies of this problem have provided a numerical solution of a system of approximate equations. In this article, a numerical solution of the system of exact equations is provided. The numerical results represent a significant improvement over those that were previously available.     

Behavior of beams under transverse impact according to higher-order beam theory

Using Reissner’s principle, we formulated an equation of motion for a beam according to higher-order beam theory. We derived the Laplace transform of the equation and investigated wave-propagation behavior under transverse impact. In other words, we studied the effect of the nonlinear component of axial-warping, which cannot be determined by a conventional approach such as the Timoshenko beam theory. Specifically, we derived the transfer matrices for finite and semi-infinite beams. By choosing the appropriate state quantities, arrangement as a vector, and definition of sign convention, we were able to derive a perfect “reciprocal relation.” In spite of the complicated Laplace inverse transform, we obtained an accurate and rapid solution by investigating appropriate branch points and poles and setting branch cuts. The extent of the nonlinear warping effect and its region of influence were also clearly demonstrated.     

Low-energy impact response of composite and sandwich composite plates with piezoelectric sensory layers

An efficient model reduction based methodology is presented for predicting the global (impact force, plate deflection and electric potential) and through-thickness local (interfacial strains and stresses) dynamic response of pristine simply-supported cross-ply composite and sandwich composite plates with piezoelectric sensory layers subjected to low-energy impact. The through-thickness response of the laminate is modelled using coupled higher-order layerwise displacement-based piezoelectric laminate theories. Linearized contact laws are implemented for simulating the impactor–target interaction during impact. The stiffness, mass, piezoelectric and permittivity matrices of the plate are formulated from ply to structural level and reduced by applying a Guyan reduction technique to yield the structural system in state space. This reduction technique enables the formulation of a plate–impactor structural system of minimum size (1 term per vibration mode for composite plates – 2 terms for sandwich plates) and reduces computational cost, thus facilitating applicability for real-time impact and vibration control.     

Determining transverse impact force on a composite laminate by signal deconvolution

Dynamic impact forces on a composite structure were recovered by using experimentally generated Green's functions and signal deconvolutions. The signal processing is straightforward. Extra windowing and filtering the recorded signals are unnecessary. The Green's functions account for boundary conditions, material properties and structure geometry. This approach can be applied to linearly elastic structures with different boundary conditions. It is realistic and convenient to use for the recovery of impact force on anisotropic or isotropic solid structures.     

双层夹芯复合材料结构横向冲击响应实验

采用玻璃纤维增强环氧树脂复合材料层合板作为内、外面板,以PVC泡沫作为芯材,构造了双层夹芯复合材料结构。采用落锤冲击实验,得到了冲击过程的撞击力历史;研究了在不同的冲击能量下,双层夹芯结构的冲击响应及内面板位置对双层夹芯结构冲击响应的影响。实验结果表明,内面板的引入及内面板的位置显著影响双层夹芯结构的撞击力历史,根据该撞击力历史可以优化设计出抗冲击性能优异的新型双层夹芯复合材料结构。     

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.     

Effects of joining techniques on impact perforation resistance of assembled composite plates

A previous study on impact response of composite laminates concluded that impact perforation was the most important damage stage in composite laminates subjected to impact loading, since impact characteristics (peak force, contact duration and absorbed energy) and mechanical properties degradation of composite laminates reached critical points once perforation took place. It was also found that thickness had a greater influence on impact perforation resistance than did in-plane dimensions. However, as the composite laminates became very thick, the manufacturing cost for obtaining high-quality composite laminates increased. In an effort to meet design requirements and reduce manufacturing costs, assembled composite plates, which were organized by assembling multiple thin composite laminates, were considered as alternatives for thick single-laminate composite plates. Various joining techniques including mechanical riveting, adhesive bonding and stitch joining, and their combinations, were used in assembling two- and three-laminate plates. Experimental results revealed that adhesive bonding outperformed other joining techniques. Although good bonding resulted in higher joining (bending) stiffness and subsequently higher perforation thresholds, increasing the laminate thickness or the number of laminates was found to be more efficient in raising perforation threshold than in improving the joining stiffness. The assembled three-laminate plates were found to have higher perforation thresholds than their thick single-laminate counterpart.     

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.     

含脱层损伤复合材料层合梁的冲击动力屈曲

针对含初始缺陷和脱层损伤的复合材料层合梁的轴向冲击动力屈曲问题进行了分析。基于Hamilton原理导出了考虑初始缺陷、轴向和横向惯性、横向剪切变形以及转动惯性影响时含脱层损伤复合材料梁的非线性动力屈曲控制方程;基于B-R准则,采用有限差分方法求解了受轴向冲击载荷作用下含脱层损伤复合材料梁的动力屈曲问题;讨论了冲击速度、初始几何缺陷、铺层角度以及脱层长度等因素对复合材料层合梁动力屈曲的影响。     

Analytical model for delamination growth during small mass impact on plates

An analytical model is presented for delamination initiation and growth and the resulting response during small mass impact on orthotropic laminated composite plates, which typically is caused by runway debris and other small objects. The solution is obtained by a fast stepwise numerical solution of a single integral equation. Delamination size, load and deflection history are predicted by extension of an earlier elastic impact model by the author. Good agreement is demonstrated in comparisons with finite element simulations and experiments.     

Dynamic response and damage analysis of fiber-reinforced composite laminated plates under low-velocity oblique impact

Fiber-reinforced composite laminates (FRCL) is susceptible to the external impacting. Understanding the crack propagation and structural mechanical properties of the damaged FRCL under low-velocity oblique impact is of great value in practical application. A new analytical dynamic model is developed in this work to research the dynamic response and damage property of FRCL under oblique impacting. The displacement field and strain–displacement relations of the FRCL are established by utilizing higher-order shear plate theory. The matrix damage and fiber rupture in FRCL under oblique impacting are captured by an internal variable-based continuum damage constitutive relation. To accurately predict the oblique impacting force, an analytical dynamic impacting model is proposed basing on a developed contact model, where normal and tangential contact is coupled and solved simultaneously. The whole initial boundary value problem is iteratively solved by synthetically using finite differential method and Newmark-\(\beta \) method. The solving convergence and accuracy of the model is demonstrated and validated. Simulations show that the matrix damage is more easily to appear in FRCL under shear force due to oblique contact when under oblique impacting, and the damage profile is different from normal impacting. The dynamic responses of the FRCL plate under oblique impacting differ also greatly from normal impacting. The current research provides a theoretical basis for FRCL design and its engineering application when under low-velocity impacting.