Interlaminar stresses in antisymmetric angle-ply cylindrical shell panels

Layerwise theory of Reddy is utilized for investigating free-edge effects in antisymmetric angle-ply laminated shell panels under uniform axial extension. Following some physical arguments, governing displacement field is divided into local and global parts. The former is discretized through the shell thickness by a zig-zag interpolation function while the latter is calculated by a first-order shear deformation theory. Local equilibrium equations are then solved through a state space approach. Accuracy of the proposed technical solution is subsequently verified by a novel analytical elasticity solution. For this end, the problem is analytically solved for specific boundary conditions along the edges. The numerical results show excellent agreement between two theories for various composite shell panels.     

Finite element modeling of low-velocity impact on laminated composite plates and cylindrical shells

In this study, composite laminates and shell structures subjected to low-velocity impact are investigated by numerical analysis using ABAQUS finite element code. In order to model the impact phenomena by commercial finite element codes, various procedures are available. Accurate modeling requires the appropriate selection of element type, solution method, impactor modeling method, meshing pattern and contact modeling. In this investigation, by considering several case studies with various conditions, validity of the existed modeling processes is examined. In each case, by comparing the results of various methods with the related available experimental test results in existing literature, the best procedure is proposed which can serve as benchmark method in low-velocity impact modeling of composite structures for future investigations.     

Forced vibration and low-velocity impact of laminated composite plates

The layerwise theory of Reddy is used to study the low velocity impact response of laminated plates. Forced-vibration analysis is developed by the modal superposition technique. Six different models are introduced for representation of the impact pressure distribution. The first five models, in which the contact area is assumed to be known, result in a nonlinear integral equation similar to the one obtained by Timoshenko in 1913. The resulting nonlinear integral equation is discretised using a time-element scheme. Two different interpolation functions, namely: (i) Lagrangian and (ii) Hermite, are used to express the impact force. The Hermitian polynomial-based representation, obviously more sophisticated, is introduced to verify the Lagrangian-based representation. Due to its modular nature the present numerical technique is preferable to the existing numerical methods in the literature. The final loading model, in which the time dependence of the contact area is taken into account according to the Hertzian contact law, resulted in a relatively more complicated but more realistic, nonlinear integral equation. The analytical developments concerning this model are all new and are reported for the first time in this paper. Also a simple, but accurate, numerical technique is developed for solving our new nonlinear integral equation which results in the time-history of the impact force. Our numerical results are first tested with a series of existing example problems. Then a detailed study concerning all the response quantities, including the in-plane and interlaminar stresses, is carried out for symmetric and antisymmetric cross-ply laminates and important conclusions are reached concerning the usefulness and accuracy of the various plate theories. This paper, presented at a symposium to celebrate the Golden Jubilee of the Aerospace Department, Indian Institute of Science, Bangalore first appeared as a paper inComputational Mechanics (1994) 13: 360–379     

Interlaminar shear stresses in laminated composite plates under thermal and mechanical loading

The combined effects of thermal and mechanical loadings on the distribution of interlaminar shear stresses in composite laminated thin and moderately thick composite plates are investigated numerically using the commercially available software package MSC NASTRAN/PATRAN. The validity of the present finite element analysis is demonstrated by comparing the interlaminar shear stresses evaluated using the experimental measurement. Various parametric studies are also performed to investigate the effect of stacking sequences, length to thickness ratio, and boundary conditions on the interlaminar shear stresses with identical mechanical and thermal loadings. It is observed that the effect of thermal environment on the interlaminar shear stresses in carbon-epoxy fiber-reinforced composite laminated plates are much higher in asymmetric cross-ply laminate and anti-symmetric laminate compared to symmetric cross-ply laminate and unidirectional laminate under identical loadings and boundary conditions.     

Interlaminar stresses analysis for laminated composite plates based on a local high order lamination theory

To evaluate the interlaminar stresses in composited laminates, a local high order lamination theory is proposed in this paper. The local displacement fields are expanded in terms of high order polynomial series through thickness within each ply. The displacement continuity constraints at the interface between layers are introduced into the potential energy functional by Lagrange multiplier method. Euler-Lagrange's equations and appropriate boundary conditions associated with the modified potential energy functional are derived. An infinite symmetric cross-ply laminate under cylindrical bending is examined to validate the credibility of the present theory.     

Interlaminar stresses in general thick rectangular laminated plates under in-plane loads

Three-dimensional multi-term extended Kantorovich method (3-D multi-term EKM) is employed to investigate analytically the interlaminar stresses near free edges of finite length general composite laminates subjected to axial and shearing loads. Using the principle of minimum total potential energy, three systems of coupled ordinary differential equations are obtained. Then an iterative procedure is established to achieve analytical solution. Laminates with finite dimensions are considered and full three-dimensional stresses in the interior and the boundary-layer regions are calculated. The results obtained from this theory are compared with those of analytical results existing in the literature or with those obtained by the finite element method when there exist no available results. It is found that interlaminar stresses have excellent agreements with those obtained by layerwise theory in the interior region and near the free edges of both finite and long laminates. Convergence study of the method reveals that the 3-D multi-term EKM converges within only four terms of trial functions and the single-term EKM is not able to estimate the local interlaminar stresses near the boundaries of laminates. Finally, the power of the present approach in obtaining the interlaminar stresses in finite length thick laminated plates with general lay-ups is examined.     

Experimental study of low-velocity impacts on glass-epoxy laminated composite plates

In this investigation, glass-epoxy laminated plates were subjected to crush experimental tests in a SHIMATSU universal traction machine and low-velocity crash impact tests in a drop test IMATEK machine. The results are shown and the dimensions of the damage are evaluated. The characterization of the damage is done in relation to the type of test, the ply stacking sequence, the plate dimensions and the maximum force achieved in the impact. In order to verify if low-velocity impacts can be modelled by crush tests on laminated composites, the results are compared and several relevant ideas are developed.     

Non-linear free vibrations of symmetrically laminated, slightly compressible cylindrical shell panels

A geometrically non-linear dynamic shell theory presented by the authors in an earlier work is used to study the non-linear free vibrations of symmetrically laminated cylindrical shell panels. The theory accounts for arbitrary lamination constructions, anisotropy, and slight compression across the thickness. In this paper, this theory is used to derive the equation of motion of the panel with quadratic and cubic non-linearities and symmetric lamination schemes. The symbolic manipulator Mathematica™ is used to perform the Rayleigh-Ritz procedure and derive a single-mode approximation to the vibration of the panel. The Lindstedt-Poincare perturbation technique is used to analyze the resulting non-linear differential equation of motion and study the effects of non-linearities on the dynamics of free vibrations of the panel. A numerical example of a symmetrically laminated graphite/epoxy shell panel is used to demonstrate the procedure. The numerical example shows that non-linearities are of the hardening type and are more pronounced for smaller opening angles. Moreover, it shows that the larger-amplitude motions are dominated by the lower modes.     

Ballistic impact response of laminated composite panels

Laminated ballistic composite panels are an important part of hard-plate protective body armour and may be subjected to a wide variety of impact conditions depending on the projectile, impact velocity and armour construction, to name a few.     

Damage prediction in composite sandwich panels subjected to low-velocity impact

The paper illustrates the application of a finite element tool for simulating the structural and damage response of foam-based sandwich composites subjected to low-velocity impact. Onset and growth of typical damage modes occurring in the composite skins, such as fibre fracture, matrix cracking and delaminations, were simulated by the use of three-dimensional damage models (for intralaminar damage) and interfacial cohesive laws (for interlaminar damage). The nonlinear behaviour of the foam core was simulated by a crushable foam plasticity model. The FE results were compared with experimental data acquired by impact testing on sandwich panels consisting of carbon/epoxy facesheets bonded to a PVC foam. Good agreement was obtained between predictions and experiments in terms of force histories, force–displacement curves and dissipated energy. The proposed model was also capable of simulating correctly nature and size of impact damage, and of capturing the key features of individual delaminations at different depth locations.     

A study on nanostructured laminated plates behavior under low-velocity impact loadings

The influence of montmorillonite (MMT) silicate layers on glass-fiber-epoxy laminated composites behavior has been investigated by low-velocity impact and X-ray diffraction tests. The glass-fiber-epoxy-nanoclay laminate composites have 16 layers and 65% fiber volume fraction is manufactured by vacuum-assisted wet lay-up. Fibers have a plain-weave configuration with density of 200 g/m², while the epoxy resin system is made of diglycidyl ether of bisphenol A resin with aliphatic amine as the curing agent. The nanoclay (Nanomer I30E) is dispersed into the epoxy system in a 1%, 2%, 5% and 10% ratio in weight with respect to the matrix. X-ray diffraction tests indicate that rather than exfoliated, these nanostructures are mostly in intercalated form, with a possible presence of immiscible nanosystems at 10% concentration. The methodology used for the impact test is based on the ASTM D5628-01 standard. The results have shown that for the four edges clamped condition not only the delamination phenomenon is reduced, but also the damping is increased during the rebounds. Moreover, for the 20 J impact energy condition the energy absorption by delamination increases close to 48%, while for larger energies, i.e. 60 J, the average improvement into energy absorption is around 15%. Even for larger energies close to total perforation, i.e. 80 J, the use of nanoclays leads to an average increase in energy absorption by delamination close to 4%. Finally, the failure mechanism seems to be affected by the nanoclay presence, as the interlaminar failure shifts to a mostly intralaminar failure with the increase of nanoclay content.     

Interlaminar shear stresses around an internal part-through hole in a laminated composite plate

The equilibrium/compatibility method, which is a semi-analytical post-processing method, is employed for computation of hitherto unavailable through-thickness variation of interlaminar (transverse) shear stresses in the vicinity of the bi-layer interface circumferential re-entrant corner line of an internal part-through circular cylindrical hole weakening an edge-loaded laminated composite plate. A C^o${\text{C}}^{\text{o}}$-type triangular composite plate element, based on the assumptions of transverse inextensibility and layer-wise constant shear-angle theory (LCST), is utilized to first compute the in-plane stresses and layer-wise through-thickness average interlaminar shear stresses, which serve as the starting point for computation of through-thickness distribution of interlaminar shear stresses in the vicinity of the bi-layer interface circumferential re-entrant corner line of the part-through hole. The same stresses computed by the conventional equilibrium method (EM) are, in contrast, in serious error in the presence of the bi-layer interface circumferential re-entrant corner line singularity arising out of the internal part-through hole, and are found to violate the interfacial compatibility condition. The computed interlaminar shear stress can vary from negative to positive through the thickness of a cross-ply plate in the neighborhood of this kind of stress singularity.     

Limits of asymptotic solutions in low-velocity impact of composite plates

This paper presents a study of the low velocity impact response of composite plates by using dimensional analysis and lumped-parameter models. Simple analytical expressions are obtained and used to construct a characterization diagram that shows the relationship between the maximum normalized impact force and two non-dimensional parameters. For a given impact event, it is shown that the non-dimensional parameters can be obtained by analytical, computational or experimental methods. Once these parameters are determined, the characterization diagram can be used to estimate the type of response, and maximum impact force. Furthermore, the diagram can be used to select adequate simple models for a given impact situation. It is shown that these simple models provide very good approximations of the impact response for a pre-determined wide range of impact parameters.     

On the stress analysis of cross-ply laminated plates and shallow shell panels

This paper extends the applicability of a new stress analysis method (Soldatos KP, Watson PA. Acta Mech 1997;123:163–186) towards the accurate prediction of stresses within cross-ply laminated doubly curved shell segments having a rectangular plan-form. The method is based on the successful incorporation of three-dimensional elasticity information for stress distributions into a two-dimensional five-degrees-of-freedom shallow shell theory. This successful matching is achieved by means of a set of two shape functions, which are incorporated within the two-dimensional shell model whereas their form depends on the particular problem considered. In the present case, two different sets of shape functions are developed and tested, one of which is more accurate than the other is, the later being however simpler than the former.     

Out-of-plane stresses in composite shell panels: layerwise and elasticity solutions

Boundary-layer effects in lengthy cross-ply laminated circular cylindrical shell panels under uniform axial extension are investigated by two analytical solutions. First, Reddy??s layerwise theory with state-space approach is utilized to determine the local interlaminar stresses. In this method, the general displacement field is discretized through the shell thickness by a linear shape function. When the shell panel is subjected to an axial force, the axial strain is estimated by an equivalent single-layer theory. Second, the stress-function approach along with Fourier series expansion is applied to develop a novel elasticity solution. The elasticity solution, which is based on simply-support edge conditions, is presented to show the effectiveness of the first solution. The numerical results show good agreements. Interlaminar stresses within various symmetric and unsymmetric cross-ply composite shell panels are then calculated and discussed. It is shown that the normal out-of-plane stress can get high magnitudes along the physical interfaces.     

Interlaminar stresses in thick rectangular laminated plates with arbitrary laminations and boundary conditions under transverse loads

In the present study, interlaminar stresses resulting from bending of thick rectangular laminated plates with arbitrary laminations and boundary conditions are analyzed analytically based on a three-dimensional multi-term extended Kantorovich method (3DMTEKM). Using the principle of minimum total potential energy, three systems of coupled ordinary differential equations with non-homogeneous boundary conditions are obtained. Then an iterative procedure is established to achieve analytical solution. The results obtained from this theory are compared with those of analytical solutions existing in the literature. It is found that the present results have excellent agreements with those obtained by layerwise theory. The results show that the multi-term EKM converges within only three terms of trial functions and the single-term EKM is not able to estimate the local interlaminar stresses near the boundaries of laminates. Finally, the power of the present approach in obtaining the interlaminar stresses in thick rectangular laminated plates with general types of boundary conditions and lay-ups is examined.     

Interlaminar stress distribution of composite laminated plates with functionally graded fiber volume fraction

Various functionally graded design methods have been proposed recently for fiber reinforced composite plates. The laminates with variable fiber spacing along the thickness direction are focused on in this paper. Fiber volume ratio distribution functions are defined separately in each single layer. Classic state space method as well as differential quadrature state space method are utilized here for different boundary and plied conditions. For the latter method, a sub-layer based scheme, which has both high accuracy and less numerical capacity, is suggested for functionally graded plates. Numerical examples indicate that the non-uniform distribution of fibers rearranges the stress field, of which the in-plane stresses are sensitive to the fibers’ distribution, while the transverse stresses are not affected so much. In-plane stresses near interfaces would decrease if the fiber ratio reduces in this region, which provides a method to resolve the interfacial stress concentration problems.     

Computation of moments and stresses in laminated composite plates by the boundary element method

This paper presents a boundary element formulation for the computation of moments and stresses at internal and boundary points of laminated composite plates. Integral equations for second transversal displacement derivatives are developed and all derivatives of the fundamental solution are computed analytically. These integral equations are used to compute moments and stresses at internal points. Stresses on the boundary are computed by a procedure that uses integral equations for the first transversal displacement derivatives, derivatives of shape functions, and constitutive relations. The obtained results are in good agreement with finite element results available in literature.     

Continuum damage mechanics approach to composite laminated shallow cylindrical/conical panels under static loading

Static response characteristics and failure load of laminated composite shallow cylindrical and conical panels subjected to internal/external lateral pressure are investigated using continuum damage mechanics approach considering geometric nonlinearity and damage evolution. The damage model is based on a generalized macroscopic continuum theory within the framework of irreversible thermodynamics and enables to predict the progressive damage and failure load. Damage variables are introduced for the phenomenological treatment of the state of defects and its implications on the degradation of the stiffness properties. The analysis is carried out using finite element method based on the first order shear deformation theory. The nonlinear governing equations are solved using Newton–Raphson iterative technique coupled with the adaptive displacement control method to efficiently trace the equilibrium path. The detailed parametric study is carried out to investigate the influences of geometric nonlinearity, evolving damage, span-to-thickness ratio, lamination scheme and semi-cone angle on the static response and failure load of laminated cylindrical/conical panels. It is revealed that the membrane forces due to geometric nonlinearity significantly influence the damage distribution and failure load.