Indentation Law Research Articles

Prediction of impact damage in composite sandwich plates

This study tracks the development of an analytical model, designed to investigate the dynamic response of a composite sandwich plate subjected to a low-velocity impact. The model is based on the thick anisotropic plate theory, developed by Reddy and Pagano, which takes into account a through-thickness shear. An indentation law, as proposed by Hertz, is used to model the local response under the impactor. The good correlation between the shock and the simulation results testifies the validity of the method as far as the prediction of the contact force and the shear deformation in the core are concerned.

A study of indentation behaviour of sandwich panels supported rigidly

PurposeThe indentation behaviour of sandwich panels is significant to incipient damage and is known to be affected by a number of dominant parameters. However, it is challenging not only to demonstrate how those few dominant parameters influence the indentation behaviour but also to ascertain that such influence was coupled to the variation of the other dominant parameters. The paper aims to discuss these issues.Design/methodology/approachIn this work, the authors adopted a controllable quasi-static testing to carry out a diagnostic interrogation on the nature of incipient damage in laminate-skinned sandwich panels using hemispherical indenter and used photographs taken from the cross-sections of all the cut-up tested specimens, which were stopped both just before and after the initial critical loads, respectively, to confirm the mechanism of the incipient damage. Sandwich panels with aluminium honeycomb core had carbon/epoxy skins of two different thicknesses and lay-ups and hemispherical nosed indenter had three different diameters.FindingsThe authors found that: the incipient damage mechanism in all the panels was combined delamination in the skin and core crushing without debonding; doubling the skin thickness had the significant enhancement on critical load and indentation and this enhancement became greater for the larger indenter diameters; the indenter diameter had the moderate effect on critical load in the thick panels from 8 to 14 mm but had the negligible effect on thin panels and no effect on the thick panels from 14 to 20 mm; varying the skin lay-up or support had little effect on the indentation behaviour.Research limitations/implicationsThese findings were limited to the constant core density and core thickness. Varying the former significantly could alter the findings accordingly.Practical implicationsThe results of this work should be tremendously useful to design and analysis in industrial applications of sandwich structures in aircraft, vehicles, marine vessels and transport carriages for situations involving localised loading and deformation.Originality/valueThe results of this research work is one of the very few that demonstrated a systematic understanding of the indentation behaviour characteristics of sandwich construction, which is vital to the establishment of indentation law for sandwich structures in future.

Analysis of the dynamic behavior of composite plates subjected to impact

This paper presents a simple analytical model of the dynamic response of a structure subjected to a low speed impact, in order to predict global damage. The model is based on thick plate theory developed by Reddy and takes into account the through thickness shear. An indentation law based on the theory of Hertz is determined experimentally to model the local response. A numerical integration method is used to calculate the contact force versus time, by combining the indentation with the dynamic response of the structure. The validation has been performed by comparing results with measurements obtained on a falling weight set-up, and a satisfactory correlation is obtained. This model leads to the identification of parameters inducing damage. The modal damping coefficient is taken into account.

The Fracture Properties of Sandwich Structures Based on a Metal Foam Core

The fracture properties of a series of metal foam sandwich structures based on glass fiber-reinforced polyamide 6,6 composite (GF/PA6,6) skins have been investigated. The open cell core materials were manufactured using the Lost Carbonate Sintering (LCS) process, a recently-developed technique for manufacturing metal foams. Initially, the effect of varying the compaction pressure used in producing the metal foams as well as the density of the samples were investigated through a series of compression tests. Here, it was shown that the compressive strength and the elastic modulus of the foams varied with density and compaction pressure, in spite of the fact that the average size of the cells in these foams were insensitive to either of these two parameters. The resistance of sandwich structures to localized loading was investigated through a series of indentation tests. Here, it was shown that the indentation response of sandwich structures could be characterized using a simple indentation law, the parameters of which did not exhibit any clear dependency on the density of the foam. Finally, three point bend tests on the sandwich structures have shown that their loading-bearing properties were sensitive to foam density.

An indentation law for doubly curved composite sandwich panels with rigid-plastic core subjected to flat-ended cylindrical indenters

In this paper, an indentation law (force – indentation relation) is derived for a doubly curved composite sandwich panel subjected to a flat-ended cylindrical indenter using the principle of minimum potential energy. The sandwich panel is composed of two laminated facesheets and a rigid-plastic core. The analysis is nonlinear due to nonlinear strain-displacement relations. In contrast to existing analytical models for the indentation of composite sandwich panels, the geometry of the sandwich panel is a doubly curved shell which includes flat, cylindrical and spherical panels as three special cases. Moreover, both bending and membrane effects are simultaneously taken into account for the indented facesheet. Additionally, the stacking sequence of the facesheets is completely arbitrary, so that all the coupling stiffnesses, are also included in the model. The validity of the derived indentation law is confirmed through comparison with the existed experimental and present FEM (finite element method) results. The effects of some important geometrical and mechanical parameters on the derived indentation law are studied and discussed. It is observed that the stacking sequence of the facesheet has a little influence on the indentation response. Furthermore, for the same indentation force, the indentation depth increases with increasing radii of curvature.

Parametric study of the influence of the structure dimensions and the material properties on the global behavior of plates made of composite materials

This paper considers the effect of the damage caused by transverse stresses due to a low velocity impact loading on composite plates. The dynamic response of a plate is calculated using the modal superposition technique based on the classical plate theory of sandwich plates (stratified) taking into account transverse shear effects. An indentation law, based on the Hertz theory and verified experimentally, governs the behavior of the plate subsequent to impact. The numerical time integration scheme is used to calculate the contact force at any instant by combining the indentation and the dynamic response of the structure. The obtained results show a good correlation between the shock results and their prediction. This method is equally applied to the contact force as well as to the deformation of the plate. The simulation permits a parametric study of the structure dimensions and the material properties on the global behavior of the plate.

Preparation and properties of thermoplastic honeycomb core sandwich structure with aluminum skin

The low-velocity impact response of thermoplastic honeycomb core sandwich structure with aluminum skin has been investigated by conducting drop-weight impact tests using an instrumented drop-weight impact tower. At first, the shear modulus of the thermoplastic honeycomb core and flexural modulus of the skin were investigated through a series of flexural test. Later in the study, the indentation test was conducted to the specimen in order to investigate the indentation characteristic. It was found that the indentation characteristic of this sandwich structure can be analyzed using a Meyer indentation law. The impact response of the thermoplastic honeycomb core sandwich structures was predicted using a simple energy-balance model which meant for energy absorption in bending, shear and contact effects. In addition, the energy-balance model was also used to identify the energy partitioning during the impact event. The result shows that the energy absorption profile was relatively constant over the range of impact energies, suggesting that the fraction of energies does not vary with impact conditions.

복합적층판의 초기응력에 의한 충격거동 특성

복합적층판은 다른 금속재료에 비해 높은 비강도, 비강성 등의 우수한 역학적 특성을 지니므로 최근 다양한 분야에서 사용하고 있다. 그러나 이러한 복합적층판은 일반 금속재료에 비해 충격에 약하다는 단점이 있다. 이 점을 해결하기 위한 연구는 많은 연구자들에 의해 다양하게 시도되고 있다. 본 연구는 복합적층판에 초기응력을 도입하여 충격거동 특성을 파악한다. 고전적인 이론식인 Hertz의 접촉법칙과 실험식인 Sun과 Tan의 압입법칙을 포함한 유한요소프로그램을 이용하여 복합적층판의 초기응력에 의한 충격거동 특성을 조사한다. Laminated composite plates have shown their superiority over metals in applications requiring high specific strength, high specific modulus, and so on. Therefore, they have used in various industry. However, they have poor resistance to impact compared to typical metal materials. To resolve this problem by many researchers for a variety of studies have been attempted. This study investigates characteristic of impact behavior of laminated composite plates due to initial stress. Using finite element program which involved the indentation law, we investigate characteristic of impact behavior of laminated composite plates due to initial stress.

Strain rate effects in the indentation behavior of foam-based sandwich structures

Indentation tests have been undertaken at quasi-static and impact rates of strain on sandwich structures based on six different types of polymer foam core and two types of composite skin. Initially, the influence of strain rate on the compression properties of the sandwich structures has been investigated, where it has been shown that the plastic collapse strength of the foam cores is rate sensitive, increasing by approximately 100% over the range of strain rates considered. The indentation behavior of the sandwich structures was characterized using a Meyer indentation law of the form P = Cαn, where P is the applied force, α is the resulting indentation, and C, the contact stiffness and n, the indentation exponent, are constants for a given system. It has been shown that the value of the exponent, n, does not vary significantly with the properties of the core or the skin, typically being close to unity for all tests. The contact stiffness C varied greatly and was found to depend on the plastic collapse strength of the foam and the properties of the skin. It has been shown that a plot of C vs. plastic collapse strength containing all of the quasi-static and dynamic data appears to yield a unique relationship for the systems considered here. Tests have also shown that C varies with indentor radius, with the greatest sensitivity to indentor radius being observed in the higher modulus foams. These results highlight the importance of using the appropriate dynamic contact properties when modeling the impact response of foam-based sandwich structures.

Indentation law for lightweight sandwich plates

Sandwich structures consisting of high-modulus thin skins and a low-modulus thick core are sensitive to local loading, which causes local indentation. Classical approaches determine the deflection of sandwich structures by a superposition method where the local compression of the core for a panel on a rigid foundation is added to global sandwich bending without core compression. In this article, a sandwich plate is analyzed using a higher-order sandwich panel theory (HOSPT) where the faces are modelled by classical laminate theory and the core by a three-dimensional elasticity solution. The results are obtained from 14 partial differential equations for displacements of the faces and core. HOSPT determines core compression with a second-order distribution of displacement through the core thickness. The force versus core compression relations in the fully backed and clamped states are compared and it is shown that the core compression assumed in the classical approach differs from the core compression in a clamped state. Results are verified by ANSYS® FEM software and experimental data. A few sandwich panel specimens were made with glass reinforced plastic (GRP) skins and a polyurethane foam core. Results from indentation tests on the panels are compared with finite element, HOSPT, and other theories.

Non-Linear Indentation Law for Lightweight Sandwich Beam

Indentation law is the relationship between force and deflection under the load on a sandwich beam in a fully backed state (on rigid ground). In the classical method, the Winkler model, which is based on analysing a beam on elastic foundation, is used for verifying the indentation law and for measuring core compression yielding load. This load is assumed to be the same as the one in the three-point bending (3PB) state. The typical formula of indentation is F= Kα n, where K and n are determined by an indentation test and α is the sandwich compression. In the present article, the sandwich beam is analysed by the higher order sandwich panel theory (HOSPT), and the indentation law is determined for fully backed and 3PB states separately. The effect of contact width on non-linearity of the indentation law is investigated, and the contact width is determined for cylindrical indentors. Foam, laminate, and sandwich beam contact pressure profiles are presented, and the indentation law is obtained in accordance to the radius of the indentor and contact width. It presents an analytical solution for the factors K and n in the indentation problem for simply supported and fully backed states. It is shown that the core compression in the 3PB state is lower than fully backed indentation. However, the classical theory predicts lower value for it than experimental value. It is shown that K increases as the indentor radius increases in the indentation law. Results are compared with the Winkler foundation method, ANSYS FEM software, and the previous article. Good agreements are found between HOSPT, ANSYS, and the previous work.

Indentation in lightweight composite sandwich beams

Indentation law is the relationship between force and deflection under the load on a sandwich beam in a fully backed state (on rigid ground), which plays an important role in low-velocity impact models. In the classical method, the Winkler model, which is based on analysing a beam on elastic foundation, is used for measuring the linear part of indentation law and core compression yielding load. This load is assumed to be the same as three-point bending (3PB) state core compression yielding load. In the present article, the sandwich beam is divided into three regions: two faces and a core. Faces are modelled with classical beam theory separately and the core is modelled with two-dimensional elasticity theory, which is known as sandwich panel higher-order theory (SPHOT). A simply supported and fully backed sandwich beam with concentrated loading is solved with Fourier series, and closed-form solutions for core compression, shear, and normal stresses of the core are obtained. The obtained core yielding load by SPHOT is compared with the classical method using the commercial finite element code ANSYS and also with recently reported researches in the literature. The results show that SPHOT predicts more accurate core yielding load and better indentation law compared with other theories and are in good agreement with the results of finite element simulation. In addition, differences between SPHOT theory and the classical method for indentation are investigated in terms of geometry and material properties of sandwich elements. The bottom face effect is determined in core compression in the 3PB state. It is shown that core compression in the 3PB state is lower than fully backed indentation and fully backed indentation is lower than classical prediction.

Contact force history analysis of composite sandwich plates subjected to low-velocity impact

Usually the modified Hertzian contact law or experimental static indentation law has been used to analyze low-velocity impact response of composite laminates. In composite laminated plates subjected to low-velocity impact, usually indentation by impact is very small and also energy absorption by indentation is negligible, so ‘spring element method’, which proposed by author recently, can be well applied to investigate impact response. In the present study ‘lumped mass method’ also had been proposed by author to approximately calculate contact force history of composite laminates will be conceptually described as well as the spring element method. And it will be discussed that how the spring element method can be applied to composite sandwich plates. Finally numerical results easily obtained from finite element analysis based on the spring element method using general-purpose commercial FEM software is compared with experimental results. The comparison shows overall agreement.

Low-velocity Impact Response of High-performance Aluminum Foam Sandwich Structures

The impact response of a range of novel sandwich structures based on fiber-reinforced thermoplastic and fiber-metal laminate (FML) skins is studied. Indentation tests on these structures show that the indentation constants in a generalized indentation law exhibit a rate-sensitive response over the range of loading conditions examined here. Low-velocity impact tests show that these systems are capable of absorbing energy through localized plastic deformation and crushing in the metal core. An energy-balance model accounting for energy dissipation in bending, shear, and indentation effects is used to predict the maximum force during the impact event. It is found that the model accurately predicts the low-velocity impact response of the plain sandwich structures up to energies close to 30 J. In contrast, the model is only capable of predicting the low-energy response of the FML sandwich structures (typically up to 2 J). At higher energies, a horizontal shear crack initiates in the metal core causing the maximum force to drop below that predicted by the model. Using an energy-partitioning approach, it is shown that indentation effects account for over half of the energy absorbed in the FML-based sandwich structures.

Low-velocity impact analysis of composite laminates using linearized contact law

Usually many researchers have used the modified Hertzian contact law or experimental static indentation law to analyze impact response of composite laminates subjected to low-velocity impact. In this study, physical meaning of the analytical method using the laws was investigated and the difference between the analytical results obtained using the two laws was also investigated. Furthermore parametric study on contact coefficient and exponent of the contact law was performed. Finally it could be shown that linearized contact law could be well applied to the low-velocity impact analysis of composite laminates. If this concept is used, any general-purpose finite element method software can be used to solve impact problem without direct developing any FEM code by each researcher. In this paper, some analytical results analyzed using a general-purpose commercial FEM software were also presented.

The low velocity impact response of an aluminium honeycomb sandwich structure

The low velocity impact response of two aluminium honeycomb sandwich structures has been investigated by conducting drop-weight impact tests using an instrumented falling-weight impact tower. Initially, the rate-sensitivity of the glass fibre reinforced/epoxy skins and aluminium core was investigated through a series of flexure, shear and indentation tests . Here, it was found that the flexural modulus of the composite skins and the shear modulus of the aluminium honeycomb core did not exhibit any strain-rate sensitivity over the conditions investigated here. In addition, it was found that the indentation characteristics of this lightweight sandwich structure can be analysed using a Meyer indentation law, the parameters of which did not exhibit any sensitivity to crosshead displacement rate. The impact response of the aluminium honeycomb sandwich structures was modelled using a simple energy-balance model which accounts for energy absorption in bending, shear and contact effects. Agreement between the energy-balance model and the experimental data was found to be good, particularly at low energies where damage was localised to the core material immediate to the point of impact. The energy balance was also used to identify energy partitioning during the impact event. Here, it was shown that the partition of the incident energy depends strongly on the geometry of the impacting projectile.

The low velocity impact response of foam-based sandwich structures

The low velocity impact response of a range of foam-based sandwich structures has been investigated using an instrumented falling-weight impact tower. Initially, the rate-sensitivity of the skin and core materials was investigated through a series of flexure and indentation tests. Here, it was shown that the flexural modulus of the skins and all 11 foam materials did not exhibit any sensitivity to crosshead displacement rate over the conditions studied here. In addition, it was shown that the indentation response of the sandwich structures could be modelled using a simple indentation law, the parameters of which did not exhibit any sensitivity to loading rate. Low velocity impact tests on the sandwich structures resulted in a number of different failure modes. Here, shear fracture was found to occur in the PVC/PUR systems based on brittle core materials. In contrast, buckling failures in the uppermost composite skin were observed in the intermediate modulus systems, whereas initial damage in the higher modulus PVC/PUR systems took the form of delamination within the top surface skin. It has been shown that a simple energy-balance model based on the dissipation of energy during the impact event can be used to successfully model the elastic response of foam-based sandwich structures. The energy-balance model is particularly useful since it can be used to establish the partition of energy during the impact process.

Nonlinear finite element analysis of composite shell under impact

Large deflection dynamic responses of laminated composite cylindrical shells under impact are analyzed by the geometrically nonlinear finite element method based on a generalized Sander’s shell theory with the first order transverse shear deformation and the von-Karman large deflection assumption. A modified indentation law with inelastic indentation is employed for the contact force. The nonlinear finite element equations of motion of shell and an impactor along with the contact laws are solved numerically using Newmark’s time marching integration scheme in conjunction with Akay type successive iteration in each step. The ply failure region of the laminated shell is estimated using the Tsai-Wu quadratic interaction criteria. Numerical results, including the contact force histories, deflections and strains are presented and compared with the ones by linear analysis. The effect of the radius of curvature on the composite shell behaviors is investigated and discussed.

The significance of indentation in the inspection of carbon fibre-reinforced plastic panels damaged by low-velocity impact

Low-velocity impact tests were carried out at different energy levels on three types of T400/934 angle-ply laminates, by means of an instrumented drop-weight apparatus. After impact, the indentation depth and the residual tensile strength were measured as a function of impact energy. From the results obtained and experimental data available in the literature, an indentation law, allowing for the prediction of the impact energy from the depth of indentation, was developed. This indentation law appears to have a quite general applicability, being scarcely affected by fibre type and orientation or matrix type. A previous formula, modelling the residual tensile strength decay as a function of impact energy, was proved to be capable of correctly predicting the residual strength results for all the laminates tested. Combining the indentation model and the residual strength model, a closed-form model, explicitly correlating the residual strength and the indentation depth, was obtained. The theoretical predictions were in very good agreement with the experimental results. It is shown that the new model assumes a simple analytical form for a given laminate, permitting material characterisation by a minimum amount of test data.

Impact analysis of composite laminates with multiple delaminations

This paper concerns with the transient response of composite laminates with multiple delaminations subjected to low-velocity impact by a rigid ball. The finite element method based on the Mindlin plate theory is employed to describe the motion and deformation of the laminates. A Hertzian-type indentation law is adopted to calculate the impact force between the laminates and the rigid ball. To deal with the dynamic contacts between delaminated layers effectively, a modified Lagrange multiplier technique is employed. For multi-body dynamic contacts, the computation of contact force between delaminated layers is addressed. Numerical results provide much information for understanding the impact phenomenon of the composite laminates with multiple delaminations.