A plastic indentation model for sandwich beams with metallic foam cores

Light weight high performance sandwich composite structures have been used extensively in various load bearing applications.Experiments have shown that the indentation significantly reduces the load bearing capacity of sandwiched beams.In this paper,the indentation behavior of foam core sandwich beams without considering the globally axial and flexural deformation was analyzed using the principle of virtual velocities.A concisely theoretical solution of loading capacity and denting profile was presented.The denting load was found to be proportional to the square root of the denting depth.A finite element model was established to verify the prediction of the model.The load-indentation curves and the profiles of the dented zone predicted by theoretical model and numerical simulation are in good agreement.     

Structural modeling of sandwich structures with lightweight cellular cores

An effective single layered finite element (FE) computational model is proposed to predict the structural behavior of lightweight sandwich panels having two dimensional (2D) prismatic or three dimensional (3D) truss cores. Three different types of cellular core topology are considered: pyramidal truss core (3D), Kagome truss core (3D) and corrugated core (2D), representing three kinds of material anisotropy: orthotropic, monoclinic and general anisotropic. A homogenization technique is developed to obtain the homogenized macroscopic stiffness properties of the cellular core. In comparison with the results obtained by using detailed FE model, the single layered computational model can give acceptable predictions for both the static and dynamic behaviors of orthotropic truss core sandwich panels. However, for non-orthotropic 3D truss cores, the predictions are not so well. For both static and dynamic behaviors of a 2D corrugated core sandwich panel, the predictions derived by the single layered computational model is generally acceptable when the size of the unit cell varies within a certain range, with the predictions for moderately strong or strong corrugated cores more accurate than those for weak cores. The project supported by the National Basic Research Program of China (2006CB601202), the National Natural Science Foundation of China (10328203, 10572111, 10572119, 10632060), the National 111 Project of China (B06024), the Program for New Century Excellent Talents in University (NCET-04-0958), the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment, and the Doctorate Foundation of Northwestern Polytechnical University.     

The impulsive response of sandwich beams: Analytical and numerical investigation of regimes of behaviour

An analytical model is developed to classify the impulsive response of sandwich beams based on the relative time-scales of core compression and the bending/stretching response of the sandwich beam. It is shown that an overlap in time scales leads to a coupled response and to the possibility of an enhanced shock resistance. Four regimes of behaviour are defined: decoupled responses with the sandwich core densifying partially or completely, and coupled responses with partial or full core densification. These regimes are marked on maps with axes chosen from the sandwich beam transverse core strength, the sandwich beam aspect ratio and the level of blast impulse. In addition to predicting the time-scales involved in the response of the sandwich beam, the analytical model is used to estimate the back face deflection, the degree of core compression and the magnitude of the support reactions. The predictions of the analytical model are compared with finite element (FE) simulations of impulsively loaded sandwich beams comprising an anisotropic foam core and elastic, ideally plastic face-sheets. The analytical and numerical predictions are in good agreement up to the end of core compression. However, the analytical model under-predicts the peak back face deflection and over-predicts the support reactions, especially for sandwich beams with high strength cores. The FE calculations are employed to construct design charts to select the optimum transverse core strength that either minimises the back face deflections or support reactions for a given sandwich beam aspect ratio or blast impulse. Typically, the value of the transverse core strength that minimises the back face deflection also minimises the support reactions. However, the optimal core strength depends on the level of blast impulse, with higher strength cores required for greater blasts.     

An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams

Summary An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions. The first author is grateful to DST, Government of India, for financial support for this work.     

Nonlinear behavior of composite sandwich beams in three-point bending

The load-deflection behavior of a composite sandwich beam in three-point bending was investigated. The beam was made of unidirectional carbon/epoxy facings and a polyvinyl chloride closed-cell foam core. The load-deflection curves were plotted up to the point of failure initiation. They consist of an initial linear part followed by a nonlinear portion. A nonlinear mechanics of materials analysis that accounts for the combined effect of the nonlinear behavior of the facings and core materials (material nonlinearity) and the large deflections of the beam (geometric nonlinearity) was developed. The theoretical predictions were in good agreement with the experimental results. It was found that the effect of material nonlinearity on the deflection of the beam is more pronounced for shear-dominated core failures in the case of short span lengths. It is due to the nonlinear shear stress-strain behavior of the core. For long span lengths, the observed nonlinearity is small and is attributed to the combined effect of the facings nonlinear stress-strain behavior and the large deflections of the beam.     

Analytical modeling and finite element simulation of the plastic collapse of sandwich beams with pin-reinforced foam cores

Analytical predictions are presented for the plastic collapse strength of lightweight sandwich beams having pin-reinforced foam cores that are loaded in 3-point bending. Both polymer and aluminum foam cores are considered, whilst the facesheet and the pins are made of either composite or metal. Four different failure modes are account for: metal facesheet yield or composite facesheet microbuckling, facesheet wrinkling, plastic shear of the core, and facesheet indentation beneath the loading rollers. A micromechanics-based model is developed and combined with the homogenization approach to calculate the effective properties of pin-reinforced foam cores. To calculate the elastic buckling strength of pin reinforcements, the pin-reinforced foam core is treated as assemblies of simply supported columns resting upon an elastic foundation. Minimum mass design of the sandwich is then obtained as a function of the prescribed structural load index, subjected to the constraint that none of the above failure modes occurs. Collapse mechanism maps are constructed and compared with the failure maps of foam-cored sandwich beams without pin reinforcements. Finite element simulations are carried out to verify the analytical model and to study the performance and failure mechanisms of the sandwich subject to loading types other than 3-point bending. The results demonstrate that the weaker the foam is, the more optimal the pin-reinforced foam core becomes, and that sandwich beams with pin-reinforced polymer foam cores are structurally more efficient than foam- or truss-cored sandwich beams.     

A blast-tolerant sandwich plate design with a polyurea interlayer

This paper presents a study of both conventional and modified sandwich plate designs subjected to blast loads. The conventional sandwich Design (1) consists of thin outer (loaded side) and inner facesheets made of fibrous laminates, separated by a layer of structural foam core. In the modified Design (2), a thin polyurea interlayer is inserted between the outer facesheet and the foam core. Comparisons of the two designs are made during a long time period of 5.0 ms, initiated by a pressure impulse lasting 0.05 ms applied to a single span of a continuous plate. In the initial response period the overall deflections are limited and significant foam core crushing is caused in the conventional design by the incident compression wave. This type of damage is much reduced in the modified design, by stiffening of the polyurea interlayer under shock compression, which provides support to the outer facesheet and alters propagation of stress waves into the foam core. This benefits the long term, bending response and leads to significant reductions in facesheet strains and overall deflection. The total kinetic energy of the modified sandwich plate is much lower than that of a conventionally designed plate, and so is the stored and dissipated strain energy. Similar reductions are found when the conventional and the enhanced sandwich plates have equal overall thickness or equal total mass.     

Cell size effects for vibration analysis and design of sandwich beams

In this work, sandwich beams are studied to reveal the underlying size effects of the periodic core cells for the first time within the framework of free vibration analysis of such an advanced lightweight structure. The energy equivalence method is formulated as a theoretical approach that takes into account the cell size effect. It is compared with the asymptotic homogenization method and direct finite element method systematically to show their consistence and applicability. The accuracy of free vibration responses predicted by the detailed finite element model is used as the standard of comparison. It is shown that the cell size is an important parameter characterizing the cellular core rigidities that influence vibration responses. The homogenization model agrees exactly with the asymptotic solution of the analytical expression of the beam model only whenever the cell size tends to be infinitely small.     

Plastic failure analysis of an auxetic foam or inverted strut lattice under longitudinal and shear loads

Auxetic materials possess negative Poisson's ratios. As such, they can be applied in situations where traditional materials perform poorly or cannot perform. We investigate the plastic failure of a 3D auxetic strut lattice under uniaxial and transverse loads in order to complement ongoing research in miniaturized strut-based sandwich cores. The chosen lattice is also representative of an auxetic foam. Plastic failure models derived with respect to two physical parameters (packing parameter and relative density) which control the negative Poisson's ratio compare well with numerical data. Microscopic failure modes differ depending on the loading state: shear failure is due to global plastic yielding while plastic localization occurs under uniaxial loads. This observation suggests among others that it is advisable to use auxetic cores when structural softness under normal loads and hardness under transverse loads are both critical design conditions.     

Functionally graded rod with small concentration of inclusions: Homogenization and optimization

This work is devoted to strain analysis and optimal design of a Functionally Graded (FG) rods and beams with small inclusions. The homogenization procedure plays a key role in our investigations. The method is illustrated using an example of the rod longitudinal deformation and bending of a beam. We consider the cases of FG inclusion sizes and FG steps between inclusions separately. Particular problems of optimal design are discussed in some details. The mathematical model of the bending beam, which adapts to the external load action, is proposed and an illustrative example of the adaptation process is given.     

Experimental evidence of parasitic effects in the shear test on sandwich beams

The present work is an experimental investigation of the standard shear test ASTM C273 carried out on sandwich structures. The goal is to highlight and to quantify some parasitic effects that occur during this test. A suitable optical method providing whole-field measurements has been used to capture the displacement and strain fields during the test. Some parasitic effects have been detected: the steel plates bend during the test, the shear strain reaches zero near the free edges and compressive/tensile strains occur in this zone.     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

Impulsive loading of clamped monolithic and sandwich beams over a central patch

An analytical model is developed for the response of clamped monolithic and sandwich beams subjected to impulse loading over a central loading patch. A number of topologies of sandwich core are investigated, including the honeycomb core, pyramidal core, prismatic diamond core and metal foam. The various cores are characterised by their dependencies of through-thickness compressive strength and longitudinal tensile strength upon relative density. Closed-form expressions are derived for the deflection of the beam when the ratio r of length of loading patch to the beam span exceeds 0.5. In contrast, an ordinary differential equation needs to be solved numerically for the choice r<0.5. Explicit finite element calculations show that most practical shock loadings can be treated as impulsive and the accuracy of the impulsive analytical predictions is confirmed. The analytical formulae are employed to determine optimal geometries of the sandwich beams that maximise the shock resistance of the beams for a given mass. The optimisation reveals that sandwich beams have a superior shock resistance relative to monolithic beams of the same mass, with the prismatic diamond core sandwich beam providing the best performance. Further, the optimal sandwich beam designs are only mildly sensitive to the length of the loading patch.     

On the screw dislocation in a functionally graded material

This paper presents the stress field of a screw dislocation in a medium graded in y-direction. The medium is exponentially graded. For such a graded material theories of elasticity as well as gradient elasticity are applied. By means of the stress function technique we found exact analytical solutions of the corresponding master equations. Using the stress field, the Peach–Koehler force is given. The axial symmetry of a screw dislocation is lost due to the gradation in the y-direction.     

Finite element analysis of the dynamic response of clamped sandwich beams subject to shock loading

The finite element (FE) method is employed to analyse the response of clamped sandwich beams subject to shock loadings. Pressure versus time histories representative of shock loadings are applied uniformly to the outer face of the sandwich beam; an impulse applied uniformly to the outer face of the sandwich beam is shown to model adequately shock loadings. Material elasticity and strain hardening representative of structural steels have only a minor effect upon the beam response. Further, the magnitude of the compressive strength of the core has only a limited influence upon the dynamic response of the sandwich beam for the representative range of core strengths considered. The FE results for the deflections and structural response time agree well with the rigid ideally-plastic analytical predictions of Fleck and Deshpande (J. Appl. Mech. (2003), in press).     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.     

Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation

This paper presents a geometric nonlinear analysis formulation for beams of functionally graded cross-sections by means of a Total Lagrangian formulation. The influence of material gradation on the numerical response is investigated in detail. Two examples are given that illustrate the main features of the formulation, in which the behavior of beams of graded cross-sections is compared with homogeneous material beams. A motivation for this work is the potential development of functionally graded risers for the offshore oil exploration industry.     

Multiobjective optimization for design of multifunctional sandwich panel heat pipes with micro-architected truss cores

A micro-architected multifunctional structure, a sandwich panel heat pipe with a micro-scale truss core and arterial wick, is modeled and optimized. To characterize multiple functionalities, objective equations are formulated for density, compressive modulus, compressive strength, and maximum heat flux. Multiobjective optimization is used to determine the Pareto-optimal design surfaces, which consist of hundreds of individually optimized designs. The Pareto-optimal surfaces for different working fluids (water, ethanol, and perfluoro(methylcyclohexane)) as well as different micro-scale truss core materials (metal, ceramic, and polymer) are determined and compared. Examination of the Pareto fronts allows comparison of the trade-offs between density, compressive stiffness, compressive strength, and maximum heat flux in the design of multifunctional sandwich panel heat pipes with micro-scale truss cores. Heat fluxes up to 3.0 MW/m² are predicted for silicon carbide truss core heat pipes with water as the working fluid.