Bhavani V. Sankar

An elasticity solution for functionally graded beams

An elasticity solution is obtained for a functionally graded beam subjected to transverse loads. The Young’s modulus of the beam is assumed to vary exponentially through the thickness, and the Poisson ratio is held constant. The exponential variation of the elastic stiffness coefficients allow an exact solution for the elasticity equations. A simple Euler–Bernoulli type beam theory is also developed on the basis of the assumption that plane sections remain plane and normal to the beam axis. The stresses and displacements are found to depend on a single non-dimensional parameter for a given variation of Young’s modulus in the functionally graded direction. It is found that the beam theory is valid for long, slender beams with slowly varying transverse loading. Stress concentrations occur in short or thick beams. The stress concentrations are less than that in homogeneous beams when the softer side of the functionally graded beam is loaded. The reverse is true when the stiffer side is loaded. # 2001 Elsevier Science Ltd. All rights reserved.

Thermal Stresses in Functionally Graded Beams

Thermoelastic equilibrium equations for a functionally graded beam are solved in closed-form to obtain the axial stress distribution. The thermoelastic constants of the beam and the temperature were assumed to vary exponentially through the thickness. The Poisson ratio was held constant. The exponential variation of the elastic constants and the temperature allow exact solution for the plane thermoelasticity equations. A simple Euler ‐ Bernoulli-type beam theory is also developed based on the assumption that plane sections remain plane and normal to the beam axis. The stresses were calculated for cases for which the elastic constants vary in the same manner as the temperature and vice versa. The residual thermal stresses are greatly reduced, when the variation of thermoelastic constants are opposite to that of the temperature distribution. When both elastic constants and temperature increasethrough the thickness in the samedirection, they causea signie cant raise in thermal stresses. For the case of nearly uniform temperature along the length of the beam, beam theory is adequate in predicting thermal residual stresses.

Analytical method for micromechanics of textile composites

Abstract An analytical method called the selective averaging method (SAM) is proposed for prediction of the thermoelastic constants of textile composite materials. The unit cell of the composite is divided into slices (mesoscale), and the slices are subdivided into elements (microscale). The elastic constants of the homogenized medium are found by averaging the elastic constants of the elements selectively for both isostress and isostrain conditions. For thin textile composites where there are fewer unit cells in the thickness direction, SAM is used to compute directly the [A], [B] and [D] matrices of the composite plate. The results obtained by the SAM are compared with available finite-element-based micro-mechanical methods and analytical solutions.

Analytical Electroacoustic Model of a Piezoelectric Composite Circular Plate

This paper presents an analytical two-port, lumped-element model of a piezoelectric composite circular plate. In particular, the individual components of a piezoelectric unimorph transducer are modeled as lumped elements of an equivalent electrical circuit using conjugate power variables. The transverse static deflection field as a function of pressure and voltage loading is determined to synthesize the two-port dynamic model. Classical laminated plate theory is used to derive the equations of equilibrium for clamped circular laminated plates containing one or more piezoelectric layers. A closed-form solution is obtained for a unimorph device in which the diameter of the piezoelectric layer is less than that of the shim. Methods to estimate the model parameters are discussed, and model verification via finite-element analyses and experiments is presented. The results indicate that the resulting lumped-element model provides a reasonable prediction (within 3%) of the measured response to voltage loading and the natural frequency, thus enabling design optimization of unimorph piezoelectric transducers.

Indentation-flexure and low-velocity impact damage in graphite epoxy laminates

Static indentation-flexure and low-velocity impact tests were performed on quasi-isotropic and cross-ply graphite/epoxy composite laminates. The load-deflection relations in static tests and impact force history in impact tests were recorded. The damage was assessed by using ultrasonic C-scanning and photo-micrographic techniques. Some features of the static behavior were explained by simple analytical models. A good correlation existed between the load-deflection curves for static and impact loading. It was found that results from a few static indentation-flexure tests can be used to predict the impact force history and delamination radius in composite laminates due to low-velocity impact.

Evaluation of Failure Criteria for Fiber Composites Using Finite Element Micromechanics

A micromechanical analysis of the unit cell of a unidirectional composite is performed using the finite element method. The circular fibers are assumed to be packed in a periodic square array. Assuming that the failure criteria for the fiber and matrix materials and also for the fiber-matrix interface are known, the failure envelope of the composite is developed using the microstresses computed in the unit cell analysis. This method is referred to as the Direct Micromechanics Method (DMM). The micromechanical methods were also used to simulate different tests to determine the strength coefficients in phenomenological failure criteria such as maximum stress, maximum strain and Tsai-Wu theories. The failure envelopes from the phenomenological failure criteria are compared with those of the DMM for the cases of biaxial and off-axis loading of a model unidirectional composite material. It is found that none of the phenomenological criteria compare well with the DMM in the entire range. A conservative failure envelope obtained using a combination of maximum stress and Tsai-Wu criteria seems to be the best choice for predicting the failure of unidirectional fiber composites.

Micromechanical Analysis of Composite Corrugated-Core Sandwich Panels for Integral Thermal Protection Systems

termsA44 andA55 werecalculatedusinganenergyapproach.Usingtheshear-deformableplatetheory,aclosed-form solution of the plate response was derived. The variation of plate stiffness and maximum plate deflection due to changing the web angle are discussed. The calculated results, which require significantly less computational effort and time, agree well with the three-dimensional finite element analysis. This study indicates that panels with rectangular webs resulted in a weak extensional, bending, and A55 stiffness and that the center plate deflection was minimum for a triangular corrugated core. The micromechanical analysis procedures developed in this study were used to determine the stresses in each component of the sandwich panel (face and web) due to a uniform pressure load.

A Micromechanical Model for Textile Composite Plates

A novel finite element based micromechanical method is developed for computing the plate stiffness coefficients (A, B, D matrices) and coefficients of thermal expansion (αs and βs) of a textile composite modeled as a homogeneous plate. Periodic boundary conditions for the plate model, which are different from those for the continuum model, have been derived. The micromechanics methods for computing the coefficients of thermal expansion are readily extended to compute the thermal residual stresses due to curing. The methods are first verified by applying to several examples for which solutions are known, and then applied to the case of woven composites. The plate stiffness coefficients computed from direct micromechanics are compared with those derived from the homogenized elastic constants in conjunction with the classical plate theory. It is found that the plate stiffness coefficients of textile composites, especially the B and D matrices, cannot be predicted from the homogenized elastic constants and the plate thickness.

Compressive Failure of Sandwich Beams with Debonded Face-Sheets:

Axial compression tests were performed on debonded sandwich composites made of graphite/epoxy face-sheets and aramid fiber honeycomb core. The sandwich beams were manufactured using a vacuum bagging process. The face-sheet and the sandwich beam were cocured. Delamination between one of the face-sheets and the core was introduced by using a Teflon® layer during the curing process. Axial compression tests were performed to determine the ultimate load carrying capacity of the debonded beams. Flatwise tension tests and Double Cantilever Beam tests were performed to determine, respectively, the strength and fracture toughness of the face-sheet/core interface. From the test results semi-empirical formulas were derived for the fracture toughness and ultimate compressive load carrying capacity in terms of the core density, core thickness, face-sheet thickness and debond length. Four different failure modes and their relation to the structural properties were identified. Linear buckling analysis was found to be inadequate in predicting the compressive load carrying capacity of the debonded sandwich composites.

A unit-cell model of textile composite beams for predicting stiffness properties

Flexural stiffness properties of a textile composite beam are obtained from a finite-element model of the unit cell. Three linearly independent deformations, namely, pure extension, pure bending and pure shear, are applied to the unit cell. The top and bottom surfaces of the beam are assumed to be traction free. Periodic boundary conditions on the lateral boundaries of the unit cell are enforced by multi-point constraint elements. From the forces acting on the unit cell, the flexural stiffness coefficients of the composite beam are obtained. The difficulties in determining the transverse shear stiffness are discussed, and a modified approach is presented. The methods are first verified by applying them to isotropic and bimaterial beams for which the results are known, and then illustrated for a simple plain-weave textile composite.