Estimation of low-velocity impact damage in laminated composite circular plates using nonlinear finite element analysis

Nonlinear finite element analysis is used for the estimation of damage due to low-velocity impact loading of laminated composite circular plates. The impact loading is treated as an equivalent static loading by assuming the impactor to be spherical and the contact to obey Hertzian law. The stresses in the laminate are calculated using a 48 d.o.f. laminated composite sector element. Subsequently, the Tsai-Wu criterion is used to detect the zones of failure and the maximum stress criterion is used to identify the mode of failure. Then the material properties of the laminate are degraded in the failed regions. The stress analysis is performed again using the degraded properties of the plies. The iterative process is repeated until no more failure is detected in the laminate. The problem of a typical T300/N5208 composite [45 °/0 °/ − 45 °/90 °]_s circular plate being impacted by a spherical impactor is solved and the results are compared with experimental and analytical results available in the literature. The method proposed and the computer code developed can handle symmetric, as well as unsymmetric, laminates. It can be easily extended to cover the impact of composite rectangular plates, shell panels and shells.     

Multi-fidelity optimization of laminated conical shells for buckling

Optimum laminate configuration for minimum weight of filament-wound laminated conical shells is investigated subject to a buckling load constraint. In the case of a composite laminated conical shell, due to the manufacturing process, the thickness and the ply orientation are functions of the shell coordinates, which ultimately results in coordinate dependence of the stiffness matrices (A,B,D). These effects influence both the buckling load and the weight of the structure and complicate the optimization problem considerably. High computational cost is involved in calculating the buckling load by means of a high-fidelity analysis, e.g. using the computer code STAGS-A. In order to simplify the optimization procedure, a low-fidelity model based on the assumption of constant material properties throughout the shell is adopted, and buckling loads are calculated by means of a low-fidelity analysis, e.g. using the computer code BOCS. This work proposes combining the high-fidelity analysis model (based on exact material properties) with the low-fidelity model (based on nominal material properties) by using correction response surfaces, which approximate the discrepancy between buckling loads determined from different fidelity analyses. The results indicate that the proposed multi-fidelity approaches using correction response surfaces can be used to improve the computational efficiency of structural optimization problems.     

Three-dimensional finite element analysis of interlaminar stresses in thick composite laminates

The three-dimensional finite element computer program has been developed to investigate interlaminar stresses in thick composite laminates. The finite element analysis is based on displacement formulation employing curved isoparametric 16-node elements. By using substructure technique, the program developed is capable of handling any composite laminates which consist of any number of orthotropic laminae and any orientations. In this paper, solid laminates and laminates with a circular hole were taken to study interlaminar stresses at the straight edge and the curved edge, respectively. Various solid laminates such as [45_n/0_n − 45_n/90_n]_s, [45/0/ − 45/90]_ns, and [45/0/ − 45/90]_sn (n = 1˜4) were analyzed. Also, [45/0/ − 45/90]_sn laminates with a circular hole were studied for n = 1 ˜ 20. The effect of laminate thickness and stacking sequence on the interlaminar stresses near the free edge was investigated. Interlaminar stresses were governed by stacking sequence rather than laminate thickness. The boundary layer width did not increase with laminate thickness but with the number of plies in the repeating unit.     

Thermodynamics of the cao-si02 system

In order to maintain consistency, analytical expressions for the free energy of mixing of phases should reproduce not only the phase diagrams but also the experimentally determined activities. Information on the partial molar free energies and the phase boundaries, in turn, can be used to estimate the free energy of formation of compounds.

An examination of thermochemical data in the CaO-SiO₂ system showed that ΔGδf values for -Ca₂Si0₄, which are stable at temperatures above 1710°K, are limited a maximum of 1800°K. The free energy of formation in a temperature range from about 1700 to 2400°K was estimated from the phase boundary and the activity of silica to be as follows:

2Ca0(s) + Si0₂(cristo.) = Ca2Si0₄() ΔG°f = −86303.50 − 34.338 Tjoules

An analytical expression for the free energy of mixing of the liquid phase was obtained for the entire composition range in the CaO-Si0₂ system. Confidence in the estimated G‡f for -Ca₂Si0₄ was demonstrated by good agreement of the calculated phase diagram and the experimentally determined activity of silica.     

Minimum weight design of symmetric angle-ply laminates under multiple uncertain loads

An angle-ply laminated plate is optimized with the objective of minimizing the weight of the plate taking into account uncertainties in the multiple transverse loads. The weight is proportional to the laminate thickness which is minimized subject to deflection and buckling constraints under the least favourable loading with the ply angles taken as design variables. The convex modelling approach is employed to analyse the uncertain loading with the uncertain quantities allowed to vary arbitrarily around their average values subject to the requirements that these variations are bounded inL ₂ norm and represented by a finite number of eigenmodes. The effect of uncertainty on the optimal design is investigated quantitatively. It is shown that the minimum weight increases with increasing level of uncertainty and the optimal ply angles also depend on the level of uncertainty.     

Nonlinear thermal and mechanical analysis of edge effects in angle-ply laminates

A method has been developed for the thermal analysis of materially nonlinear Carbon Fibre Reinforced Plastic (CFRP) laminated plates using a quasi-three-dimensional iso-parametric finite element. The variation of the linear expansitivity of CFRP lamina with temperature in the transverse direction is included. The nonlinear initial thermal stresses resulting from thermal cooling from the stress-free temperature of 132.22 to 20°C in the [0/90]_s and [±45]_s laminates were found. These two types of laminates were then subject to a uniform applied strain ε_x until failure was detected inside the laminate according to a maximum strain failure criterion. This nonlinear analysis was based on an initial stress iteration method formulated in a previous paper. The [±45]_s laminate analysis was carried out with and without resin layers between the laminae modelled. All results obtained were compared with those of previous investigators. It was found that with initial thermal stresses included the [0/90]_s laminate failed at an earlier stage than without initial thermal stresses included. The [±45], laminate with initial thermal stresses included failed at a later stage than that without initial thermal effects and the inclusion of resin layers delayed the failure even further.     

Theoretical calculation of the phase diagram between one-dimensional long-period structures in the quasi binary sections: Pd3xRh3(1−x)V, Pt3xRh3(1−x)V, and Pt3VxTi(1−x)

The relative stabilities of L1₂, D0₂₂, D0₂₃, 21, and 3 structures in the Pd₃V, Pt₃V, Rh₃V, and Pt₃Ti compounds are investigated employing the Vienna ab initio simulation package. In the pseudobinary Pd_3xRh_3(1−x)V, Pt_3xRh_3(1−x)V, and Pt₃V_xTi_(1−x) alloys, the energy differences from L1₂ of D0₂₂, D0₂₃, 21, and 3 structures are assumed to be linear as function of the number of electrons per atom. At T=0K, the resulting energy diagram shows that the equilibrium between the limiting binary phases is the most stable state. At high temperature, the Gibbs energy curves are computed assuming a Bragg and Williams entropy of mixing in the pseudobinary sections. The D0₂₃ and 21 structures are stabilized in the pseudobinary Pd_3xRh_3(1−x)V, Pt_3xRh_3(1−x)V, and Pt₃V_xTi_(1−x) alloys. The phase diagram between the various structures is calculated in each pseudobinary section and compared with the experimental one.     

Combined effects of in-plane boundary conditions and stiffening on buckling of eccentrically stringer-stiffened cylindrical panels

Donnel type stability equations for buckling of stringer stiffened cylindrical panels under combined axial compression and hydrostatic pressure are solved by the displacement approach of [6], The solution is employed for a parametric study over a wide range of panel and stringer geometries to evaluate the combined influence of panel configurations and boundary conditions along the straight edges on the buckling behavior of the panel relative to a complete “counter” cylinder (i.e. a cylinder with identical skin and stiffener parameters).

The parametric studies reveal a “sensitivity” to the “weak in shear”, N_x = N_xφ = 0, along the straight edges, SS1 boundary conditions type where the panel buckling loads are always smaller than those predicted for a complete “counter” cylinder. In the case of “classical”, SS3 B.Cs., there always exist values of panel width, 2φ₀, for which ρ = 1, i.e. the panel buckling load equals that of the complete “counter” cylinder. For SS2 and SS4 B.Cs. types, the nature by which the panel critical load approaches that of the complete cylinder appears to be panel configuration dependent.

Utilization of panels for the experimental determination of a complete cylinder buckling load is found to be satisfactory for relatively very lightly and heavily stiffened panels, as well as for short panels, (L/R) = 0.2 and 0.5. Panels of moderate length and stiffening have to be debarred, since they lead to nonconservative buckling load predictions.     

Behavior of the beta-splines with values of the parameters beta2 negative

In this paper we study the behavior of the beta-spline functions in the case the parameter β₂(i) is negative. We prove that a negative value exists so that if , the beta-spline functionsN_i(u) are positive. Moreover, if the control vertices are such that x₀ x_m−1, we have proved that the design curve keeps the properties already proved in the case β₂(i) 0.     

Piecewise hierarchical p-version axisymmetric solid element for laminated composites

This paper presents a piecewise hierarchical p-version finite element formulation for laminated composites axisymmetric solids for linear static analysis. The element formulation incorporates higher order deformation theories and is in total agreement with the physics of deformation in laminated composites. The element geometry is defined by eight nodes located on the boundaries of the element. The lamina thicknesses are used to create a nine-node p-version configuration for each lamina of the element. The displacement approximation for the element is piecewise hierarchical and is developed by first establishing a hierarchical displacement approximation for the nine-node configuration of each lamina of the laminate and then imposing interlamina continuity condition of displacements at the interfaces between laminas. The hierarchical approximation functions and the corresponding nodal variables for each lamina are derived from the Lagrange family of interpolation functions and can be of arbitrary polynomial order p_c and ^kp_η in the ε and ^kη directions for a typical lamina k. The formulation ensures C⁰ continuity, i.e., continuity of displacement across interelement as well as interlamina boundaries.

The element properties are constructed by assembling individual lamina properties which are derived using the principle of virtual work and the hierarchical displacement approximation for the laminas. Transformation matrices, formed based on interlamina continuity conditions, are used to transform each lamina's degrees of freedom into the degrees of freedom for the laminate. Thus, each individual lamina stiffness matrix and equivalent load vector are transformed and then summed to establish the laminate stiffness matrix and equivalent load vector. There is no restriction on either the number of laminas or their lay-up pattern. Each lamina can be generally orthotropic and the material directions and the layer thickness may vary from point to point within each lamina.

Numerical examples are presented to demonstrate the effectiveness, modeling convenience, accuracy, and overall superiority of the present formulation for laminated composite axisymmetric solids and shells.     

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature of composite laminates

Design and control optimization is presented to minimize the thermal postbuckling dynamic response and to maximize the buckling temperature level of composite laminated plates subjected to thermal distribution varying linearly through the thickness and arbitrarily with respect to the in-plane coordinates. The total elastic energy of the laminates is taken as a measure of the dynamic response. The optimization control problem is solved under constraints on the laminate thickness and the control energy produced by a transverse dynamic load distributed over the upper surface of the laminate. The constrained control objective is expressed as the sum of the total elastic energy and penalty term involving the control force, which may be considered as a measure of the control energy. The thickness of layers and the fibers orientation angles are taken as optimization design variables. The design and control objectives are formulated based on shear deformation theory accounting for the von-Karman nonlinearity. The displacements are chosen as the sum of time-independent displacements due to the static thermal load and time-dependent displacements due to the initial disturbances and the applied control force. Liapunov–Bellman theory is used to obtain the optimal control force, buckled deflections and controlled elastic energy. Numerical examples are presented for angle-ply antisymmetric laminates with simply supported edges. Graphical studies are carried out to show the advantages of the present design and control procedures.     

Derivative based approximation for predicting the effect of changes in laminate stacking sequence

Derivatives of buckling loads of stiffened panels with respect to ply thicknesses are easy to calculate. Consequently, such derivatives are often available in computer programs that calculate buckling loads of composite structures. These derivatives can be used to construct approximations of the dependence of the buckling load on ply thicknesses. The present work demonstrates the use of derivatives of buckling loads with respect to ply thicknesses to approximate the effects of changes in stacking sequence and ply orientations on buckling load of a laminate. Examples of unstiffened and stiffened panels are used to demonstrate the effectiveness of the proposed approximation.     

Sensitivity analysis and optimization of thin laminated structures with a nonsmooth eigenvalue based criterion

This paper presents a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects. To overcome the nondifferentiability of multiple eigenvalues, which may occur during a structural optimization involving free vibrations or buckling design situations, a nonsmooth eigenvalue based criterion is implemented. Angle-ply design variables and vectorial distances from the laminated midle surface to the upper surface of each layer are considered as design variables. The design sensitivities and the directional derivatives are evaluated analytically. The efficiency and accuracy of the model developed is discussed with two illustrative cases which show the need to compute sensitivities of multiple eigenvalues as directional derivatives for laminated composite structures.     

Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM

In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)₂]_s and angle-ply [(15°/−75°)₂]_s, [(30°/−60°)₂]_s, [(45°/−45°)₂]_s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates.     

Thermal stress analysis of laminated composite plates and shells

In this paper, Semiloof shell finite element formulation has been extended to thermal stress analysis of laminated plates and shells. The accuracy of the formulation has been verified using sample problems available in the literature. Thermal stresses in cross-ply and angle-ply laminated plates and shells subjected to thermal gradients across the thickness are presented for different boundary conditions, taking into account the temperature dependence of the material properties. The behaviour of laminates under thermal load is found to be different from that under mechanical loads in certain respects.     

Effects of random system properties on the thermal buckling analysis of laminated composite plates

This paper examines the effect of random system properties on thermal buckling load of laminated composite plates under uniform temperature rise having temperature dependent properties using HSDT. The system properties such as material properties, thermal expansion coefficients and thickness of the laminate are modeled as independent random variables. A C⁰ finite element is used for deriving the eigenvalue problem. A Taylor series based first-order perturbation technique is used to handle the randomness in the system properties. Second-order statistics of the thermal buckling load are obtained. The results are validated with those available in the literature and Monte Carlo simulation.     

Searching for optimal sensitivity of single-mode D-type optical fiber sensor in the phase measurement

To improve the sensitivity of a single-mode D-type optical fiber sensor, we selected a D-type optical fiber sensor with 4 mm long and 4 μm core thickness made of a single-mode fiber, a Au-coating on the sensor with a thickness range of 15–32 nm, a light wavelength of 632.8 nm, and an incident angle of 86.5–89.5° for different refractive index (1.33–1.40) sensing. These simulations are based on the surface plasmon resonance (SPR) theory using the phase method which shows that the sensitivity is proportional to the refractive index, Au film thickness and lower incident angle on the sensing interface. The sensitivity is higher than 4000 (degree/RIU), and the resolution is better than 2.5 × 10⁻⁶(RIU) as the minimum phase variation is 0.01°. This device is used to detect the refractive index or gas or liquid concentration in real-time. The proposed sensor is small, simple, inexpensive, and provides an in vivo test.     

Local quality of smoothening based a-posteriori error estimators for laminated plates under transverse loading

The local and global quality of various smoothening based a-posteriori error estimators is tested in this paper, for symmetric laminated composite plates subjected to transverse loads. Smoothening based on strain recovery and displacement-field recovery is studied here. Effect of ply orientation, laminate thickness, boundary conditions, mesh topology, and plate model is studied for a rectangular plate. It is observed that for interior patches of elements, both the estimators based on strain or displacement smoothening are reliable. For element patches at the boundary of the domain, all estimators tend to be unreliable (especially for angle-ply laminates). However, the strain recovery based estimator is clearly more robust for element patches at the boundary, as compared to displacement-recovery based error estimators. Globally, all the estimators tested here were found to be very robust.     

Buckling load enhancement of damaged composite plates under hygrothermal environment using unified particle swarm optimization

Optimization with Unified Particle Swarm Optimization (UPSO) method is performed for the enhancement of buckling load capacity of composite plates having damage under hygrothermal environment which has received little or no attention in the literature. Numerical results are presented for effect of damage in buckling behavior of laminated composite plates using an anisotropic damage model. Optimized critical buckling temperature of laminated plates with internal flaw is computed with the fiber orientation as the design variable by employing a UPSO algorithm and results are compared with undamaged case for various aspect ratios, ply orientations, and boundary conditions. FEM formulation and programming in the MATLAB environment have been performed. The results of this work will assist designers to address some key issues concerning composite structures. It is observed that the degradation of buckling strength of a structural element in hygrothermal environment as a result of internal flaws can be avoided to a large extent if we use these optimized ply orientations at design phase of the composite structure. This specific application proves the contribution of present work to be of realistic nature.     

Simulation and fabrication of piezoresistive membrane type MEMS strain sensors

Two piezoresistive (n-polysilicon) strain sensors on a thin Si₃N₄/SiO₂ membrane with improved sensitivity were successfully fabricated by using MEMS technology. The primary difference between the two designs was the number of strips of the polysilicon patterns. For each design, a doped n-polysilicon sensing element was patterned over a thin 3 μm Si₃N₄/SiO₂ membrane. A 1000×1000 μm² window in the silicon wafer was etched to free the thin membrane from the silicon wafer. The intent of this design was to fabricate a flexible MEMS strain sensor similar in function to a commercial metal foil strain gage. A finite element model of this geometry indicates that strains in the membrane will be higher than strains in the surrounding silicon. The values of nominal resistance of the single strip sensor and the multi-strip sensor were 4.6 and 8.6 kΩ, respectively. To evaluate thermal stability and sensing characteristics, the temperature coefficient of resistance [TCR=(ΔR/R₀)/ΔT] and the gage factor [GF=(ΔR/R₀)/] for each design were evaluated. The sensors were heated on a hot plate to measure the TCR. The sensors were embedded in a vinyl ester epoxy plate to determine the sensor sensitivity. The TCR was 7.5×10⁻⁴ and 9.5×10⁻⁴/°C for the single strip and the multi-strip pattern sensors. The gage factor was as high as 15 (bending) and 13 (tension) for the single strip sensor, and 4 (bending) and 21 (tension) for the multi-strip sensor. The sensitivity of these MEMS sensors is much higher than the sensitivity of commercial metal foil strain gages and strain gage alloys.