Plate Of Finite Thickness Research Articles

Determination of the aeroelastic transfer functions for streamlined bodies by means of a Navier–Stokes solver

This paper proposes a method to determine the flutter derivatives of two-dimensional streamlined cylinders by means of a modified indicial approach adapted to a Navier–Stokes solver using an Arbitrary Lagrangian Eulerian formulation. The method relies on heave or pitch motion imposed onto the section according to smoothed-ramp time-histories and on the computational evaluation of the transient forces arising on the obstacle. Hence, the indicial transfer function relating the plate motion to the induced force in the frequency domain is obtained. Application to a flat plate of finite thickness and length is proposed. The steady viscous flow simulated around the motionless plate is compared with the well-known Blasius solution. The computed flutter derivatives are compared both with those obtained from the Theodorsen function in the frame of the thin airfoil theory and with those resulting from previous methods in the frame of the computational approach.

RADIATION HEAT TRANSFER WITH ABLATION IN A FINITE PLATE

The phenomenon of ablation is a process of thermal protection with several applications, mainly, in mechanical and aerospace engineering. Ablative thermal protection is applied using special materials (named ablative materials) externally on the surface of a structure in order to isolate it against thermal effects. The ablative phenomenon is a complex process involving phase changes with partial or total loss of the material. So the position of the boundary is initially unknown. The governing equations of the process form a non-linear system of coupled partial differential equations. The one-dimensional analysis of an ablative process on the plate is performed by using the generalized integral transform technique – GITT for solution of the system of governing equations. By application of this solution technique, the system of partial differential equations is transformed into a system of infinite ordinary differential equations that can be solved after the truncation of that system by numerical techniques codes available. The plate of finite thickness at constant properties is subjected to a time-dependent prescribed radiation heat flux at one face, initially with a uniform temperature T0, and insulated on the other face. After an initial heating period, ablation starts at the heated surface through melting and continuous removal of the plate material. The results of interest are the thickness and the loss rate of the ablative material. The obtained results are compared with available results from other solution techniques in the literature.

RADIATION HEAT TRANSFER WITH ABLATION IN A FINITE PLATE

The phenomenon of ablation is a process of thermal protection with several applications, mainly, in mechanical and aerospace engineering. Ablative thermal protection is applied using special materials (named ablative materials) externally on the surface of a structure in order to isolate it against thermal effects. The ablative phenomenon is a complex process involving phase changes with partial or total loss of the material. So the position of the boundary is initially unknown. The governing equations of the process form a non-linear system of coupled partial differential equations. The one-dimensional analysis of an ablative process on the plate is performed by using the generalized integral transform technique – GITT for solution of the system of governing equations. By application of this solution technique, the system of partial differential equations is transformed into a system of infinite ordinary differential equations that can be solved after the truncation of that system by numerical techniques codes available. The plate of finite thickness at constant properties is subjected to a time-dependent prescribed radiation heat flux at one face, initially with a uniform temperature T0, and insulated on the other face. After an initial heating period, ablation starts at the heated surface through melting and continuous removal of the plate material. The results of interest are the thickness and the loss rate of the ablative material. The obtained results are compared with available results from other solution techniques in the literature.

The effects of thickness on the impedance of a rectangular aperture in the presence of a grazing flow

This paper presents an analytical investigation into the effect of a grazing mean flow upon the acoustic impedance of a rectangular aperture in a plate of finite thickness. Previous similar analytical works each made some simplifying approximation about the variation of the acoustic potential throughout the thickness of the aperture, such that they were valid only in either the thin or thick plate limits. In addition to providing a theoretical framework to correctly allow for a plate of any thickness, the results given here indicate the range of validity of the previous approximate theories. Two different forms of matching conditions across the shear layer have been proposed in previous works. Equations and boundary element method (BEM) solutions for both variants are given here. Results from both forms are compared against some existing measured results. As the approximations regarding aperture thickness have now been removed, it is possible to be definitive as to which form of matching conditions gives the most accurate results for a given application. The analysis is shown to predict the reactance of the aperture very well, although the results for resistance show greater discrepancy against experimental values.

Estimations of the T-stress for small cracks at notches

The T-stress is increasingly being recognized as an important additional stress field characterizing parameter in the analyses of cracked bodies. Using T-stress as the constraint parameter, the framework of failure assessments including the constraint effect has been established; and the effect of T-stress on fatigue crack propagation rate has been investigated by several researchers. In this paper, a simple method for determining the T-stress for small notch-emanating cracks is presented. First, the background on the T-stress calculation using the superposition principle and the similarities between the elastic notch-tip stress fields described by two parameters: the stress concentration factor ( K t) and the notch-tip radius ( ρ), are summarized. Then, the method of estimating T-stress for both short and long cracks at the notches is presented. The method is used to predict T-stress solutions for cracks emanating from an internal hole in a wide plate, and cracks emanating from an U-shaped edge notch in a finite thickness plate. The results are compared to the T-stress results in the literature, and the T-stresses solutions obtained from finite element analysis. Excellent agreements have been achieved for small cracks. The method presented here can be used for a variety of notch crack geometries and loading conditions.

Inverse design of phononic crystals by topology optimization

Abstract Band gaps, i.e frequency ranges for which waves cannot propagate, can be found in most elastic structures if the material or structure has a specific periodic modulation of material properties. In this paper, we maximize phononic band gaps for infinite periodic beams modelled by Timoshenko beam theory, for infinite periodic, thick and moderately thick plates, and for finite thick plates. Parallels are drawn between the different optimized crystals and structures and several new designs obtained using the topology optimization method.

Numerical investigations of maximum stress concentration at elliptic holes in finite thickness piezoelectric plates

The through-thickness variations of stress-concentration factors along the wall of elliptic holes in finite thickness plates of transversely isotropic piezoelectric materials subjected to uniaxial remote tensile stress and applied electric field have been systematically analyzed using the finite element method. The three-dimensional stress concentration factor K t is found to be a function of the thickness to root radius ratio B/ ρ and the aspect ratio t (short to long axial length) of the elliptic holes under tensile loading. It is found that the maximum stress-concentration factor through the thickness, ( K t ) max, is 20–150% higher than the value on the free surface ( K t ) surf when t changes from 1 to 0.01, and the ratio of the surface value ( K t ) surf to the corresponding planar solution ( K t ) p− σ at the roots is only 0.84–0.44 when t ranges from 1 to 0.01 if B/ ρ is large enough. When B/ ρ is decreasing to 1, both the ratios are approaching unity. Simple empirical formulae for the relationships between the three values were obtained by fitting the numerical results with good engineering accuracy for large range of B/ ρ (from 1 to 100,000) and t (from 0.01 to 1). The proposed formulae will be useful for strength and fatigue designs of engineering structures with notches and holes. Additional applied electric field can cause higher opening stress in the interior and lower opening stress at the free surface of the plate near the hole, and enhance the out-of-plane constraint significantly. Therefore, the three-dimensional effects can be much stronger in piezoelectric ceramics than in metallic materials.

Weight functions for T-stress for semi-elliptical surface cracks in finite-thickness plates

This paper presents the application of the weight function method for the calculation of elastic T-stress for semi-elliptical surface cracks. First, the weight function method for the calculation of T-stress previously developed for two-dimensional crack problems was extended for the T-stress calculation for three-dimensional crack problems. Then, the T-stress weight functions for the deepest point (corresponding to the parametric angle ϕ = 90°) and for any general point (5° ≤ ϕ < 90°) along the crack front of semi-elliptical surface cracks in finite-thickness plates for wide ranges of crack aspect ratios a/c and relative depths a/t were derived. The resulting weight functions were validated using available finite element results for non-linear stress fields and remote tension and bending cases, and very good agreement was achieved. The weight functions are suitable for the calculation of the T-stress under complex loading conditions for any general point (5° ≤ ϕ ≤ 90°) of surface cracks with wide ranges of aspect ratios, 0 ≤ a/c ≤ 1, and relative depths, 0 ≤ a/t ≤ 0.8.

Contact deformation and stress in orthotropic plates

Contact loading of orthotropic plates is of interest in impact problems of composite materials. A review of the literature shows that the major features of contact loading of an orthotropic half space can be determined, but that the connection between these results and the problem of contact loading of finite thickness plates is relatively unexplored. It is found that the pressure distribution of the half space problem can be used in conjunction with the elasticity solution of Pagano and Srinivas and Rao for transverse pressure loading of simply supported orthotropic plates, with an empirical modification of the peak pressure value. The entire stress, strain, and deformation field can then be calculated throughout the plate. Examples show the relative importance of the contact stresses relative to the overall bending stresses.

Analysis of interacting semi-elliptical surface cracks in finite thickness plates under remote bending load

Interaction effects of two coplanar self-same shallow and deep semi-elliptical surface cracks in finite thickness plates subjected to remote tension have been previously investigated by Sethuraman et al. Using the finite element based force method. In the present study, the effect of remote bending load on interacting semi-elliptical surface cracks in a finite thickness plate is analyzed. Stress intensity factors are evaluated along the entire crack front using a modified force method based on the three-dimensional finite element solution. The line spring model has also been used to evaluate stress intensity factors at the deepest point of a crack using shell finite element analysis. Parametric studies involving a wide ranges of geometric dimensions and crack configurations viz. crack shape aspect ratio (0.2≤ a/ c≤1.0), crack depth ratio (0.2≤ a/ t≤0.9), relative crack location (0.2≤2 c/ d≤0.9) and normalized location on the crack front (0≤2 ϕ/π≤2) are carried out for numerical estimation of crack interaction factors. Due to the crack interaction, the stress intensity factor distribution is observed to be asymmetric along the crack front. The interaction is also observed to cease when the distance between two cracks is more than five times the crack width (i.e. 2 c/ d less than 0.2) irrespective of crack shape aspect ratio. Finally, an empirical relation is proposed for the evaluation of crack interaction crack interaction factors for the range of parameters considered. For the ranges considered, the proposed empirical relation predicts the crack interaction factors at the inner and outer surface points of the crack within ±4% of the three-dimensional finite element solutions.

Fatigue growth of a surface crack in a welded T-joint

Structural component failures due to fatigue loading often occur in welded T-joints, the geometry of which gives raise to a local stress increase that makes them preferential sites for crack initiation and propagation. In the present paper, the fatigue behaviour of a T-jointed blade in a hydraulic turbine runner is analysed both experimentally and numerically. For relatively small cracks, it is shown that the structural component geometry does not remarkably influence the stress-intensity factor (SIF) values, provided that the stress field in the vicinity of the crack is approximately the same. Therefore, in order to simplify the problem, a cracked finite-thickness plate is examined instead of the actual T-jointed blade. The SIFs along the front of a semi-elliptical surface flaw in such a plate are determined through a finite element analysis for different elementary loading conditions that can be employed to model the actual stress field at the expected crack location in the examined T-joint. Then, by applying the superposition principle and the power series expansion of the actual stress field due to the load applied to the T-joint being considered, an approximate SIF for the cracked T-joint is evaluated and used for fatigue crack growth simulations by also employing a two-parameter model based on the Paris law. The numerical results deduced are compared with those obtained from experimental tests. The agreement between the different results is quite satisfactory.

Uniform Elliptic Estimate for an Infinite Plate in Linear Elasticity

We present a new study of linear elasticity for an infinite three-dimensional plate of finite thickness Ω = ℝ2 × (−1, 1). We first characterize the kernel of the operator of elasticity as polynomials which can be build from the kernel of the classical Kirchhoff–Love model of plate. Using this characterization, we get optimal uniform elliptic estimates W k, p , C k, α on the solution as a function of the exterior forces. We also give an interior estimate.

THERMOELASTIC LAMB WAVES IN ELECTRICALLY SHORTED TRANSVERSELY ISOTROPIC PIEZOELECTRIC PLATE

The propagation of Lamb waves in a homogeneous, transversely isotropic, piezothermoelastic plate, which is stress free, electrically shorted, and thermally insulated (or isothermal), is investigated. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave mode propagation are derived in completely separate terms. It is shown that the motion of the purely transverse shear horizontal (SH) mode gets decoupled from the rest of the motion and remains unaffected due to piezoelectric, pyroelectric, and thermal effects. The secular equations for stress-free piezoelectric, thermoelastic, and elastic plates are deduced as special cases in the current analysis. At short wavelength limits the secular equations for symmetric and skew symmetric modes reduce to Rayleigh surface wave frequency equation, because a finite-thickness plate in such a situation behaves like a semi-infinite medium. The amplitudes of dilatation, electrical potential, and temperature change are also computed during the symmetric and skew symmetric motion of the plate. Finally, numerical solutions of various secular equations and other relevant relations are carried out for cadmium selenide (6 mm class) material. The dispersion curves, attenuation coefficients and amplitudes of dilatation, temperature change, and electrical potential for symmetric and antisymmetric wave modes are presented graphically to illustrate and compare the analytical results. The theory and numerical computations are found to be in close agreement. The coupling between the thermal/electric/elastic fields in piezoelectric materials provides a mechanism for sensing thermomechanical disturbances from measurements of induced electric potentials and for altering structural responses via applied electric fields. Therefore, the analysis will be useful in the design and construction of Lamb wave sensors, temperature sensors, and surface acoustic wave filter devices.

On the flexural and extensional thermoelastic waves in orthotropic plates with two thermal relaxation times

Analysis for the propagation of plane harmonic thermoelastic waves in an infinite homogeneous orthotropic plate of finite thickness in the generalized theory of thermoelasticity with two thermal relaxation times is studied. The frequency equations corresponding to the extensional (symmetric) and flexural (antisymmetric) thermoelastic modes of vibration are obtained and discussed. Special cases of the frequency equations are also discussed. Numerical solution of the frequency equations for orthotropic plate is carried out, and the dispersion curves for the first six modes are presented for a representative orthotropic plate. The three motions, namely, longitudinal, transverse, and thermal, of the medium are found dispersive and coupled with each other due to the thermal and anisotropic effects. The phase velocity of the waves gets modified due to the thermal and anisotropic effects and is also influenced by the thermal relaxation time. Relevant results of previous investigations are deduced as special cases.

Finite element based evaluation of stress intensity factors for interactive semi-elliptic surface cracks

A finite thickness plate with two coplanar self-same shallow and deep semi-elliptical surface cracks subjected to remote tensile surface traction is considered for fracture analysis. Based on three-dimensional (3D) finite element solutions, stress intensity factors (SIFs) are evaluated along the entire crack front using a force method. The line spring model has also been used to evaluate crack depth point SIFs using shell finite element analysis. A wide range of geometric dimensions and crack configurations viz. crack shape aspect ratio (0.3≤ a/ c≤1.2), crack depth ratio (1.25≤ t/ a≤6), relative crack location (0.33≤2 c/ d≤0.9) and normalized location on the crack front (0≤2 φ/π≤2) are considered for numerical estimation of crack interaction factors. SIFs evaluated at the depth point using the force method from the 3D finite element results are compared with SIFs evaluated using the line spring model. Finally, using finite element results, an empirical relation is proposed for the evaluation of crack interaction factors. For the ranges considered, the proposed empirical relation predicts crack interaction factors at critical locations within ±2% of the 3D finite element solutions.

A study of erosion through plate targets

Computational experiments have been performed to study projectile erosion during the various phases of perforation of finite thickness steel plate targets. Studies were conducted at several impact velocities for thin, three-plate, spaced targets separated by one plate thickness. For tungsten alloy projectiles (Wi-Ni-Co), it is shown that the primary phase erosion is quantitatively similar irrespective of impact velocity and results in a perforation efficiency, η p , normally attained only in semi-infinite target, hypervelocity penetration cases. The indifference with respect to velocity is due to the close proximity of the plate rear surface and results in a similarity of the instantaneous perforation efficiency profile through the plate thickness. Further, it is shown that values for the semi-infinite penetration efficiency, as a function of impact velocity, vary during the penetration event and only attain their respective reference values at late-time. In short, P/L is not as velocity differentiated at early-time as it is at late-time. Bulge phase and residual phase erosion results are consistent with the notion that these erosion mechanisms are sensitive to impact velocity. For comparative purposes, simulations were also performed for a single-plate, equivalent line-of-sight (LOS) thickness plate target. In this case, the primary phase only perforation efficiency increases slightly with impact velocity. Taken together, perforation efficiency results for these two finite thickness plate targets and archival data for the penetration efficiency of semi-infinite thick targets suggest a small geometric effect when corresponding data are compared.

Rayleigh–Lamb waves in magneto-thermoelastic homogeneous isotropic plate

The propagation of magnetic-thermoelastic plane wave in an initially unstressed, homogeneous isotropic, conducting plate under uniform static magnetic field has been investigated. The generalized theory of thermoelasticity is employed, by assuming electrical behaviour as quasi-static and the mechanical behaviour as dynamic, to study the problem. The secular equations for both symmetric and skew-symmetric waves have been obtained. The magneto-elastic shear horizontal (SH) mode of wave propagation gets decoupled from rest of the motion and it is not influenced by thermal variations and thermal relaxation times. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation, because a finite thickness plate in such a situation behaves like a semi-infinite medium. Thin plate results are also deduced at the end. Dispersion curves are represented graphically for various modes of wave propagation in different theories of thermoelasticity. The amplitudes of displacement, perturbed magnetic field and temperature change are also obtained analytically and computed numerically. The result in case of elastokinetics, magneto-elasticity and coupled magneto-elasticity has also been deduced as special cases at appropriate stages of this work.

On the stability of planar fluid interfaces under van der Waals surface forces

This paper concerns an analysis of the stability of infinite planar interfaces between two immiscible fluids. The interface is subjected to a two-dimensional stress arising from the van der Waals interaction with a solid, infinite plate of finite thickness, positioned edge-on to the interface. The physical problem models the setup of a Wilhelmy plate experiment just prior to plate immersion through a fluid interface and three-phase contact line formation. The van der Waals surface force is attractive, causing the fluid interface to deform. We investigate the stability of the fluid interface and establish an analytical condition which is sufficient to ensure a stable interface as well as a condition for which the interface will not be stable to arbitrary disturbances. Both conditions focus on a sole, key function of the equilibrium profile, the latter a solution of the Euler–Lagrange equation for the system.

Mixed mode stress intensity factors for deflected and inclined surface cracks in finite-thickness plates

Using three-dimensional enriched finite elements, mixed mode stress intensity factor solutions are presented for deflected and inclined surface cracks in finite-thickness plates under uniform tensile remote loading. It is demonstrated that the enriched finite element technique provides fracture solutions for these types of problems in a very direct and convenient manner without a need for post-processing the finite element solution. Mixed mode stress intensity factor solutions are generated for semi-circular surface cracks with various deflection and inclination angles ranging from 0° to 75°. The effect of plate thickness on the mixed mode fracture solution is also investigated. In the analyses presented, crack length/plate thickness ( a/ t) ratio ranges from 0.2 to 0.8. It is shown that decreasing the plate thickness results in magnification of mixed mode stress intensity factors along the crack front. Based on the computed mixed mode stress intensity factors, crack propagation angles along the deflected and inclined crack fronts are also determined.

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates subject to non-uniform stress distributions

This paper deals with the T-stress calculation for surface cracks in plates. Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. The T-stress solutions are presented along the crack front for cracks with a/ t values of 0.2, 0.4, 0.6 or 0.8 and a/ c values of 0.2, 0.4, 0.6 or 1.0. Four stress distributions: uniform, linear, parabolic and cubic were applied to the crack face. The superposition method for the calculation of T-stress for three-dimensional cracks is presented. Using the obtained solutions for the four polynomial stress distributions, the application of the superposition method to calculate the T-stress for other stress distributions is demonstrated. These current T-stress solutions together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components under non-uniform stress distributions.