A finite element for the nonlinear analysis of laminated circular plates

Recently, a new finite element for modelling axisymmetric circular plates was developed.¹ The elements were based on Mindlin's shear-deformable plate theory, and unlike most conventional plate elements, these new elements could be stacked on top of one another to model laminated plates. The elements assured continuity of the displacements between the layers, but not continuity of the traction vectors. Neither interlaminar slip nor debonding between the layers was considered.

In this paper, these new plate elements are incorporated into a Von Karman-type plate theory. The elements may be used to model problems where the membrane stresses play an important role in determining the plate's deflection. For example, it is well-known that for a homogeneous, isotropic plate the membrane stresses begin to affect the deflections when the deflections reach about half the thickness of the plate. For laminated plates, regardless of the magnitude of the deflections, the membrane stresses are present in each layer to balance the interlaminar shear stresses. In this paper, the behavior of these membrane stresses and their influence on the deflections of laminated circular plates are investigated.     

Springback of circular clamped armor steel plates subjected to spherical air-blast loading

The development of protective elements (plates) that are impulsively loaded requires comprehensive experiments in which the transient dynamic and residual deflections are measured. Whereas the residual plastic deflections are easily measured upon completion of the experiment, transient deflection measurements involve more elaborate setups. Relying on the residual deflection only can be misleading when one tries to assess the peak transient deflection which poses a real hazard to the protected subject. In this study, the peak transient and the residual deflections are compared for a clamped circular armor steel plate subjected to large close-range spherical air-blast loading. The difference between the maximal transient and the residual plastic deflection is addressed here as the elastic springback.     

Impulsive loading of fibre-reinforced rigid-plastic plates

A theory for the dynamic deformations of ideal fibre-reinforced rigid-plastic plates was formulated in [1]. This theory is here applied to the problems of loading of circular and rectangular plates by uniformly distributed impulsive loads. The resulting permanent deflections are compared with the corresponding deflections which are predicted by the theory for unreinforced plates. It is found that the effect of ideal reinforcement is to reduce the maximum deflection by a factor which is the order of magnitude of the aspect ratio of the plate.     

Williams meets von Karman: Mode coupling and nonlinearity in the fracture of thin plates

The stress field near the tip of a crack in a plate subjected to membrane and bending loads and undergoing large deflections, is investigated by performing an asymptotic analysis in the context of von Karman plate theory. It is demonstrated that the character of the near tip fields is identical to those of the linear plate theory. However, the determination of the crack tip stress intensity factors requires the solution of a large deflection, and hence nonlinear, problem due to the coupling of the membrane and bending modes. This effect is illustrated through the solution of three fracture problems involving plates of simple geometries loaded by pressure, tension and shearing. In two of these problems, the energy release rate is obtained exactly. Nonlinear finite element computations are performed to obtain the stress intensity factors and energy release rate associated with tension, bending and shearing. These results are compared to the theoretical results for energy release rate and stress intensity factors. This revised version was published online in July 2006 with corrections to the Cover Date.     

Deflection relationships between classical and third-order plate theories

Summary In this paper the differential relationship between the deflections of the classical Kirchhoff and third-order Reddy plate theories is developed, and it is used to determines the relationship between the deflections of polygonal plates with simply supported boundary conditions. As an example, the deflection of a simply supported rectangular plate using the third-order plate theory is obtained from the relationship developed herein. The relationship indicates that the third-order theory yields virtually the same solution as the first-order theory for simply supported rectangular plates.     

FLOW OF A DENSE PARTICULATE MIXTURE USING A MODIFIED FORM OF THE MIXTURE THEORY

The objective of this work is to study the effect of the relative densities of the components in fluid-solid mixture using the formulation presented in Johnson et al., (1991 a,b). The mixture theory is modified to include the effect of buoyancy forces; for this purpose we introduce a term, A₆ v gradρ, in the mechanical interaction between mixture components. When coefficient A₆ is zero, we recover the results presented in Johnson et al., (1991 a,b). It is expected to see settling of the solid particles toward the lower plate which is normally related to the “buoyancy effect” in solid-fluid mixtures. Equations for two-component flows are used to analyze Poiseuille flow between two parallel plates. A review of the basic principles of the mixture theory is presented. The equations for the stress tensor of each component of the mixture and the interactions between the components are given. Flow of a fluid-solid mixture between plates is presented with numerical methods and results.     

Damage detection of surface cracks in composite laminates using modal analysis and strain energy method

This paper presents an approach to detect surface cracks in various composite laminates. Carbon/epoxy composite AS4/PEEK was used to fabricate laminated plates, [0]₁₆, [90]₁₆, [(0/90)₄]_S and [±45/0/90]_2S. Surface crack damage was created on one side of the plate using a laser cutting machine. Modal analysis was performed to obtain the mode shapes from both experimental and finite element analysis results. The mode shapes were then used to calculate strain energy using the differential quadrature method (DQM). Consequently, the strain energies of laminated plates before and after damaged were used to define a damage index which successfully identified the surface crack location.     

Beam analysis of angle-ply laminate end-notched flexure specimens

Analysis and experiments on quasi-unidirectional and angle-ply laminate end-notched flexure specimens are presented. The analysis is based on laminated beam theory incorporating first-order shear deformation theory. Compliance and strain-energy release rate determined for relatively thin unidirectional and angle-ply laminate ENF specimens were in good agreement with a previous classical plate theory formulation. For thicker laminates, however, effects of shear deformation on the compliance of the ENF specimen become significant. An experimental study on glass/polyester quasi-unidirectional and angle-ply laminate ENF specimens was conducted. Specifically, [0]₆, [±30]₅ and [±45]₅ laminates with mid-plane delaminations were considered. Experimental compliance data agreed well with analytical predictions. The fracture toughness increased with increased angle θ at the ±θ interface. This is attributed to the fracture work associated with the debonding of transversely oriented fiber bundles in the quasi-unidirectional plies. The angle-ply laminates displayed more yarn debonding than the quasi-unidirectional laminate. For all laminates it was observed that the crack propagated in a non-uniform manner which is correlated with elastic coupling effects with cracked regions of the laminate beams.     

Preparation of cobalt oxide nanoparticles and cobalt powders by solvothermal process and their characterization

Co₃O₄ nanoparticles and cobalt (fcc-Co) powders were successfully synthesized by solvothermal process from a single precursor. The reaction of Co(Ac)₂ with sodium dodecylbenzenesulfonate (SDBS) shows evident-dependent temperature effect. At 180 °C, Co(Ac)₂ reacts with SDBS to produce precursor CoCO₃ plate structures, which are assembled by small nanoparticles. At the temperature of 250 °C, the precursor CoCO₃ can be gradually decomposed to form Co₃O₄ nanoparticles with diameter of ca. 70 nm. While, at 250 °C, the reaction of Co(Ac)₂ with SDBS also produce precursor CoCO₃ nanoparticles/plates, but the CoCO₃ nanoparticles/plates would only decompose to give metal Co. In this process, SDBS acts as not only a surfactant but also a reagent. Magnetic measurements reveal that the as-prepared Co₃O₄ nanoparticles exhibit weak ferromagnetic properties and Co powders show ferromagnetic properties. In addition, a possible formation mechanism was elaborately discussed.     

Experimental and theoretical investigations on stiffened and unstiffened composite panels under uniform transverse loading

An experimental investigation on fibre-reinforced stiffened and unstiffened panels under transverse uniform pressure has been carried out. The deflections and strains measured inside the laminate are compared with a finite element analysis. The effect of one or two stiffeners within the panel is small since the stiffener fail at small loads. In contrary to this, the effect of a different lay-up is big, because the angle-ply panels [+45°/−45°]₂ carried higher loads than the cross-ply panels [0°/90°]₂. The failure of the panels near the centre of the long edge at the clamping is correctly predicted by the FEA. In addition, calculated and predicted stresses are close to each other.     

Stress relaxation in the interface between creep particle and elastic matrix-strengthening mechanism

The effect of a power law creep particle on interface behavior between the particle and elastic matrix is investigated by stress analysis. Using the results obtained through the stress analysis, the forces due to interaction of an applied stress and stress concentration with an edge dislocation are determined. The direct interaction between the edge dislocation and the creeping particle is studied under fully relaxed stress conditions. Through the investigation the following results are derived. Stress relaxation in the interface can be caused by power law creep along or by diffusion, or a combination of both mechanisms. The degree of stress relaxation caused by diffusion can be defined in terms of the relaxation time for both boundary diffusion and volume diffusion. The amount of stress relaxation caused by the power law creep particle is characterized by the quantity ₂ which is a function of Γ₀ = 2(1/√3)^{1 + m} × (σ_∞/2μ)^m(σ₀/σ_∞t^m), where m is strain rate hardening exponent, σ_∞ is applied stress, μ is the shear modulus, σ₀ is the material constant of the power law creep particle, and t is elapsed time. The value ₂ = 1.0 corresponds to the fully relaxed condition and ₂ = −0.6 corresponds to the initial state. The time to reach a fully relaxed condition is very sensitive to the strain rate exponent, with the smaller m values leading to longer times. The stress state of complete relaxation in the elastic matrix is equivalent to the solution of a void in an elastic matrix superposed on the solution of positive surface traction on the void. This result is identical to that obtained by Srolovitz et al. [Acta. Metall.32, 1979 (1984)]. When the stress is completely relaxed in the particle, all stress components (σ_r, σ_θand σ_rθ) are relaxed, while in the matrix relaxations are observed only for σ_rand σ_θ. The critical resolved shear stress and critical stress to climb the dislocation in the neighborhood of the particle exceed the Orowan stress. Also, the particle attracts the dislocation. Therefore the strengthening of a power law creep particle in an elastic matrix is caused by the Orowan mechanism and by attraction of the dislocation.     

Analysis of plates reinforced with beams

 In this paper a solution to the problem of plates reinforced with beams is presented. The adopted model takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems which are nonlinearly coupled, is achieved using the analog equation method (AEM). The adopted model describes better the actual response of the plate–beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. The resulting deflections are considerably smaller than those obtained by other models. Received 21 April 1999     

An analysis of crack propagation velocity in delayed failure under repeating load using an internal friction model

Frequency and temperature dependency of crack propagation velocity in delayed failure under superposed repeating load was analyzed using an internal friction model which assumes the interaction between hydrogen atoms and the cyclic moving of the position with tri-axial tensile stress at crack tip.

The decrease of crack propagation velocity (da/dt)_R by the superposition of repeating load, the appearance of minimum value in (da/dt)_R at a certain frequency ƒ₀, and the shift of ƒ₀ by the change of temperature, are well explained by the internal friction model. Another reason for the decrease of (da/dt)_R by the superposition of repeating load appears to be the decrease of effective stress intensity at crack tip, though this cannot explain the appearance of minimum value of (da/dt)_R.     

Solid-Phase Syntheses of Two Deacetyl-Thymosin β4 Analogues with Substitution at Position 12 and Their Effects on Impaired Blastogenic Response of T Lymphocytes of Uremic Patients

Two deacetyl-thymosin β₄ analogues containing Phe(4Br) or D-Phe(4Br) as position 12 were synthesized by the manual solid-phase method, and their immunological effects on the impaired blastogenic response of phytohemagglutinin-stimulated T lymphocytes of uremic patients with infectious diseases were studied. Bromination of the p-position of Phe¹² resulted in a marked restorative effect on the impaired blastogenic response of T lympocytes compared with that of our synthetic deacetyl-thymosin β₄ The synthetic [Phe(4Br)¹²]deacetyl-thymsin β₄ was approximately equal in potency to our synthetic [Phe(4F)¹²]deacetyl-thymosin β₄ in uremic patients, but the other analogue, [D-Phe(4Br)¹²]deacetyl-thymosin β₄, had no effect.     

Analysis of thickness locking in classical, refined and mixed multilayered plate theories

This paper discusses the thickness locking (TL) mechanism, also known as Poisson locking, which is caused by the use of simplified kinematic assumptions in the plate analysis. Bending and vibration problems have been analyzed for isotropic, orthotropic and multilayered, composite plates. TL has been investigated for a large variety of plate theories: thin plate theory (TPT), First order shear deformation theory (FSDT), higher order theories (HOT), mixed theories and layer-wise (LW) theories. Transverse normal stress σ_zz and strain _zz zero conditions are discussed. Penalty numbers have been introduced to force _zz = 0 condition in the three-dimensional solution and refined plate theories. The unified formulation has been used to implement the whole considered plate modelings.

Analytical closed form solutions have been considered. A comprehensive numerical investigation has been performed. The following main conclusions have been acquired. (1) TL is strongly due to the coupling between transverse normal strain and in-plane strain in the constitutive law (Poisson effect). (2) TL appears if and only if transverse normal strains _zz are assumed constant in the thickness directions (that happens for TPT, FSDT and HOT with constant and linear transverse displacement expansion in the thickness direction). (3) TL can lead to large error (about 25% for deflections and 15% for circular frequency) in thin, isotropic plate analysis. (4) TL reduces significantly in orthotropic and laminated plates. (5) The use of LW models introduces benefits vs TL. (6) Mixed methods do not make any improvements with respect TL. (7) Penalties technique on elastic coefficients can be efficiently used to enforce _zz = 0 conditions in 3D solutions as well as in HOT, mixed and layer-wise plate theories.     

Nonlinear analysis of laminated,antisymmetric, flat plates subjected to eccentric compression

A nonlinear analysis of simply supported, antisymmetric, angle-ply rectangular plates under uniform axial compression is presented. The solution is based on von Karman's equations for large deflections. A double Fourier series solution is assumed for both the stress function and the transverse deflection. Numerical results are obtained for plates made of graphite/epoxy. Results obtained for square plates are compared favorably with those obtained by a different solution scheme. The effect of plate aspect ratio and the effect of load eccentricity on the response characteristics of the plate are investigated. In some of the studies the number of layers is varied.     

Effect of ferroic domain pattern changes on the Raman spectra of some ferroic crystals

The influence of changes in the pattern of ferroic domain structure on the Raman spectra of β-LiNH₄SO₄ and (NH₄)₃H(SO₄)₂ single crystals were studied. It was shown that the Raman spectra of β-LiNH₄SO₄ passed from the ferroelastic phase differ from those of “as-grown” crystal and those of the crystal, which was in the paraelectric phase. Significant changes could be observed in the Raman bands related to triply degenerated ν₃ and ν₄ vibrations of the SO₄ tetrahedron. Detailed temperature studies of the Raman spectra of β-LiNH₄SO₄ close to the paraelectric–ferroelectric phase transition, exhibit anomaly of some internal vibrations of SO₄ in the temperature range where a regular large-scale structure is observed. Different types of evolution of the ferroelastic domain structure and temperature behaviour of the donor and acceptor vibrations were shown while heating and cooling the (NH₄)₃H(SO₄)₂ crystal. Different values of temperature hysteresis were found in temperature studies of the ferroelastic domain structure (ΔT_S  3–5 K) and in Raman spectra studies (ΔT_S  12 K). No changes were observed in the pattern of ferroelastic domain structure at the temperature T_II–III  265 K, at which C2/c → P2/n structural phase transition takes place. On the other hand, at T_III–IV  135 K additional domains with W′-type of domain wall orientation were found.     

Geometrically nonlinear axisymmetric analysis of thick laminated annular plate using FSDT

This paper presents geometrically nonlinear, axisymmetric, static and transient, moderately large deflection response of a laminated, composite, moderately thick annular plate subjected to uniformly distributed ring loads. The layers are considered to be polar orthotropic. Rotational inertia is included but the inplane inertia is neglected. First-order shear deformation theory is used in terms of the deflection w, rotation ψ of the midplane and the stress function Φ. The governing equations are solved by the orthogonal point collocation method and Newmark-β scheme. The effect of the radius-to-thickness ratio of the plate on the deflection and stress response is studied for symmetrically and unsymmetrically laminated plates.     

The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise models

A continually growing interest in the response of unsymmetric multilayered plates is apparent. Analyses were recently completed addressing the load-deflection behaviour of these plate geometries. The characteristic feature of the analyses is the use of nonlinear strain-displacement relations, even at low loading levels, in reaction to the large-deflection effect enhanced by the bending-extension and twisting-shearing coupling. Approaches where use is made of Higher Order Shear Deformation Theories (HSDT) for predicting global quantities, such as deflections and critical loads, are not found in the open literature. Such modelling approaches, in particular those of the layerwise type, are reserved to predict distributions across the thickness. Thus, a further assessment of the influence of the transverse shear effect on global quantities should be required. To give some preliminary contributions on this subject, the load-deflection behaviour of a [90₄/0₄] cross-ply plate with pinned edges, subjected to cylindrical bending under uniform transverse pressure, is investigated. Use is made of the Layerwise Higher Order Shear Deformation Layerwise Theory (RHSD) to serve this purpose. From the numerical results presented, the influence of modelling is enhanced or reduced, depending on the sign of loading. It is concluded that, depending on the loading, boundary conditions and lay-up, higher-order approaches can be used for predicting global quantities in unsymmetric multilayered plates. In order to investigate stability, nonlinear equations are developed where critical points are located under boundary and combined loading conditions which vary during perturbation.     

A three-dimensional analysis of laminated orthotropic plates

A mathematical apparatus is developed for the analysis of the three-dimensional stress–strain state with small deflections of laminated orthotropic plates of any thickness and non-symmetrical layer structure through the thickness. This theory is based on the representation of the displacement vector in terms of products of the given functions in the direction of the axes x₁, x₂ and the unknown functions over the axis x₃. Both the real and complex roots of the characteristic equation describing the representation of the sought functions through the thickness of the plate are taken into account. Such an approach allows us to expand the scope of physical and mechanical relationships among the material characteristics.