First-order Theory Research Articles

Free Vibration Analysis of Functionally Graded Porous Cylindrical Panels Reinforced with Graphene Platelets.

The free vibration of functionally graded porous cylindrical shell panels reinforced with graphene platelets (GPLs) was numerically investigated. The free vibration problem was formulated using the first-order shear deformation shell theory in the framework of the 2-D natural element method (NEM). The effective material properties of the GPL-reinforced shell panel were evaluated by employing the Halpin-Tsai model and the rule of mixtures and were modified by considering the porosity distribution. The cylindrical shell surface was transformed into the 2-D planar NEM grid to avoid complex computation, and the concept of the MITC3+shell element was employed to suppress shear locking. The numerical method was validated through benchmark experiments, and the free vibration characteristics of FG-GPLRC porous cylindrical shell panels were investigated. The numerical results are presented for four GPL distribution patterns (FG-U, FG-X, FG-O, and FG-Λ) and three porosity distributions (center- and outer-biased and uniform). The effects of GPL weight, porosity amount, length-thickness and length-radius ratios, and the aspect ratio of the shell panel and boundary condition on the free vibration characteristics are discussed in detail. It is found from the numerical results that the proposed numerical method accurately predicts the natural frequencies of FG-GPLRC porous cylindrical shell panels. Moreover, the free vibration of FG-GPLRC porous cylindrical shell panels is significantly influenced by the distribution pattern as well as the amount of GPLs and the porosity.

Free Vibration Responses of Functionally Graded CNT-Reinforced Composite Conical Shell Panels.

Functionally graded CNT (carbon nanotube)-reinforced composites (FG-CNTRCs) are intensively studied because the mechanical behaviors of conventional composites can be dramatically improved. Only a small amount of CNTs are appropriately distributed through the thickness. However, the studies on conical shell panels have been poorly reported when compared with beams, plates and cylindrical shells, even though more parameters are associated with the mechanical behaviors of conical shell panels. In this context, this study intends to profoundly investigate the free vibration of FG-CNTRC conical shell panels by developing an effective and reliable 2-D (two-dimensional) numerical method. The displacement field is expressed using the first-order shear deformation shell theory, and it is approximated by the 2-D planar natural element method (NEM). The conical shell surface is transformed into the 2-D planar NEM grid, and the approach for MITC3+shell element is employed to suppress the shear locking. The developed numerical method is validated through the benchmark experiments, and the free vibration responses of FG-CNTRC conical shell panels are investigated with respect to all the associated parameters. It is found from the numerical results that the free vibration of FG-CNTRC conical shell panels is significantly influenced by the volume fraction and distribution pattern of CNTs, the geometry parameters of the conical shell, and the boundary condition.

Porosity-dependent impact study of a plate with Winkler-Pasternak elastic foundations reinforced with agglomerated CNTs

PurposeThe purpose of this article is to investigate the porosity-dependent impact study of a plate with Winkler–Pasternak elastic foundations reinforced with agglomerated carbon nanotubes (CNTs).Design/methodology/approachBased on the first-order shear deformation plate theory, the strain energy related to elastic foundations is added to system strain energy. Using separation of variables and Lagrangian generalized equations, the nonlinear and time-dependent motion equations are extracted.FindingsVerification examples are fulfilled to prove the precision and effectiveness of the presented model. The impact outputs illustrate the effects of various distribution of CNTs porosity functions along the plate thickness direction, Winkler–Pasternak elastic foundations and different boundary conditions on the Hertz contact law, the plate center displacement, impactor displacement and impactor velocity.Originality/valueThis paper investigates the effect of Winkler–Pasternak elastic foundations on the functionally graded porous plate reinforced with agglomerated CNTs under impact loading.

Quantum Kerr learning

Quantum machine learning is a rapidly evolving field of research that could facilitate important applications for quantum computing and also significantly impact data-driven sciences. In our work, based on various arguments from complexity theory and physics, we demonstrate that a single Kerr mode can provide some ‘quantum enhancements’ when dealing with kernel-based methods. Using kernel properties, neural tangent kernel theory, first-order perturbation theory of the Kerr non-linearity, and non-perturbative numerical simulations, we show that quantum enhancements could happen in terms of convergence time and generalization error. Furthermore, we make explicit indications on how higher-dimensional input data could be considered. Finally, we propose an experimental protocol, that we call quantum Kerr learning, based on circuit QED.

Algebro-geometric Algorithms for Template-Based Synthesis of Polynomial Programs

Template-based synthesis, also known as sketching, is a localized approach to program synthesis in which the programmer provides not only a specification, but also a high-level "sketch" of the program. The sketch is basically a partial program that models the general intuition of the programmer, while leaving the low-level details as unimplemented "holes". The role of the synthesis engine is then to fill in these holes such that the completed program satisfies the desired specification. In this work, we focus on template-based synthesis of polynomial imperative programs with real variables, i.e. imperative programs in which all expressions appearing in assignments, conditions and guards are polynomials over program variables. While this problem can be solved in a sound and complete manner by a reduction to the first-order theory of the reals, the resulting formulas will contain a quantifier alternation and are extremely hard for modern SMT solvers, even when considering toy programs with a handful of lines. Moreover, the classical algorithms for quantifier elimination are notoriously unscalable and not at all applicable to this use-case.In contrast, our main contribution is an algorithm, based on several well-known theorems in polyhedral and real algebraic geometry, namely Putinar's Positivstellensatz, the Real Nullstellensatz, Handelman's Theorem and Farkas' Lemma, which sidesteps the quantifier elimination difficulty and reduces the problem directly to Quadratic Programming (QP). Alternatively, one can view our algorithm as an efficient way of eliminating quantifiers in the particular formulas that appear in the synthesis problem. The resulting QP instances can then be handled quite easily by SMT solvers. Notably, our reduction to QP is sound and semi-complete, i.e. it is complete if polynomials of a sufficiently high degree are used in the templates. Thus, we provide the first method for sketching-based synthesis of polynomial programs that does not sacrifice completeness, while being scalable enough to handle meaningful programs. Finally, we provide experimental results over a variety of examples from the literature.

First-Order Theory of Rewriting for Linear Variable-Separated Rewrite Systems: Automation, Formalization, Certification

The first-order theory of rewriting is decidable for linear variable-separated rewrite systems. We present a new decision procedure which is the basis of FORT, a decision and synthesis tool for properties expressible in the theory. The decision procedure is based on tree automata techniques and verified in Isabelle. Several extensions make the theory more expressive and FORT more versatile. We present a certificate language that enables the output of FORT to be certified by the certifier FORTify generated from the formalization, and we provide extensive experiments.

Hilbert spaces and C⁎-algebras are not finitely concrete

We show that no faithful functor from the category of Hilbert spaces with injective linear contractions into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital C ⁎ -algebras with ⁎-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.

Reconciling transparency, low Δ0-complexity and axiomatic weakness in undecidability proofs

Abstract In a first-order theory $\varTheta $, the decision problem for a class of formulae $\varPhi $ is solvable if there is an algorithmic procedure that can assess whether or not the existential closure $\varphi ^{\exists }$ of $\varphi $ belongs to $\varTheta $, for any $\varphi \in \varPhi $. In 1988, Parlamento and Policriti already showed how to tailor arguments à la Gödel to a very weak axiomatic set theory, referring them to the class of $\varSigma _{1}$-formulae with $(\forall \exists \forall )_{0}$-matrix, i.e. existential closures of formulae that contain just restricted quantifiers of the forms $(\forall x \in y)$ and $(\exists x \in y)$ and are writable in prenex form with at most two alternations of restricted quantifiers (the outermost quantifier being a ‘$\forall $’). While revisiting their work, we show slightly less weak theories under which incompleteness for recursively axiomatizable extensions holds with respect to existential closures of $(\forall \exists )_{0}$-matrices, namely formulae with at most one alternation of restricted quantifiers.

Vibration characteristics of smart laminated carbon nanotube-reinforced composite cylindrical shells resting on elastic foundations with open circuit

This study analyzes the vibration of smart laminated carbon nanotube (CNT)-reinforced composite cylindrical shells integrated with piezoelectric materials and resting on elastic foundations. The effect of open circuit electrical boundary condition on vibration characteristics is investigated when a quadratic variation is considered for the electrical potential. Furthermore, the elastic foundations are included in the mathematical modeling to see their effects when the open circuit electrical boundary condition is applied. The first-order shear deformation shell theory and Maxwell’s static electricity equation are employed to derive the governing equations, and natural frequencies are calculated by solving an eigenvalue problem. Vibration characteristics with the open circuit electrical boundary condition are compared with those of closed (short) circuit electrical boundary condition for various mechanical boundary conditions and shell and piezoelectric geometrical parameters. The results demonstrate that the open circuit electrical boundary condition leads to higher frequencies than the closed circuit condition and it must be noticed in the design of smart composite structures with embedded or surface-bonded piezoelectric materials.

First-order perturbation theory of eigenmodes for systems with interfaces

We present an exact first-order perturbation theory for eigenmodes in systems with interfaces causing material discontinuities. We show that when interfaces deform, higher-order terms of the perturbation series can contribute to the eigenmode frequencies in first order in the deformation depth. In such cases, the first-order approximation is different from the usual diagonal approximation and its single-mode result. Extracting additional first-order corrections from all higher-order terms enables us to recover the diagonal formalism in a modified form. A general formula for the single-mode first-order correction to electromagnetic eigenmodes in systems with interfaces is derived, capable of treating dispersive, magnetic, and chiral materials of arbitrary shape.

Vibration response of the exponential functionally graded material plate with variable thickness resting on the orthotropic Pasternak foundation

The manuscript aims to perform the free and forced vibration analysis considering the porosity and orthotropic foundation effect. For the first time, the study is conducted on a variable thickness FGM plate resting on an orthotropic foundation. This study uses the first-order shear deformation plate theory (FSDT) model to present an exact solution for free and forced vibration analysis of the functionally graded material (FGM) plate with variable thickness. The FGM plate, having exponentially varied material properties in the thickness direction, was restrained on simply supported (S-S-S-S) and clamped-clamped (C-C-C-C) boundary conditions. Three different types of microstructural defects (porosity) are taken into consideration during the establishment of the mathematical modeling. In addition, the effect of the orthotropic Pasternak foundation has been incorporated during the formulation. The governing differential equation for thick to thin plates is derived using a variational approach, and further, the Galerkin method is applied to solve the equation of motion. The results have been compared and validated with the existing literature available for a rectangular plate on an elastic foundation. The parametric studies have been done to explore the detailed understanding of the E-FGM (Exponentially graded material) plate including microstructural defects. Even the benchmark results have been investigated for free and forced vibration analysis of the E-FGM plate on the orthotropic elastic foundation.

Scaling of bi-material sandwich panels subjected to impact loadings

Scaling is a widely used concept in impact and explosion engineering which is mainly utilized to predict the response behaviour of large-scale mechanical structures by conducting experiments on scaled trial models. However, the practical application of currently available conventional scaling methodologies for impact and explosive processes are significantly restricted by their inability to properly capture the scale effects of strain rate and strain hardening. Also, they are suffering from an insufficient number of independent degrees of freedom. Furthermore, the zeroth-order finite similitude theory, which can annihilate the non-scalability of rate and hardening effects, doesn’t provide us with efficient scaling methods when the investigated structure is made out of different materials. More precisely, the zeroth-order finite similitude theory and the classical dimensional analysis, which are suffering from a limited number of independent degrees of freedom, cannot scale the elastic-plastic behaviour represented by different parts of a structure made out of different materials. In other words, classical scaling methodologies including dimensional analysis and even the zeroth-order finite similitude theory cannot fix two different features having the same unit using different degrees of freedom, which make them useless for structures made of more than one material such as bi-material sandwich panels. In the presented paper, a new scaling methodology is developed based on the recently proposed first-order finite similitude theory which for the first time enables us to capture the rate dependent plastic behaviour of an impacted structure made out of two different materials. A sandwich panel made of two different materials which is subjected to an impact loading is considered as the case study in order to investigate the efficiency of the proposed first-order based scaling technique. The numerical results are determined using Abaqus finite element software in which the Johnson-Cook constitutive equation is utilized to take into account the rate and hardening effects on the initial yield stress. The results reveal that the behaviour of impacted full-scale bi-material sandwich panels can be predicted with a high accuracy when the extra degree of freedom provided us with the first-order finite similitude (i.e. R1) is set to vary between −1 and 1 (i.e. [Formula: see text]). Furthermore, it is found that the pure mechanical and thermo-mechanical behaviour of the impacted core honeycomb are highly accurately predicted using scaled-down trial models which are made from dissimilar materials and designed based on the zeroth-order finite similitude theory.

Nonlinear dynamics analysis of a graphene laminated composite plate based on an extended Rayleigh–Ritz method

Linear and nonlinear vibration characteristics of the graphene platelet-reinforced composite (GPLRC) laminated plates are investigated. Based on the first-order shear theory and von Karman geometric nonlinearity, the energy expressions of the GPLRC laminates are established. The boundary elastic potential energy is established by penalty function method to simulate different boundary conditions. The linear and nonlinear frequencies of the GPLRC laminated plate are calculated by introducing boundary potential energy into Rayleigh–Ritz method. The convergence and accuracy of the method are verified by numerical examples, and the effects of different parameters on frequency are analyzed. Considering the cantilever boundary conditions, the nonlinear motion governing equations of the GPLRC laminated plate are obtained by Hamilton principle. The two-degree-freedom ordinary differential motion equations of the laminates are derived by Galerkin method. Considering the fundamental parameter resonance and 1:1 internal resonance, the amplitude–frequency response curves of the structure under transverse excitation are obtained. The effects of transverse excitation and damping coefficient on nonlinear vibration characteristics of the GPLRC laminated plates are investigated by numerical simulation.

Buckling analysis of laminated composite thin-walled I-beam under mechanical and thermal loads

Despite the extensive use of thin-walled structures, the studies on their behaviours when exposed to extreme thermal environment are relatively scarce. Therefore, this paper aims to present the buckling analysis of thin-walled composite I-beams under thermo-mechanical loads. The thermal effects are investigated for the case of studied beams undergoing a uniform temperature rise through their thickness. The theory is based on the first-order shear deformation thin-walled beam theory with linear variation of displacements in the wall thickness. The governing equations of motion are derived from Hamilton's principle and are solved by series-type solutions with hybrid shape functions. Numerical results are presented to investigate the effects of fibre angle, material distribution, span-to-height's ratio and shear deformation on the critical buckling load and temperature rise. These results for several cases are verified with available references to demonstrate the present beam model’s accuracy.

Acoustic Radiation From Stiffened Double Concentric Large Cylindrical Shells: Part I Circumferential Harmonic Waves

Abstract Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically investigated via circumferential harmonic waves driven by a point force. The vibration of double isotropic circular cylindrical shells is described by the first-order shear deformation shell theory. One set of uniformly spaced annular bulkheads connects the inner and outer cylindrical shells. In-plane motion equations of the annular bulkheads are expressed by two displacement potential functions. Sound pressure loadings of periodic cavities exerting on the inner and outer cylindrical shells are derived according to the Fourier transform and Poisson summation formula. Far-field sound pressure of the stiffened double cylindrical shells is obtained using the stationary phase method and acoustic radiation features of stiffened double concentric large cylindrical shells with periodic cavities are analyzed in terms of sound pressure power spectra and sound pressure level. Acoustic propagation features of stiffened double cylindrical shells with or without acoustic cavities are shown.

Low-velocity impact, buckling, and vibrations of the piezoelectric doubly curved system via coupled Galerkin and Newmark method

ABSTRACT In the present study, buckling and vibrational analysis of a double-curved piezoelectric panel structure under low-velocity impact force is presented. In this regard, the first-order deformation theory of shell structures is utilized to approximate the displacement of the shell, and the linear strain-displacement relations are presented in a constitutive equation in small deformation elastic region. Finally, the principle of virtual work is exploited to obtain equations of motion under electrical and mechanical loads. The complexity of the equations of double-curved shell structure under electrical and impact loads necessitates using semi-analytical methods in obtaining the effects of different factors on the frequency, contact force, and indentation of the impactor. The results are presented in a detailed parametric study. Effects of moduli of elasticity of both panel and impactor on the contact force and indenter velocity are investigated. As a desirable result, it is determined that in a certain range of applied external voltage, the vibration of the panel is suppressed completely.

Higher-Order Effects in Biaxial Flexure of GFRP I-Section Beams

A theoretical study of Glass Fiber Reinforced Polymer (GFRP) beams subjected to biaxial bending moments is presented with a focus on the influence of higher-order effects on maximum normal stresses. It is shown that the biaxial bending type of loading causes a dramatic increase in the maximum normal stress for a GFRP beam when induced torsional effects are included. The study demonstrates that the traditional first-order theory can grossly underestimate the maximum normal stress in a GFRP beam. Based on the numerical results presented using a higher-order theory which also accounts for induced warping normal stresses, the maximum normal stress is found to be about two to three times larger than that determined using the first-order theory.

Stability Analyses of Cracked Functionally Graded Graphene-Platelets Reinforced Composite Beam Covered with Piezoelectric Layers

As cracks are unavoidable and always reduce structural local stiffness and strength, this paper pays attention to the effect of cracks on the stability of the cracked functionally graded (FG) graphene-nanoplates reinforced composite (GRC) beam covered with piezoelectric layers. Both the critical buckling loads and postbuckling paths of the novel structures with cracks are considered. The massless rotational spring model is employed to calculate the bending stiffness of the cracked section. Three different graphene platelets (GPLs) distribution patterns along the thickness direction of the FG-GRC core beam are studied. The effective material properties of the FG-GRC core beam are calculated by Halpin–Tsai model and the rule of mixture. The governing equations of stability of the cracked FG-GRC piezoelectric beam are established within the framework of the first-order shear deformation beam theory, von Kármán geometric nonlinearity and Ritz method. The direct iteration method is used to examine the effects of boundary conditions, crack parameters, piezoelectric layers and GPL parameters on the critical buckling loads and postbuckling responses of the cracked FG-GRC piezoelectric beams. Results clearly illustrate that GPLs can significantly improve the stability of the cracked FG-GRC piezoelectric beams, while the increasing crack depth has the opposite effect.

THEORIES WITH DISTAL SHELAH EXPANSIONS

AbstractWe show that a complete first-order theory T is distal provided it has a model M such that the theory of the Shelah expansion of M is distal.

Cortico-limbic disruption, material-specificity, and deficits in cognitive-affective theory of mind.

The Theory of Mind deficit due to cognitive-affective disintegration is a poorly understood cognitive consequence of cortical and subcortical disruption in right temporal lobe epilepsy. Following Marr's trilevel approach, we used the material-specific processing model to understand the Theory of Mind deficit in drug-resistant epilepsy (N = 30). We examined pre- and post-surgery changes in first-order (somatic-affective, non-verbal component) and second-order Theory of Mind (cognitive-verbal component) in three groups formed using: (i) seizure side (right versus left), (ii) right temporal epilepsy (right temporal lobe epilepsy versus non-right temporal lobe epilepsy), and (iii) right temporal lobe epilepsy with amygdalohippocampectomy (right temporal lobe epilepsy versus left temporal lobe epilepsy amygdalohippocampectomy versus non-amygdalohippocampectomy). We observed a marked deficit in the first-order Theory of Mind in the right temporal lobe amygdalohippocampectomy group; we mapped this deficit to decline in the non-verbal component of Theory of Mind (somatic-affective component). Preliminary results support using a material-specific processing model to understand the Theory of Mind deficits in right temporal lobe epilepsy amygdalohippocampectomy. Malleability of verbal processing in presence of deterioration of non-verbal processing might have clinical relevance for post-surgery recovery in right temporal lobe epilepsy amygdalohippocampectomy. Documenting the material-specific nature of deficits (verbal versus non-verbal) in non-western, linguistically, and socioeconomically diverse country enables us to understand the problem of heterogeneity in post-surgery cognitive consequences in the right amygdalohippocampectomy.