Modulus Theory Research Articles

АНАЛИЗ ВЛИЯНИЯ ПОВЕРХНОСТНЫХ НАПРЯЖЕНИЙ НА ЭФФЕКТИВНЫЕ СВОЙСТВА НАНОПОРИСТЫХ ПЬЕЗОКОМПОЗИТОВ

The present paper considers the homogenization problems for mixed piezoelectric composite materials with stochastic distributions of inclusions or pores and with taking into account the mechanical imperfect interphase boundaries. The accepted interface statements correspond to the Gurtin - Murdoch model and give a significant contribution only for nanostructured composites. To determine the effective properties, an integrated approach was used, based on the theory of effective moduli, on the modelling of representative element volumes and on the finite element method. An aggregate of boundary value problems was described, which allow one to find a complete set of effective stiffness moduli, piezomoduli, and dielectric constants for a piezocomposite of arbitrary anisotropy class. The numerical solution of homogenization problems was carried out in the ANSYS finite-element package, which was used both for modelling of representative element volumes and for computation of the effective properties of composite material. The representative volume consisted of a regular cubic array of piezoelectric finite elements with the material properties of the two phases. The contact boundaries between materials of different phases were covered with elastic membrane elements that simulated interface surface stresses. Specific implementation was performed for nanoporous piezoceramic composites, for which both the initial phases and the homogeneous material were materials of the hexagonal symmetry class, and the pores were considered as piezoelectric material with negligibly small stiffness moduli and piezomoduli. For this composite the membrane elements inherited the anisotropy structure of volume elements on their common edges. As an example, the results of calculations of effective moduli for porous ferroelectric soft piezoceramics PZT-5H were presented. It was noted that the surface stresses on the boundaries of the pores can significantly increase the values of the effective stiffness moduli. However, they had a weak influence on the values of the effective piezomoduli and dielectric constants.

New and old results on spherical varieties via moduli theory

Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme MΓ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an ‘adjoint torus’ Tad and has a distinguished Tad-fixed point X0. In this paper, we obtain a complete description of the Tad-module structure in the tangent space of MΓ at X0 for the case where Γ is saturated. Using this description, we prove that the root monoid of any affine spherical G-variety is free. As another application, we obtain new proofs of uniqueness results for affine spherical varieties and spherical homogeneous spaces first proved by Losev in 2009. Furthermore, we obtain a new proof of Alexeev and Brion's finiteness result for multiplicity-free affine G-varieties with a prescribed weight monoid. At last, we prove that for saturated Γ all the irreducible components of MΓ, equipped with their reduced subscheme structure, are affine spaces.

Moduli of Galois Representations

We develop a theory of moduli of Galois representations. More generally, for an object in a rather general class \mathfrak A of noncommutative topological rings, we construct a moduli space of its absolutely irreducible representations of a fixed degree as a (so we call) "f- \mathfrak A scheme". Various problems on Galois representations can be reformulated in terms of such moduli schemes. As an application, we show that the "difference" between the strong and weak versions of the finiteness conjecture of Fontaine–Mazur is filled in by the finiteness conjecture of Khare–Moon.

Rigidity of integral coisotropic submanifolds of contact manifolds

Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.

Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method

The finite element (FE) method offers an efficient framework to investigate the evolution of phononic crystals which possess materials or geometric nonlinearity subject to external loading. Despite its superior efficiency, the FE method suffers from spectral distortions in the dispersion analysis of waves perpendicular to the layers in infinitely periodic multilayered composites. In this study, the analytical dispersion relation for sagittal elastic waves is reformulated in a substantially concise form, and it is employed to reproduce spatial aliasing-induced spectral distortions in FE dispersion relations. Furthermore, through an anti-aliasing condition and the effective elastic modulus theory, an FE modeling general guideline is provided to overcome the observed spectral distortions in FE dispersion relations of infinitely periodic multilayered composites, and its validity is also demonstrated.

Improvement of Dynamic Buckling Behavior of Intermediate Aluminized Stainless Steel Columns

This paper experimentally investigated the dynamic buckling behavior of AISI 303 stainless steel aluminized and as received intermediate columns. Twenty seven specimens without aluminizing (type 1) and 75 specimens with hot-dip aluminizing at different aluminizing conditions of dipping temperature and dipping time (type 2), were tested under dynamic compression loading (compression and torsion), dynamic bending loading (bending and torsion), and under dynamic combined loading (compression, bending, and torsion) by using a rotating buckling test machine. The experimental results werecompared with tangent modulus theory, reduced modulus theory, and Perry Robertson interaction formula. Reduced modulus was formulated to circular cross-section for the specimens of type (1).The experimental results obtained showed an advantageous influence of hot-dip aluminizing treatment on the dynamic buckling behavior of AISI 303 stainless steel intermediate columns. The improvements based on the average value of critical stress were19.4 % for intermediate columns type (2) compared with columns type (1) under dynamic compression loading, 8.7 % for intermediate columns type (2) compared with columns type (1) under dynamic bending loading, and 16.5 % for intermediate columns type (2) compared with columns type (1) under dynamic combined loading.

Moduli spaces and Modular forms

The roots of both moduli spaces and modular forms go back to the theory of elliptic curves in the 19th century. Both topics have seen an enormous growth in the second half of the 20th century, but the interaction between the two remained limited. Recently there have been new developments that led to new points of contact between the two topics. One is the theory of K3 surfaces that is rapidly gaining a lot of new interest. Here the link with modular forms on orthogonal groups has led to progress on the Kodaira dimension of the moduli spaces of K3 surfaces. Another new development has been the use of moduli spaces of curves to gather new information about Siegel modular forms. The workshop intended to bring representatives from both the theory of moduli and the theory of modular forms together to further the interaction between the two topics as the time seemed ripe to do this.

Thermal Stress Analysis for Bi-modulus Foundation Beam under Nonlinear Temperature Difference

It is shown that many materials in practical engineering have different moduli in tension and compression. Especially, graphene, a magical material with the highest strength, is a kind of bi-modulus material. In this paper, mechanical response of bi-modulus Winkler foundation beam under nonlinear temperature distribution along height of the beam is studied. A new and convenient method, “intersecting line criterion”, is proposed to certify number of neutral axis in foundation beam and governing equation (set) of position of neutral axis is established, which is solved by iteration procedure of Newton’s method in Matlab. Semi-analytical solutions of stress, deflection and bending moment are subsequently obtained. Meanwhile, a finite element calculation procedure for calculating the temperature stress in bi-modulus structures is developed. Return the analytical solution to the result of the same modulus theory, and compare the analytical solution with finite element procedure solution and common finite element software Abaqus simulation solution. It shows that both of semi-analytical solution and calculation procedure proposed are reliable to use. Finally, discrepancies between nonlinear temperature effect and linear temperature and external force effect are discussed, which may supply some suggestions for calculation and optimization of such structures and members.

BEHAVIOR OF SLENDER RC COLUMNS WITH INITIAL DEFORMATION IN FIRE

Reinforced concrete columns formed with high-strength concrete have been developed for use as axial force carrying members. They have a reduced cross section compared with conventional “short” columns and are known as “slender” columns. These columns realize larger building interior spaces and offer architectural space with good visibility. However, under fire conditions, bending stiffness of RC columns decreases considerably due to thermal degradation of the cover concrete. It is anticipated that slender RC columns would be more vulnerable to buckling failure than to cross sectional failure. The authors have proposed an evaluation method based on tangent modulus theory for the buckling strength of straight RC columns subjected to centric axial load in fire, and shown that this proposed method offered a good approximation of both experimental fire-resistance time and failure mode. However, existing members have initial deformation formed in manufacturing process. For this reason, it is anticipated that slender RC columns with initial deformation fails earlier than straight ones. This paper presents the main results from an analytical study of slender RC columns with initial deformation shaped sine wave under fire conditions, leading to a discussion of fire performance, which includes deformation behavior and fire-resistance time. All analytical models consist of concrete with a compressive strength of approximately 120N/mm2 and have a square section where the length of one side is 250mm and 5000mm in height. From thermal stress analysis in consideration of geometrical nonlinearity, the followings are obtained: 1. Fire-resistance time of slender RC columns whose the initial deformation α/h was less than approximately 1/103 decreased sharply. 2. There are two failure modes of slender RC columns with initial deformation which are buckling and three-hinged mode. 3. The time, while columns resist an axial force after they start to bend at the time evaluated by the analytical method based on tangent modulus theory, is approximately 10~15 minutes. 4. The larger transient strain was, the shorter fire-resistance time was because the large additional moment occurred with bending deformation increases in fire. 5. The directions of initial deformation have little influence on fire-resistance time. 6. The mechanical boundary conditions at edge of columns have large impact on fire-resistance time.

Composite gas-turbine blades

Theoretical analysis of composite blades for gas-turbine engines is based on the theory of effective moduli and the Lagrange equations. The blades are regarded as twisted rods; the flexure-approximation coefficients are adopted as generalized coordinates.

On spatial aliasing in the phononic band-structure of layered composites

The conventional finite element (FE) method has been used to efficiently obtain phononic dispersion relations for a wide range of geometries and materials. The motive of this research is to draw attention to the research community and addressing that special care should be taken in interpreting the FE results of dispersion analysis for some periodic structures whose FE model possesses an artificial periodicity. Layered composites have been investigated as a representative example to study this numerical issue. Despite the simple geometry of layered composites, it is well known that the accuracy of numerical dispersion relations for waves perpendicular to the layers may be highly impaired by the existence of fictitious modes. The spectral distortion stems from an arbitrarily selected unit cell which causes an artificial finite periodicity in the direction parallel to the composite layers, and they are known to be highly dependent on the unit cell configuration. However, this issue has never been thoroughly investigated, and there has been no specific guideline for a proper unit cell configuration for numerical dispersion relations. In this study, using the classical analytical solution for wave motion in the sagittal plane, the authors show that the spectral distortion induced from the artificial finite periodicity are the manifest results of spatial aliasing in the wavevector domain. In order to prevent the spectral distortion in numerical dispersion relations for waves perpendicular to the layers, this study presents a definitive guideline based on an anti-aliasing condition and the effective elastic modulus theory for layered composites and demonstrates its validity.

Cement-stabilised rammed-earth square column in compression

The paper presents a novel experimental investigation, comprising tests on cement-stabilised rammed-earth (CSRE) prisms and columns of varying slenderness ratios, focusing on the effects of concentric axial loading and variation of the moisture content along the height of the column, and of the slenderness ratio on capacity/stress reduction factors. An attempt was also made to determine the ultimate compressive strength (σu) of columns using tangent modulus theory and the validity of using masonry design rules for the design of CSRE columns. The results showed that the load-carrying capacity of columns decreased with increasing slenderness ratio. The location/zone in the column where the difference in the moisture content was relatively higher (i.e. 3·86% and above) at the time of testing was found to be the weakest zone leading to untimely failure. Tangent modulus theory underestimated σu. The experimental reduction factors were found to be more conservative than the masonry code provisions. The characteristic strength obtained for CSRE columns yielded a relatively higher safety factor and showed that it is possible to construct a single-storey load-bearing house when designed properly.

Dynamic stability analysis for Beck’s type of inelastic Timoshenko steel columns

In this study, the dynamic stability analysis of a Beck’s type of steel column subjected to a sub-tangential follower force is presented considering the effects of gradual yielding along the length of column, the shear deformation, and the rotary inertia. The nonlinear tangent modulus-stress, tangent modulus-slenderness ratio, and compressive stress-strain relationships of an inelastic column are derived from the column strength curve given in Korea Bridge Design Code (KBDC) based on the tangent modulus theory. The equations of motion of the Timoshenko inelastic column are derived from the Hamilton’s principle. The finite element method using the Hermite cubic interpolation functions is employed to obtain the mass, inelastic stiffness, geometric stiffness, and load correction matrices, and the evaluation procedure for critical values of divergence and flutter of the non-conservative system is briefly described. The obtained results are compared with available results of other numerical or analytical methods. Finally, the effects of various structural parameters such as the material inelasticity, the shear deformation, the rotary inertia, the effective length factor, and the sub-tangentiality factor on the divergence and flutter behavior of non-conservative columns are parametrically investigated.

Manifestation of the internal symmetry of irreducible triatomic interaction in the nonlinear properties of low-symmetry metals

A theory of the elastic moduli of crystals and amorphous substances is constructed that considers the internal symmetry of the irreducible energy of triatomic clusters. Allowing for the internal symmetry of clusters allows us to build three invariants by which the potential energy of any free triatomic cluster depends on the atomic coordinates. Direct calculation of the elastic modulus using an analytic form of the dependence of invariants on atomic coordinates shows that the adopted model leads to the following relationship between independent components of the third-rank elasticity tensor: $${C_{xx,\;yy,\;zz\;}} - ({C_{xx,\;yz,yz\;}} + {C_{yy,\;xz,\;xz\;}} + {C_{zz,\;xy,\;xy}}) + 2{C_{xy,\;yz,\;zx\;}} = 0$$ This result is independent of the global symmetry of a substance and represents a generalization of the widely known Caunhy discrepancies, which follow from models that consider only pair interactions. Fifteen laws of conservation that determine the movement of a point representing the atomic coordinates of cluster atoms in 9-dimensional configuration spaces of triatomic clusters are identified. The symmetry group of a triatomic cluster that determines these laws of conservation is identified.

Topological methods in moduli theory

One of the main themes of this long article is the study of projective varieties which are K(H,1)’s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera–de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)’s, through Teichmuller theory. The main trhust of the paper is to show how in the case of K(H,1)’s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lonne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an algebraic surface S such that the Galois conjugate surface of S has fundamental group not isomorphic to the one of S.

Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields

This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigenforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark's conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.

Relative dimension of morphisms and dimension for algebraic stacks

Motivated by applications in moduli theory, we introduce a theory of lower bounds on relative dimension of morphisms of schemes and algebraic stacks. The theory is robust, applies to a wide range of situations, and has strong consequences. We thus obtain simple tools for making dimension-based deformation arguments on moduli spaces. In a complementary direction, we develop the basic properties of codimension for algebraic stacks.

Axial load-capacity of rectangular cement stabilized rammed earth column

The paper presents a novel study on structural behaviour of axially loaded cement stabilized rammed earth (CSRE) prisms and columns of square and rectangular cross-sections. The effects of slenderness ratio and aspect ratio on the load-capacity of columns and stress reduction factors were assessed. Experimental results were compared with that obtained by tangent modulus theory. Experimental capacity/stress reduction factors were compared with published codal values. The result shows that the load-capacity of column decreases as the value of slenderness ratio increases and aspect ratio of column was found to influence the load-capacity. The ultimate compressive strength of column predicted by tangent modulus theory tend to converge the experimental values at higher slenderness ratios. The stress reduction factors in earthen (NZS 4297, AS HB 195) and masonry (IS 1905) standards are found to be in close agreement with the experimental values. The characteristic strength of column yields relatively higher safety factor (∼23–35) indicating a possibility of using CSRE for construction of load-bearing houses.

Analytic elasticity solution of bi-modulus beams under combined loads

A unified stress function for bi-modulus beams is proposed based on its mechanic sense on the boundary of beams. Elasticity solutions of stress and displacement for bi-modulus beams under combined loads are derived. The example analysis shows that the maximum tensile stress using the same elastic modulus theory is underestimated if the tensile elastic modulus is larger than the compressive elastic modulus. Otherwise, the maximum compressive stress is underestimated. The maximum tensile stress using the material mechanics solution is underestimated when the tensile elastic modulus is larger than the compressive elastic modulus to a certain extent. The error of stress using the material mechanics theory decreases as the span-to-height ratio of beams increases, which is apparent when L/h ⩽ 5. The error also varies with the distributed load patterns.

Some closed-form solutions of functionally graded beams undergoing nonuniform torsion

Torsion of linearly elastic isotropic beams, with both cross-sectional and axial inhomogeneities, is analyzed. Twist (rate of torsional rotation along the beam axis) and warping of cross-sections are not uniform if arbitrary axial variations of elastic properties are considered. Composite beams undergoing nonuniform torsion are commonly investigated by finite and boundary element methods. New closed-form solutions are found in the present paper, by detecting axial distributions of longitudinal and shear moduli inducing an axially uniform warping field. The warping is evaluated by Saint-Venant beam theory, while twist and axial distribution of shear moduli are inversely proportional. Coordinate-free expressions of displacement, normal and shear stress fields are given for simply and multiply connected cross-sections. Exact solutions are obtained for elliptic and equilateral triangle beams, by assuming exponentially graded longitudinal and shear moduli. New benchmarks for numerical analyses are thus also provided.