Modulus Theory Research Articles

Noncommutative algebraic geometry

The need for a noncommutative algebraic geometry is apparent in classical invariant and moduli theory. It is, in general, impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties by introducing a noncommutative algebraic geometry, where affine “schemes” are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory proposed by the author in [10]. More generally, the geometry in the theory is represented by a swarm , i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given k -linear Abelian category ( k a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative k -algebras, locally parametrizing the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information.

The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in 'linear algebra type' situations.

SAINT-VENANT'S PRINCIPLE FOR A MICROPERIODIC COMPOSITE THERMOELASTIC SEMISPACE: THE DYNAMICAL REFINED AVERAGE THEORY

A Saint-Venant's principle associated with a one-dimensional dynamic coupled thermoelastic effective modulus theory (EMT) for a microperiodic layered semispace was presented in J. Thermal Stresses, vol. 23, pp. 1-14, 2000. It was shown there that a thermoelastic energy associated with a solution to an initial boundary value problem of the theory decays exponentially as a distance x from the thermomechanical load region goes to infinity and that its decay length depends on the time t, an effective velocity d c 1 * ¢ , an effective time unit d T* ¢ , and an effective thermoelastic coupling parameter d k * ¢ . In the present article the Saint-Venant's principle is extended to include a refined averaged theory (RAT) for a microperiodic layered thermoelastic semispace in which a microstructural length is taken into account (see [IFTR Report #25, pp. 1-158, 1995]). It is shown that for such an extended theory, a similar exponential decay estimate for a thermoelastic energy holds true. In the refined estimate the thermoelastic energy depends on a number of microstructural parameters while its decay length is independent of these parameters; and the decay length for small (large) times is comparable to that of a pure thermal (elastic) energy for a rigid (elastic) semispace for every time t > 0 .

435 円筒シェルの連続塑性座屈モードと吸収エネルギー

Experiments were performed to study the behavior of axially crushed aluminum round tubes. The mean radius to thickness (R/h) ratios of these varied from 15 to 75. The modes of collapse and the mean collapse loads obtained in these tests are presented. A simple estimation on the energy-absorption efficiency is presented by using the buckling stress expression based on the tangent modulus theory and the expression of mean collapse load considering the formation of plastic hinges. The estimation results are compared with the experiments.

Smooth indentation of a multilayer laminate by a cylindrical indentor

The smooth contact between a rigid cylindrical indentor and a composite laminate with arbitrary ply-orientation is investigated. An exact Green‘s function for the surface displacement of an anisotropic beam under cylindrical bending is derived from Jones's displacement formulation. By matching the displacements of the top surface of the anisotropic laminate and those of the rigid indentor within the contact region, we can find the contact stress distribution, the magnitude of contact force and the indentation. This exact Green's function approach is also valid for the indentation analysis between an orthotropic laminate and a cylindrical indentor. To simplify the contact analysis, the multilayered laminate is homogeneized as an orthotropic medium using the 3-D effective moduli theory, and thus forms an orthotropic solution. Numerical results for laminates with arbitrary ply-orientation subjected to cylindrical indentation are evaluated to verify the applicability of the proposed exact Green‘s function method. Simplified orthotropic solutions applying the 3-D effective moduli theory are also compared with the exact Green‘s function solution with obvious agreement.

Modeling of structures in the mechanics of composites

One possible general statement of a quasi-static problem in the mechanics of composites is considered. It is assumed that a composite is characterized not only by the heterogeneity of a regular structure, but also by the presence of imperfections, impurities, cracks, and the roughness of surfaces, which are partly taken into account by introducing appropriate couple stresses. Two statements, “in displacements” and “in stresses,” are considered together with the statement of the same problems in the case where the constitutive relations are linear integral operators. The boundary-value problem remains nonlinear due to the nonlinearity of a scattering function which enters into the heat equation. The theory of effective moduli for a nonpolar medium is discussed in more detail. The equilibrium equations for a homogeneous medium with reduced characteristics and the equation of heat inflow, introduced in nonlinear (in an explicit form) and linear variants, are examined. For a simple laminated composite, all effective mechanical and thermophysical characteristics are found in an explicit form. The effective material functions for a transversely isotropic medium are constructed on the basis of a unique dimensionless relaxation kernel with the use of several Il'ushin kernels. Based on the known solution of the boundary-value problem for the reduced medium, the stress and strain concentration tensors, at any point of a simple laminated composite, are also constructed in an explicit form. In this case, the changes in the structure are taken into account.

The significance of effective modulus theory (homogenization) in composite laminate mechanics

In this work, we consider the issues regarding the homogenization of fiber-reinforced layers in a laminate in the presence of macroscopic (ply-level) stress gradients. This is accomplished by considering a series of (free edge) boundary value problems similar to those treated by Pagano and Rybicki in 1974. Despite our inability to provide such a homogenization theory, if one truly exists, we can devise approaches to predict the fiber/matrix interfacial stresses in an arbitrary cell by applying certain displacements and/or tractions on the cell boundaries. These boundary conditions are those derived by representing each layer in the laminate by conventional effective modulus theory. It is shown that these approximations can lead to reasonably accurate interfacial stresses and offer great promise as a means of solving practical laminate problems reinforced by fibers of moderate diameter.

Mathematical models and methods of the mechanics of stochastic composites

The basic approaches used in mathematical models and general methods for solution of the equations of the mechanics of stochastic composites are generalized. They can be reduced to the stochastic equations of the theory of elasticity of a structurally inhomogeneous medium, to the equations of the theory of effective elastic moduli, to the equations of the theory of elastic mixtures, or to more general equations of the fourth order. The solution of the stochastic equations of the elastic theory for an arbitrary domain involves substantial mathematical difficulties and may be implemented only rather approximately. The construction of the equations of the theory of effective moduli is associated with the problem on the effective moduli of a stochastically inhomogeneous medium, which can be solved by the perturbation method, by the method of moments, or by the method of conditional moments. The latter method is most appropriate. It permits one to determine the effective moduli in a two-point approximation and nonlinear deformation properties. In the structure of equations, the theory of elastic mixtures is more general than the theory of effective moduli; however, since the state equations have not been strictly substantiated and the constants have not been correctly determined, theoretically or experimentally, this theory cannot be used for systematic designing composite structures. A new model of the nonuniform deformation of composites is more promising. It is constructed by performing strict mathematical transformations and averaging the output stochastic equations, all the constants being determined. In the zero approximation, the equations of the theory of effective moduli follow from this model, and, in the first approximation, fourth-order equations, which are more general than those of the theory of mixtures, follow from it

A SPATIAL DECAY ESTIMATE FOR TRANSIENT THERMOELASTIC PROCESS IN A COMPOSITE SEMISPACE

A Saint-Venant's principle associated with a one-dimensional dynamic coupled ther moelastic effective modulus theory for a microperiodic layered semispace is presented. In such a theory, the displacement u=u(x,t) and the temperature theta=theta(x,t) (x>=0,t>=0) are approximated by u(x,t)=U(x,t)+h(x)V(x,t) and theta(x,t)=THETA(x,t)+ h(x)PHI(x,t), where U(x,t) and PHI(x,t) represent a macrodisplacement and a macrotemperature, respectively; V(x,t) and PHI(x,t)denote a displacement corrector and a temperature corrector, respectively; h=h(x) is a prescribed periodic microshape function; and the pairs (U,THETA) and (V,PHI) are found by solving an initial boundary value problem described by a system of linear partial differential equations with effective thermoelastic moduli subject to suitable initial and boundary conditions. It is shown that the thermoelastic energy associated with a solution to the problem and stored in the semi-space lying beyond a distance x from the loaded boundary x=0 over the time interval [0,t] decays exponentially as x to infinity and its decay length L depends on the time t, an effective velocity of thermoelastic wave (c*l), an effective time (T*), and an effective thermoelastic coupling parameter (epsilon*). In particular, it is shown that for small (large) times the function L reveals behavior of the decay length for a pure thermal (elastic) energy of a semispace.

Numerical modeling of nonlinear deformation and buckling of composite plate-shell structures under pulsed loading

Nonlinear three-dimensional problems of dynamic deformation, buckling, and posteritical behavior of composite shell structures under pulsed loads are analyzed. The structure is assumed to be made of rigidly joined plates and shells of revolution along the lines coinciding with the coordinate directions of the joined elements. Individual structural elements can be made of both composite and conventional isotropic materials. The kinematic model of deformation of the structural elements is based on Timoshenko-type hypotheses. This approach is oriented to the calculation of nonstationary deformation processes in composite structures under small deformations but large displacements and rotation angles, and is implemented in the context of a simplified version of the geometrically nonlinear theory of shells. The physical relations in the composite structural elements are based on the theory of effective moduli for individual layers or for the package as a whole, whereas in the metallic elements this is done in the framework of the theory of plastic flow. The equations of motion of a composite shell structure are derived based on the principle of virtual displacements with some additional conditions allowing for the joint operation of structural elements. To solve the initial boundary-value problem formulated, an efficient numerical method is developed based on the finite-difference discretization of variational equations of motion in space variables and an explicit second-order time-integration scheme. The permissible time-integration step is determined using Neumann's spectral criterion. The above method is especially efficient in calculating thin-walled shells, as well as in the case of local loads acting on the structural element, when the discretization grid has to be condensed in the zones of rapidly changing solutions in space variables. The results of analyzing the nonstationary deformation processes and critical loads are presented for composite and isotropic cylindrical shells reinforced with a set of discrete ribs in the case of pulsed axial compression and external pressure.

Structural response of cylindrically curved laminated composite shells subjected to blast loading

This paper is concerned with the theoretical analysis and correlation with the numerical results of the displacement time histories of the cylindrically curved laminated composite shells exposed to normal blast shock waves. The laminated composite shell is clamped at its all edges. The dynamic equation of the cylindrical shell used in this study is valid under the assumptions made in Love’s theory of thin elastic shells. The constitutive equations of laminated composite shells are given in the frame of e!ective modulus theory. The governing equation of the cylindrical shell is solved by the Runge-Kutta method. In addition, a finite element modeling and analysis are presented and compared with the theoretical results. The peak deflections and response frequencies obtained from theoretical and numerical analyses are in agreement. The effects of material properties and geometrical properties are examined on the dynamic behaviour.

On interacting longitudinal-shear waves in large elastic deformation of a composite cylindrical laminate

We consider a large deformation of a cylindrical layer. Through the process of homogenization and the theory of effective moduli this heterogeneous medium is converted to a homogeneous but anisotropic medium. On the basis of John’s semilinear material, an energy function is identified and equations of motion obtained. Using the method of characteristics the speeds of propagating linear longitudinal and non-linear shear cylindrical waves are obtained: these “p-wave” and “s-wave” do interact, contrary to the known view, of the small deformation theory.

Anisotropic Constraint Effects on Interactive Damage Progression of General Lay-up Composites

In this paper, a five-layer model is proposed to analyze the anisotropic constraint effects on interactive damage progression. Incorporating this model with the general theory of overall moduli of a cracked body, the effective damage factors proposed by Fan and Zhang in 1993 have been extended as quantitative measures for characterizing the anisotropic constraint effects on damage-induced stiffness reduction of general lay-up composites. By applying the first-order shear deformation plate theory to each sublayer these factors are determined and applied for investigating progressive matrix cracking-delamination interaction. Results show that by this rather unique approach, the intractable problem of anisotropic constraint effects of general lay-up composites on interactive damage progression can be handled in a systematic manner.

Torsion of composite laminated bars with a large number of layers

In view of the effective elastic moduli theory[1], analyzing the thick composite laminated bars subjected to an externally applied torque are presented by theeedimensional finite element (3-D FEM) and global-local method in this paper. Numerical results involving the distribution of shearing stresses on cross-section and the torsional deformation and the interlaminar stresses near to free edges are given. If necessary elements discretization may be densely carried out only in the high stress gradient. region. Obviously, it requires less computer memory and computational time so that it offers and effective way for evaluating strength of laminated bars torsion with a great number of layers.

SAINT-VENANT TYPE DECAY ESTIMATES FOR TRANSIENT HEAT CONDUCTION IN A COMPOSITE RIGID SEMISPACE

The global and pointwise spatial decay estimates for a solution to the one-dimensional initial-boundary value problem of heat conduction in a composite rigid semispace are presented. Vie global estimates cover: (i) a transient heat conduction problem for an isotropic nonhomogeneous semispace with arbitrary variation with depth of the density, specific heat, and thermal conductivity; (ii) a transient heat conduction problem for an isotropic semispace made of a functionally graded material that composition changes from a ceramic surface at x = 0 to a metal semispacex≥.x0> 0; and (iii) a transient heat conduction problem for a microperiodic layered semispace. The pointwise spatial decay estimates are confined to homogenized transient heat conduction problems for a microperiodic layered semispace described by an effective modulus theory as well as by a refined averaged theory

Prediction of the buckling stress of intermediate wooden columns using the secant modulus

We made several buckling tests of wooden columns with intermediate slenderness ratios (λ) and examined the empirical formulas. On the basis of the examination, we formulated an equation for predicting the buckling stress (σ) of an intermediate wooden column. Sitka spruce (Picea sitchensis Carr.) and buna (Japanese beech, Fagus crenata BI.) were used for the studies. A compressive load was applied on the specimen supported with pin ends, and the buckling stress was predicted by the tangent modulus theory and two empirical equations: those of Tetmajer and Newlin-Gahagan. The predicted σ-λ relations were compared with the test results, and the applicability of these predictions were examined. Based on the comparisons, we formulated an equation that can predict the σ-λ relations of materials with various stress-strain characters in the plastic strain range.

Calculation of the goffered multilayered composite cylindrical shells

By using the effective modulus theory and Seydel assumptions the governing equations of the goffered multilayered composite cylindrical shell are formulated. Solution to the problem 1n term of displacements is given by the operator method, while the solution in term of stress function and deflection is obtained by Bubnov-Galerkin method in choosing appropriate forms of these quantities.

The homogenization method in the problem of a spherical layered elastic composite hollow

In this paper, by using the homogenization method for the partial differential equations we study the statics and dynamics problems of a spherical layered elastic composite hollow. We can determine the approximate solution with an arbitrary accuracy, by solving a recurrent series of problems with homogeneous material, where the solution of problem in zero degree (the effective modulus theory) is initial data.

Doppler-Shifted Acoustic Cyclotron Resonance and Ultrasound Velocity in Deformed Tungsten

The effect of dislocations on the Doppler-shifted acoustic cyclotron resonance and the velocity of ultrasound are studied in pure tungsten single crystals. The measurements were made as the ultrasonic frequencies of 40 to 180 MHz with both longitudinal and transverse waves propagating along the [100] axis of the crystal. The relaxation rates and scattering cross-section of electrons were estimated in deformed samples as well as in undeformed ones. A comparison was carried out between the experimental temperature dependence of the modulus defect and theory.

Tensor Products and Powerspaces in Quantitative Domain Theory

One approach to quantitative domain theory is the thesis that the underlying boolean logic of ordinary domain theory which assumes only values in the set {true, false} is replaced by a more elaborate logic with values in a suitable structure Ν. (We take Ν to be a value quantale.) So the order ⊑ is replaced by a generalised quasi-metric d, assigning to a pair of points the truth value of the assertion x ⊑ y.In this paper, we carry this thesis over to the construction of powerdomains. This means that we assume the membership relation ∈ to take its values in Ν. This is done by requiring that the value quantale Ν carries the additional structure of a semiring. Powerdomains are then constructed as free modules over this semiring.For the case that the underlying logic is the logic of ordinary domain theory our construction reduces to the familiar Hoare powerdomain. Taking the logic of quasi-metric spaces, i.e. Ν = [0, ∞] with usual addition and multiplication, reveals a close connection to the powerdomain of extended probability measures.As scalar multiplication need not be nonexpansive we develop the theory of moduli of continuity and m-continuous functions. This makes it also possible to consider functions between quantitative domains with different underlying logic. Formal union is an operation which takes pairs as input, so we investigate tensor products and their behavior with respect to the ideal completion.