Dimensional Continuum Research Articles

Evaluation of cross-ply laminate stiffness with a non-uniform distribution of transverse matrix cracks by means of a computational meso-mechanic model

This work addresses the evaluation of the stiffness of fiber-reinforced composite laminates, by means of a computational meso-mechanic model, considering two non-uniformly spaced transverse matrix cracks. Laminates with [0n/908]S and [908/0n]S, with n = 1 and 8, have been studied. The meso-mechanic model includes a three dimensional Finite Element continuum model at meso-scale and the macro-scale contains a classical thin laminated plate model. Periodic boundary conditions were used and the stress resultants were evaluated accounting for the equivalence of mechanical power between scales (Hill-Mandel principle). The results obtained with the present model showed good agreement with numerical and experimental data reported in the literature. A parametric analysis allowed identifying the stiffness components which are more influenced by a non-uniform crack distribution. The results suggest that the model with uniformly distributed cracks underestimates the in-plane and bending stiffness, while the bending-extension coupling stiffness components are overestimated.

More absorbers in hyperspaces

The family of all subcontinua that separate a compact connected n-manifold X (with or without boundary), n≥3, is an Fσ-absorber in the hyperspace C(X) of nonempty subcontinua of X. If D2(Fσ) is the small Borel class of spaces which are differences of two σ-compact sets, then the family of all (n−1)-dimensional continua that separate X is a D2(Fσ)-absorber in C(X). The families of nondegenerate colocally connected or aposyndetic continua in In and of at least two-dimensional or decomposable Kelley continua are Fσδ-absorbers in the hyperspace C(In) for n≥3. The hyperspaces of all weakly infinite-dimensional continua and of C-continua of dimensions at least 2 in a compact connected Hilbert cube manifold X are Π11-absorbers in C(X). The family of all hereditarily infinite-dimensional compacta in the Hilbert cube Iω is Π11-complete in 2Iω.

Controlling deformations of electro-active truss structures with nonlinear history-dependent response

This study presents simulations of shape changing in active truss structures that undergo non-linear and time dependent response. The truss structure comprises of slender members with electro-active (piezoelectric) components joined by pins, allowing for large deformations due to rotations at the pins. A nonlinear time-dependent electro-mechanical constitutive model is considered for the piezoelectric components in the truss systems. In order to find the actuation inputs to achieve a desired shape, the required shape is defined with respect to reference configuration for a equivalent 3 dimensional continuum. The mapping between reference and current configuration is used to calculate the macroscopic strains in the continuum. The corresponding strains along the longitudinal axes of the truss members whose ends coincide with certain material points in the continuum are then calculated. The strains corresponding to the predefined shapes are achieved by applying electric field through the piezoelectric materials, and as a result the truss system undergoes the desired shape changes. The shape changing behaviors in electro-active truss systems are analyzed using a finite element (FE) method, incorporating nonlinear material and geometry. Numerical implementation of the active truss structures is also presented. Two types of truss configurations are considered which are planar and beam configurations. The planar configuration is formed by arrangement of cubical truss elements while the beam configuration consists of several tetrahedral truss elements. Several shape configurations of the planar and beam truss structures are shown as examples. While considerable amount of work has been done with piezo-actuated truss systems under the assumption of linear piezoelectric response, we study the influence of the nonlinear hysteretic electro-mechanical behavior and highlight the differences in the deformations of the truss structures when the nonlinear and time-dependent electro-mechanical and linear electro-mechanical models are considered for the piezoelectric components.

Refined Regularity of the Blow-Up Set Linked to Refined Asymptotic Behavior for the Semilinear Heat Equation.

We consider , a solution of which blows up at some time , where , and . Define to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an -dimensional continuum for some , then S is in fact a manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable and reach significant small terms in the polynomial order for some . Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.

Finite Elements Approaches in the Solution of Field Functions in Multidimensional Space: A Case of Boundary Value Problems

An idealized two dimensional continuum region of GRP composite was used to develop an efficient method for solving continuum problems formulated for space domains. The continuum problem is solved by minimization of a functional formulated through a finite element procedure employing triangular elements and assumption of linear approximation polynomial. The assemblage of elements functional derivatives system of equations through FEM assembly procedure made possible the definition of a unique and parametrically defined model from which the solution of continuum configuration with an arbitrary number of scales is solved. The finite element method(FEM )developed is recommended to be applied in the evaluation of the function of functions in irregular shaped continuum whose boundary conditions are specified such as in the evaluation of displacement in structures and solid mechanics problems, evaluation of temperature distribution in heat conduction problems, evaluation of displacement potential in acoustic fluids evaluation of pressure in potential flows, evaluation of velocity in general flows, evaluation of electric potential in electrostatics, evaluation of magnetic potential in magnetostatics and in the solution of time dependent field problems. A unified computational model with standard error of 0.15 and correlation coefficient of 0.72 was developed to aid analysis and easy prediction of regional function with which the continuum function was successfully modeled and optimized through gradient search and Lagrange multipliers approach. Above all the optimization schemes of gradient search and Lagrangian multiplier confirmed local minimum of function as 0.006-0.00847 to confirm the predictions of FEM and constraint conditions.

A homogeneous substitute material for the core layer of photovoltaic composite structures

Abstract The present contribution is concerned with the computational homogenisation of mechanical properties of the heterogeneous photovoltaic module core layer. This is required since the mechanical approach with a novel layerwise theory used for efficient global structural analysis is based on homogeneous layers. This layerwise theory is build on the direct approach for plates, where all considerations are restricted to the deformable midsurface of the individual layer. Due to the geometric structure of the core layer, an anistropic material behavior of the effective substitute material results. The resulting orientation dependence of mechanical properties is specified and visualized. Finally, based on aforementioned work, the results of anisotropic elasticity gained for the three dimensional Cauchy continuum, are transferred to stiffness measures of surface elasticity.

A systematic approach to derive dynamic equations for homogeneous and functionally graded micropolar plates

This work considers a systematic derivation process to obtain hierarchies of dynamical equations for micropolar plates being either homogeneous or with a functionally graded (FG) material variation over the thickness. Based on the three dimensional micropolar continuum theory, a power series expansion technique of the displacement and micro-rotation fields in the thickness coordinate of the plate is adopted. The construction of the sets of plate equations is systematized by the introduction of recursion relations which relates higher order powers of displacement and micro-rotation terms with the lower order terms. This results in variationally consistent partial differential plate equations of motion and pertinent boundary conditions. Such plate equations can be constructed in a systematic fashion to any desired truncation order, where each equation order is hyperbolic and asymptotically correct. The resulting lowest order flexural plate equation is seen to be of a generalized Mindlin type. The numerical results illustrate that the present approach may render accurate solutions of benchmark type for both homogeneous and functionally graded micropolar plates provided higher order truncations are used. Moreover, low order truncations render new sets of plate equations that can act as engineering plate equations, e.g. of a generalized Mindlin type.

Post-buckling responses of functionally graded beams with porosities

The objective of this work is to analyze post-buckling of functionally graded (FG) beams with porosity effect under compression load. Material properties of the beam change in the thickness direction according to power-law distributions with different porosity models. It is known that post-buckling problems are geometrically nonlinear problems. In the nonlinear kinematic model of the beam, total Lagrangian finite element model of two dimensional (2-D) continuum is used in conjunction with the Newton-Raphson method. In the study, the effects of material distribution, porosity parameters, compression loads on the post-buckling behavior of FG beams are investigated and discussed with porosity effects. Also, the effects of the different porosity models on the FG beams are investigated in post-buckling case.

Diffusion of elastic waves in a two dimensional continuum with a random distribution of screw dislocations

We study the diffusion of anti-plane elastic waves in a two dimensional continuum by many, randomly placed, screw dislocations. Building on a previously developed theory for coherent propagation of such waves, the incoherent behavior is characterized by way of a Bethe–Salpeter (BS) equation. A Ward–Takahashi identity (WTI) is demonstrated and the BS equation is solved, as an eigenvalue problem, for long wavelengths and low frequencies. A diffusion equation results and the diffusion coefficient D is calculated. The result has the expected form D=v∗l/2, where l, the mean free path, is equal to the attenuation length of the coherent waves propagating in the medium and the transport velocity is given by v∗=cT2/v, where cT is the wave speed in the absence of obstacles and v is the speed of coherent wave propagation in the presence of dislocations.

The high density phase of the k-NN hard core lattice gas model

The k-NN hard core lattice gas model on a square lattice, in which the first k next nearest neighbor sites of a particle are excluded from being occupied by another particle, is the lattice version of the hard disc model in two dimensional continuum. It has been conjectured that the lattice model, like its continuum counterpart, will show multiple entropy-driven transitions with increasing density if the high density phase has columnar or striped order. Here, we determine the nature of the phase at full packing for k up to . We show that there are only eighteen values of k, all less than k = 4134, that show columnar order, while the others show solid-like sublattice order.

Exact Results for One Dimensional Fluids Through Functional Integration

We review some of the exactly solvable one dimensional continuum fluid models of equilibrium classical statistical mechanics under the unified setting of functional integration in one dimension. We make some further developments and remarks concerning fluids with non pairwise-additive interaction. We then apply our developments to the study of a particular non pairwise-additive Gaussian model for which we are unable to find a well defined thermodynamics.

Towards a voxel-based geographic automata for the simulation of geospatial processes

Many geographic processes evolve in a three dimensional space and time continuum. However, when they are represented with the aid of geographic information systems (GIS) or geosimulation models they are modelled in a framework of two-dimensional space with an added temporal component. The objective of this study is to propose the design and implementation of voxel-based automata as a methodological approach for representing spatial processes evolving in the four-dimensional (4D) space–time domain. Similar to geographic automata models which are developed to capture and forecast geospatial processes that change in a two-dimensional spatial framework using cells (raster geospatial data), voxel automata rely on the automata theory and use three-dimensional volumetric units (voxels). Transition rules have been developed to represent various spatial processes which range from the movement of an object in 3D to the diffusion of airborne particles and landslide simulation. In addition, the proposed 4D models demonstrate that complex processes can be readily reproduced from simple transition functions without complex methodological approaches. The voxel-based automata approach provides a unique basis to model geospatial processes in 4D for the purpose of improving representation, analysis and understanding their spatiotemporal dynamics. This study contributes to the advancement of the concepts and framework of 4D GIS.

Application of high dimensional continuum mechanics to micro-economics

Application of high dimensional continuum mechanics to micro-economics

Adaptive Hierarchical Fracture Model for Enhanced Geothermal Systems

In enhanced geothermal systems, cold working fluid (usually water) is injected into a fractured reservoir to enhance the efficiency of the power plant by increasing the conductivity of existing fractures and by creating new ones. For the modeling of single phase flow and transport in such dynamically changing fractured reservoirs, a new approach is presented. It is based on a hierarchical fracture representation that results in a network of multiple dominant fractures through which most of the mass flow occurs. These large fractures have a discrete representation, i.e., each fracture is represented by a lower dimensional continuum. A single continuum representation is also employed for the damaged matrix, which consists of many small and medium sized fractures accounted for by appropriate effective properties. The main advantage of the new approach is that no expensive remeshing of the domain is required whenever new fractures are added. Here, discretization and coupling between the different continua (damaged matrix with discrete dominant fractures) are explained. In order to remove singularities, kernel functions have been introduced in the governing equations to capture discontinuities like fractures and intersections thereof. This way transfer coefficients are well defined. Numerical verification studies involving flow and heat transport are presented and discussed. It has to be emphasized that geomechanics, rock chemistry, and upscaling are not topics of this paper.

A dimensional comparison between delusional disorder, schizophrenia and schizoaffective disorder

IntroductionSince the early description of paranoia, the nosology of delusional disorder has always been controversial. The old idea of unitary psychosis has now gained some renewed value from the dimensional continuum model of psychotic symptoms. Aims1. To study the psychopathological dimensions of the psychosis spectrum; 2. to explore the association between psychotic dimensions and categorical diagnoses; 3. to compare the different psychotic disorders from a psychopathological and functional point of view. Material and methodsThis is an observational study utilizing a sample of some 550 patients with a psychotic disorder. 373 participants had a diagnosis of schizophrenia, 137 had delusional disorder and 40 with a diagnosis of schizoaffective disorder. The PANSS was used to elicit psychopathology and global functioning was ascertained using the GAF measure. Both exploratory and confirmatory factor analyses of the PANSS items were performed to extract psychopathological dimensions. Associations between diagnostic categories and dimensions were subsequently studied using ANOVA tests. Results5 dimensions – manic, negative symptoms, depression, positive symptoms and cognitive – emerged. The model explained 57.27% of the total variance. The dimensional model was useful to explained differences and similarities between all three psychosis spectrum categories. The potential clinical usefulness of this dimensional model within and between clinical psychosis spectrum categories is discussed.

Minimization of Shear Energy in Two Dimensional Continua with Two Orthogonal Families of Inextensible Fibers: The Case of Standard Bias Extension Test

In this paper we consider Pipkin-type bi-dimensional continua with two orthogonal families of inextensible fibers. We generalize the representation formula due to Rivlin (J. Ration. Mech. Anal. 4(6):951–974, 1955) valid for planar placement fields. They are the sum of two vector functions each of which depends on one real variable only, are piece-wise $\mathscr{C}^{2}$ and may exhibit jumps of their first gradients on some inextensible fibers. Subsequently we consider a deformation energy depending only on the shear deformation relative to the inextensible families of fibers. In the suitably introduced space of configurations representing considered constrained kinematics, we formulate the relative energy minimization problem for the standard bias extension test problem, i.e., elongation of specimens which (i) have the shape of a rectangle with one side exactly three times longer than the other; (ii) are subject to a relative displacement of shorter sides in the direction parallel to the longer one. By exploiting the material and geometric symmetries, we reduce the aforementioned minimization problem to the determination of a piece-wise real function defined in a real interval. A delicate calculation of the energy first variation produces a necessary stationarity condition: it consists of an integral equation which is to be satisfied by the unknown function. The crucial points of this deduction are represented by (i) the reduction of two-dimensional integrals to one-dimensional integrals by the Fubini Theorem and (ii) the determination of the set of admissible kinematical functions on the basis of imposed boundary conditions, which implies a further integral constraint condition on the unknown function. Therefore the energy minimization problem requires the introduction of a global Lagrange multiplier. The established integral equations are solved numerically with a scheme based on a contraction type iterative process. Finally, the equilibrium shapes of the specimen undergoing large deformations, as determined by the presented model, are shown and briefly discussed.

Emergence of four dimensional quantum mechanics from a deterministic theory in 11 dimensions

We develop a deterministic theory which accounts for the coupling of a high dimensional continuum of environmental excitations (called gravonons) to massive particle in a very localized and very weak fashion. For the model presented Schrödinger's equation can be solved practically exactly in 11 spacetime dimensions and the result demonstrates that as a function of time an incoming matter wave incident on a screen extinguishes, except at a single interaction center on the detection screen. This transition is reminiscent of the wave - particle duality arising from the ’’collapse” (also called ’’process one”) postulated in the Copenhagen-von Neumann interpretation. In our theory it is replaced by a sticking process of the particle from the vacuum to the surface of the detection screen. This situation was verified in experiments by using massive molecules. In our theory this ”wave-particle transition” is connected to the different dimensionalities of the space for particle motion and the gravonon dynamics, the latter propagating in the hidden dimensions of 11 dimensional spacetime. The fact that the particle is detected at apparently statistically determined points on the screen is traced back to the weakness and locality of the interaction with the gravonons which allows coupling on the energy shell alone. Although the theory exhibits a completely deterministic ”chooser” mechanism for single site sticking, an apparent statistical character results, as it is found in the experiments, due to small heterogeneities in the atomic and gravonon structures.

Particle partition entanglement of bosonic Luttinger liquids

We consider the R\'{e}nyi entanglement entropy of bosonic Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher order corrections involve a microscopic length scale, the leading order scaling depends only on this sole dimensionless parameter which characterizes the low energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Luttinger liquid theory for the Lieb-Liniger model of $\delta$-function interacting bosons in the one dimensional spatial continuum.

A Dimensional Redefinition of the Schizophrenia Spectrum: Delusional Disorder Versus Schizophrenia Versus Schizoaffective Disorder

Introduction Since the early description of paranoia, nosology of delusional disorder has always been controversial. The idea of ??unitary psychosis is old but has now taken on new value from the dimensional continuum model of psychosis. Aims 1. To study the psychopathological dimensions of the schizophrenia spectrum. 2. To explore the relationship between the dimensions obtained and the categorical diagnoses. 3 To compare the different diagnoses of the psychosis from a psychopathological and functional point of view. Material and Methods an observational study with 550 patients was conducted. 373 patients with schizophrenia, 137 patients with delusional disorder, 40 patients with schizoaffective disorder. PANSS was used to assess the psychopathology and GAF for global functioning. Exploratory and confirmatory factor analysis of the PANSS items was performed in order to obtain a dimensional model. The relationship between diagnostic categories and dimensions was subsequently studied with ANOVA tests. Results 5 Factors,-manic, negative symptoms, depression, positive symptoms and cognition-, similar in composition to other models were obtained. The model yielded the 57.27% of the total variance. The dimensional model obtained was able to explain the differences and similarities between the different categories of the schizophrenia spectrum and the validity of the categories was questioned. The value of the model in order to help establish the diagnosis, prognosis and treatment decision-making was postulated.

Local cohomological properties of homogeneous ANR compacta

In accordance with the Bing–Borsuk conjecture, we show that if $X$ is an $n$-dimensional homogeneous metric ANR continuum and $x\in X$, then there is a local basis at $x$ consisting of connected open sets $U$ such that the cohomological properties of $\ov