Numerical Estimation of the Elastic Properties of Thin-Walled Structures Manufactured from Short-Fiber-Reinforced Thermoplastics

A model which allows us to estimate the elastic properties of thin-walled structures manufactured by injection molding is presented. The starting step is the numerical prediction of the microstructure of a short-fiber-reinforced composite developed during the filling stage of the manufacturing process. For this purpose, the Moldflow Plastic Insight^® commercial program is used. As a result of simulating the filling process, a second-rank orientation tensor characterizing the microstructure of the material is obtained. The elastic properties of the prepared material locally depend on the orientational distribution of fibers. The constitutive equation is formulated by means of orientational averaging for a given orientation tensor. The tensor of elastic material properties is computed and translated into the format for a stress-strain analysis based on the ANSYSÒ finite-element code. The numerical procedure and the convergence of results are discussed for a thin strip, a rectangular plate, and a shell of revolution. The influence of manufacturing conditions on the stress-strain state of statically loaded thin-walled elements is illustrated.     

Probabilistic homogenization of solid foams

The present study is concerned with a numerical procedure for prediction of uncertainties in the effective properties of solid foams. The approach is based on the multiple homogenization analysis of small-scale testing volume elements. Their microstructure is defined in terms of random variables with known probability distribution. Using a discretization of the space of the random variables, the probability distribution for the effective properties can easily be determined from the homogenization results and the probability for occurrence of the underlying microstructures of the testing volume elements. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some basic ideas for the reconstruction of statistically similar microstructures for multiscale simulations

In this contribution we propose a method for the generation of statistically similar representative volume elements (RVE's). These RVE's are obtained by assuming that given real microstructures may be represented by periodic ones. Then an optimization problem has to be solved where the side condition of equal spectral density of the real microstructure and the statistically similar RVE is taken into account. First numerical results show that the proposed method works in principle and leads to more efficient direct multiscale simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study on statistically similar RVEs for real microstructures based on different statistical descriptors

A method for constructing statistically similar representative volume elements (SSRVEs) for a real dual-phase (DP) steel microstructure is presented in this contribution. The advantageous material properties of such kind of steels originate from the interaction of the microstructure constituents of the material on the microscale. In order to capture these effects directly in the material modeling, the FE² method is a suitable tool, where an RVE representing the microstructure of a material is used in the microscopic boundary value problem, which is solved at each Gauss-point of the macroscopic boundary value problem. However, RVEs based on real microstructures typically implicate high computational expenses due to the complexity of the underlying microstructure and its discretization. SSRVEs, which have a lower complexity than conventional RVEs but are still able to represent the material, can be used instead. Here, different statistical measures for the construction of SSRVEs and their comparison are the main focus. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fortin operator and discrete compactness for edge elements

Summary. The basic properties of the edge elements are proven in the original papers by Nédélec [22,23] In the two-dimensional case the edge elements are isomorphic to the face elements (the well-known Raviart–Thomas elements [24]), so that all known results concerning face elements can be easily formulated for edge elements. In three-dimensional domains this is not the case. The aim of the present paper is to show how to construct a Fortin operator which converges uniformly to the identity in the spirit of [5,4]. The construction is given for any order tetrahedral edge elements in general geometries. We relate this result to the well-known commuting diagram property and apply it to improve the error estimate for a mixed problem which involves edge elements. Finally we show that this result can be applied to the analysis of the approximation of the time-harmonic Maxwell's system. Received March 22, 1999 / Revised version received September 23, 1999 / Published online July 12, 2000     

Construction of Statistically Similar RVEs for 3D Microstructures

This contribution presents a method to construct three-dimensional Statistically Similar RVEs (SSRVEs) for the simulation of dual phase steel (DP steel). Since the microstructure of DP steel strongly influences the overall material properties, it should be incorporated in numerical calculations. For this purpose the FE² method can be applied and for an efficient computation SSRVEs with a reduced complexity compared to the real microstructure have to be defined, which still represent the mechanical response of the material accurately. The construction method is based on the minimization of a least-square functional considering suitable statistical measures describing the inclusion morphology of a given real microstructure. The mechanical response of the SSRVEs is compared to the response of the real microstructure in virtual mechanical tests. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of uncertainties in the effective material behaviour of disordered structural foams

The present contribution is concerned with a numerical analysis of the uncertainties in the structural response of threedimensional structural foams with partially open cells. The effective thermo mechanical material response is determined by means of an energy based homogenization procedure. Stochastic effects in the geometry and topology of the microstructure are treated by means of a repeated analysis of small-scale representative volume elements with prescribed relative density and prescribed cell size distribution. The results are evaluated by stochastic methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity in algebraic frames

This article considers algebraic frames in which the meet of two compact elements is compact, and, in that context, when the subframe of all regular elements is itself regular. Motivated by the study of a frame of convex ?-subgroups of a lattice-ordered group, a number of relevant sufficient conditions are given for this subframe to be regular. An example is given of a frame of convex ?-subgroups for which the subframe of regular elements is not regular.     

Effect of the properties of the components and the geometric characteristics of the structure on the thermal-conductivity coefficients of glass-reinforced plastics

A method is proposed for estimating the thermal-conductivity coefficients of orthotropic composite glass-reinforced plastics with a structure composed of continually repeating units, in which the contours of the reinforcing elements can be described by a sinusoidal law. The arguments in the formulas are the relative content and thermophysical characteristics of the components and the parameters of the microstructure of the materials considered.Mekhanika Polimerov, Vol. 4, No. 1, pp. 18–23, 1968     

Comparison of Statistical Descriptors for the Construction of Statistically Similar RVEs

A method for the construction of three-dimensional statistically similar representative volume elements (SSRVEs) is presented. Since the beneficial material properties of microheterogeneous materials originate in the microstructure, its incorporation in material modeling is desired. The FE₂ method is a suitable tool to accomplish this step. Applying this method, a microscopic boundary value problem, given by a representative volume element (RVE) is attached to every integration point of the macroscale. Such RVEs taken from real microstructures exhibit a high level of complexity, thus raise computational costs. The use of a SSRVE with lower complexity is able to decrease these costs. Here, different statistical measures are compared in view of the performance in the SSRVE construction procedure. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Model for the Evolution of Laminates

We study the time evolution of a generalized standard material in elastoplasticity. Of our particular interest are the formation and the evolution of microstructure. Our aim is to prove the existence of solutions. This is a challenging task, since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with the notion of generalized standard materials. More precisely, we shall allow such forms of microstructure that are given by simple laminates. We will consider a model for the evolution of these laminates and we will state a result on the existence of solutions to the time-incremental minimization problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A sensitivity analysis for linear systems involving M-matrices and its application to the Leontif model

A sensitivity analysis is made for solutions to linear equation systems involving M-matrices. We present a theorem which tells about relative changes of elements of the solution vector when the coefficients of a given M-matrix shift. The Metzler theorem and the Morishima theorem are generalized, and applied to the Leontief model.     

In situ analysis of trace elements in metalloproteins of human liver by synchrotron radiation X-ray fluorescence

A method using synchrotron radiation X-ray fluorescence analysis toin situ determine trace elements during protein electrophoretically separating process was established. The distribution of elements in protein bands for human liver cytosolic sample separated by SDS-PAGE was analyzed along polyacrylamide gel. The results showed that the protein fraction of peak III in cytosol was mainly composed of metal ion Zn-associated proteins, being in agreement with that given by atomic absorption spectrometry. Thus, it demonstrated the feasibility of this novel technique forin situ analysis of trace elements in protein bands.     

Flow of fluids with microstructure between parallel plates—I

The basic equations for fluids with microstructure are applied to the steady flow between two parallel plates under the action of a constant pressure gradient. The flow is governed by a microstructure parameter α*. The classical flow is recovered when α* → ∞, while maximum effects of microstructure correspond to α* → 0. For a Poiseuille flow, the microstructure fluid exhibits resistance to motion greater than or equal to that of the classical flow. For a Couette flow it is shown that for a given applied velocity to the moving plate, the shearing stress at the plate is greater than or equal to that corresponding to the classical flow situation. For a Generalised Couette flow, it is shown that for a given pressure gradient in the direction of flow, the flow is retarded; while for an adverse pressure gradient the back flow is controlled.     

On the number of maximal dependencies in a data base relation of fixed order

The paper gives asymptotic bounds for the maximum number N_n of non-trivial maximal elements in a data base relation of given order. The result shows that there exist relations which are very rich in maximal elements.     

Analysis of a finite element method for pressure/potential formulation of elastoacoustic spectral problems

A finite element method to approximate the vibration modes of a structure enclosing an acoustic fluid is analyzed. The fluid is described by using simultaneously pressure and displacement potential variables, whereas displacement variables are used for the solid. A mathematical analysis of the continuous spectral problem is given. The problem is discretized on a simplicial mesh by using piecewise constant elements for the pressure and continuous piecewise linear finite elements for the other fields. Error estimates are settled for approximate eigenvalues and eigenfrequencies. Finally, implementation issues are discussed.

     

A Phase Field Approach for Martensitic Transformations in Elastoplastic Materials

A multivariant phase field model for martensitic transformations in elastoplastic materials is introduced which is in mathematical terms the regularization of a sharp interface approach. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, plasticity is considered for the austenitic phase which influences the martensitic evolution. With aid of the model these interactions are studied in detail. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The lattice of strict completions of a finite poset

For a given finite poset , we construct strict completions of P which are models of all finite lattices L such that the set of join-irreducible elements of L is isomorphic to P. This family of lattices, , turns out to be itself a lattice, which is lower bounded and lower semimodular. We determine the join-irreducible elements of this lattice. We relate properties of the lattice to properties of our given poset P, and in particular we characterize the posets P for which . Finally we study the case where is distributive. Received October 13, 2000; accepted in final form June 13, 2001.     

On natural isomorphisms of cycle permutation graphs

Two problems are approached in this paper. Given a permutation onn elements, which permutations onn elements yield cycle permutation graphs isomorphic to the cycle permutation graph yielded by the given permutation? And, given two cycle permutation graphs, are they isomorphic? Here the author deals only with natural isomorphisms, those isomorphisms which map the outer and inner cycles of one cycle permutation graph to the outer and inner cycles of another cycle permutation graph. A theorem is stated which then allows the construction of the set of permutations which yield cycle permutation graphs isomorphic to a given cycle permutation graph by a natural isomorphism. Another theorem is presented which finds the number of such permutations through the use of groups of symmetry of certain drawings of cycles in the plane. These drawings are also used to determine whether two given cycle permutation graphs are isomorphic by a natural isomorphism. These two methods are then illustrated by using them to solve the first problem, restricted to natural isomorphism, for a certain class of cycle permutation graphs.     

On polyhedra of Perron-Frobenius eigenvectors

This paper deals with the positive eigenvectors of nonnegative irreducible matrices which are merely characterized by a given upper bound u on their spectral radius and by a given matrix L of lower bounds for their elements. For any such matrix, the normalized positive left [right] eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows [columns] of the matrix $\text{(u}\text{I}\text{?}\text{L}\text{)}{}^{\mathrm{?1}}$. This polyhedron is proven to be also the smallest closed set which is guaranteed to contain the positive left [right] normalized eigenvector; its vertices are therefore the best componentwise bounds one can obtain on the positive eigenvectors of these matrices. A less general result has also been obtained for the symmetrical case, when the matrices are only characterized by a given lower bound l on their spectral radius and by a given matrix U of upper bounds for their elements.