Time-continuous modeling of microstructures in single-slip finite elasto-plasticity based on energy relaxation

The non-(quasi)convexity of the free energy of single crystals in finite elasto-plasticity gives rise to the formation of fine-scale fluctuations of the displacement field, which can be interpreted as material microstructures. We outline a numerical procedure to model the origin and subsequent evolution of laminate microstructures. This approach incrementally solves the evolution equations by employing relaxed energy potentials. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformation concentration for martensitic microstructures in the limit of low volume fraction

We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of \(\Gamma \)-convergence. The limit functional turns out to be similar to the Mumford–Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for \(SBV^p\) functions whose jump sets have a prescribed orientation.     

Formulation and application of models for anisotropic hardening in sheet metals subject to complex loading-path changes

In this work material models for the simulation of forming processes involving non-uniform loading path changes are investigated. Materials exhibiting an evolution of the yield surface due to the formation and interaction of dislocation structures on the micro structural level are modeled by employing evolving structural tensors. These account for the evolution of anisotropy and hardening behavior known from technical alloys on a phenomenological basis. In a first step material parameters are identified from experiments for non-uniform loading conditions. In a second step, the evolution of structural tensors for varying loading conditions is investigated and related to the evolution of kinematic quantities. Finally applications relevant to the simulation of sheet metal forming processes are presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dusty shock wave interaction – formation of fully dispersed waves and particle focusing effect

The problem of regular (symmetric and asymmetric) interaction of plane shock waves in a steady-state dusty gas flow is considered. For near-sonic flows with a fairly high particle mass loading, the possibility of the formation of wave structures is revealed, in which either all or only some of the incident or reflected waves degenerate into so-called fully dispersed waves, i.e. zones in which no discontinuities appear in the parameters of each phase. For stronger shock waves and low particle mass concentration, the effect of aerodynamic particle focusing and the formation of a narrow high-concentration beam of particles behind the point of the interaction of the waves are detected on the basis of parametric numerical calculations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of microstructures via higher‐order rank‐one relaxation

Calculation of inelastic material behaviour by means of relaxation algorithms provides an effective way to deal with fine‐scale microstructures resulting from non‐quasiconvex potentials. Especially the rank‐one relaxation admits a straight forward algorithmical realization and results in valuable information that enables the reconstruction of the underlying microstructure. This article is concerned with the application of higher–order rank–one relaxation to a single–slip plasticity model and focusses especially on the numerical algorithm. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantization in Geometric Pluripotential Theory

The space of Kähler metrics, on the one hand, can be approximated by subspaces of algebraic metrics, while, on the other hand, it can also be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite-dimensional spaces of Fubini-Study metrics of Kähler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of Kähler potentials can be quantized. More precisely, given a Kähler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of Kähler potentials. This has a number of applications, among them a new Lidskii-type inequality on the space of Kähler metrics, a new approach to the rooftop envelopes and Pythagorean formulas of Kähler geometry, and approximation of finite-energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects. © 2019 Wiley Periodicals, Inc.     

Analytical solutions for vibrating fractal composite rods and beams

Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating composite rods and beams. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions are limited to beams and rods constructed from an idealised-composite material consisting of relatively large rigid particles embedded in an infinitely thin pliable matrix. Although, as a result, the fractal composite system is not representative of a realistic physical system the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. Static loading is restricted to spatially invariant axial forces and bending moments as solutions on a unified state of continuum stress are sought which then serve as initial conditions for the vibratory problem. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.     

Numerical homogenization of foam-like structures based on the FE2-approach

The computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE²-approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point. In this contribution, we present a two–dimensional geometrically nonlinear FE²-framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p-version. The p-version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures. In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Design of anisotropic structures: Calculation,optimization, and tests

Conclusions A computational procedure based on a limiting state has been worked out which proposes the performance of tests of samples, small-scale models, and full-scale structures and a statistical analysis of the experimental data. Advantages of the method are shown in comparison with the computational method based on fracturing loads which pertain to estimating the accuracy of the computational scheme of structures and determining their safety coefficients. An optimality criterion is formulated within the framework of the method which is related to ensuring a maximum of the limiting (fracturing or critical) load with a constant mass of the structure.Applications of the method are given to estimation of the strength and stability of structures made out of polymeric and metal CM with the use of the Gol'denblat-Kopnov criterion and solutions based on linear shell theory. The conditions for realization of the optimality criterion of structures are determined. Recommendations are made for the rational reinforcement of structures made out of filamentary CM, and a procedure is proposed for the rational design of reinforced structures.As an example of the tests for stability of cylindrical shells made out of aluminum-magnesium alloy under external pressure, the legitimacy of the cause of the disagreement between the calculated and experimental critical loads, which consists of a discrepancy in the nature of wave formation at the instant of stability loss and initial imperfections, and the effectiveness of the computational method based on a limiting state are confirmed. Refined computational formulas for the critical loads of cylindrical shells made out of fiberglass, carbon fiber, and boraluminum under typical kinds of loading are proposed within the framework of the method and on the basis of an analysis of the results of tests, and a comparison is made of the effectiveness of the reinforcement of shells made out of aluminum-magnesium alloy and carbon fiber. The values of the safety coefficients of structures which guarantee their reliability are determined.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 262–271, March–April, 1980.     

Surface Waves on a Cavity in a Semiinfinite Elastic Medium

Biot [5] examined the propagation of waves along the free surface of a cylindrical cavity in an elastic body of infinite extent and obtained a dispersion relation for the velocity of this wave in terms of the ratio of the wavelength to the cavity diameter. This paper contains solutions for waves in a semiinfinite elastic medium with a cylindrical cavity with axially symmetric harmonic loading of the plane surface. The solutions are expressed in terms of Lame potentials which are represented by combinations of integrals containing trigonometric kernels and kernels of Weber transforms. A solution is obtained for volume waves and Biot waves. The relative velocity and relative length of surface waves are studied as functions of the loading frequency.     

Determination of the stress state of a circular disk with a crack

With the use of complex potentials from the solution of a contact problem for slits in a multiply-connected region, a solution is found for a problem of the theory of elasticity for an isotropic circular disk with an arbitrary radial crack. The case of an edge crack is among the cases for which a solution is found. The types of loading examined are uniform tension on an outside edge, internal pressure on the edges of cracks, and concentrated forces at arbitrary points of a disk. The unknown coefficients in the complex potentials are found from the boundary conditions on the outside edge of the disk by the series method, the colocation method, or the least squares method. Detailed numerical studies are conducted to determine the effect of the geometric characteristics and the points of application of concentrated forces on the character of the stress distribution and the stress intensity factor. A comparative characteristic of the methods used to find the coefficients is presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19 pp. 50–61, 1988.     

New neighborhood structures for the Double Traveling Salesman Problem with Multiple Stacks

The changing requirements in transportation and logistics have recently induced the appearance of new vehicle routing problems that include complex constraints as precedence or loading constraints. One of these problems that have appeared during the last few years is the Double Traveling Salesman Problem with Multiple Stacks (DTSPMS), a vehicle routing problem in which some pickups and deliveries must be performed in two independent networks, verifying some precedence and loading constraints imposed on the vehicle. In this paper, four new neighborhood structures for the DTSPMS based on reinsertion and permutation of orders to modify both the routes and the loading planning of the solutions are introduced and described in detail. They can be used in combination with any metaheuristic using local search as a subprocedure, guiding the search to unexplored zones of the solution space. Some computational results obtained using all proposed neighborhood structures are presented, providing good quality solutions for real sized instances.      

Limiting stresses in spatially reinforced composites with components-having different structural geometries nd elastic properties

A model which is proposed for calculating structural stresses in spatially reinforced composites and an invariant-polynomial criterion for evaluating their limiting values are used to predict the effect of the elastic and strength properties of the components and their relative content on the limiting stress-strain state of composites of different structures. Emphasis is given to tri-orthogonal and 4D cubic structures, in addition to structures with hexagonal and angle-ply fiber reinforcement schemes in the plane and perpendicular to it. The types of composite loading typical of standard tests are examined in separate numerical experiments for shear, tension, compression, and their proportional combination. A computational variant of a criterional estimate of the limiting stresses is substantiated for an anisotropic composite of variable strength. The limiting-stress surface is obtained along with contour maps showing stress isolines as a function of the properties of the components and the geometry of the structure. The maps are suitable for practical use. The cases of maximum resistance to shear, tension, compression, and combination loading of 3D and 4D composites are compared to the analogous cases for two-dimensional structures.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October, 1995).Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 616–639, September–October, 1995.     

Micromechanics of polymer fracture

Small-angle x-ray scattering has been used to investigate the formation of embryonic submicroscopic cracks in polymers under a load. The main characteristics of crack formation in various loading regimes are analyzed. It is shown that there is a relation between the submicrocrack concentration and the deformation of the loaded polymer. The principal parameters of crack formation determining the strength properties of the polymer are found to be the transverse dimension of the initial submicrocracks relative to the loading axis, which is determined by the structural heterogeneity of the material, and the submicrocrack concentration in the prefracture state. The principles of the micromechanics of polymer fracture are formulated on the basis of the results of an analysis of the quantitative relationship between these parameters. The dominant role of the surface in the fracture process is demonstrated by comparing the parameters of crack formation in the interior and at the surface of the loaded polymer.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 792–801, September–October, 1974.     

Variational approach to formation of misoriented microstructures in plastic deformation

The formation of misoriented microstructures in plastic deformation is explained within the framework of continuum mechanics as a result of the reduction of the energetically costly hardening in multislip by local lattice rotations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution of the Stress–Strain State of a Three-Dimensional Plate Weakened by a Central Crack under Impulsive Tension

A method is developed to describe the formation of the stress–strain state in the vicinity of the tip of a stationary crack in a three-dimensional plate under dynamic loading. The energy model used to describe the formation of the stress concentration zone around the crack tip is modified to take into account the transient character of the loading process and the influence of the free surfaces of the plate on the stress–strain state of the central part of the sample. The method is useful for describing static and dynamic brittle fracture from a unified point of view.     

On the variational formulation and implementation of Allen-Cahn and Cahn-Hilliard-type phase field theories

Phase field theory is a promising framework for analyzing evolving microstructures in materials. Phenomena like those related to microstructures in Ni-based superalloys, twin structures in martensites or precipitation in Al-alloys can be predicted by phase field theory. While phase transformations such as those characterizing twinning are captured by an Allen-Cahn-type approach, a Cahn-Hilliard-type formulation is used, if the respective interface motion is driven by the concentration of the species. Although the Allen-Cahn and the Cahn-Hilliard formulation are indeed different, they do share some similarities. To be more precise, a Cahn-Hilliard model is obtained by enforcing balance of mass in the Allen-Cahn approach. Within an energy-based formulation this can be implemented by adding additional energy terms to the underlying Allen-Cahn energy. Such a universal energy-based framework is elaborated in this presentation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-field simulation of the electro-mechanical response of ferroelectrics during Piezoresponse Force Microscopy

The contribution adresses the simulation of ferroelectric matrials in the framework of the Piezoresponse Force Microscopy (PFM). Based on the PFM, ferroelectric domain structures can be analyzed in great detail by measuring the electrically induced mechanical deformations of the surface of a ferroelectric. We employ a flexible continuum-mechanical model based on the phase-field method in order to analyze the behavior of ferroelectric microstructures numerically. Since ferroelectric materials are often highly anisotropic, the phase-field formulation will account for transversely isotropic symmetry. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adapted heuristic approach for a clustered traveling salesman problem with loading constraints

The joint optimization of routing and loading operations is crucial to fully optimize the overall planning process in logistics, and as a result routing problems with side constraints are becoming more and more important during the last years. This paper approaches the design of optimal routes for pickup and delivery operations considering in addition some capacity and loading constraints on the vehicles to be used. It is focused on exploiting new ideas to deal with real life situations in which the customers are not uniformly distributed on the pickup or delivery regions of the problem. An adapted and effective heuristic based on a Variable Neighborhood Search framework using improved neighborhood structures is proposed and discussed. The algorithm is applied to several new sets of instances with special structures to better represent real life situations, providing computational results to evaluate its performance.