Identification of optimal material models and parameters in finite strain plasticity

The numerical simulation of the behaviour of a workpiece during manufacturing depends to a large extent on the quality of the applied material model. In this work, a method for the identification of constitutive models and material parameters in engineering applications is proposed. The presented method is used in the setting of optimal experimental design and is based on successive optimization of a set of finite strain plasticity models with kinematic and/or isotropic hardening. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularization of optimal design problems for bars and plates,part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables.Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.     

Optimal Experimental Design to Identify the Average Stress-Strain Response in Short Fiber-Reinforced Plastics

We show how to use optimal experimental design methods for the parameter identification of short fiber reinforced plastic (SFRP) materials. The experimental data is given by computer simulations of representative volume elements (RVE) of the SFRP material. The experiments are designed such that a minimal number of RVE simulations is required and that the model response attains a minimal variance for a class of strains and fiber orientations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Stability of Sizing Optimization Problems for a Class of Nonlinearly Elastic Materials

In this work we deal with a stability aspect of sizing optimization problems for a class of nonlinearly elastic materials, where the underlying state problem is nonlinear in both the displacements and the stresses. In [14] it is shown under which conditions there exists a unique solution of discrete design problems for a body made of the considered nonlinear material, if the nonlinear state problem is solved exactly. In numerical examples the nonlinear state problem has to be solved iteratively, and therefore it can be solved only up to some small error \eps . The question of interest is how this affects the optimal solution, respectively the set of solutions, of the design problem. We show with the theory of point-to-set mappings that if the material is not too nonlinear, then the optimal design depends continuously on the error \eps . Accepted 15 March 2001. Online publication 14 August 2001.     

An Optimal Design Method Based on Small Amplitude Homogenization

An optimal design method for two materials based on small amplitude homogenization is presented. The method allows to use quite general objective functions at the price that the two materials should have small contrasts in their relevant physical parameters. The following two applications are shown: Stress constrained compliance minimization and defect location in elastic bodies.     

Applied optimal shape design

This paper is a short survey of optimal shape design (OSD) for fluids. OSD is an interesting field both mathematically and for industrial applications. Existence, sensitivity, correct discretization are important theoretical issues. Practical implementation issues for airplane designs are critical too.The paper is also a summary of the material covered in our recent book, Applied Optimal Shape Design, Oxford University Press, 2001.     

The Optimization of Slender Reinforced Concrete Columns

Slenderness is an important issue in design of reinforced concrete (RC) columns. Especially for long columns, second order effects may be not so small to neglect, but the calculation of second order effects may take too much time. For that reason, ACI 318 design code includes a simple approach in order to increase the flexural moment of columns according to their slenderness. Thus, second order effects are considered. In optimization, the effect of slenderness can be considered by using the factored design flexural moments. In this paper, harmony search (HS) algorithm is employed to find the optimum design variables of slender RC columns. These design variables are web width, height, diameter and number of reinforcements. The optimization objective is total cost of materials including concrete and steel. The developed method is effective to find the optimal design for axial force, flexural moment and shear force values. As numerical examples, optimum design of columns with different lengths, but with the same loadings and material properties were investigated. Thus, the effect of slenderness was seen on the optimum costs. By the increase of column length, increase of total material cost is more than a linear increase. This situation shows us the effect of slenderness on optimum RC columns (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some estimates for the torsional rigidity of composite rods

A well‐known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain Ω, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are mixed in a microscopic scale. Then, the optimal torsional rigidity becomes difficult to compute. In this paper we give some different theoretical upper and lower bounds for the optimal torsional rigidity, and we compare them on some significant domains. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evaluation of three production planning procedures for the use of recipe flexibility

Process industries often obtain their raw materials from mining or agricultural industries. These raw materials usually have variations in quality, which often lead to variations in the recipes used for manufacturing a product. Another reason for varying the recipe is to minimize production costs by using the cheapest materials that still lead to a satisfactory quality in the product. A third reason for using recipe flexibility is that it may occur that at the time of production not all materials for the standard recipe are available. In earlier research we showed under what conditions the use of this type of recipe flexibility should be preferred to the use of high materials stock to avoid materials shortages. We also showed that the use of recipe flexibility to account for material shortages can be justified if the material replenishment leadtime is long, the demand uncertainty is high and the required service level is high. In this paper we assume that these conditions are satisfied and we investigate three different production planning procedures that make use of recipe flexibility to cope with the uncertainty in demand and supply. We assume that the customer order leadtime is much smaller than the material replenishment leadtime, and therefore demand uncertainty is high. The optimal procedure optimizes material use over a planning horizon equal to the material replenishment leadtime, taking into account the customers orders and knowledge of the distribution function of future demand. The deterministic procedure also optimizes the material use over the material replenishment leadtime, but it assumes a deterministic demand level for unknown orders. The simplest, myopic procedure optimizes material use over only the accepted customer orders. These three procedures are investigated via an experimental design of computer simulations of an elementary small scale model of the production planning situation. The results show that the optimal procedure outperforms the other two procedures. Furthermore, for a realistic cost structure in feed industry under certain circumstances the use of the optimal procedure may lead to a 4% increase in profit. However, this improvement must be weighted against the cost incurred by the operational use of this complex procedure. Based on these considerations and the numerical results in this paper, we may expect that for some situations in practice the use of the simplest myopic procedure, optimizing material use only over the available customer orders, will be justified from an overall cost point of view.     

Multicriterion optimization of hybrid composite plates with various supports under transverse load

In this paper applications of multicriteria optimization to design hybrid laminates is shown. In the considered numerical example, the obtained preferred solution is 30% heavier than Carbon/Epoxy laminate and use a third of the more expensive material. Moreover, a small number of layers is needed to build an optimal solution. In the case of scalar optimization three layers are required. In the case of bicriteria optimization one E-Glass/Epoxy layer placed between four Carbon/Epoxy layers is enough to build an optimal laminate. Two criteria may lead to a set of preferred equal solutions. Thus, connection of Pareto concept with a lexicographic order by introduction of an additional criterion is useful for ordering them. A similar approach may be used for designing hybrid laminates with a requirement of a compromise between the stiffness and cost of a plate. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frictional contact of elastomer materials on rough rigid surfaces

The frictional behavior of elastomer materials is still unexplored, but very important for many industrial applications. Special attention is turned to rubber friction on rough road tracks. Due to the non‐rigid material characteristics of the rubber it is not sufficient to use a constant friction coefficient like Coulombs law. The frictional qualities depend on many different influences like sliding velocity, applied normal stress, surface roughness, material properties and the temperature in the contact zone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aspects of computational homogenisation of microheterogeneous materials including decohesion at finite strains

During the last years, the development and application of new composite materials gained more and more importance. For engineering applications it is necessary to get effective material properties of such materials. In this contribution we present some aspects of computational homogenisation procedures of microheterogeneous materials which can show decohesion in a cohesive zone around the particles. Due to the decohesion we get finite deformations and .nite strains within the RVE. The geometrical and material nonlinearities cause the main dif.culties. The homogenization procedure leads to an effective stress strain curve for the RVE, and for the nonlinear elastic case one can also obtain effective material parameters. It is necessary to do statistical tests in order to get a representative result. Here we set a special focus on the adaptive numerical model, the statistical testing procedure and the different boundary conditions (pure tractions and pure displacements) applied on the RVE. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear constitutive behavior of orthotropic materials

The problem of optimal material orientation is studied in the case of nonlinear elastic materials. Optimal orientations corresponding to extreme (maximal or minimal) energy density are obtained for orthotropic materials. The material behavior (the stress-strain relation) is simulated in a general form, which includes, as particular cases, different versions of the power law stress-strain relations. The optimality conditions are derived for the general cases. Local and global extremums are determined for particular cases.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Tartu University, Estonia. Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 445–454, March–April, 2000.     

Gradient methods for multiple state optimal design problems

We optimise a distribution of two isotropic materials α I and β I (α < β) occupying the given body in R ^d . The optimality is described by an integral functional (cost) depending on temperatures u ₁, . . . , u _m of the body obtained for different source terms f ₁, . . . ,f _m with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by the homogenisation theory. The necessary conditions of optimality are derived via the Gateaux derivative of the cost functional. Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal design. In the case m < d we show that there exists an optimal design which is a rank-m sequential laminate with matrix material α I almost everywhere on Ω. Contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems (although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient methods for minimising the cost functional.      

Shape optimization for the Helmholtz equation

The aim of our work is to develop optimal dielectric composite structures with specific qualities. The task is to design interfaces of given material components such that the originated structure attains certain optical properties. Propagation of the electromagnetic waves in the composite is described by the Helmholtz equation. Success of the structure is enumerated by the objective function which is to be minimized. Interfaces of the given materials are parametrized by the cubic B–spline curves. The design variables are afterwards the positions of B–spline control points. For objective function evaluation one forward computation of the Helmholtz equation is needed. To get the sensitivity of the objective function we solve the backward (adjoint) equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficiency of the Tresca yield criterion in modeling of annular plates with rigid constraint

Correct prediction of the load carrying capacity and initiation of failure of compound annular plates (with or without inner solid disk) embedded into a rigid container are crucial for achieving their optimal structural design and reliable service conditions. In this paper an analytical study devoted to mathematically and physically rigorous stress/strain analysis of such structures has been performed. The decohesive carrying capacity criterion based on the radial strains is applied and discussed. The material of the plate is considered to be elastic/perfectly-plastic, and the inner insert is modeled as purely elastic. Two principle engineering criteria for material yielding, namely, the Tresca and the Mises, are employed, discussed and compared. The range of validity of each criterion is analyzed, and the choice of the suitable criterion for particular engineering applications is suggested. The advantages of the Tresca yield criterion both in mathematical modeling and engineering design are outlined.     

Simulation model for anisotropic fibrous materials

Paper and paper–based materials such as cardboard are used in a wide variety of applications and in the development of new applications such as boxes an accurate simulation model is of major importance. Industrially made paper material typically has an orthotropic fibrous structure, due to the manufacturing process, where the fibers tend to align in the direction of motion in the machine. In this work a plasticity–based material model allowing for finite strains is developed. The model is suitable for materials with an anisotropic fibrous structure such as paper. The general framework is based on separate mappings describing the deformations of the continuum and the substructure and a multiplicative split of these mappings into elastic and plastic parts. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Smoothness Results for Classical Problems in Optimal Design and Applications

The author considers two classical problems in optimal design consisting in maximizing or minimizing the energy corresponding to the mixture of two isotropic materials or two-composite material. These results refer to the smoothness of the optimal solutions. They also apply to the minimization of the first eigenvalue.     

Methods of design of extremal experiments in problems of vector optimization of composite shells

Using the methods of the theory of design of extremal experiments and vector optimization we solve a problem of optimal design of cylindrical shells of composite materials with respect to the criteria of minimal weight and maximum critical load. The use of the three-dimensional scheme of reinforcement of material leads to improvements in the weight of the shell in comparison with the two-dimensional reinforcement scheme.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 70–74.