Computational Homogenisation using Reduced-Order Modelling applied to Hyperelasticity

Within this study nonlinear reduced-order modelling for hyperelastic material is applied for the boundary value problem on the micro-scale which arises in the context of computational homogenisation. This involves the Proper Orthogonal Decomposition and the Discrete Empirical Interpolation Method for the nonlinear term. Considered error measures are the errors of the displacement field, the averaged stresses and the effective elasticity tensor. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Homogenisation of Polycrystalline Elastoplastic Microstructures at Finite Deformation

During sheet bulk metal forming processes both, flat geometries and three-dimensional structures change their shape significantly while undergoing large plastic deformations. As for forming processes, FE-simulations are often done before in situ experiments, a very accurate material model is required, performing well for a huge variety of different geometrical characteristics. Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically observable plastic deformation is traced back to dislocations within considered slip systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level. A finite crystal plasticity model is used to model the behaviour of polycrystalline materials in representative volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the elastic contribution. In order to circumvent singularities stemming from the linear dependency of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the plastic slips and slip resistances. The model is validated with experimental microstructural data under deformation. Through homogenisation and optimisation techniques, effective stress-strain curves are determined and can be compared to results from real forming processes leading to a suitable effective material model. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material Forces in elasto-plastic Materials

In this contribution the concept of configurational forces, also called material forces, is applied to rate–independent, elasto–plastic materials. The theory of configurational forces is briefly recast. Zones of plastic deformation can be interpreted as distributed inhomogeneities. With this background the theory of configurational forces can be applied in many situations, including plastic zones at crack tips, elastic inclusions in elasto–plastic materials and localized deformation. The numerical evaluation is done with the Finite Element Method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two formulations of an elasto-plastic problem

Solutions of two elasto-plastic problems are compared when all the stress tensor components are continuous on the elasto-plastic boundary and when the tangential component undergoes a discontinuity.     

Homogenisation of random composites via the multiscale finite‐element method

The transition between the chosen microstructure and microvariables and the material properties on the macrolevel is always a sensitive point in the theory of homogenisation. In this talk we will observe the transfer of data between the scales based on the multiscale finite element method where in each Gauss point of the macromesh a micromesh is attached. For a given deformation gradient provided from the macroscale one calculates microfluctuations satisfying periodic boundary conditions and from those the effective first Piola‐Kirchhoff stress tensor for each Gauss point. The latter provides a possibility to calculate the elasticity tensor on the macrolevel. We study a microstructure containing elliptical cracks of random aspect ratio and orientation. The results based on such procedure show the dependence of the macrovariables on the crack ellipticity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenisation method to capture the non-linear behaviour of intervertebral discs in multi-body systems

The present contribution is motivated by the desire to compute physiological loads on the intervertebral discs (IVD) of a human lumbar spine during activities like standing, bending and falling. Following this, a mechanical multi-body system (MBS) is utilised to capture the overall mechanical behaviour of a human, whereas an inhomogeneous, anisotropic, multi-phasic finite-element model (FEM) is applied to resolve the resulting field quantities inside an IVD. In order to couple the FEM of the IVD with the numerically diverse MBS, a homogenisation procedure has to be applied such that field quantities can be converted into discrete quantities. In particular, the MBS captures the mechanical behaviour of an IVD using a bushing element, which provides discrete force-displacement and moment-rotation relations. The goal of this contribution is to present a homogenisation method for the IVD as well as a possibility to include the homogenised results in the MBS without the need for embedded FE computations in the MBS. Instead, certain deformation modes of the IVD are pre-computed and represented using a non-linear constitutive equations. This task becomes even more challenging, as the resulting discrete DOF of a motion segment appear in a coupled fashion due to the structure of the IVD, i. e., a rotation in the sagittal plane triggers a resulting moment and a resulting force. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions

A two-phase elastic composite with weakly compressible elastic inclusions is considered. The homogenised two-scale limit problem is found, via a version of the method of two-scale convergence, and analysed. The microscopic part of the two-scale limit is found to solve a Stokes type problem and shown to have no microscopic oscillations when the composite is subjected to body forces that are microscopically irrotational. The composites spectrum is analysed and shown to converge, in an appropriate sense, to the spectrum of the two-scale limit problem. A characterisation of the two-scale limit spectrum is given in terms of the limit macroscopic and microscopic behaviours.     

The generalized plane strain of an elasto-plastic material

Zusammenfassung Unter der Voraussetzung, dass der hydrostatische Druck nicht gross ist, tritt die Hauptspannung normal zur Ebene des verallgemeinerten ebenen Verzerrungszustandes in der Fliessbedingung vonTresca nicht auf. Man kann zeigen, dass die Spannungs- und Geschwindigkeitsgleichungen für ein elastisch-ideal-plastisches Material hyperbolisch sind, wobei die beiden Charakteristikenscharen paarweise koinzident sind. Die auf die Charakteristiken bezogenen Gleichungen unterscheiden sich von denjenigen des starr-plastischen Materials nur durch die Gegenwart von vier Zusatztermen in jeder der Geiringer-Gleichungen.Für ein Material mit Verfestigung werden nur kleine Spannungen berücksichtigt, und es wird angenommen, dass die Gleichungen zwischen den Spannungen und den Gesamtdehnungen verwendet werden können. Unter diesen Voraussetzungen werden die Gleichungen für die Spannungen und Verschiebungen elliptisch. Da alle Ableitungen stetig sind, existiert im plastischen Bereich eine Airysche Spannungsfunktion samt speziellen Ableitungen aller Ordnungen, und es kann gezeigt werden, dass sie einer nichtlinearen partiellen Differentialgleichung vierter Ordnung genügt. Für einige einfache Probleme, bei denen die abhängigen Veränderlichen nur Funktionen einer der beiden unabhängigen Variablen sind, werden die Lösungen angegeben.Der Einfluss von Temperaturschwankungen kann sowohl bei idealplastischen Materialien wie auch bei solchen mit Verfestigung mitberücksichtigt werden.     

On the implementation of elasto-plastic boundary element analysis

This paper discusses efficient methods for implementing elasto-plastic boundary element programs. A solution strategy based on minimizing the number of computer operations for each iteration of the elasto-plastic algorithm is described for both the direct and indirect formulations. Interior strains may be computed from the integral equations but this is shown to be an order of magnitude greater in terms of operations (excluding evaluation of the kernel functions) than obtaining strains using a finite difference procedure. Cells existing for the volume integrations associated with the initial stress terms may be used for the differencing process. The assumption of constant stress across cells has been used and found adequate for many problems but a linear variation, with stresses being computed at cell nodes, may prove better in certain cases. All integrations must be performed numerically. A simplified flow chart of the algorithm is presented and typical solutions for two test problems are given. Finally a comparison between finite element and boundary element solution times for several problems is attempted, it seems that solution times are similar.     

Geometrical super-elements for elasto-plastic shells with large deformation

This paper presents the application of the so-called Geometrical Elements Method (Lukasiewicz and Szyszkowski, 1974; Pogorelov, 1967) to the solution of elasto-plastic problems of shells. The approach is based on the observation that, during large deformations, the shell structure deforms in a nearly isometrical manner. Therefore, its deformed shape can be determined and analysed making use of the Gauss theorem according to which the Gaussian curvature of the isometrically deformed surface remains unchanged. The shell structure is subdivided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. The equilibrium of the whole structure is defined by the stationary value of the Hamiltonian function which requires the calculation of the strain energy in the elements. This can easily be obtained if we recognize that the isometrically deformed elements contain only bending energy. Using the method described, we are able to significantly the number of unknown values defining the shape of the deformed structure. The problem is reduced to the numerical evaluation of the minimum of a function of many variables. The elasto-plastic state of stress of the plastic material in the structure canbe determined by using the deformation theory of plasticity or the theory of plastic flow. Also, the strains and stresses in the plastic regions are the only functions of the assumed displacements field. The corresponding energy of the plastic deformation can easily be evaluated and added to the minimized functionals. For example, the elasto-plastic behaviour of a spherical shell under a concentrated load is studied. The solution obtained defines the large deformation behaviour and the motion of the plastic zones on the surface of the shell.     

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)     

Tensile behavior of plastic foams

The tensile stress-strain diagram of plastic foams is calculated on the basis of a previously proposed 14-faced cell model. An S-shaped inflection due to the flexural deformations of the ties is detected on the initial section of the diagram. It is shown that for closed-celled foams this inflection may degenerate. The results of testing PVC and polyurethane foams (volume content of polymer base about 3–6%) prove to be in satisfactory agreement with the calculations.Vladimir Scientific-Research Institute of Synthetic Resins. Translated from Mekhanika Polimerov, No. 4, pp. 670–675, July–August, 1971.     

Micromorphic Homogenisation of a Porous Medium: Application to Size Effects and Quasi-Brittle Damage

The classical theory of continuum mechanics requires that distances over which gradients occur are much larger than the length scales of the microstructure. This requirement is violated if deformation localizes due to material softening which is why the predictions of classical damage models feature certain unrealistic aspects. In the present contribution, a micromorphic model for quasi-brittle damage, derived through homogenisation, is employed in FEM simulations to overcome this problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the Sinc method to a dynamic elasto-plastic problem

This paper presents the application of Sinc bases to simulate numerically the dynamic behavior of a one-dimensional elastoplastic problem. The numerical methods that are traditionally employed to solve elastoplastic problems include finite difference, finite element and spectral methods. However, more recently, biorthogonal wavelet bases have been used to study the dynamic response of a uniaxial elasto-plastic rod [Giovanni F. Naldi, Karsten Urban, Paolo Venini, A wavelet-Galerkin method for elastoplasticity problems, Report 181, RWTH Aachen IGPM, and Math. Modelling and Scient. Computing, vol. 10, 2000]. In this paper the Sinc–Galerkin method is used to solve the straight elasto-plastic rod problem. Due to their exponential convergence rates and their need for a relatively fewer nodal points, Sinc based methods can significantly outperform traditional numerical methods [J. Lund, K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992]. However, the potential of Sinc-based methods for solving elastoplasticity problems has not yet been explored. The aim of this paper is to demonstrate the possible application of Sinc methods through the numerical investigation of the unsteady one dimensional elastic-plastic rod problem.     

Microscopic and macroscopic modelling of foams

The macroscopic behaviour of foams is influenced by the size and structure of the pores. Therefore, a Cosserat theory is motivated by a micromechanical beam model. A homogenization strategy is applied to the micromechanical model leading to results that are comparable to the predictions of the macroscopic Cosserat model.     

Coupled problems of the optimization of the elasto-plastic deformation of metals

The mathematical formulation of a coupled optimization problem, which includes several optimal problems which are interrelated via the optimization parameters and functions of state, is analysed. The conditions for the existence of a solution are formulated and a technique for reducing the problem, to a sequence of optimal problems with unknown functions in the optimality and bounding criteria is described. A special minimizing sequence is constructed and its convergence conditions are derived. An example is given to demonstrate the method.     

Acoustics of gas-saturated foams: Modelling comparisons

We show results obtained by Biot's equations and a simple Theory of Porous Media-based hybrid model describing acoustical waves in a gas-saturated high-porous foam. The resulting dispersion relations for the longitudinal mode and phase velocities vs. porosity show the extraordinary effect of waves in gas-saturated porous media. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The influence of semi-rigid connections on the shakedown of elasto-plastic frames

The main structural elements of steel framed multi-storey structures are the columns, the beams and their connections. The assumption has been widely applied in the past that the connections are either rigid or pinned. The actual behaviour of the connections is however somewhere between these limits, they are semi-rigid. The aim of this paper is to analyze the effect of the semi-rigid connections on the shakedown of steel structures under multi-parameter static loading. In the analysis, to control the plastic behavior of the structure, bound on the complementary strain energy of the residual forces is applied. The formulation of the problem yields to nonlinear mathematical programming. The results of the numerical calculation show that the semi-rigid connections can influence significantly the magnitude of the shakedown parameter. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)