Martensitic transformations and damage: A combined phase field approach

A combined continuum phase field model for martensitic transformations and damage is introduced. The present approach considers the eigenstrain within the martensitic phase which leads in the surrounding material to both tensile and compressive stresses. The damage model needs to account for an appropriate differentiation thereof, since compressive stresses should not promote fracture. Interactions between micro crack propagation and the formation of the martensitic phases are studied in two dimensions. In agreement with experimental observations, martensite forms at the crack tip and influences the crack formation. For the numerical implementation finite elements are used while for the transient terms an implicit time integration scheme is employed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational model for the functional fatigue in polycrystalline shape memory alloys

Shape memory alloys show a very complex material behavior associated with a diffusionless solid/solid phase transformation between austenite and martensite. Due to the resulting (thermo-)mechanical properties – namely the effect of pseudoelasticity and pseudoplasticity – they are very promising materials for the current and future technical developments. However, the martensitic phase transformation comes along with a simultaneous plastic deformation and thus, the effect of functional fatigue. We present a variational material model that simulates this effect based on the principle of the minimum of the dissipation potential. We use a combined Voigt/Reuss bound and a coupled dissipation potential to predict the microstructural developments in the polycrystalline material. We present the governing evolution equations for the internal variables and yield functions. In addition, we show some numerical results to prove our model's ability to predict the shape memory alloys' complex inner processes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of a Hybrid-Forming Process using Temperature- and Rate-Dependent Viscoplastic Material Models

Hybrid-forming processes for graded structures are quite innovative methods for the production of components with tailored properties, particularly tailored material properties and geometrical shape. In this contribution a hybrid-forming process based on the utilization of locally varying thermo-mechanical effects is investigated [1]. For process optimization and improvement of the resulting work piece the simulation of the entire forming process is necessary in modern engineering. The main topics of this contribution are the simulation of the cyclic thermal loaded forming tool and the simulation of the work piece treated at large deformations with phase transformations. For both materials temperature- and rate-dependent viscoplastic material models are applied and parameter identification using cyclic tensile-compression tests for the forming tool material and phase transformation tests for a low-alloy steel similar to the work piece material is presented. For validation of finite-element-calculations for the forming tool thermal shock experiments are performed with optical deformation measurements. For validation of finite-element-calculations for the work piece numerical results of geometry and structure after heating, forming and cooling are compared to experimental micro sections. Results concerning the forming tool will be used for future lifetime prediction and results concerning the work piece will be used for future specific setting of graded material properties. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical relationships for the mechanical properties of additively manufactured porous biomaterials based on octahedral unit cells

Additively manufacturing (AM) techniques make it possible to fabricate open-cell interconnected structures with precisely controllable micro-architectures. It has been shown that the morphology, pore size, and relative density of a porous structure determine its macro-scale homogenized mechanical properties and, thus, its biological performance as a biomaterial. In this study, we used analytical, numerical, and experimental techniques to study the elastic modulus, Poisson`s ratio, and yield stress of AM porous biomaterials made by repeating the same octahedral unit cell in all spatial directions. Analytical relationships were obtained based on both Euler-Bernoulli and Timoshenko beam theories by studying a single unit cell experiencing the loads and boundary conditions sensed in an infinite lattice structure. Both single unit cells and corresponding lattice structures were manufactured using AM and mechanically tested under compression to determine the experimental values of mechanical properties. Finite element models of both single unit cell and lattice structure were also built to estimate their mechanical properties numerically. Differences in the bulk mechanical properties of struts built in different directions were observed experimentally and were taken into account in derivation of the analytical solutions. Although the analytical and numerical results were generally in good agreement, the mechanical properties obtained by the Timoshenko beam theory were closer to numerical results. The maximum difference between analytical and numerical results for elastic modulus and Poisson's ratio was below 6%, while for yield stress it was about 13%, both occurring at the relative density of 50%. The maximum difference between the analytical and experimental values of the elastic modulus was <15% (relative density = 50%).     

Modeling of bainitic phase transformation

In this work, we present a macroscopic material model for simulation of austenite to bainite and of austenite to martensite transformations accompanied by transformation-induced plasticity (TRIP), which is an important phenomenon in metal forming processes. In order to account for the incubation time the model considers nucleation of the bainite phase. When this quantity attains a barrier term, growth of bainite volume fraction is started. The model formulation allows for individual evolutions of upper and lower bainite. Furthermore, the numerical implementation of the constitutive equations into a finite element program is described. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenfrequencies of a tube bundle immersed in a fluid

In this paper we study a simplified version of a mathematical model that describes the eigenfrequencies and eigenmotions of a coupled system consisting of a set of tubes (or a tube bundle) immersed in an incompressible perfect fluid. The fluid is assumed to be contained in a rectangular cavity, and the tubes are assumed to be identical, and periodically distributed in the cavity. The mathematical model that governs this physical problem is an elliptic differential eigenvalue problem consisting of the Laplace equation with a nonlocal boundary condition on the holes, and a homogeneous Neumann boundary condition on the boundary of the cavity. In the simplified model that we study in this paper, the Neumann condition is replaced by a periodic boundary condition. Our goal in studying this simple version is to derive some basic properties of the problem that could serve as a guide to envisage similar properties for the original model. In practical situations, this kind of problem needs to be solved for tube bundles containing a very large number of tubes. Then the numerical analysis of these problems is in practice very expensive. Several approaches to overcome this difficulty have been proposed in the last years using homogenization techniques. Alternatively, we propose in this paper an approach that consists in obtaining an explicit decomposition of the problem into a finite family of subproblems, which can be easily solved numerically. Our study is based on a generalized notion of periodic function, and on a decomposition theorem for periodic functions that we introduce in the paper. Our results rely on the theory of almost periodic functions, and they provide a simple numerical method for obtaining approximations of all the eigenvalues of the problem for any number of tubes in the cavity. We also discuss a numerical example.     

Development of porous implants with non-uniform mechanical properties distribution based on CT images

In this study, the mechanical properties (elastic modulus, yield stress, and Poisson's ratio) of rhombic dodecahedron (RD) unit cell has been studied analytically and numerically. For the analytical study, two well-known beam theories, namely Euler Bernoulli and Timoshenko, have been implemented. For validating the analytical relationships, finite element model of unit cell with repetitive boundary condition has been created. Moreover, the experimental results of recent studies have been used for validation. The results showed that the presented analytical relationships for RD lattice structure have good agreement with numerical and experimental results in all the relative densities particularly in lower relative densities. Besides, the analytical relationships based on Timoshenko theory showed closer results with numerical/experimental data. The derived analytical relationships for RD as well as the data extracted from CT scan images of a femur bone, were combined and used to create a porous femur implant model. The stress and strain distributions of the porous femur model under typical static compressive load due to human weight as well as axial rigidity of the model in the same loading conditions have been obtained and compared with the experimental results from other studies. The stress and strain distributions of the porous femur implant model based on RD unit cells, as well as its axial rigidity, showed good agreement with the results obtained for human femur.     

A model updating approach based on design points for unknown structural parameters

Finite element analysis has become an essential tool to estimate structural responses under static and dynamic loads. However, there are a lot of uncertainties in structural properties. For this reason, in many cases, the outcomes of the theoretical and experimental modal analyses do not match. Therefore, the analytical models of the structures need to be updated according to the experimental test results. The commonly used method to get parameters for model updating is experimental modal analysis which provides structural dynamic characteristic (natural frequencies, mode shapes and modal damping ratio). There are many methods available for the updating process. This study addresses an updating algorithm to modify the numerical models by using the design points for unknown structural properties. The proposed method aims to minimize the difference between the analytical and experimental natural frequencies by updating uncertain parameters for each mode and combine them to get an optimum solution. The algorithm is tested on a column and a 2D frame models. These models are investigated by taking the connection rigidity and elasticity modulus as unknown parameters. It is observed that the proposed algorithm gives better results for unknown structural properties compared to the initial values.     

Simulation of pseudo-plasticity in shape-memory-alloys

The name shape memory originates from the material's capability to recover its original shape after an apparent plastic deformation. The secret of this property lies in the specific microstructure. During mechanical loading, alloys of this particular kind change their crystallographic structure from randomly orientated martensite to ordered martensite. With austenite as high-temperature inter-state induced by heat supply, a recovery from the ordered to the unordered martensite is possible. This is accompanied by a macroscopic "healing" process. We apply our material model for shape-memory alloys to this special property and present numerical results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of deformation-induced phase transformation during very high cycle fatigue (VHCF) using the boundary element method

A microstructural simulation model is proposed which accounts for damage accumulation in shear bands and deformation-induced martensite formation in a metastable austenitic stainless steel (AISI304). The model is numerically solved using the two-dimensional (2-D) boundary element method. By using this method, sliding displacements can be directly evaluated in shear bands and austenite grains as well as generated martensite domains with their individual mechanical properties and shape deformation can be considered. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A three-phase FE-model for the simulation of partially saturated soils

While for many numerical simulations in geotechnics use of a two-phase model is sufficient, separate consideration of all three phases is mandatory for numerical simulations of partially saturated soils subjected to compressed air. This is a common technique frequently applied for temporary ground support in tunnelling. For the numerical simulation of tunnelling using compressed air, a multiphase model for soft soils is developed, in which the individual constituents of the soil – the soil skeleton, the fluid and the gaseous phase – and their interactions are considered. The three phase model is formulated within the framework of the Theory of Porous Media (TPM), based upon balance equations and constitutive relations for the soil constituents and their mixture. Water is modelled as an incompressible and air as a compressible phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of Single Crystal Magnetostriction Based on Numerical Energy Relaxation Techniques

In this work, an incremental energy minimization technique is proposed to simulate the magnetomechanically-coupled, nonlinear, anisotropic and hysteretic response of single crystalline magnetic shape memory alloys (MSMA). The model captures the three key physical mechanisms that cause this characteristic behavior, namely the field- or stress-induced martensite variant reorientation (twin boundary motion), magnetic domain wall motion, and local magnetization rotation, through an (incremental) energy minimizing evolution of internal state variables. Representative numerical response predictions are presented, compared to experimental observations, and discussed with respect to the associated microstructure evolution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulations Of Three Dimensional Glottis Flows

A detailed numerical investigation of the transient three dimensional flow downstream of the human glottis is performed. For the present work a threefoldly scaled up realistic static model of the vocal folds and a transient inflow function, which is obtained from experimental data is applied to the CFD model. In the numerical simulations transient flow features like forming ring vortexes and axis switching are resolved which are also known for elliptical and rectangular jets. Similar structures can also be identified in the results of the time–resolved PIV measurements. The influence of the ventricular folds and the geometrical variation of the glottis contacts are also considered in the numerical study. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of short fiber reinforced polymers

This contribution presents ideas, how composite materials can be characterized with respect to experimental testing. The material properties of the investigated short glass fiber reinforced polymer are obtained by providing results from the experiment in order to seperate different material effects, such as elasticity, plasticity, damage, viscoelasticity, compressibility and anisotropy. Therefore, at first, linear uniaxial tensile tests with cyclic loadings have been realized. The application of the material in this work is the machining by a three-dimensional forming process. Hence, multiaxial loadings have to be additionally taken into account matching these conditions. In order to provide more information, biaxial tensile tests have to be realized using a testing device supplying the additional necessary experimental data [1, 2]. A final aim of this work is to develope a verification experiment representing the three-dimensional forming process as realistically as possible, e. g. a Nakajima test [4]. For this case, a three-dimensional optical analysis in order to get the necessary measurement data, is indispensable to realize an inverse method [3] by comparing the information of the complete deformation field as well as the force data of the experiment using a given material model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Verification and validation of the foredrag coefficient for supersonic and hypersonic flow of air over a cone of fineness ratio 3

The foredrag coefficient resulting from the supersonic and hypersonic flow of air over a cone was calculated numerically using a finite volume approach based on the compressible Euler and Navier-Stokes equations with constant and variable thermophysical properties. No turbulence model was considered. Simulations were carried out for a cone of fineness ratio 3 under the free-stream Mach numbers 2.73, 3.50, 4.00, 5.05 and 6.28 (the Reynolds number, based on cone length, is within 0.45 and 2.85 million). Up to six grids were employed for numerical calculations, with 60 × 60 to 1920 × 1920 volumes. The numerical error was estimated to be less than 0.01% of the numerical solution for all models. Comparisons of the numerical foredrag coefficients of the three models with the experimental data showed that the Navier–Stokes model with variable thermophysical properties agreed better with the experimental foredrag for the entire Mach number interval studied, taking into account the validation standard uncertainty.     

Comparison of experimental and numerical results on metallic plates subjected to explosions

A comparative study of dynamic response including damage and rupture processes of thin metallic plates subjected to shock-wave impulses – explosions is presented. The results of the finite element numerical analysis are related to experiments. Due to high strain rate during explosions the elasto-viscoplastic Chaboche's constitutive law including damage effects has been applied. For the assumed model proper material parameters identification has been done. In the dynamic, geometrically non-linear analysis the MSC.Marc system has been used. A good correlation between numerical and experimental results has been observed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical framework for rheological models based on the decomposition of the deformation rate tensor

In this contribution, the rheological models based on the decomposition of the deformation rate tensor, as proposed by Palmow (1984), is combined with the numerical solution of objective tensorial ODEs as published by Rashid (1993). The resulting framework is suitable to model the complex inelastic properties of many materials at large strains. As an example, the Schwedoff model, which is appropriate for the simulation of metal forming processes, is analyzed within a cyclic simple shear test. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Irreversible Phase Transitions in Steel

We present a mathematical model for the austenite–pearlite and austenite–martensite phase transitions in eutectoid carbon steel. The austenite–pearlite phase change is described by the Additivity Rule. For the austenite–martensite phase change we propose a new rate law, which takes into account its irreversibility. We investigate questions of existence and uniqueness for the three-dimensional model and finally present numerical calculations of a continuous cooling transformation diagram for the eutectoid carbon steel C1080. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Numerical simulation of rotational symmetric tube bulging with inside pressure and axial compression

The contribution treats the process of rotational symmetric tube bulging with inside pressure and axial compression. This process enables the standard tubes to be formed into different rotational symmetric hollow parts in such a way that their central part is expanded into a desired shape while the ends remain unchanged. The superposition of axial compression contributes to a more favourable forming stress state, which is reflected in larger forming limits and smaller wall thinning in the widened area. The problems characterizing the process are a limited range of compression stability and difficulties met when establishing and optimizing the technological parameters of the process whose course cannot be defined in an analytical way. Based on a physical model of the forming process, a numerical model was built. Using ABAQUS code the model was simulated over the entire stress/forming region. A comparison of the computer simulated forming process with the experimentally obtained results showed that the model was highly accurate.     

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.