Authors:Claas Hüter; Jörg Neugebauer; Guillaume Boussinot; Bob Svendsen; Ulrich Prahl; Robert Spatschek Pages: 895 - 911 Abstract: We discuss the modelling of grain boundary dynamics within an amplitude equations description, which is derived from classical density functional theory or the phase field crystal model. The relation between the conditions for periodicity of the system and coincidence site lattices at grain boundaries is investigated. Within the amplitude equations framework, we recover predictions of the geometrical model by Cahn and Taylor for coupled grain boundary motion, and find both \({\langle100\rangle}\) and \({\langle110\rangle}\) coupling. No spontaneous transition between these modes occurs due to restrictions related to the rotational invariance of the amplitude equations. Grain rotation due to coupled motion is also in agreement with theoretical predictions. Whereas linear elasticity is correctly captured by the amplitude equations model, open questions remain for the case of nonlinear deformations. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0424-7 Issue No:Vol. 29, No. 4 (2017)

Authors:Wolfgang Dreyer; Clemens Guhlke Pages: 913 - 934 Abstract: Diffuse and sharp interface models represent two alternatives to describe phase transitions with an interface between two coexisting phases. The two model classes can be independently formulated. Thus there arises the problem whether the sharp limit of the diffuse model fits into the setting of a corresponding sharp interface model. We call a diffuse model admissible if its sharp limit produces interfacial jump conditions that are consistent with the balance equations and the second law of thermodynamics for sharp interfaces. We use special cases of the viscous Cahn–Hilliard equation to show that there are admissible as well as non-admissible diffuse interface models. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0434-5 Issue No:Vol. 29, No. 4 (2017)

Authors:Kerstin Weinberg; Christian Hesch Pages: 935 - 945 Abstract: Phase-field approaches to fracture allow for convenient and efficient simulation of complex fracture pattern. In this paper, two variational formulations of phase-field fracture, a common second-order model and a new fourth-order model, are combined with a finite deformation ansatz for general nonlinear materials. The material model is based on a multiplicative decomposition of the principal stretches in a tensile and a compressive part. The excellent performance of the new approach is illustrated in classical numerical examples. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0440-7 Issue No:Vol. 29, No. 4 (2017)

Authors:Tim Dally; Kerstin Weinberg Pages: 947 - 956 Abstract: In a phase-field approach to fracture crack propagation is modeled by means of an additional continuous field. In this paper, two problems of linear elastic fracture mechanics are studied experimentally and numerically in order to evaluate the practicability of the phase-field approach and to validate the measured parameters. At first, a three-point bending experiment of silicon dies is simulated assuming static plate bending. Then, wave propagation and spallation in a Hopkinson bar test are analyzed in a dynamic regime. The simulations show that phase-field fracture reproduces the experimental results with high accuracy. The results are comparable to other fracture simulations, e.g., the cohesive element technique. In total, the phase-field approach to fracture is capable of tracking crack evolution in a very convenient and quantitatively correct way. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0443-4 Issue No:Vol. 29, No. 4 (2017)

Authors:Regina Schmitt; Charlotte Kuhn; Ralf Müller Pages: 957 - 968 Abstract: A continuum phase field model for martensitic transformations is introduced, including crystal plasticity with different slip systems for the different phases. In a 2D setting, the transformation-induced eigenstrain is taken into account for two martensitic orientation variants. With aid of the model, the phase transition and its dependence on the volume change, crystal plastic material behavior, and the inheritance of plastic deformations from austenite to martensite are studied in detail. The numerical setup is motivated by the process of cryogenic turning. The resulting microstructure qualitatively coincides with an experimentally obtained martensite structure. For the numerical calculations, finite elements together with global and local implicit time integration scheme are employed. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0446-1 Issue No:Vol. 29, No. 4 (2017)

Authors:Guillaume Boussinot; Efim A. Brener; Claas Hüter; Robert Spatschek Pages: 969 - 976 Abstract: We present a non-diagonal phase field model for phase transformations with unequal but finite diffusivities in the two phases. This model allows to recover the desired boundary conditions at the diffuse interface, and especially the elimination of the artificially enhanced surface diffusion effect. The model is non-diagonal since it incorporates the kinetic cross-coupling between the non-conserved and the conserved fields that was recently introduced (Brener and Boussinot in Phys Rev E 86:060601, 2012). We test numerically this model for the two-dimensional relaxation of a weakly perturbed interface towards its flat equilibrium. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0447-0 Issue No:Vol. 29, No. 4 (2017)

Authors:Alexander Schlüter; Charlotte Kuhn; Ralf Müller; Marilena Tomut; Christina Trautmann; Helmut Weick; Carolin Plate Pages: 977 - 988 Abstract: This work presents a continuum mechanics approach to model fracturing processes in brittle materials that are subjected to rapidly applied high-temperature gradients. Such a type of loading typically occurs when a solid is exposed to an intense high-energy particle beam that deposits a large amount of energy into a small sample volume. Given the rapid energy deposition leading to a fast temperature increase, dynamic effects have to be considered. Our existing phase field model for dynamic fracture is thus extended in a way that allows modelling of thermally induced fracture. A finite element scheme is employed to solve the governing partial differential equations numerically. Finally, the functionality of our model is illustrated by two examples. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0456-z Issue No:Vol. 29, No. 4 (2017)

Authors:Michael Kutter; Christian Rohde; Anna-Margarete Sändig Pages: 989 - 1016 Abstract: Epitaxy, a special form of crystal growth, is a technically relevant process for the production of thin films and layers. It can generate microstructures of different morphologies, such as steps, spirals or pyramids. These microstructures are influenced by elastic effects in the epitaxial layer. There are different epitaxial techniques, one being liquid phase epitaxy. Thereby, single particles are deposited out of a supersaturated liquid solution on a substrate where they contribute to the growth process. This article studies a two-scale model including elasticity, introduced in Eck et al. (Eur Phys J Special Topics 177:5–21, 2009) and extended in Eck et al. (2006). It consists of a macroscopic Navier–Stokes system and a macroscopic convection–diffusion equation for the transport of matter in the liquid, and a microscopic problem that combines a phase field approximation of a Burton–Cabrera–Frank model for the evolution of the epitaxial layer, a Stokes system for the fluid flow near the layer and an elasticity system for the elastic deformation of the solid film. Suitable conditions couple the single parts of the model. As the main result, existence and uniqueness of a solution are proven in suitable function spaces. Furthermore, an iterative solving procedure is proposed, which reflects, on the one hand, the strategy of the proof of the main result via fixed point arguments and, on the other hand, can be the basis for a numerical algorithm. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0462-1 Issue No:Vol. 29, No. 4 (2017)

Authors:Carlos Alberto Hernandez Padilla; Bernd Markert Pages: 1017 - 1026 Abstract: Nowadays crack initiation and evolution play a key role in the design of mechanical components. In the past few decades, several numerical approaches have been developed with the objective to predict these phenomena. The objective of this work is to present a simplified, nonetheless representative phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional elasto-plastic crystal plasticity model and a phase-field model modified to predict ductile fracture. A two-dimensional initial boundary value problem of ductile fracture is introduced considering a single-crystal setup and Nickel-base superalloy material properties. The model is implemented into the finite element context subjected to a quasi-static uniaxial tension test. The results are then qualitatively analyzed and briefly compared to current benchmark results in the literature. PubDate: 2017-07-01 DOI: 10.1007/s00161-015-0471-0 Issue No:Vol. 29, No. 4 (2017)

Authors:Nicolas Charalambakis; George Chatzigeorgiou; Yves Chemisky; Fodil Meraghni Abstract: In this paper, a review of papers on mathematical homogenization of dissipative composites under small strains and on the interplay between homogenization procedure and dissipation due to mechanical work is presented. Moreover, a critical survey on the links between mathematical homogenization and computational homogenization is attempted. PubDate: 2017-07-13 DOI: 10.1007/s00161-017-0587-5

Authors:Danil A. Kozhevnikov; Mikhail A. Sheremet Abstract: The effect of surface tension on laminar natural convection in a vertical cylindrical cavity filled with a weak evaporating liquid has been analyzed numerically. The cylindrical enclosure is insulated at the bottom, heated by a constant heat flux from the side, and cooled by a non-uniform evaporative heat flux from the top free surface having temperature-dependent surface tension. Governing equations with corresponding boundary conditions formulated in dimensionless stream function, vorticity, and temperature have been solved by finite difference method of the second-order accuracy. The influence of Rayleigh number, Marangoni number, and aspect ratio on the liquid flow and heat transfer has been studied. Obtained results have revealed that the heat transfer rate at free surface decreases with Marangoni number and increases with Rayleigh number, while the average temperature inside the cavity has an opposite behavior; namely, it growths with Marangoni number and reduces with Rayleigh number. PubDate: 2017-07-07 DOI: 10.1007/s00161-017-0586-6

Authors:Hejie Li; Andreas Öchsner; Prasad K. D. V. Yarlagadda; Yin Xiao; Tsuyoshi Furushima; Dongbin Wei; Zhengyi Jiang; Ken-ichi Manabe Abstract: Most of hexagonal close-packed (HCP) metals are lightweight metals. With the increasing application of light metal products, the production of light metal is increasingly attracting the attentions of researchers worldwide. To obtain a better understanding of the deformation mechanism of HCP metals (especially for Mg and its alloys), a new constitutive analysis was carried out based on previous research. In this study, combining the theories of strain gradient and continuum mechanics, the equal channel angular pressing process is analyzed and a HCP crystal plasticity constitutive model is developed especially for Mg and its alloys. The influence of elevated temperature on the deformation mechanism of the Mg alloy (slip and twin) is novelly introduced into a crystal plasticity constitutive model. The solution for the new developed constitutive model is established on the basis of the Lagrangian iterations and Newton Raphson simplification. PubDate: 2017-07-03 DOI: 10.1007/s00161-017-0583-9

Authors:F. Welsch; J. Ullrich; H. Ossmer; M. Schmidt; M. Kohl; C. Chluba; E. Quandt; A. Schütze; S. Seelecke Abstract: The exploitation of the elastocaloric effect in superelastic shape memory alloys (SMA) for cooling applications shows a promising energy efficiency potential but requires a better understanding of the non-homogeneous martensitic phase transformation. Temperature profiles on sputter-deposited superelastic \({\mathrm {Ti_{55.2}Ni_{29.3}Cu_{12.7}Co_{2.8}}}\) shape memory alloy thin films show localized release and absorption of heat during phase transformation induced by tensile deformation with a strong rate dependence. In this paper, a model for the simulation of the thermo-mechanically coupled transformation behavior of superelastic SMA is proposed and its capability to reproduce the mechanical and thermal responses observed during experiments is shown. The procedure for experiment and simulation is designed such that a significant temperature change from the initial temperature is obtained to allow potential cooling applications. The simulation of non-local effects is enabled by the use of a model based on the one-dimensional Müller–Achenbach–Seelecke model, extended by 3D mechanisms such as lateral contraction and by non-local interaction, leading to localization effects. It is implemented into the finite element software COMSOL Multiphysics, and comparisons of numerical and experimental results show that the model is capable of reproducing the localized transformation behavior with the same strain rate dependency. Additionally to the thermal and the mechanical behavior, the quantitative prediction of cooling performance with the presented model is shown. PubDate: 2017-06-26 DOI: 10.1007/s00161-017-0582-x

Authors:Marin Marin; Andreas Öchsner Abstract: This study is concerned with the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. For the solutions of this problem we prove a result of Hölder’s-type stability on the supply terms. We impose middle restrictions on the thermoelastic coefficients, which are common in continuum mechanics. For the same conditions we propose a continuous dependence result with regard to the initial data. PubDate: 2017-06-24 DOI: 10.1007/s00161-017-0585-7

Authors:Ivan Argatov; Alexei Iantchenko; Vitaly Kocherbitov Abstract: A large amplitude oscillatory shear (LAOS) is considered in the strain-controlled regime, and the interrelation between the Fourier transform and the stress decomposition approaches is established. Several definitions of the generalized storage and loss moduli are examined in a unified conceptual scheme based on the Lissajous–Bowditch plots. An illustrative example of evaluating the generalized moduli from a LAOS flow is given. PubDate: 2017-06-24 DOI: 10.1007/s00161-017-0584-8

Authors:C. S. K. Raju; P. Sanjeevi; M. C. Raju; S. M. Ibrahim; G. Lorenzini; E. Lorenzini Abstract: A theoretical analysis is performed for studying the flow and heat and mass transfer characteristics of Maxwell fluid over a cylinder with Cattaneo–Christov and non-uniform heat source/sink. The Brownian motion and thermophoresis parameters also considered into account. Numerical solutions are carried out by using Runge–Kutta-based shooting technique. The effects of various governing parameters on the flow and temperature profiles are demonstrated graphically. We also computed the friction factor coefficient, local Nusselt and Sherwood numbers for the permeable and impermeable flow over a cylinder cases. It is found that the rising values of Biot number, non-uniform heat source/sink and thermophoresis parameters reduce the rate of heat transfer. It is also found that the friction factor coefficient is high in impermeable flow over a cylinder case when compared with the permeable flow over a cylinder case. PubDate: 2017-06-21 DOI: 10.1007/s00161-017-0580-z

Authors:L. F. R. Espath; A. F. Sarmiento; L. Dalcin; V. M. Calo Abstract: We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation. PubDate: 2017-06-10 DOI: 10.1007/s00161-017-0581-y

Authors:Sebastian Glane; Felix A. Reich; Wolfgang H. Müller Abstract: This study is dedicated to continuum-scale material modeling of isotropic permanent magnets. An affine-linear extension to the commonly used ideal hard model for permanent magnets is proposed, motivated, and detailed. In order to demonstrate the differences between these models, bar and horseshoe magnets are considered. The structure of the boundary value problem for the magnetic field and related solution techniques are discussed. For the ideal model, closed-form analytical solutions were obtained for both geometries. Magnetic fields of the boundary value problems for both models and differently shaped magnets were computed numerically by using the boundary element method. The results show that the character of the magnetic field is strongly influenced by the model that is used. Furthermore, it can be observed that the shape of an affine-linear magnet influences the near-field significantly. Qualitative comparisons with experiments suggest that both the ideal and the affine-linear models are relevant in practice, depending on the magnetic material employed. Mathematically speaking, the ideal magnetic model is a special case of the affine-linear one. Therefore, in applications where knowledge of the near-field is important, the affine-linear model can yield more accurate results—depending on the magnetic material. PubDate: 2017-06-09 DOI: 10.1007/s00161-017-0578-6

Authors:Vamshi Krishna Chillara Abstract: We develop a thermodynamic framework for modeling nonlinear ultrasonic damage sensing and prognosis in materials undergoing progressive damage. The framework is based on the internal variable approach and relies on the construction of a pseudo-elastic strain energy function that captures the energetics associated with the damage progression. The pseudo-elastic strain energy function is composed of two energy functions—one that describes how a material stores energy in an elastic fashion and the other describes how material dissipates energy or stores it in an inelastic fashion. Experimental motivation for the choice of the above two functionals is discussed and some specific choices pertaining to damage progression during fatigue and creep are presented. The thermodynamic framework is employed to model the nonlinear response of material undergoing stress relaxation and creep-like degradation. For each of the above cases, evolution of the nonlinearity parameter with damage as well as with macroscopic measurables like accumulated plastic strain is obtained. PubDate: 2017-06-06 DOI: 10.1007/s00161-017-0575-9

Authors:K. S. Surana; A. D. Joy; J. N. Reddy Abstract: This paper presents a non-classical continuum theory for fluent continua in which the conservation and balance laws are derived by incorporating both internal rotation rates arising from the velocity gradient tensor and the rotation rates of the Cosserats. Specifically, in this non-classical continuum theory we have (1) the usual velocities ( \(\bar{ \pmb {\varvec{v }}}\) ), (2) the three internal rotation rates ( \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) ) about the axes of a fixed triad whose axes are parallel to the x-frame arising from the velocity gradient tensor \((\bar{ \pmb {\varvec{L }}})\) that are completely defined by the antisymmetric part of the velocity gradient tensor, and (3) three additional rotation rates ( \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) ) about the axes of the same triad located at each material point as additional three unknown degrees of freedom, referred to as Cosserat rotation rates. This gives rise to \(\bar{ \pmb {\varvec{v }}}\) and \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) as six degrees of freedom at a material point. The internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) , often neglected in classical fluid mechanics, exist in all deforming fluent continua as these are due to velocity gradient tensor. When the internal rotation rates \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) are resisted by deforming fluent continua, conjugate moment tensor arises that together with \({}_i^t\bar{ \pmb {\varvec{\Theta }}}\) may result in energy storage and/or dissipation, which must be considered in the conservation and balance laws. The Cosserat rotation rations \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) also result in conjugate moment tensor that together with \({}_e^t\bar{ \pmb {\varvec{\Theta }}}\) may also result in energy storage and/or dissipation. The main focus of this paper is a consistent derivation of conservation and balance laws for fluent continua that incorporate the aforementioned physics and associated constitutive theories for thermofluids using the conditions resulting from the entropy inequality. The material coefficients derived in the constitutive theories are clearly defined and discussed. PubDate: 2017-05-31 DOI: 10.1007/s00161-017-0579-5