A non-local gradient-enhanced damage-plasticity formulation is proposed, which prevents the loss of well-posedness of the governing field equations in the post-critical damage regime. The non-locality of the formulation then manifests itself in terms of a non-local free energy contribution that penalizes the occurrence of damage gradients. A second penalty term is introduced to force the global damage field to coincide with the internal damage state variable at the Gauss point level. An enforcement of Karush–Kuhn–Tucker conditions on the global level can thus be avoided and classical local damage models may directly be incorporated and equipped with a non-local gradient enhancement. An important part of the present work is to investigate the efficiency and robustness of different algorithmic schemes to locally enforce the Karush–Kuhn–Tucker conditions in the multi-surface damage-plasticity setting. Response simulations for representative inhomogeneous boundary value problems are studied to assess the effectiveness of the gradient enhancement regarding stability and mesh objectivity. © 2017, © The Author(s) 2017.

Phenomenological constitutive models of inelastic responses based on the methods of classical plasticity provide several advantages, especially in terms of computational efficiency. For this reason, they are attractive for the analysis of complex boundary value problems comprising large computational domains. However, for the analysis of problems dominated by single crystal behavior (e.g., inclusion, granular interaction problems or inter-granular fracture), such approaches are often limited by the symmetry assumptions inherent in the stress invariants used to form yield-type criteria. On the other hand, the high computational effort associated with micro-mechanical or crystal plasticity-type models usually prevents their use in large structural simulations, multi-scale analyses, or design and property optimization computations. The goal of the present work is to establish a modeling strategy that captures micro-scale single-crystalline sma responses with sufficient fidelity at the computational cost of a phenomenological macro-scale model. Its central idea is to employ an anisotropic transformation yield criterion with sufficiently rich symmetry class—which can directly be adopted from the literature on plasticity theory—at the single crystal level. This approach is conceptually fundamentally different from the common use of anisotropic yield functions to capture tension-compression asymmetry and texture-induced anisotropy in poly-crystalline SMAs. In our model, the required anisotropy parameters are calibrated either from experimental data for single crystal responses, theoretical considerations or micro-scale computations. The model thus efficiently predicts single crystal behaviors and can be applied to the analysis of complex boundary value problems. In this work we consider the application of this approach to the modeling of shape memory alloys (SMAs), though its potential utility is much broader. Example analyses of SMA single crystals that include non-transforming precipitates and poly-crystalline aggregates are considered and the effects of both elastic and transformation anisotropy in these materials are demonstrated. © 2017 Elsevier Ltd

The aim of the application of open cell ceramic foam filters during casting of metals is the reduction of non-metallic inclusions and turbulences in the melt flow. Hence, an improvement of the quality of the cast products is achieved. The integrity of the filter at mechanical loading under elevated temperatures requires a mechanical characterization of the bulk material of the filter. In particular, fracture toughnesses have to be determined for a new generation of filter materials. The presented work describes an experimental method to measure fracture toughnesses of the filter materials. The mechanical testing is performed with the help of 4-point-bending tests using miniaturized chevron-notched specimens at different temperatures. Additionally, the geometry function of the test set-up is calculated and compared with an empirical formula by Munz [1]. At the end, the fracture toughness is determined at room temperature and 800°C. Further results characterize the influence of different geometrical parameters of the test set-ups on the maximum tensile stresses in the specimen and the load-displacement curves. © 2018 Elsevier Ltd and Techna Group S.r.l.

This work presents a first investigation of the small punch test (SPT) as a possible method to identify material parameters for shape memory alloy (SMA) behavior. In comparison to more common tests, the SPT has advantages in setup simplicity, small sample size, uncomplicated shape, and the possibility of specimen clamping, while offering controlled multi axial loading. Different loading scenarios are described and executed. The parameters of an established SMA model are subsequently (partially) calibrated from the measured SPT force-deflection curves. For some loading regimes, the effective response curves suggest the occurrence of damage events. To investigate the underlying mi-croscale failure mechanisms, a first SEM study was conducted. These first results underline that the SPT is a promising efficient and inexpensive characterization method to support SMA constitutive model development under multiaxial loading — including aspects of damage, fracture and fatigue. Copyright © 2018 ASME.

This contribution is concerned with the embedding of constitutive relations for magneto-active polymers (MAP) into finite element simulations. To this end, a recently suggested, calibrated, and validated material model for magneto-mechanically coupled and rate-dependent MAP response is briefly summarized in its continuous and algorithmic settings. Moreover, the strongly coupled field equations of finite deformation magneto-mechanics are reviewed. For the purpose of numerical simulation, a finite element model is then established based on the usual steps of weak form representation, discretization and consistent linearization. Two verifying inhomogeneous numerical examples are presented in which a classical 'plate with a hole' geometry is equipped with MAP properties and subjected to different types of time-varying mechanical and magnetic loading. © 2016 IOP Publishing Ltd.

In this contribution, a micro-mechanically motivated, energy relaxation-based constitutive model for phase transformation, martensite reorientation and twin formation in shape memory alloys is proposed. The formulation builds on an idealized parametrization of the austenite-twinned martensite microstructure through first- and second-order laminates. To estimate the effective rank-one convex energy density of the phase mixture, the concept of laminate-based energy relaxation is applied. In this context, the evolution of the energetic and dissipative internal state variables, that describe characteristic microstructural features, is computed via constrained incremental energy minimization. This work also suggests a first step towards the continuous modeling of twin formation within the framework of energy relaxation and can be viewed as a generalization of earlier models suggested by Bartel and Hackl (2009) and Bartel et al. (2011). More specifically, in the current model the orientation of martensitic variants in space is not pre-assigned. Variants are rather left free to arrange themselves relative to the martensite-martensite interface in an energy-minimizing fashion, where, however, it is assumed that they form crystallographically-twinned pairs. The formulation also eliminates the need to introduce specific expressions for the Bain strains in each of the martensitic variants, by relating them to a master variant and utilizing the information about their absolute orientation. The predictive capabilities of the proposed modeling framework are demonstrated in several representative numerical examples. In the first part of the results section, the focus is placed on purely energetic analysis, and the particular influence of the different microstructural degrees of freedom on the relaxed energy densities and the corresponding stress-strain responses is investigated in detail. In the second part, macro-homogeneous uniaxial strain and shear loading cases are analyzed for the dissipative case. It is shown, that the proposed model, which, compared to purely phenomenological macro-scale models, has the advantage of strong micro-mechanical motivation, is capable of qualitatively predicting central features of single crystal shape memory alloy behavior, such as the phase diagram in stress-temperature space, and pseudo-elastic and pseudo-plastic responses, while simultaneously providing valuable insight into the underlying micro-scale mechanisms. © The Author(s) 2014.

Strains produced by active materials embedded in plates have been extensively used to manipulate the shape of surface-like engineering structures. Shape memory alloys (SMAs) are active materials that provide a significant amount of strain under large stresses, a characteristic of great utility in such morphing structures. In this work, an analytical approach to approximate the deformation of plates with SMA constituents is developed via the Rayleigh-Ritz method. An additive set of kinematically admissible displacement fields with unknown coefficients is used to describe the plate displacement field. The total potential energy is then calculated using the displacement field, loading conditions, and constitutive relations for the plate layer(s) composed of SMA wire meshes, dense SMA films, and/or elastic material. The unknown coefficients are then found via minimization of the total potential energy. This approach provides closed-form expressions for the approximate deformation of the plates including multistable configurations. The response of circular SMA-based plates is studied herein. The results show that temperature fields with a linear variation in the radial direction induce multistable configurations in which the plate Gaussian curvature is determined by the direction of the temperature gradient. An alternative inversion of the proposed approach is used to directly compute the temperature field required to morph a plate towards a prescribed goal shape. The obtained closed-form expressions show good agreement with detailed non-linear finite element analysis simulations. The proposed approach provides analysts with a low computational cost and relatively simple implementation to assess the potentially stable configurations of SMA-based plates under given loading conditions. Knowledge of such stable configurations is very valuable in the design of SMA-based morphing structures. © 2015 Elsevier Ltd.

The magneto-electric (ME) coupling of multiferroic materials is of high interest for a variety of advanced applications like in data storage or sensor technology. Since the ME coupling of single-phase multiferroics is too low for technical applications, the manufacturing of composite structures becomes relevant. These composites generate the effective ME coupling as a strain-induced product property. Several experiments on composite multiferroics showed remarkable ME coefficients that are orders of magnitudes higher than those of single-phase materials. The present paper investigates the arising effective product properties of two-phase ME composites by simulating the coupling behavior using a two-scale finite element (FE2) homogenization approach. By means of this method, microstructures with different volume fractions of the individual phases and associated macroscopic ME coupling coefficients are considered. We investigate the influence of different magnetization states by means of the non-linear dissipative magnetostriction material model originally established in [1]. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In virtual design of the hot stamping process, a reliable description of the material flow behaviour is an important input to ensure accurate estimations of the part's feasibility. Currently, to characterise the hot stamping material's flow behaviour at elevated temperatures, tensile and upsetting tests are available. The measurement of the flow behaviour out of such tests, which is generally temperature and strain rate dependent, still remains a complex task. Therefore traditional methods to measure flow curves out of such measurements are not necessarily precise enough. In this contribution the authors focus on non-isothermal conductive tensile tests of the manganeseboron steel MBW® 1500 in order to understand its flow behaviour at elevated temperature. Numerical calculations using Finite Element Method (FEM) of the tests itself with correct boundary conditions as well as for all necessary phenomena are used to identify accurately the material's flow curves by using inverse optimisation. Finally, for validation purpose the identified flow curves out of the optimisation method were used to simulate the hot stamping of two different parts. Both geometries were chosen such that various strain paths are covered i.e. uniaxial tension to plane strain. © 2015 Trans Tech Publications, Switzerland.

This paper proposes energy relaxation-based approaches for the modeling of magnetostriction, with a particular focus on single crystalline magnetic shape memory alloy response. The theoretical development relies on concepts of energy relaxation in the context of nonconvex free energy landscapes whose wells define preferred states of spontaneous straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] represents the point of departure for the model development, and its capabilities, but also limitations, are demonstrated by means of representative numerical examples. The key features that characterize the extended approach are (i) the incorporation of elastic deformations, whose distribution among the individual phases occurs in an energy minimizing fashion, (ii) a finite magnetocrystalline anisotropy energy, that allows magnetization rotations away from easy axes, and (iii) dissipative effects, that are accounted for in an incremental variational setting for standard dissipative materials. In the context of introducing elastic strain energy, two different relaxation concepts, the convexification approach and the rank-one relaxation with respect to first-order laminates, are considered. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the pre-variant reorientation regime, and the stress dependence of the maximum field induced strain, can be captured, which are prohibited by the inherent assumptions of the constrained theory. The enhanced modeling capabilities of the extended approach are demonstrated by several representative response simulations and comparison to experimental results taken from literature. These examples particularly focus on the response of single crystals under cyclic magnetic field loading at constant stress and cyclic mechanical loading at constant magnetic field. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

This paper presents an energy relaxation-based approach for the modeling of single crystalline magnetic shape memory alloy response under general two-dimensional magnetomechanical loading. It relies on concepts of energy relaxation in the context of non-convex free energy landscapes whose wells define preferred states of straining and magnetization. The constrained theory of magnetoelasticity developed by DeSimone and James [1] forms the basis for the model development. The key features that characterize the extended approach are (i) dissipative effects, accounted for in an incremental variational setting, and (ii) finite magnetocrystalline anisotropy energy. In this manner, important additional response features, e.g. the hysteretic nature, the linear magnetization response in the prevariant reorientation regime, and the stress dependence of the maximum field induced strain, can be captured, which are prohibited by the inherent assumptions of the constrained theory. The enhanced modeling capabilities of the extended approach are demonstrated by several representative response simulations and comparison to experimental results taken from literature. These examples particularly focus on the response of single crystals under cyclic magnetic field loading at constant stress, and cyclic mechanical loading at constant magnetic field. © 2014 by ASME.

In the present study, the effective elastic properties of multifunctional carbon nanotube composites are derived due to the agglomeration of straight circular carbon nanotubes dispersed in soft polymer matrices. The agglomeration of CNTs is common due to the size of nanotubes, which is at nanoscales. Furthermore, it has been proved that straight circular CNTs provide higher stiffness and elastic properties than any other shape of the nanofibers. Therefore, in the present study, the agglomeration effect on the effective elastic moduli of the CNT polymer nanocomposites is investigated when circular CNTs are aligned straight as well as distributed randomly in the matrix. The Mori-Tanaka micromechanics theory is adopted to newly derive the expressions for the effective elastic moduli of the CNT composites including the effect of agglomeration. In this direction, analytical expressions are developed to establish the volume fraction relationships for the agglomeration regions with high, and dilute CNT concentrations. The volume of the matrix in which there may not be any CNTs due to agglomeration is also included in the present formulation. The agglomeration volume fractions are subsequently adopted to develop the effective relations of the composites for transverse isotropy and isotropic straight CNTs. The validation of the modeling technique is assessed with results reported, and the variations in the effective properties for high and dilute agglomeration concentrations are investigated. Copyright © 2014 by ASME.

In this contribution, a micro-mechanically motivated constitutive model for phase transformation, martensite reorientation and twin formation in shape memory alloys is proposed. The formulation builds on an effective parametrization of the austenite-twinned martensite microstructure through first- And second-order laminates. To define the effective energy density of the phase mixture, the concept of energy relaxation is applied. The values of the dissipative internal state variables that describe the microstructure evolution are computed via constrained incremental energy minimization. This work also suggests a first step towards the continuous modeling of twin formation embedded into the concept of energy relaxation and can be viewed as a generalization of earlier models suggested in [1-3]. More specifically, in the current model the orientation of martensitic variants in space is not pre-assigned. Variants are rather left free to arrange in an energy-minimizing fashion and are only distinguished by their rotation in reference to a master variant. Finally, macro-homogeneous uniaxial strain and pure shear loading cases are analyzed to demonstrate the capabilities of the proposed modeling framework. Copyright © 2013 by ASME.

Several constitutive models for magnetic shape memory alloys (MSMAs) have been proposed in the literature. The implementation of numerical integration schemes, which allow the prediction of constitutive response for general loading cases and ultimately the incorporation of MSMA response into numerical solution algorithms for fully coupled magneto-mechanical boundary value problems, however, has received only very limited attention. In this work, we establish two algorithmic implementations of the internal variable model for MSMAs proposed in(Kiefer and Lagoudas 2005 Phil. Mag. Spec. Issue: Recent Adv. Theor. Mech. 85 4289-329, Kiefer and Lagoudas 2009 J. Intell. Mater. Syst. 20 143-70), where we restrict our attention to pure martensitic variant reorientation to limit complexity. The first updating scheme is based on the numerical integration of the reorientation strain evolution equation and represents a classical predictorcorrector-type general return mapping algorithm. In the second approach, the inequality-constrained optimization problem associated with internal variable evolution is converted into an unconstrained problem via Fischer-Burmeister complementarity functions and then iteratively solved in standard Newton-Raphson format. Simulations are verified by comparison to closed-form solutions for experimentally relevant loading cases. © 2012 IOP Publishing Ltd.

This paper outlines a new variational-based modeling and computational implementation of macroscopic continuum magneto-mechanics involving non-linear, inelastic material behavior, with a special focus on dissipative magnetostriction. It is based on a constitutive variational principle that optimizes a generalized incremental work function with respect to the internal state variables. In an incremental setting at finite time steps, this variational problem defines a quasi-hyper-magnetoelastic potential for the stresses and the magnetic induction, and incorporates energy storage as well as dissipative mechanisms. The existence of this potential further allows the incremental boundary-value problem of quasi-static inelastic magneto-mechanics to be recast into a principle of stationary incremental energy. The second focus of this paper is on the careful construction of the energy storage and dissipation functions for the model problem of hysteretic magnetostriction at the macroscopic level. It is then demonstrated that the proposed model is capable of predicting the ferromagnetic and field-induced strain hysteresis curves characteristic of magnetostrictive material response in good agreement with experiments. The numerical solution of the coupled non-linear boundary-value problem is based on a monolithic multi-field finite element implementation. As a consequence of the proposed incremental variational principle, the discretization of the multi-field problem appears in a compact symmetric format. In this sense, the proposed formulation provides a canonical framework for the simulation of boundary-value-problems in dissipative magnetostriction at the macro-level. The performance of the proposed algorithm is tested by application to relevant numerical examples. © 2011 Elsevier Ltd. All rights reserved.

This paper is concerned with the finite element analysis of boundary value problems involving nonlinear magnetic shape memory behavior, as might be encountered in experimental testing or engineering applications of magnetic shape memory alloys (MSMAs). These investigations mainly focus on two aspects: first, nonlinear magnetostatic analysis, in which the nonlinear magnetic properties of the MSMA are predicted by the phenomenological internal variable model previously developed by Kiefer and Lagoudas, is utilized to investigate the influence of the demagnetization effect on the interpretation of experimental measurements. An iterative procedure is proposed to deduce the true constitutive behavior of MSMAs from experimental data that typically reflect the shape-dependent system response of a sample. Secondly, the common assumption of a homogeneous Cauchy stress distribution in the MSMA sample is tested. This is motivated by the expectation that the influence of magnetic body forces and body couples caused by field matter interactions may not be negligible in MSMAs that exhibit blocking stresses of well below 10 MPa. To this end, inhomogeneous Maxwell stress distributions are first computed in a post-processing step, based on the magnetic field and magnetization distributions obtained in the magnetostatic analysis. Since the computed Maxwell stress fields, though allowing a first estimation of the influence of the magnetic force and couple, do not satisfy equilibrium conditions, a finite element analysis of the coupled field equations is performed in a second step to complete the study. It is found that highly non-uniform Cauchy stress distributions result under the influence of magnetic body forces and couples, with magnitudes of the stress components comparable to externally applied bias stress levels. © 2011 Taylor & Francis.

The research work presents the modeling of effective properties and thermo-mechanical behavior of shape memory fiber (SMF) and shape memory polymer (SMP) composite laminates using micromechanical approaches based on the method of mixtures (MOM) and method of cells (MOC). The fiber is made of a nickel-titanium (Ni-Ti) shape memory alloy (SMA), while the matrix consists of a shape memory thermoset epoxy polymer (SMP). The use of an SMP matrix provides large strain compatibility with the SMA fiber, while being active at high temperatures without losing its elastic properties. Additionally, the SMP matrix is also able to produce similar pseudoelastic and shape memory effects, which are noticed in SMAs. In the analysis, a two step homogenization scheme is followed. In the first step the effective properties of each layer are determined via a micromechanics approach with iso-strain conditions. In the second step the effective properties of the SMF-SMP composite are computed making a thin plate theory assumption, which takes into account the transverse shear deformations. The possible elastic couplings for SMF-SMP laminates are discussed, and the laminate force and moment resultants are computed for various laminate configurations. The analysis takes into account the effects of phase transformations and the resulting change in the fiber-matrix modulus. The results have been compared by considering different fiber volume fractions, temperatures, fiber orientations, and lamina stacking sequences. The results show that adaptive SMA-SMP composites laminates can be developed that provide shape controllability via tunable laminate stiffnesses leading to optimal response. Furthermore, the work presents the necessary framework for a reliable and efficient analysis of SMA-SMP laminates for practical applications. The theory can be directly used in established plate and shell formulations of finite element analysis. Finally, the variations in force and moment resultants with respect to fiber orientations and stacking sequences are presented, which are useful to study the bending and buckling characteristics of active composites for shape control of adaptive structures. The work concludes that efficient adaptive laminate development for high performance composite applications, exhibiting large shape adaptivity, high stresses, and increased stiffness, are feasible as compared to SMA composites without active matrix. POLYM. COMPOS., 2011. © 2011 Society of Plastics Engineers.

This paper presents a general framework for the macroscopic, continuum-based formulation and numerical implementation of dissipative functional materials with electro-magneto-mechanical couplings based on incremental variational principles. We focus on quasi-static problems, where mechanical inertia effects and time-dependent electro-magnetic couplings are a priori neglected and a time-dependence enters the formulation only through a possible rate-dependent dissipative material response. The underlying variational structure of non-reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so-called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum-based modeling in macroscopic electro-magneto-mechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle-point principles. Next, the implications on the formulation of boundary-value problems are addressed, which appear in energy-based formulations as minimization principles and in enthalpy-based formulations in the form of saddle-point principles. Furthermore, the material stability of dissipative electro-magneto-mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro-magneto-mechanics, an energy-based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy-based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright © 2011 John Wiley & Sons, Ltd.