Energetic modelling for carbon fibre reinforced carbon

In this contribution an energetic model for multi-phase materials is developed describing the influence of microstructure on different length scales as well as the evolution of phase changes. Restrictions on the energy functional are discussed. In such a non-convex framework, interfacial contributions serve for relaxing the total energy. Such models can be applied to describe the macroscopic material properties of carbon fibre reinforced carbon where phase transitions between regions of different texture of the carbon matrix are observed on nanoscale as well as columnar microstructures on microscale [2]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat Conduction in a Phase Field Model for Martensitic Transformation

We study the martensitic transformation with a phase field model, where we consider the Bain transformation path in a small strain setting. For the order parameter, interpolating between an austenitic parent phase and martensitic phases, we use a Ginzburg-Landau evolution equation, assuming a constant mobility. In [1], a temperature dependent separation potential is introduced. We use this potential to extend the model in [2], by considering a transient temperature field, where the temperature is introduced as an additional degree of freedom. This leads to a coupling of both the evolution equation of the order parameter and the mechanical field equations (in terms of thermal expansion) with the heat equation. The model is implemented in FEAP as a 4-node element with bi-linear shape functions. Numerical examples are given to illustrate the influence of the temperature on the evolution of the martensitic phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Regularized Sharp Interface Model for Martensitic Phase Transformations at Large Strains

The macroscopic mechanical behavior of many functional materials crucially depends on the formation and evolution of their microstructure. When considering martensitic shape memory alloys, this microstructure typically consists of laminates with coherent twin boundaries. We suggest a variational-based phase field model for the dissipative evolution of microstructure with coherence-dependent interface energy and construct a suitable gradient-extended incremental variational framework for the proposed dissipative material. We use our model to predict laminate microstructure in martensitic CuAlNi. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanical Approach of Martensitic Transformation Problem in Steels

To determine the mechanical behavior of material involving the martensitic phase transformation (for example, steels like 100Cr6), a representative volume element (RVE) model including phase transformation criterion is desireable at micromechanical approach. A framework combining the Eshelby's inclusion theory as well as continuum mechanics with phase-transformation (PT) critical condition at RVE model is presented briefly. And application of this model to estimate the critical aspect ratio of martensitic plate or lath inside homogeneneous stress field is also included, where the RVE can be under uniaxial tension/compression or pure shear loading case. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal forest rotation age when carbon captured is considered: theory and applications

Carbon dioxide (CO₂) is the gas most responsible for the greenhouse effect. Since trees absorb CO₂ during growth, some authors have argued for the importance of considering forests not only as timber producers but also as carbon pools. The consideration of carbon uptake as a public good generates a divergence between the private and social optima. This paper presents a methodology to determine optimal forest rotation ages in this context of multiple use and to remove the divergence between these two optima. A theoretical framework, based upon compromise programming, is applied to a case study of a beech forest in Spain, within the context of the current European financial aid for afforestation programs.     

A coupled phase-field – Cahn-Hilliard model for lower bainitic transformation

The lower bainitic transformation is highly dependent on carbon diffusion. Bainite consists of bainitic ferrite, residual austenite and carbides. The numerical modeling of the interaction between these phases and the carbon is extremely demanding. The goal of this work is to describe the formation of carbides in lower bainite. To model the evolution of a bainitic sheaf a phase-field model is coupled with a Cahn-Hilliard equation simulating the diffusion. The system of equations is solved using the finite element method. Numerical examples show the growth of the ferrite and the following uphill diffusion within this phase. At accumulation points of carbon, carbides are precipitated. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transformation strains for bainitic variant evolution in steel

In our work, we supplement a thermodynamically consistent multi-scale model for the bainitic phase transformation in a low alloy steel, which takes into account the mechanisms of elasto-viscoplasticity, phase transformations and heat conduction as well as the poly-crystalline structure of steel. In order to obtain realistic simulation results, the microscopic conversion procedures for the austenite-to-bainite transformation have to be described in an appropriate way. To this end, the transformation strains for the crystallographic variants have been adjusted. For calculation of transformation strains a theory for the formation of dislocated martensite, more precisely a hierarchical packet-block structure, is used. The calculated transformation strains are used in a simulation of bainite phase transformation in a polycrystalline RVE. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of numerical models for one-dimensional Stefan problems

In this paper, we present a critical comparison of the suitability of several numerical methods, level set, moving grid and phase field model, to address two well-known Stefan problems in phase transformation studies: melting of a pure phase and diffusional solid-state phase transformations in a binary system. Similarity solutions are applied to verify the numerical results. The comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques. Finally, it is shown both numerically and analytically that the solid-solid phase transformation is a limiting case of the solid–liquid transformation.     

Numerical analysis of quenching – Heat conduction in metallic materials

Metallic materials present a complex behavior during heat treatment processes. In a certain temperature range, change of temperature induces a phase transformation of metallic structure, which alters physical properties of the material. Indeed, measurements of specific heat and conductivity show strong temperature-dependence during processes such as quenching of steel. Several mathematical models, as solid mixtures and thermal–mechanical coupling, for problems of heat conduction in metallic materials, have been proposed. In this work, we take a simpler approach without thermal–mechanical coupling of deformation, by considering the nonlinear temperature-dependence of thermal parameters as the sole effect due to those complex behaviors. The above discussion of phase transformation of metallic materials serves only as a motivation for the strong temperature-dependence as material properties. In general, thermal properties of materials do depend on the temperature, and the present formulation of heat conduction problem may be served as a mathematical model when the temperature-dependence of material parameters becomes important. For this mathematical model we present the error estimate using the finite element method for the continuous-time case.     

A Regularized Sharp Interface Model for Phase Transformation Accounting for Prescribed Sharp Interface Kinetics

The macroscopic mechanical behavior of multi-phasic materials depends on the formation and evolution of their microstructure by means of phase transformation. In case of martensitic transformations, the resulting phase boundaries are sharp interfaces. We carry out a geometrically motivated discussion of the regularization of such sharp interfaces by use of an order parameter/phase-field and exploit the results for a regularized sharp interface model for two-phase elastic materials with evolving phase boundaries. To account for the dissipative effects during phase transition, we model the material as a generalized standard medium with energy storage and a dissipation function that determines the evolution of the regularized interface. Making use of the level-set equation, we are thereby able to directly translate prescribed sharp interface kinetic relations to the constitutive model in the regularized setting. We develop a suitable incremental variational three-field framework for the dissipative phase transformation problem. Finally, the modeling capability and the associated numerical solution techniques are demonstrated by means of a representative numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prediction of temperature distribution and phase transformation on the run-out table in the process of hot strip rolling

In this paper a mathematical model based on the finite element method and the Scheil additivity rule is presented for predicting the temperature distribution and phase transformation behavior on the run-out table during the hot strip rolling of a low carbon steel. The model considers the austenite to ferrite and pearlite transformations, the temperature-dependent material properties of the cooling austenite as well as the austenite work hardening effect on the kinetics of austenite transformation. To determine the validity of the model predictions, the time-temperature histories of a low carbon steel rod in different cooling media were measured and also hot rolling experiments were performed. Good agreement between the predictions and the experimental results indicates the reliability of the model.     

Buckling of a non-Euclidean annular plate

The effect of free edges of a monoatomic graphene sheet leads to excess edge energy due to the reconstruction of dangling bonds. Molecular static calculations show, that individual carbon atoms near the edge are displaced out of plane for relaxed nanoribbons [1]. In this work we are considering the effect of excess edge energy for almost circular graphene patches. To tackle this problem in the framework of continuum mechanics we are modelling the edge effect with a non-Euclidean plate model. A linear stability analysis of the flat configuration leads to the stability boundary in the parameter plane. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems

Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.     

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.     

市场自治与低碳认证情形下企业低碳生产行为研究

消费者愿意为产品支付低碳溢价的低碳消费趋势及政府环境政策驱动企业审视自身的低碳生产决策问题。通过将影响企业低碳生产决策的因素纳入有限理性的博弈框架,分析市场自治与政府介入低碳认证情形下企业群体低碳生产行为。结果表明,企业低碳生产决策受低碳收益及公平效用影响;市场初始阶段,低碳成本较高,具备低碳知识的理性消费者较少,企业与消费者关于低碳信息的不对称导致企业不能获得合适的低碳溢价收益,在完全自治的市场情形下,仅靠企业自身决策很难达到主动进行低碳生产的良性状态;低碳认证作为政府介入手段,当市场中低碳认证的产品达到一定比例,能够加速推动市场向企业进行低碳生产的良性状态转化。     

A computational two scaled model for the simulation of micro-heterogeneous low-alloyed TRIP steels

In transformation induced plasticity (TRIP) steel a diffusionless austenitic-martensitic phase transformation induced by plastic deformation can be observed, resulting in excellent macroscopic properties. In particular low-alloyed TRIP steels, which can be obtained at lower production costs than high-alloyed TRIP steel, combine this mechanism with a heterogeneous arrangement of different phases at the microscale, namely ferrite, bainite, and retained austenite. The macroscopic behavior is governed by a complex interaction of the phases at the micro-level and the inelastic phase transformation from retained austenite to martensite. A reliable model for low-alloyed TRIP steel should therefore account for these microstructural processes to achieve an accurate macroscopic prediction. To enable this, we focus on a multiscale method often referred to as FE² approach, see [6]. In order to obtain a reasonable representative volume element, a three-dimensional statistically similar representative volume element (SSRVE) [1] can be used. Thereby, also computational costs associated with FE² calculations can be significantly reduced at a comparable prediction quality. The material model used here to capture the above mentioned microstructural phase transformation is based on [3] which was proposed for high alloyed TRIP steels, see also e.g. [8]. Computations based on the proposed two-scale approach are presented here for a three dimensional boundary value problem to show the evolution of phase transformation at the microscale and its effects on the macroscopic properties. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Multi-phase-field model including inelastic deformation for solid state transformations

A multi phase field model is presented in order to take the plastic deformation during a solid state transformation into account and to investigate its effect on the transformation kinetics and morphology in a multi phase material. The model is formulated consistently with the multi phase field model for diffusional and surface driven phase transformations [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Peridynamics model for flexoelectricity and damage

A flexoelectric peridynamic (PD) theory is proposed. In the PD framework, the formulation introduces a nanoscale flexoelectric coupling that entails non-uniform strain in centrosymmetric dielectrics. This potentially enables PD modeling of a large class of phenomena in solid dielectrics involving cracks, discontinuities etc. wherein large strain gradients are present and the classical electromechanical theory based on partial differential equations do not directly apply. PD electromechanical equations, derived from Hamilton's principle, satisfy the global balance laws. Linear PD constitutive equations reflect the electromechanical coupling effect, with the mechanical force state affected by the polarization state and the electrical force state in turn by the displacement state. An analytical solution to the PD electromechanical equations is presented for the static case when a point mechanical force and a point electric force act in an infinite 3D solid dielectric. A parametric study on how different length scales influence the response is undertaken. In addition, the model is extended to incorporate damage using phase field – an order parameter, supplemented with a PD bond breaking criterion to study flexoelectric effects in damage and fracture problems. To demonstrate the performance of our proposal, we first simulate, considering small flexoelectricity effect and no damage, an externally pressured 2D flexoelectric disk subjected to a potential difference between the inner and outer surfaces and compare the results with existing solutions in the literature. Next, we simulate a plate with a central pre-crack under tension considering damage and flexoelectricity effects, and study the effect of various constitutive parameters on the damage evolution. We also furnish a classical derivation of phase field based flexoelectricity in Appendix I.     

Extending a finite strain hyperelastic micro-sphere framework towards phase transformations

A finite strain micro-sphere framework for hyperelastic solids elaborated by Carol et al. is extended towards the modelling of phase transformations in order to simulate polycrystalline solids under large deformations such as, e.g., shape memory alloys and shape memory polymers. The implemented phase transformation mechanism is based on statistical physics and is not restricted in terms of the number of solid material phases that can be considered, though we restrict the provided examples to two phases for the sake of conceptual clarity. The specifically chosen non-quadratic format of the Helmholtz free energy functions considered on the micro-plane level includes Bain-type transformation strains for each of the phases considered. Following the Voigt assumption on the micro-scale, identical total micro-stretches act in each of the material phases, where a multiplicative decomposition into elastic and transformation-related contributions is applied. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Disparate data fusion for protein phosphorylation prediction

New challenges in knowledge extraction include interpreting and classifying data sets while simultaneously considering related information to confirm results or identify false positives. We discuss a data fusion algorithmic framework targeted at this problem. It includes separate base classifiers for each data type and a fusion method for combining the individual classifiers. The fusion method is an extension of current ensemble classification techniques and has the advantage of allowing data to remain in heterogeneous databases. In this paper, we focus on the applicability of such a framework to the protein phosphorylation prediction problem.