A Phase Field Model for Martensitic Transformations

The martensitic transformation is described using a phase field model which is in mathematical terms the regularization of a sharp interface approach. In this work, up to two martensitic orientation variants are considered. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, the global energy evolution during the martensitic transformation and the influence of external loads on the formation of the different martensitic phases are studied in 2d. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A condensed microelectromechanical constitutive and damage model for tetragonal ferroelectrics

A condensed model for ferroelectric solids with tetragonal unit cells is presented. The approach is microelectromechanically and physically motivated, considering discrete switching processes on the level of unit cells and quasi-continuous evolution of inelastic fields on the domain wall level. To calculate multiple grain interactions an interaction tensor is introduced. Hysteresis loops are simulated for pure electric and electromechanical loading, demonstrating e.g. the influence of a compressive preload on the poling process and interaction between statistically arranged crystallits. The residual stresses and the corresponding principle stresses are used to simulate fatigue damage in ferroelectric materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field simulation of thermomechanical fracture

In Francfort and Marigo's variational free-discontinuity formulation of brittle fracture [1] cracking is regarded as an energy minimization process, where the total energy is minimized with respect to any admissible crack set and displacement field. No additional criterion is needed to determine crack paths, branching of cracks and crack initiations. However, a direct discretization of the model is faced with significant technical problems, as it involves minimizations in a set of possibly discontinuous functions. A regularized version of the model has been introduced by Bourdin [2] and based on this, we use the concept of a continuum phase field model to simulate cracking processes. Cracks are indicated by the order parameter of the phase field model and cracking can be regarded as a phase transition problem. Additionally, introducing the heat equation into the model, it is capable to also take account of crack propagation due to thermal stresses. In the numerical implementation, crack parameter as well as temperature are treated as additional degrees of freedom and the coupled field equations are solved using the finite element method together with an implicit time integration scheme. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microcrack Evolution in Functionally Graded Material under Dynamic Loading

The damage process of metal-ceramic functionally graded material (FGM) is investigated. The microcrack evolution in a layered structure is analyzed using a numerical simulation of stresses and configurational forces. The modelling of an FGM of alumina ceramic and a metallic phase with gradually changing volume fraction of alumina is performed. A structure of two different layers bonded to a substrate is simulated. The stiffness and density of the three materials are varying. The evolution of configurational forces is simulated. The influence of the crack length on the crack driving force is studied for the case of a stress wave loading. The stress loading is applied in the horizontal direction as a dead load. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plastification and Damage in Wheel-Rail Rolling Contact – Case Study on a Crossing

A fully three-dimensional, dynamic model for a wheel running over a crossing is developed using an explicit finite element program. The full mass of the wheel and the crossing and elastic-plastic material behaviour are considered. The damage in the contact area is investigated with a very dense mesh taken from the dynamic model using a submodelling technique. With this kind of calculations the stresses and strains produced in the wheel and the crossing during the cross-over process can be determined, as well as the respective reaction forces in the bedding and the axle. Calculations for different crossing-geometries are performed. Finally a damage indicator is introduced to identify the probable location of crack initiation. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damage model for ferroelectric materials and its applications to multilayer actuators

In this paper a damage model for ferroelectric materials is presented. It is implemented in terms of a user element in the commercial FEM-code Abaqus. The model is based on micromechanical considerations of domain switching and its interaction with microcrack growth and coalescence. Finite element analysis of a multilayer actuator is performed, showing principal stresses leading to crack initiation and damage of the actuator. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Approach for Martensitic Transformations in Elastoplastic Materials

A multivariant phase field model for martensitic transformations in elastoplastic materials is introduced which is in mathematical terms the regularization of a sharp interface approach. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, plasticity is considered for the austenitic phase which influences the martensitic evolution. With aid of the model these interactions are studied in detail. (© 2013 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)     

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)     

Two-dimensional elastic phase-field simulation of fcc to bcc martensitic phase transformations in polycrystals

Numerical results of two-dimensional elastic phase-field simulations of martensitic phase transformations (fcc-bcc) in polycystals are presented. The stresses and strains in the diffuse interface domain are described by following Khachaturyan's approach of microelasticity. A fixed-point iteration algorithm in Fourier space is used to solve the mechanical equilibrium condition for the microscopic, inhomogeneous strain field and apply mechanical loadings to the system. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An experimental-analytical hybrid model for the elasto-plastic stresses at a crack

The primary obstacle preventing the analytical determination of physically sensible stresses at a crack tip is the presence of a mathematical singularity there. This singularity is best known in its elastic form; however it persists even in elasto-plastic crack-tip stresses. To overcome the difficulty we adopt the following strategy: we attempt to capture initial elastic stresses experimentally, than track subsequent elasto-plastic stress distributions analytically.We infer a finite stress at a crack tip from the experimental behaviour of cracked specimens at fracture when the specimens are made of a truly brittle material. Given a size-independent result, we argue that the crack-tip stress at fracture must equal the ultimate stress for such a material; thus dividing by the applied stress at the same point gives a measure of the stress concentration factor, K_T. The approach is checked for size independence and against hole configurations with known theoretical, yet physically reasonable, K_T. Then the effective experimental KT are taken as inputs for the second phase of the study in which we model the crack as being a smooth notch having the same stress concentration factor as found experimentally. In this way our configuration initially shares the same stresses at the crack tip as we inferred physically. Next we track effects of incremental plastic flow on a set of finite element grids. Satisfactory resolution in return for modest computational effort is obtained by employing a substructuring method. The accuracy in both the elastic and the elasto-plastic regime is checked against trial problems with exact solutions. Thereafter, physically interpretable stress distributions ahead of the crack are determined for a range of materials and for varying load levels.     

A phase field model for martensitic transformations with a temperature-dependent separation potential

Metallic materials often exhibit a complex microstructure with varying material properties in the different phases. Of major importance in mechanical engineering is the evolution of the austenitic and martensitic phases in steel. The martensitic transformation can be induced by heat treatment or by plastic surface deformation at low temperatures. A two dimensional elastic phase field model for martensitic transformations considering several martensitic orientation variants to simulate the phase change at the surface is introduced in [1]. However here, only one martensitic orientation variant is considered for the sake of simplicity. The separation potential is temperature dependent. Therefore, the coefficients of the Landau polynomial are identified by results of molecular dynamics (MD) simulations for pure iron [1]. The resulting separation potential is applied to analyse the mean interface velocity with respect to temperature and load. The interface velocity is computed by use of the dissipative part to the configurational forces balance as suggested in [3]. The model is implemented in the finite element code FEAP using standard 4-node elements with bi-linear shape functions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prediction of the growth of fatigue cracks taking environmental factors into account

A model of corrosion fatigue is proposed which takes into account the main phenomena of a mechanical nature, namely, the transfer of the active agent from the month of the crack to its tip, the accumulation of mechanical damage due to cyclic loads and the breakdown of the stability conditions in the body-with-cracks — load — environment system as the reason for the propagation of the crack tip. Particular attention is devoted to the mechanism by which active agent is transferred. In addition to diffusion, convective transfer as a consequence of the change in the shape and dimensions of the crack cavity is taken into account. The results of modelling are supplemented with diagrams, which relate the rate of growth of the crack with the range of the stress intensity factor, the concentration of the agent at the mouth of the crack and the characteristics of the cycle, equal to the ratio of the extremal values of the applied stresses in each cycle. The results are compared with experimental data.     

Numerical analysis of the effect of membrane preloads on the low-speed impact response of composite laminates

This article presents a comprehensive study on the mechanical behaviour of composite laminated plates undergoing a low-speed impact of an external body while they are subjected to in-plane preloads. The effect of such preloading was investigated by means of finite-element analysis of several impact events on laminates with three different span-to-thickness ratios. Tensile and compressive preloads, both uniaxial and biaxial, were considered; in the case of compression, the impact on buckled specimens was also studied. The results obtained show that the span-to-thickness ratio is a fundamental parameter in determining the effect of initial strains. Under a tensile preload, the impact-caused peak stresses were higher than in the case of no preload, and their increment was higher in thicker laminates. Under compression, the most dangerous influence of initial stresses was found at medium span-to-thickness ratios for preloads comparable with the buckling load, whereas, in other cases, negligible or even beneficial effects were observed. These results can justify some experimental findings from the existing literature, even if they were obtained without modelling the material degradation due to damage. Also, they allow us to conclude that the explanation of other phenomena strictly related to damage, as well as an accurate prediction of the extent of damage, requires a failure model.     

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)     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

Coupled chemomechanics and phase field modeling of failure in electrode materials of Li-ion batteries

We propose a canonical finite strain theory for diffusion-mechanics coupling for the intercalation induced stress generation in Li-ion electrode particles. The intrinsic coupling arises from both mechanical pressure gradient-induced diffusion of Li-ion particles and diffusion induced swelling/shrinkage leading to mechanical stresses. In addition, we extend the finite strain theory for diffusion-mechanics coupling to chemomechanical fracture of electrode particles by introducing a nonlocal crack phase field which replaces a sharp crack topology with a smooth diffuse interpolation between the intact and broken states of the material. We employ a semi-implicit Galerkin-type finite element method for the solution of resulting set of differential equations. In addition to the mechanical, chemical and crack phase field, we introduce the pressure as an independent field variable in order to reduce the smoothness requirements on the interpolation functions. We illustrate characteristic features of the proposed model by means of representative initial-boundary value problems. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fibre microbuckling model for predicting the notched compressive strength of composite sandwich panels

Cohesive-zone models have been successfully applied to predicting the damage from notches in engineering materials loaded intension. They have also been used to determine the growth of fibre microbuckling from a hole in a composite laminate under compression. The usual strategy is to replace the in elastic deformation associated with plasticity or microbuckling with a line crack and to assume some form of stress-displacement bridging law across the crack faces. This paper examines recent published experimental data for notched glass-fibre epoxy/honey comb sand wich panels loaded in uniaxial compression. A plastic fibre kinking analysis and a linear softening cohesive-zone model are used for the prediction of the unnotched and open-hole compressive strength and the theoretical results are found to be in a good agreement with experimental data. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 73–84, January–February, 2007.