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Indentation into a metastable austenite may induce the phase transformation to the bcc phase. We study this process using
atomistic simulation. At temperatures low compared to the equilibrium transformation temperature, the indentation triggers the
transformation of the entire crystallite: after starting the transformation, it rapidly proceeds throughout the simulation crystallite.
The microstructure of the transformed sample is characterized by twinned grains. At higher temperatures, around the equilibrium
transformation temperature, the crystal transforms only locally, in the vicinity of the indent pit. In addition, the indenter
produces dislocation plasticity in the remaining austenite. At intermediate temperatures, the crystal continuously transforms
throughout the indentation process.
Lattice Boltzmann method for antiplane shear deformation: non-lattice-conforming boundary conditions
(2022)
In this work, two different approaches to treat boundary conditions in a lattice Boltzmann method (LBM) for the wave equation are presented. We interpret the wave equation as the governing equation of the displacement field of a solid under simplified deformation assumptions, but the algorithms are not limited to this interpretation. A feature of both algorithms is that the boundary does not need to conform with the discretization, i.e., the regular lattice. This allows for a larger flexibility regarding the geometries that can be handled by the LBM. The first algorithm aims at determining the missing distribution functions at boundary lattice points in such a way that a desired macroscopic boundary condition is fulfilled. The second algorithm is only available for Neumann-type boundary conditions and considers a balance of momentum for control volumes on the mesoscopic scale, i.e., at the scale of the lattice spacing. Numerical examples demonstrate that the new algorithms indeed improve the accuracy of the LBM compared to previous results and that they are able to model boundary conditions for complex geometries that do not conform with the lattice.
Recently, phase field modeling of fatigue fracture has gained a lot of attention from many researches and studies, since the fatigue damage of structures is a crucial issue in mechanical design. Differing from traditional phase field fracture models, our approach considers not only the elastic strain energy and crack surface energy, additionally, we introduce a fatigue energy contribution into the regularized energy density function caused by cyclic load. Comparing to other type of fracture phenomenon, fatigue damage occurs only after a large number of load cycles. It requires a large computing effort in a computer simulation. Furthermore, the choice of the cycle number increment is usually determined by a compromise between simulation time and accuracy. In this work, we propose an efficient phase field method for cyclic fatigue propagation that only requires moderate computational cost without sacrificing accuracy. We divide the entire fatigue fracture simulation into three stages and apply different cycle number increments in each damage stage. The basic concept of the algorithm is to associate the cycle number increment with the damage increment of each simulation iteration. Numerical examples show that our method can effectively predict the phenomenon of fatigue crack growth and reproduce fracture patterns.
Adaptive numerical integration of exponential finite elements for a phase field fracture model
(2021)
Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.
In this contribution a phase field model for ductile fracture with linear isotropic hardening is presented. An energy functional consisting of an elastic energy, a plastic dissipation potential and a Griffith type fracture energy constitutes the model. The application of an unaltered radial return algorithm on element level is possible due to the choice of an appropriate coupling between the nodal degrees of freedom, namely the displacement and the crack/fracture fields. The degradation function models the mentioned coupling by reducing the stiffness of the material and the plastic contribution of the energy density in broken material. Furthermore, to solve the global system of differential equations comprising the balance of linear momentum and the quasi-static Ginzburg-Landau type evolution equation, the application of a monolithic iterative solution scheme becomes feasible. The compact model is used to perform 3D simulations of fracture in tension. The computed plastic zones are compared to the dog-bone model that is used to derive validity criteria for KIC measurements.
Phase field modeling of fracture has been in the focus of research for over a decade now. The field has gained attention properly due to its benefiting features for the numerical simulations even for complex crack problems. The framework was so far applied to quasi static and dynamic fracture for brittle as well as for ductile materials with isotropic and also with anisotropic fracture resistance. However, fracture due to cyclic mechanical fatigue, which is a very important phenomenon regarding a safe, durable and also economical design of structures, is considered only recently in terms of phase field modeling. While in first phase field models the material’s fracture toughness becomes degraded to simulate fatigue crack growth, we present an alternative method within this work, where the driving force for the fatigue mechanism increases due to cyclic loading. This new contribution is governed by the evolution of fatigue damage, which can be approximated by a linear law, namely the Miner’s rule, for damage accumulation. The proposed model is able to predict nucleation as well as growth of a fatigue crack. Furthermore, by an assessment of crack growth rates obtained from several numerical simulations by a conventional approach for the description of fatigue crack growth, it is shown that the presented model is able to predict realistic behavior.
Within this work, we utilize the framework of phase field modeling for fracture in order to handle a very crucial issue in terms of designing technical structures, namely the phenomenon of fatigue crack growth. So far, phase field fracture models were applied to a number of problems in the field of fracture mechanics and were proven to yield reliable results even for complex crack problems. For crack growth due to cyclic fatigue, our basic approach considers an additional energy contribution entering the regularized energy density function accounting for crack driving forces associated with fatigue damage. With other words, the crack surface energy is not solely in competition with the time-dependent elastic strain energy but also with a contribution consisting of accumulated energies, which enables crack extension even for small maximum loads. The load time function applied to a certain structure has an essential effect on its fatigue life. Besides the pure magnitude of a certain load cycle, it is highly decisive at which point of the fatigue life a certain load cycle is applied. Furthermore, the level of the mean load has a significant effect. We show that the model developed within this study is able to predict realistic fatigue crack growth behavior in terms of accurate growth rates and also to account for mean stress effects and different stress ratios. These are important properties that must be treated accurately in order to yield an accurate model for arbitrary load sequences, where various amplitude loading occurs.
One technique to describe the failure of mechanical structures is a phase field model for fracture. Phase field models for fracture consider an independent scalar field variable in addition to the mechanical displacement [1]. The phase field ansatz approximates crack surfaces as a continuous transition zone in which the phase field variable varies from a value that indicates intact material to another value that represents cracks. For a good approximation of cracks, these transition zones are required to be narrow, which leads to steep gradients in the fracture field. As a consequence, the required mesh density in a finite element simulation and thus the computational effort increases. In order to circumvent this efficiency problem, exponential shape functions were introduced in the discretization of the phase field variable, see [2]. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the steep transition with less elements. Unfortunately, the exponential shape functions are not symmetric, which requires a certain orientation of elements relative to the crack surfaces. This adaptation is not uniquely determined and needs to be set up in the correct way in order to improve the approximation of smooth cracks. The issue is solved in this work by reorientating the exponential shape functions according to the nodal value of phase field gradient in a particular element. To be precise, this work discusses an adaptive algorithm that implements such a reorientation for 2d and 3d situations.
Machining is very common in industry, e.g. automotive industry and aerospace industry, which is a nonlinear dynamic problem including large deformations, large strain, large strain rates and high temperatures, that implies some difficulties for numerical methods such as Finite element method. One way to simulate such kind of problems is the Particle Finite Element Method (PFEM) which combines the advantages of continuum mechanics and discrete modeling techniques. In this work we introduce an improved PFEM called the Adaptive Particle Finite Element Method (A-PFEM). The A-PFEM introduces particles and removes wrong elements along the numerical simulation to improve accuracy, precision, decrease computing time and resolve the phenomena that take place in machining in multiple scales. At the end of this paper, some examples are present to show the performance of the A-PFEM.
In cold regions of the earth, like Antarctica, Greenland or mountains at high altitude, the annual amount of deposited snow exceeds the amount of snow melting. Snow, which is more than one year old, is called firn. Over time firn transforms into ice by a sintering process, mainly driven by overburden pressure and temperature. This ultimately leads to the formation of glaciers and ice sheets.
We simulate firn densification based on the processes of sintering. The constitutive law represents grain boundary sliding, dislocation creep and diffusion. These mechanisms sum up to the overall densification which leads to the transformation of snow to ice. The model aims at obtaining a physics driven simulation tool for firn densification which provides data for a wider range of areas. It will contribute to develop better models and better understanding of the cryosphere.