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In the last decades, the phase field method has drawn much attention for its application in fracture mechanics because it offers a simple unified framework for crack propagation. The core idea of phase field models for fracture is to introduce a continuous scalar field representing the discontinuous crack. Recently, a phase field model for fatigue has been proposed along this path. The fatigue failure differs from the other fracture scenarios since cracks only occur after a considerable number of load cycles. As fracturing happens, changes of the material microstructure are involved, which causes the evolution of the structural configuration. Thus, a new mathematical description not based on traditional spatial coordinates but the material manifold is desired, which will serve as an elegant analysis tool to understand the energetic forces for crack propagation. Configurational forces are a suitable choice for this purpose, as they describe the energetic driving forces associated with phenomena changing the material itself. In this work, we present a phase field model for fatigue. Furthermore, the phase field fatigue model is analyzed within the concept of configurational forces, which provides a straightforward way to understand the phase field simulations of fatigue fracture.
Firn describes the interstage product between snow and ice in cold regions of the earth, where annual snow fall exceeds the amount of snow melting. The continuing accumulation of snow leads to its densificiation due to overburden stress until it becomes ice. In the field of glaciology various attempts on simulating firn densification have been made and new models are still developed, as the knowledge of the firn column's density structure allows important derivations.
The presented study reassesses a model description for low density firn based on the process of grain boundary sliding presented by Alley in 1987 [1] using an optimisation approach. By comparing simulation results to 159 measured firn density profiles from Greenland and Antarctica it finds a possible additional dependency of the constitutive relation on the mean surface mass balance. This result is interpreted as an insufficient description of the stress regime.
Lattice Boltzmann methods [1] have been extended beyond their initial usage in transport problems, and can be used to solve a broader range of partial differential equations, e.g. the wave equation [2]. Thereby they can be utilized for fracture mechanics [3]. In the context of antiplane shear deformation we previously examined a stationary crack [4, 5] with a finite width. In this work we present two implementation strategies for non-mesh conforming boundary conditions, for which the bounding geometry does not need to adhere to the underlying lattice. This rectifies problems in modeling the crack. A numerical example shows the improvement compared to the previous results.
Metastable austenitic CrNi steels undergo phase transformation when loaded or deformed plastically. In the current work a macroscopic and phenomenological constitutive model is presented to model the strain induced transformation of austenite to martensite. The approach is based on the previous works of Olsen and Cohen [1] & Stringfellow et al. [2]. The kinetics of the phase transformation is modelled based on the assumption that the intersections of the shear bands in the austenitic phase, act as potential martensite nucleation locations. Evolution of the shear band density and their intersections are modelled using the plastic strain in the austenitic phase. The probability of the intersection creating martensite is given by a Gaussian cumulative distribution, which in turn depends on the temperature and stress triaxiality. The resulting stress- strain behavior considers the volume fraction, plastic strains and the strain hardening parameters of the individual phases as internal variables. An explicit formulation of the material model is implemented as a user subroutine in a bi-linear element formulation of FEM. Some of the required material parameters are estimated by fitting experimental stress-strain and martensite volume evolution curves. For the purpose of illustrating the model's behavior, boundary value problems of components with structured surfaces are presented.
The Lattice Boltzmann Method (LBM), e.g. in [1] and [2], can be interpreted as an alternative method for the numerical solution of certain partial differential equations that is not restricted to its origin in computational fluid mechanics. The interpretation of the LBM as a general numerical tool allows to extend the LBM to solid mechanics as well, see e.g. [3], which is concerned with the simulation of elastic solids under simplified deformation assumptions, and [4] as well as [5] which propose LBMs for the general plane strain case. In previous works on a LBM for plain strain such as [5], the treatment of practically relevant boundary conditions like Neumann and Dirichlet type boundary conditions is not the main focus and thus periodic conditions or absorbing layers are specified to simulate numerical examples. In this work, we show how Neumann and Dirichlet type boundary conditions are implemented in our LBM for plane strain from [4].
Surface wetting can be described by using phase field models [1]. In these models, often either the contact angle or the surface tensions between the solid and the fluid are prescribed directly on the wall in order to represent the solid-fluid interaction. However, the interaction of the wall and the fluid are not strictly local. The influence of the wall, which can be described by wall potentials [2], reaches out into the fluid, which is the reason for the formation of adsorbate layers. The investigation shows how such a wall potential can be included into a phase field model of wetting. It is found that by considering this energy contribution, the model is able to capture the adsorbate layer.
Fracture phenomena can be described by a phase field model in which an independent scalar field variable in addition to the mechanical displacement is considered [3]. This field approximates crack surfaces as a continuous transition zone from a value that indicates intact material to another value that represents the crack. For an accurate approximation of cracks, narrow transition zones resulting in steep gradients of the fracture field are required. This necessitates a high mesh density in finite element simulations, which leads to an increased computational effort. In order to circumvent this problem without forfeiting accuracy, exponential shape functions were introduced in the discretization of the phase field variable, see [4]. These special shape functions allow for a better approximation of steep gradients of the phase field with less elements as compared to standard Lagrange elements. Unfortunately, the orientation of the exponential shape functions is not uniquely determined and needs to be set up in the correct way in order to improve the approximation of smooth cracks. This work solves the issue by adaptively reorientating the exponential shape functions according to the nodal values of the phase field gradient in each element. Furthermore, a local approach is pursued that uses exponential shape function only in the vicinity of the crack, whereas standard bilinear shape function are used away from the crack.
Manufacturing techniques that can produce surfaces with a defined microstructure are in the focus of current research efforts. The ability to manufacture such surfaces gives rise to the need for numerical models that can predict the wetting properties of a given microstructure and can help to optimize these surfaces with respect to certain wetting properties. The present phase field (PF) model for wetting is linked to molecular dynamics (MD) simulations by the usage of the MD based perturbed Lennard-Jones truncated and shifted (PeTS) equation of state as well as a MD based viscosity correlation. The lower computational effort of the PF simulations compared to MD simulations enables the model to simulate wetting scenarios on the microscale.
In this work we illustrate the ability of a phase field model for fatigue crack growth in terms of extension capability to various amplitude loading and mean stress effects. The additional energy density contribution accounting for the energy associated with fatigue is modified in order to provide a more general model. Results obtained from numerical fatigue crack growth simulations are briefly presented and discussed.
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.
The Lattice Boltzmann Method (LBM), e.g. in [3] and [4], can be interpreted as an alternative method for the numerical solution of partial differential equations. The LBM is usually applied to solve fluid flows. However, the interpretation of the LBM as a general numerical tool, allows to extend the LBM to solid mechanics as well. In this spirit, the LBM has been studied in recent years. First publications [5], [6] present a LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. For so-called anti-plane shear deformation, the only non-zero displacement component is governed by the two-dimensional wave equation. In this work, the existing LBM for the two-dimensional wave equation is extended to more general plane strain problems. The algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non-zero component of the rotation vector, respectively.
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.
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.
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.
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.
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.
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.
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.