Phase field approaches of bone remodeling based on TIP

AbstractThe process of bone remodeling includes a cycle of repair, renewal, and optimization. This adaptation process, in response to variations in external loads and chemical driving factors, involves three main types of bone cells: osteoclasts, which remove the old pre-existing bone; osteoblasts, which form the new bone in a second phase; osteocytes, which are sensing cells embedded into the bone matrix, trigger the aforementioned sequence of events. The remodeling process involves mineralization of the bone in the diffuse interface separating the marrow, which contains all specialized cells, from the newly formed bone. The main objective advocated in this contribution is the setting up of a modeling and simulation framework relying on the phase field method to capture the evolution of the diffuse interface between the new bone and the marrow at the scale of individual trabeculae. The phase field describes the degree of mineralization of this diffuse interface; it varies continuously between the lower value (no mineral) and unity (fully mineralized phase, e.g. new bone), allowing the consideration of a diffuse moving interface. The modeling framework is the theory of continuous media, for which field equations for the mechanical, chemical, and interfacial phenomena are written, based on the thermodynamics of irreversible processes. Additional models for the cellular activity are formulated to describe the coupling of the cell activity responsible for bone production/resorption to the kinetics of the internal variables. Kinetic equations for the internal variables are obtained from a pseudo-potential of dissipation. The combination of the balance equations for the microforce associated to the phase field and the kinetic equations lead to the Ginzburg–Landau equation satisfied by the phase field with a source term accounting for the dissipative microforce. Simulations illustrating the proposed framework are performed in a one-dimensional situation showing the evolution of the diffuse interface separating new bone from marrow.

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Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation

Archive of Applied Mechanics ◽

10.1007/s00419-020-01759-3 ◽

2020 ◽

Cited By ~ 1

Author(s):

Arne Claus Hansen-Dörr ◽

Jörg Brummund ◽

Markus Kästner

Keyword(s):

Crack Propagation ◽

Phase Field ◽

Strain Energy ◽

Heterogeneous Media ◽

Phase Field Model ◽

Diffuse Interface ◽

Jump Conditions ◽

Modeling Framework ◽

Material Interfaces ◽

Mechanical Jump Conditions

Abstract In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the regularized crack. The key novelty is the combination of a strain energy split with a partial rank-I relaxation in the vicinity of the diffuse interface. The former is necessary to account for physically meaningful crack kinematics like crack closure, the latter ensures the mechanical jump conditions throughout the diffuse region. The model is verified by a convergence study, where a circular bi-material disc with and without a crack is subjected to radial loads. For the uncracked case, analytical solutions are taken as reference. In a second step, the model is applied to crack propagation, where a meaningful influence on crack branching is observed, that underlines the necessity of a reasonable homogenization scheme. The presented model is particularly relevant for the combination of any variational strain energy split in the fracture phase-field model with a diffuse modeling approach for material heterogeneities.

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Viscoelastic phase-field fracture using the framework of representative crack elements

International Journal of Fracture ◽

10.1007/s10704-021-00522-1 ◽

2021 ◽

Author(s):

Bo Yin ◽

Johannes Storm ◽

Michael Kaliske

Keyword(s):

Phase Field ◽

Consistency Condition ◽

Field Method ◽

Anisotropic Elasticity ◽

Phase Field Method ◽

Internal Variables ◽

Crack Model ◽

Material Degradation ◽

Discrete Crack ◽

Deformation State

AbstractThe promising phase-field method has been intensively studied for crack approximation in brittle materials. The realistic representation of material degradation at a fully evolved crack is still one of the main challenges. Several energy split formulations have been postulated to describe the crack evolution physically. A recent approach based on the concept of representative crack elements (RCE) in Storm et al. (The concept of representative crack elements (RCE) for phase-field fracture: anisotropic elasticity and thermo-elasticity. Int J Numer Methods Eng 121:779–805, 2020) introduces a variational framework to derive the kinematically consistent material degradation. The realistic material degradation is further tested using the self-consistency condition, which is particularly compared to a discrete crack model. This work extends the brittle RCE phase-field modeling towards rate-dependent fracture evolution in a viscoelastic continuum. The novelty of this paper is taking internal variables due to viscoelasticity into account to determine the crack deformation state. Meanwhile, a transient extension from Storm et al. (The concept of representative crack elements (RCE) for phase-field fracture: anisotropic elasticity and thermo-elasticity. Int J Numer Methods Eng 121:779–805, 2020) is also considered. The model is derived thermodynamic-consistently and implemented into the FE framework. Several representative numerical examples are investigated, and consequently, the according findings and potential perspectives are discussed to close this paper.

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Phase Field Model for Influence of Edge Dislocation on Precipitation

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.415-417.1482 ◽

2011 ◽

Vol 415-417 ◽

pp. 1482-1485

Author(s):

Chuang Gao Huang ◽

Ying Jun Gao ◽

Li Lin Huang ◽

Jun Long Tian

Keyword(s):

Edge Dislocation ◽

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Field Method ◽

Phase Field Method ◽

Second Phase ◽

Free Energy Function ◽

Phase Nucleation ◽

Good Agreement

The second phase nucleation and precipitation around the edge dislocation are studied using phase-field method. A new free energy function is established. The simulation results are in good agreement with that of theory of dislocation and theory of non-uniform nucleation.

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Grain Refinement by Second Phase Particles under Applied Stress in ZK60 Mg Alloy with Y through Phase Field Simulation

Materials ◽

10.3390/ma11101903 ◽

2018 ◽

Vol 11 (10) ◽

pp. 1903 ◽

Cited By ~ 2

Author(s):

Yuhao Song ◽

Mingtao Wang ◽

Yaping Zong ◽

Ri He ◽

Jianfeng Jin

Keyword(s):

Grain Refinement ◽

Applied Stress ◽

Phase Field ◽

Phase Particle ◽

Alloy System ◽

Second Phase ◽

Recrystallization Process ◽

Time Space ◽

Second Phase Particles ◽

Zk60 Alloy

Based on the principle of grain refinement caused by the second-phase particles, a phase field model was built to describe the recrystallization process in the ZK60 alloy system with Y added under applied stress between temperatures 573 and 673 K for 140 min duration. The simulation of grain growth with second phase particles and applied stress during annealing process on industrial scale on the condition of real time-space was achieved. Quantitative analysis was carried out and some useful laws were revealed in ZK60 alloy system. The second phase particles had a promoting effect on the grain refinement, however the effect weakened significantly when the content exceeded 1.5%. Our simulation results reveal the existence of a critical range of second phase particle size of 0.3–0.4 μm, within which a microstructure of fine grains can be obtained. Applied stress increased the grain coarsening rate significantly when the stress was more than 135 MPa. The critical size of the second phase particles was 0.4–0.75 μm when the applied stress was 135 MPa. Finally, a microstructure with a grain size of 11.8–13.8 μm on average could be obtained when the second phase particles had a content of 1.5% and a size of 0.4–0.75 μm with an applied stress less than 135 Mpa after 30 min annealing at 573 K.

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Modelling Mortgage Insurance Claims Experience: A Case Study

Astin Bulletin ◽

10.2143/ast.24.1.2005084 ◽

1994 ◽

Vol 24 (1) ◽

pp. 97-129 ◽

Cited By ~ 1

Author(s):

Greg Taylor

Keyword(s):

Geographic Location ◽

Real Data ◽

Internal Variables ◽

Functional Dependencies ◽

Data Set ◽

Mortgage Insurance ◽

Section 8 ◽

Sequence Of Events ◽

Claim Frequency ◽

Credit Worthiness

AbstractMortgage insurance indemnifies a mortage lender against loss on default by the borrower. The sequence of events leading to a claim under this type of insurance is relatively complex, depending not only on the credit worthiness of the borrower but also on a number of external economic factors.Prominent among these external factors are the loan to valuation ratio of the insured loan, the disposable income of the borrower, and movements in property values. A broad theoretical model of the functional dependencies of claim frequency and average claim size on these variables is established in Sections 6 and 7. Section 8 fits these models, extended by other “internal” variables such as the geographic location of the mortgaged property, to a real data set.Section 9 compares the fitted model with the data, and finds an acceptable fit despite extreme fluctuations in the claims experience recorded in the data set.

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(In)compressibility and parameter identification in phase field models for capillary flows

Theoretical and Applied Mechanics ◽

10.2298/tam170803009d ◽

2017 ◽

Vol 44 (2) ◽

pp. 189-214 ◽

Cited By ~ 6

Author(s):

M. Dehsara ◽

H. Fu ◽

S.Dj Mesarovic ◽

D.P. Sekulic ◽

M. Krivilyov

Keyword(s):

Phase Field ◽

Triple Line ◽

Slip Boundary Condition ◽

Diffuse Interface ◽

Slip Boundary ◽

Phase Field Models ◽

Capillary Flows ◽

Wide Range ◽

Mobility Parameter ◽

Wetting Process

Phase field (diffuse interface) models accommodate diffusive triple line motion with variable contact angle, thus allowing for the no-slip boundary condition without the stress singularities. We consider two commonly used classes of phase field models: the compositionally compressible (CC) model with compressibility limited to the fluid mix within the diffuse interface, and the incompressible (IC) model. First, we show that the CC model applied to fluids with dissimilar mass densities exhibits the computational instability leading to the breakup of the triple line. We provide a qualitative physical explanation of this instability and argue that the compositional compressibility within the diffuse interface is inconsistent with the global incompressible flow. Second, we derive the IC model as a systematic approximation to the CC model, based on a suitable choice of continuum velocity field. Third, we benchmark the IC model against sharp interface theory and experimental kinetics. The triple line kinetics is well represented by the triple line mobility parameter. Finally, we investigate the effects of the bulk phase field diffusional mobility parameter on the kinetics of the wetting process and find that within a wide range of magnitudes the bulk mobility does not affect the flow.

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