Orthotropic bone remodeling: case of plane stresses

Cancellous bone is constituted by a porous solid matrix filled with fluid. Matrix microstructure gives bone most of its mechanical strength properties. In our macroscopic approach, bone is seen as a continuous medium with a local (at our scale) time-dependent linearly elastic orthotropic behavior. Remodeling consists, by matrix material apposition or resorption, in microstructure modifications in order to optimize its mechanical characteristics. The proposed model is built on a time iterative procedure where the compliance tensor evolves such that, depending on the applied stresses, principal strains tend to fall within an admissible domain. The suggested remodeling laws in this work modify the elasticity “constants” as well as the orthotropy directions. The first results presented here correspond to the plane stresses case.     

A fully anisotropic hierarchical hybrid cellular automaton algorithm to simulate bone remodeling

Computational models of the bone remodeling process have been utilized to further our understanding of the adaptation of bone architecture to changes in its mechanical environment. The hierarchical hybrid cellular automata (HHCA) algorithm is a multi-scale approach for the simulation of the adaptation of bone. Currently, this remodeling algorithm utilizes the apparent material properties of the trabecular architecture. The objective of this work is to increase the fidelity of the HHCA algorithm by incorporating the local anisotropic properties of these structures. Preliminary analyses display improved efficiency and a more consistent material distribution when incorporating anisotropic properties into the HHCA methodology.     

Effect of magnetic field on poroelastic bone model for internal remodeling

This paper studies the effects of the magnetic field and the porosity on a poroelastic bone model for internal remodeling. The solution of the internal bone remodeling process induced by a magnetic field is presented. The bone is treated as a poroelastic material by Biot’s formulation. Based on the theory of small strain adaptive elasticity, a theoretical approach for the internal remodeling is proposed. The components of the stresses, the displacements, and the rate of internal remodeling are obtained in analytical forms, and the numerical results are represented graphically. The results indicate that the effects of the magnetic field and the porosity on the rate of internal remodeling in bone are very pronounced.     

Un modèle viscoélastique de remodelage osseux : approche unidimensionnelleA viscoelastic model of bone remodeling: unidimensional approach

A bone remodeling model taking into account the viscoelastic properties of the material is proposed. The nonlinear equations governing the evolution of the bone apparent density are solved by a finite difference method in the unidimensional case of a n-unit elements model. The results show the effects of the viscous damping on the structure for a controlled mechanical loading. To cite this article: S. Baïotto, M. Zidi, C. R. Mecanique 332 (2004).     

Parametric study of control mechanism of cortical bone remodeling under mechanical stimulus

The control mechanism of mechanical bone remodeling at cellular level was investigated by means of an extensive parametric study on a theoretical model described in this paper. From a perspective of control mechanism, it was found that there are several control mechanisms working simultaneously in bone remodeling which is a complex process. Typically, an extensive parametric study was carried out for investigating model parameter space related to cell differentiation and apoptosis which can describe the fundamental cell lineage behaviors. After analyzing all the combinations of 728 permutations in six model parameters, we have identified a small number of parameter combinations that can lead to physiologically realistic responses which are similar to theoretically idealized physiological responses. The results presented in the work enhanced our understanding on mechanical bone remodeling and the identified control mechanisms can help researchers to develop combined pharmacological-mechanical therapies to treat bone loss diseases such as osteoporosis.     

A mathematical model of cortical bone remodeling at cellular level under mechanical stimulus

A bone cell population dynamics model for cortical bone remodeling under mechanical stimulus is developed in this paper. The external experiments extracted from the literature which have not been used in the creation of the model are used to test the validity of the model. Not only can the model compare reasonably well with these experimental results such as the increase percentage of final values of bone mineral content (BMC) and bone fracture energy (BFE) among different loading schemes (which proves the validity of the model), but also predict the realtime development pattern of BMC and BFE, as well as the dynamics of osteoblasts (OBA), osteoclasts (OCA), nitric oxide (NO) and prostaglandin E₂ (PGE₂) for each loading scheme, which can hardly be monitored through experiment. In conclusion, the model is the first of its kind that is able to provide an insight into the quantitative mechanism of bone remodeling at cellular level by which bone cells are activated by mechanical stimulus in order to start resorption/formation of bone mass. More importantly, this model has laid a solid foundation based on which future work such as systemic control theory analysis of bone remodeling under mechanical stimulus can be investigated. The to-be identified control mechanism will help to develop effective drugs and combined nonpharmacological therapies to combat bone loss pathologies. Also this deeper understanding of how mechanical forces quantitatively interact with skeletal tissue is essential for the generation of bone tissue for tissue replacement purposes in tissue engineering.     

A theoretical model of magnetoelastic bucking for soft ferromagnetic thin plates

As an essential model of magnetoelastic interaction between magnetic field and mechanical deformation, the study on magnetoelastic buckling phenomenon of soft ferromagnetic plates in a magnetic environment has been conducted. One of the key steps for the theoretical prediction of the critical magnetic field is how to formulate magnetic force exerted on the magnetized medium. Till today, the theoretical predictions, from theoretical models in publications, of the magnetoelastic buckling of ferromagnetic cantilevered beam-plate in transverse magnetic field are all higher than their experimental data. Sometimes, the discrepancy between them is as high as 100%. In this paper, the macroscope formulation of the magnetic forces is strictly obtained from the microscope Amperion current model. After that, a new theoretical model is established to describe the magnetoelastic buckling phenomenon of ferromagnetic thin plates with geometrically nonlinear deformation in a nonuniform transverse magnetic field. The numerical method for quantitative analysis is employed by combining the finite elemental method for magnetic fields and the finite difference method for deformation of plates. The numerical results obtained from this new theoretical model show that the theoretical predictions of critical values of the buckling magnetic field for the ferromagnetic cantilevered beam-plate are in excellent agreement with their experimental data. By the way, the region of applicability to the Moon-Pao's model, or the couple model, is checked by quantitative results. This project was supported in part by the National Natural Science Foundation of China and the Foundation of the SEdC of China for Returned Chinese Scholars from Abroad.     

A Theoretical Model for the Simulation of Microdamage Accumulation and Repair in Compact Bone*

This paper describes a new theoretical approach to bone microdamage, in which a population of cracks is explicitly modelled. A given sample of bone is assumed to contain a certain number of cracks, whose growth characteristics are described with an equation containing stochastic variables to create statistical differences from one crack to another. This type of model allows us to predict a wide variety of data. The present paper illustrates the different types of prediction which can be made, including: (i) standard damage parameters such as the number and length of cracks and the reduction in stiffness; (ii) fatigue test data such as the number of cycles to failure as a function of stress level, including scatter; (iii) effects due to the living system, including repair, remodelling and adaptation. A useful feature of the model is our ability to examine the statistics of the crack population in detail to find, for example, the number of cracks which are potentially dangerous as opposed to those which are dormant, and to investigate the reasons for increased crack numbers in the bones of older people. The potential also exists to use the model to investigate different theories of bone remodelling and adaptation.     

A simple model for myocardial changes in a failing heart

In a simplified setting, a multi-network model for remodeling in the left ventricle (LV) is developed that can mimic various pathologies of the heart. The model is an extension of the simple model introduced by Nardinocchi and Teresi [9], Nardinocchi et al. [10], [11] that results in an algebraic relation for LV pressure–volume–contraction. We considered two networks, the original tissue and a new tissue, each of which has its own volume fraction, stress-free reference configuration, elastic properties, and contractility. This is used to explore the consequences of microstructural changes in the muscle tissue on LV function in terms of the pressure–volume loop during a single cardiac cycle. Special attention is paid to the stroke volume, which is directly related to cardiac output, and changes in LV wall stress caused by various disease states, including wall thinning (dilated cardiomyopathy), wall thickening (hypertrophic cardiomyopathy), contractility degradation, and stiffness changes (scarring). Various scenarios are considered that are of clinical relevance, and the extent and nature of remodeling that could lead to heart failure are identified.     

A theoretical model of matrix cracking in composite laminates under thermomechanical loading

A theoretical approach is presented for analyzing the ply cracking in general symmetric laminates subjected to any combination of in-plane mechanical loading and uniform temperature changes. The equivalent constraint model proposed by the authors in a previous work is used to account for the cracking interaction between laminae in the laminates. By using a superposition scheme and the stress field solutions the energy release rate for a ply cracking is explicitly expressed as a function of stiffness reduction parameters of the laminates. The ratio of mode I to mode II is introduced for construction of the fracture criterion. The effects of the laminate parameters and the crack spacing on the energy release rate and the mode mixity are illustrated. Finally, the model is used to predict the thermomechanical load for the first-ply-cracking. Project supported by the National Natural Science Foundation of China (No. 19972076) and the Germen Research Foundation (DFG).     

Energy dissipation mechanism and experiment of particle dampers for gear transmission under centrifugal loads

As a passive means of vibration reduction, particle damping is mainly applied to the horizontal or vertical steady field. However, it is seldom applied to centrifugal fields. Under high speed and heavy loading, the vibration of tooth surfaces of gear transmissions becomes more severe shortening gear service life and augmenting noise. Under centrifugal loading, the particle system exhibits different characteristics, for example, particles are extruded at the end farthest from the center. We investigated gears with drilled via holes filled with damping particles. Using the discrete-element method, we developed an energy dissipation model for the particle system accounting for friction and inelastic collisions. Energy dissipation and damping characteristics of this system were analyzed. Experiments were also conducted with the gear system having different particle filling rates. The results show that this filling rate is an important parameter associated with particle damping in a centrifugal field. An unsuitable filling rate would significantly reduce damping effectiveness. With changes in rotation speed and load, the gear transmission system has different optimal filling rates. The results provide guidelines for the application of particle damping in centrifugal fields of gear transmissions.     

A theoretical model for tissue growth in confined geometries

It is known that cells proliferate and produce extracellular matrix in response to biochemical and mechanical stimuli. Constitutive models considering these phenomena are needed to quantitatively describe the process of tissue growth in the context of tissue engineering and regenerative medicine. In this paper we re-examine the theoretical framework provided by Ambrosi and Guana (2007) and Ambrosi and Guillou (2007). We show how a volumetric growth rate term can be obtained (both in a large and small strain setting), which is consistent with the laws of thermodynamics and then apply the model to a simple geometry of tissue growth within a circular pore. The model, despite its simplicity, is comparable with experimental measurements of tissue growth and highlights the contribution of the mechanical stresses produced during tissue growth on the growth rate itself.     

旋转弹体表面红外辐射干扰补偿模型

针对地球红外辐射姿态测试方法中弹体表面红外辐射干扰较大的问题,建立旋转弹体自身红外辐射补偿模型,从而提高弹体姿态测量精度。首先,采用瞬态热平衡微分方程推导出旋转弹体表面温度变化曲线,获取弹体自身红外辐射亮度。然后,根据地球红外辐射及其在大气中的传播规律,建立了弹体自身红外辐射补偿数学模型。最终结合红外传感器输出和所推导的补偿模型,估算红外辐射补偿参数。结果表明补偿后姿态角解算误差显著降低,俯仰角和横滚角解算误差最终分别保持在±0.2°和±0.4°以内。本方法对姿态测量误差的减小具有显著效果,补偿方法简单有效,对旋转弹体的红外姿态测试技术的研究具有借鉴意义。     

A fluid flow model in the lacunar-canalicular system under the pressure gradient and electrical field driven loads

The lacunar-canalicular system (LCS) is acknowledged to directly participate in bone tissue remodeling. The fluid flow in the LCS is synergic driven by the pressure gradient and electric field loads due to the electro-mechanical properties of bone. In this paper, an idealized annulus Maxwell fluid flow model in bone canaliculus is established, and the analytical solutions of the fluid velocity, the fluid shear stress, and the fluid flow rate are obtained. The results of the fluid flow under pressure gradient driven (PGD), electric field driven (EFD), and pressure-electricity synergic driven (P-ESD) patterns are compared and discussed. The effects of the diameter of canaliculi and osteocyte processes are evaluated. The results show that the P-ESD pattern can combine the regulatory advantages of single PGD and EFD patterns, and the osteocyte process surface can feel a relatively uniform shear stress distribution. As the bone canalicular inner radius increases, the produced shear stress under the PGD or P-ESD pattern increases slightly but changes little under the EFD pattern. The increase in the viscosity makes the flow slow down but does not affect the fluid shear stress (FSS) on the canalicular inner wall and osteocyte process surface. The increase in the high-valent ions does not affect the flow velocity and the flow rate, but the FSS on the canalicular inner wall and osteocyte process surface increases linearly. In this study, the results show that the shear stress sensed by the osteocyte process under the P-ESD pattern can be regulated by changing the pressure gradient and the intensity of electric field, as well as the parameters of the annulus fluid and the canaliculus size, which is helpful for the osteocyte mechanical responses. The established model provides a basis for the study of the mechanisms of electro-mechanical signals stimulating bone tissue (cells) growth.

     

A theoretical model for electroelastic analysis in piezoelectric fibre push-out test

A theoretical model is presented to study the elastic deformation process and frictional sliding behavior in single piezoelectric fibre push-out tests. Based on the theoretical model and some necessary simplifications, stress and electric fields are obtained for push-out tests of a circular piezoelectric fibre embedded in an elastic matrix. Numerical results of a piezoelectric fibre/expoxy matrix system are presented to verify the proposed formulation. The study shows that there is a significant effect of the piezoelectric parameter and embedded fibre length on stress transfer, electric field distribution and load-displacement curve of the frictional sliding process. This study also indicates that the piezoelectric effect has a distinct influence on the mechanical behavior and properties of the interface in a fibre/matrix system.     

A theoretical mode for the instability of turbulent boundary layer over compliant surface

A theoretical model for the instability of turbulent boundary layer over compliant surfaces is described. The investigation of instability is carried out from a time-asymptotic space-time perspective that classifies instabilities as either convective or absolute. Results are compared against experimental observations of surface waves on elastic and viscoelastic compliant layers.     

A theoretical model of reversible adhesion in shape memory surface relief structures and its application in transfer printing

Transfer printing is an important and versatile tool for deterministic assembly and integration of micro/nanomaterials on unusual substrates, with promising applications in fabrication of stretchable and flexible electronics. The shape memory polymers (SMP) with triangular surface relief structures are introduced to achieve large, reversible adhesion, thereby with potential applications in temperature-controlled transfer printing. An analytic model is established, and it identifies two mechanisms to increase the adhesion: (1) transition of contact mode from the triangular to trapezoidal configurations, and (2) explicit enhancement in the contact area. The surface relief structures are optimized to achieve reversible adhesion and transfer printing. The theoretical model and results presented can be exploited as design guidelines for future applications of SMP in reversible adhesion and stretchable electronics.     

A modified basis reduction method for limited kinematic hardening shakedown analysis under complex loads

A novel extension of the basis reduction method for kinematic hardening shakedown problem is presented. Firstly, the basis reduction method is implemented based on the modified Newton–Raphson (N-R) method. Then a new technique for the construction of back stress field is introduced, where the simultaneous influence of multiple load corners in shakedown is taken into consideration. Finally, two typical numerical examples are investigated. The results compared with previous works in literatures demonstrated that the proposed method is accurate and the performance in reducing of computation time is significant.     

Phase field model for coupled displacive and diffusive microstructural processes under thermal loading

A non-isothermal phase field model that captures both displacive and diffusive phase transformations in a unified framework is presented. The model is developed in a formal thermodynamic setting, which provides guidance on admissible constitutive relationships and on the coupling of the numerous physical processes that are active. Phase changes are driven by temperature-dependent free-energy functions that become non-convex below a transition temperature. Higher-order spatial gradients are present in the model to account for phase boundary energy, and these terms necessitate the introduction of non-standard terms in the energy balance equation in order to satisfy the classical entropy inequality point-wise. To solve the resulting balance equations, a Galerkin finite element scheme is elaborated. To deal rigorously with the presence of high-order spatial derivatives associated with surface energies at phase boundaries in both the momentum and mass balance equations, some novel numerical approaches are used. Numerical examples are presented that consider boundary cooling of a domain at different rates, and these results demonstrate that the model can qualitatively reproduce the evolution of microstructural features that are observed in some alloys, especially steels. The proposed model opens a number of interesting possibilities for simulating and controlling microstructure pattern development under combinations of thermal and mechanical loading.