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Natural and artificial chiral materials such as deoxyribonucleic acid (DNA), chromatin fibers, flagellar filaments, chiral nanotubes, and chiral lattice materials widely exist. Due to the chirality of intricately helical or twisted microstructures, such materials hold great promise for use in diverse applications in smart sensors and actuators, force probes in biomedical engineering, structural elements for absorption of microwaves and elastic waves, etc. In this paper, a Timoshenko beam model for chiral materials is developed based on noncentrosymmetric micropolar elasticity theory. The governing equations and boundary conditions for a chiral beam problem are derived using the variational method and Hamilton’s principle. The static bending and free vibration problem of a chiral beam are investigated using the proposed model. It is found that chirality can significantly affect the mechanical behavior of beams, making materials more flexible compared with nonchiral counterparts, inducing coupled twisting deformation, relatively larger deflection, and lower natural frequency. This study is helpful not only for understanding the mechanical behavior of chiral materials such as DNA and chromatin fibers and characterizing their mechanical properties, but also for the design of hierarchically structured chiral materials.

Chemo-mechanical coupling exists in a lot of intelligent materials including hydrogels, biological tissues and other soft materials. These materials are able to respond to external stimulus, such as temperature, chemical concentration, and pH value. In this paper, a one-dimensional theoretical model for chemo-mechanical coupling is proposed for analyzing the uniaxial stress/strain state of coupling materials. Based on the chemo-mechanical coupled governing equation, the displacement function and concentration function are derived and the stress and chemical potential are obtained. It is shown that the present chemo-mechanical theory can characterize the chemo-mechanical coupling behavior of intelligent materials.

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green's functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii's formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.

Bifurcation of limit cycles in two given planar polynomial systems is investigated by using both qualitative analysis and numerical exploration. The investigation is based on detection functions which are particularly effective for the perturbed planar polynomial systems. The study reveals that each of the two systems has 8 limit cycles. By using method of numerical simulation, the distributed orderliness of the 8 limit cycles is observed, and their nicety places are determined. The study also indicates that each of the 8 limit cycles passes the corresponding nicety point. The results presented here are helpful for further investigating the Hilbert's 16th problem.

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.

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green's functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper.

Carbon nanotube fibers can be fabricated by the chemical vapor deposition spinning process. They are promising for a wide range of applications such as the building blocks of high-performance composite materials and micro-electrochemical sensors. Mechanical twisting is an effective means of enhancing the mechanical properties of carbon nanotube fibers during fabrication or by post processing. However, the effects of twisting on the mechanical properties remain an unsolved issue. In this paper, we present a two-scale damage mechanics model to quantitatively investigate the effects of twisting on the mechanical properties of carbon nanotube fibers. The numerical results demonstrate that the developed damage mechanics model can effectively describe the elastic and the plastic-like behaviors of carbon nanotube fibers during the tension process. A definite range of twisting which can effectively enhance the mechanical properties of carbon nanotube fiber is given. The results can be used to guide the mechanical twisting of carbon nanotube fibers to improve their properties and help optimize the mechanical performance of carbon nanotube-based materials.

Special hole elements are presented for analyzing the stress behavior of an isotropic elastic solid containing an elliptical hole. The special hole elements are constructed using the special fundamental solutions for an infinite domain containing a single elliptical hole, which are derived based on complex conformal mapping and Cauchy integrals. During the construction of the special elements, the interior displacement and stress fields are assumed to be the combination of fundamental solutions at a number of source points, and the frame displacement field defined over the element boundary is independently approximated with conventional shape functions. The hybrid finite element model is formulated based on a hybrid functional that provides a link between the two assumed independent fields. Because the fundamental solutions used exactly satisfy both the traction-free boundary conditions of the elliptical hole under consideration and the governing equations of the problems of interest, all integrals can be converted into integrals along the element boundary and there is no need to model the elliptical hole boundary. Thus, the mesh effort near the elliptical hole is significantly reduced. Finally, the numerical model is verified through three examples, and the numerical results obtained for the prediction of stress concentration factors caused by elliptical holes are extremely accurate.

A novel hybrid graded element model is developed in this paper for investigating thermal behavior of functionally graded materials (FGMs). The model can handle a spatially varying material property field of FGMs. In the proposed approach, a new variational functional is first constructed for generating corresponding finite element model. Then, a graded element is formulated based on two sets of independent temperature fields. One is known as intra-element temperature field defined within the element domain; the other is the so-called frame field defined on the element boundary only. The intra-element temperature field is constructed using the linear combination of fundamental solutions, while the independent frame field is separately used as the boundary interpolation functions of the element to ensure the field continuity over the inter-element boundary. Due to the properties of fundamental solutions, the domain integrals appearing in the variational functional can be converted into boundary integrals which can significantly simplify the calculation of generalized element stiffness matrix. The proposed model can simulate the graded material properties naturally due to the use of the graded element in the finite element (FE) model. Moreover, it inherits all the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional FEM and boundary element method (BEM). Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show a good numerical accuracy.

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest's algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.

The present paper develops a new type of hybrid finite element model with regular and special fundamental solutions (also known as Green's functions) as internal interpolation functions for analyzing plane elastic problems in structures weakened by circular holes. A variational functional used in the proposed model is first constructed, and then, the assumed intra-element displacement fields satisfying a priori the governing partial differential equations of the problem under consideration is constructed using a linear combination of fundamental solutions at a number of source points outside the element domain, aswas done in themethod of fundamental solutions. To ensure continuity of fields over inter-element boundaries conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The linkage of these two independent fields and the element stiffness equations in terms of nodal displacements are enforced by the minimization of the proposed variational functional. Special-purpose Green's functions associated with circular holes are used to construct a special circular hole element to effectively handle stress concentration problems without complicated local mesh refinement or mesh regeneration around the hole. The practical efficiency of the proposed element model is assessed via several numerical examples.

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.

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.

Using a recently established liquid crystal model for vesicles, we present a theoretical method to analyze the morphological stability of liquid crystal vesicles in an electric field. The coupled mechanical-electrical effects associated with elastic bending, osmotic pressure, surface tension, Maxwell pressure, as well as flexoelectric and dielectric properties of the membrane are taken into account. The first and second variations of the free energy are derived in a compact form by virtue of the surface variational principle. The former leads to the shape equation of a vesicle embedded in an electric field, and the latter allows us to examine the stability of a given vesicle morphology. As an illustrative example, we analyze the stability of a spherical vesicle under a uniform electric field. This study is helpful for understanding and revealing the morphological evolution mechanisms of vesicles in electric fields and some associated phenomena of cells.

A new type of hybrid finite element formulation with fundamental solutions as internal interpolation functions, named as HFS-FEM, is presented in this paper and used for solving two dimensional heat conduction problems in single and multi-layer materials. In the proposed approach, a new variational functional is firstly constructed for the proposed HFS-FE model and the related existence of extremum is presented. Then, the assumed internal potential field constructed by the linear combination of fundamental solutions at points outside the elemental domain under consideration is used as the internal interpolation function, which analytically satisfies the governing equation within each element. As a result, the domain integrals in the variational functional formulation can be converted into the boundary integrals which can significantly simplify the calculation of the element stiffness matrix. The independent frame field is also introduced to guarantee the inter-element continuity and the stationary condition of the new variational functional is used to obtain the final stiffness equations. The proposed method inherits the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional finite element method (FEM) and boundary element method (BEM), and avoids the difficulty in selecting appropriate terms of T-complete functions used in HT-FEM, as the fundamental solutions contain usually one term only, rather than a series containing infinitely many terms. Further, the fundamental solutions of a problem are, in general, easier to derive than the T-complete functions of that problem. Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show good numerical accuracy and remarkable insensitivity to mesh distortion.

This paper presents our new development of parallel finite element algorithms for elastic-plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic-plastic stress-strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic-plastic problems whose size is much too large for a single workstation becomes possible.

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations (BIE) and solved with the newly developed boundary point method (BPM). The model is closely derived from the concept of the equivalent inclusion of Eshelby tensors. Eigenstrains are iteratively determined for each short-fiber embedded in the matrix with various properties via the Eshelby tensors, which can be readily obtained beforehand either through analytical or numerical means. As unknown variables appear only on the boundary of the solution domain, the solution scale of the inhomogeneity problem with the model is greatly reduced. This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM. The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element (RVE), showing the validity and the effectiveness of the proposed computational modal and the solution procedure.

A theoretical model for surface bone remodeling under electromagnetic loads is proposed in this paper. In the model, surface bone remodeling is assumed to be related to growth factors. Growth factors in latent form in osteocytes are released to the bone fluid after the osteocytes are absorbed by osteoclasts, and then regulate the bone formation process. At the same time, environmental loadings can influence the generation of growth factors. This paper shows how surface bone remodeling is triggered under the influence of growth factors. Based on this hypothesis, a computational model is established that simulates the bone coupling remodeling process, including internal and surface bone remodeling. The effects of various loadings, including electrical and magnetic loadings, are simulated and compared. The interactions between internal and surface bone remodeling are investigated via the numerical method. The results indicate that an electromagnetic field can strongly influence the bone remodeling process and that the remodeling process will be altered after surface bone remodeling is triggered, compared to the sole effect of the internal remodeling process.

The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson's equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition (SVD) technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric (MQ) are fully analyzed to provide useful reference.

This paper presents a hybrid Trefftz (HT) boundary element method (BEM) by using two indirect techniques for mode III fracture problems. Two Trefftz complete functions of Laplace equation for normal elements and a special purpose Trefftz function for crack elements are proposed in deriving the Galerkin and the collocation techniques of HT BEM. Then two auxiliary functions are introduced to improve the accuracy of the displacement field near the crack tips, and stress intensity factor (SIF) is evaluated by local crack elements as well. Furthermore, numerical examples are given, including comparisons of the present results with the analytical solution and the other numerical methods, to demonstrate the efficiency for different boundary conditions and to illustrate the convergence influenced by several parameters. It shows that HT BEM by using the Galerkin and the collocation techniques is effective for mode III fracture problems.

This paper is concerned with the development of Trefftz p-element for twodimentional piezoelectric materials. Solutions in Stroh formalism for transversely isotropic piezoelectric materials are used for the intra-element displacement and electric potentia

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.

Anti-plane electroelastic problems are studied by the Trefftz boundary element method (BEM) in this paper. The Trefftz BEM is based on a weighted residual formulation and indirect boundary approach. In particular the point-collocation and Galerkin techniques, in which the basic unknowns are the retained expansion coefficients in the system of complete equations, are considered. Furthermore, special Trefftz functions and auxiliary functions which satisfy exactly the specified boundary conditions along the slit boundaries are also used to derive a special purpose element with local defects. The path-independent integral is evaluated at the tip of a crack to determine the energy release rate for a mode III fracture problem. In final, the accuracy and efficiency of the Trefftz boundary element method are illustrated by an example and the comparison is made with other methods.

Based on the interpolation technique with the aid of boundary integral equations, a new differential quadrature method has been developed (boundary integral equation supported differential quadrature method, BIE-DQM) to solve boundary value problems over

A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.

Multilayer piezoelectric ceramic displacement actuators are susceptible to cracking in the region near the edge of the internal electrode, which may cause system damage or failure. In this paper, the stress distribution of a multilayer piezoelectric composite is investigated in a working environment and the optimized geometrical configuration of the piezoelectric layer is obtained. The stress distribution in the structure and the stress concentration near the edge of the internal electrode, induced by non-uniform electric field distribution, are analyzed by moiré interferometry experiment and finite element numerical simulation. Based on the above analysis, two optimized geometrical models are presented for the purpose of geometrical configuration selection, with which stress concentration can be reduced significantly while the feasibility of the machining process and the basic structural functions occurring in the conventional model are retained. The numerical results indicate that the maximum stress in the optimized models is effectively diminished compared to the conventional model. For instance, the peak value of the principal stress in the optimized model II is 93.1% smaller than that in the conventional model. It is proved that stress concentration can be effectively relaxed in the latter of the two optimized models and thus the probability of fracture damage can be decreased.

Applications of the Trefftz finite element method to anti-plane electroelastic problems are presented in this paper. A dual variational functional is constructed and used to derive Trefftz finite element formulation. Special trial functions which satisfy