International Journal of Applied Mechanics

In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods.

Measurements of various effective stiffnesses have been suggested as non-invasive biomechanical methods to assess healing status of a fractured long bone. This paper has compared the sensitivities of five assessment methods for fracture healing of long bones: torsion, compression, 3-point bending, 4-point bending and cantilever bending. A fractured human femur is modeled by an Euler beam and Castigliano’s theorem is used to obtain the effective stiffnesses of the fractured bone. The variations of the effective stiffnesses of the fractured bone with the healing status of the callus have been investigated. The healing process of the callus is represented by gradual increases of the Young’s modulus. The callus is divided into multiple regions. The narrowing process of the fracture gap is characterized by assigning different Young’s modulus for different regions. Our findings showed that torsional, compressional, and bending stiffnesses all perform similarly with respect to the healing process. Before the rapid increase, all effective stiffness curves show a substantial creeping stage which corresponds to the narrowing process of the fracture gap. A higher value of them indicates bony bridging of healing callus. The effective stiffnesses for cantilever bending and 3-point bending are more sensitive to the fracture location, while torsional and compressional stiffness are independent from the fracture location.

To avoid procedural accidents during/after oral implant surgery in a jawbone (e.g., perforation of the lingual side due to an inadequate drilling angle or scratching the mandibular canal), it is important for dentists to develop skilled drilling techniques. We have developed a stochastic multi-scale prediction of apparent elastic moduli of the jawbone with modified first-order perturbation-based stochastic homogenization in which individual differences of drilling properties relating to bone quality can be evaluated with a force-sensitive device. To establish drilling properties in various types of jawbone and the perforation case, and for drill diameters, we propose an original computational method that avoids procedural accidents by defining the drilling angle as a parameter and quantifying individual differences and the inner distribution of bone quality in several jawbones as a drilling force. Three samples demonstrated that drilling forces changed depending on the trabecular structure, with the drilling force changing little around the mandibular canal and increasing just before perforation. Drilling force also depended on the drill diameter but did not always match its size. Estimated drilling forces of the expected value and 50% and 90% probability results suggested prominent differences in drilling properties. This information enables dentists to develop a skillful surgical technique.

The compliance matrix for a general anisotropic material is usually expressed in an arbitrarily chosen coordinate system, which brings some confusion or inconvenience in identifying independent elastic material constants and comparing elastic properties between different materials. In this paper, a unique stiffest orientation-based standardized compliance matrix is established, and 18 independent elastic material constants are clearly shown. During the searching process for the stiffest orientation, it is interesting to find from our theoretical analysis and an example that a material with isotropic tensile stiffness does not definitely possess isotropic elasticity. Therefore, the ratio between the maximum and minimum tensile stiffnesses, although widely used, is not a correct measure of anisotropy degree. Alternatively, a simple and correct measure of anisotropy degree based on the maximum shear-extension coupling coefficient in all orientations is proposed. However, for a two-dimensional constitutive relation, both the stiffness ratio and the shear-extension coupling coefficient can be adopted as proper measures of anisotropy degree.

The strain energy release rate $G$ is of prime importance in composite materials fracture mechanics. In order to experimentally and numerically evaluate this parameter in the case of quasi-isotropic and angle-ply (AP) woven-ply thermoplastic (TP) laminates, single edge notched (SEN) specimens have been subjected to monotonic tensile loading at $T>T_g$ when the toughness and the viscous behavior of the (TP) matrix are exacerbated. From the simulation standpoint, a particular attention was paid to the type of meshing as well as its refinement in the vicinity of the crack tip where the triaxiality rate leads to significant stress concentrations. For this purpose, a linear spectral viscoelastic and a generalized Norton-type viscoplastic models have been used. A comparison between two types of meshing (radiant and concentric) has been conducted. Both types of meshing allow us to define crowns in order to represent the surface of the integration ring around the crack tip. These crowns are necessary to evaluate the strain energy release rate $G_I$ in opening mode using $G_{\theta }$-integral computation. Both overstress and overstrain profiles near the crack tip were investigated and validated using theoretical stress fields derived from the linear elastic fracture mechanics (LEFM) framework and overstrain fields obtained from digital image correlation (DIC) to verify the model’s ability to provide accurate mechanical fields at singularity zones.

For the interpolating element-free Galerkin (IEFG) method set up by combining the shape function constructed with interpolating moving least-squares (IMLS) approximation and Galerkin integral weak form of partial differential equation, we show that the task of accuracy analysis of the IEFG method in solving transient heat conduction problems can be reduced to the estimation of errors in the procedure of IMLS approximation and the integral weak form of heat conduction equation. Following this practice, accuracy analysis of IEFG method is derived. To illustrate our results, numerical tests are discussed by two examples and give an optimal convergence rate with respect to two types of norm.

The present work investigates the effects of the curvature terms in the three-dimensional (3D) equilibrium equations used for the free vibration analysis of functionally graded material (FGM) structures. The 3D equilibrium equations have been written in general orthogonal curvilinear coordinates which are valid for spherical shells. They automatically degenerate in those for cylindrical shells and plates considering one of the two radii of curvature and both radii of curvature equal to infinite, respectively. The approximation of curvature terms in the 3D equilibrium equations has been evaluated by means of frequency analyses. Results obtained via 3D equilibrium equations with exact geometry have been compared with those calculated via 3D equilibrium equations written with the approximation of the curvature terms. The effects of the curvature approximations depend on the thickness and curvature of the structures, on the materials, lamination sequences and FGM laws, on the frequency orders and vibration modes. The resulting system of second order partial differential equations has been reduced into a system of first order partial differential equations redoubling the variables. Therefore, the exponential matrix method has been employed using a layer wise approach. The final 3D equations have been solved in exact form considering harmonic displacement components and simply supported structures. The approximation of the curvature terms has been introduced in the 3D equilibrium shell equations. For numerical reasons, interlaminar continuity conditions and the top and bottom boundary and loading conditions have been written including the exact geometry. The introduction of curvature approximations only in the equilibrium equations is sufficient to obtain an exhaustive qualitative analysis of the importance of curvature terms in the free vibration problems for FGM structures.

In this paper, the compression of an isolated cell by two rigid indenters is analyzed. The neo-Hookean model is employed to characterize the hyperelastic behavior of biological cells. Owing to the greatly increased ratio between surface energy density and elastic modulus, surface energy plays important roles in the mechanical performance of biological cells. Using the dimensional analysis method and a finite element approach incorporating surface energy, we study the elastic compression of hyperelastic cells at finite deformation and give the explicit relations of contact radius and indent depth depending on compressive load. Our results reveal that surface energy obviously influences both the local deformation and the overall responses of hyperelastic cells at finite deformation. The obtained results are useful to determine the elastic properties of biological cells from indent-depth curves accurately.

Due to their lightness, the aerospace structures are vulnerable to vibrations. Their amplitudes need to be mitigated through damping devices. The aim of this paper is to control the tightening force of a bolted joint in order to improve the vibration reduction. The work focuses on a single vibration mode modeled by two degrees of freedom coupled with a Jenkins’ cell. Then two control strategies are proposed in order to reduce the harmonically forced vibrations and transient vibrations. The first control law uses a constant optimum clamping for each excitation level. The second law consists in a sinusoidal variation of the clamping force adapted to each level and frequency of excitation. Finally, in the third section, the control law is modified in order to reduce transient vibrations. For each control law, the performances of the model are discussed according to the aimed application.

Bi-modulus materials have different moduli in tension and compression and the stress–strain relation depends on principal stress that is unknown before displacement is determined. Establishment of variational principle is important for mechanical analysis of materials. First, parametric variational principle (PVP) is proposed for static analysis of bi-modulus materials and structures. A parametric variable indicating state of principal stress is included in the potential energy formulation and the nonlinear stress–strain relation is evolved into a linear complementarity constraint. Convergence of finite element analysis is thus improved. Then the proposed variational principle is extended to a dynamic problem and the dynamic equation can be derived based on Hamilton’s principle. Finite element analysis of nacreous bio-composites is performed, in which a unilateral contact behavior between two hard mineral bricks is modeled using the bi-modulus stress–strain relation. Effective modulus of composites can be determined numerically and stress mechanism of “tension–shear chain” in nacre is revealed. A delayed effect on stress propagation is found around the “gaps” between mineral bricks, when a tension force is loaded to nacreous bio-composites dynamically.

This paper studies the thermoelastic damping in microbeams considering the couple stress theory with microstructure. This theory includes the microinertia effects, coming from the kinetic energy due to the velocity gradient through the differential macroelements. A Galerkin-based reduced order model and complex frequency approach have been used to determine the quality factor. For a gold microbeam as a case study, the obtained results for different ambient temperatures, beam lengths and thicknesses are compared to those obtained using the classic theory of elasticity. The comparison has been made for different values of the length scale parameter. The effects of the microinertia term on the magnitude of the thermoelastic damping have also been investigated and shown that for which conditions these effects are significant.

A coarse-grained molecular dynamics (MD) simulation approach based on a widely used fuse model is developed to study the mechanical behaviors of hierarchical brick and mortar bio-composites made from hard minerals and soft polymers. Massively parallel MD simulations are performed to investigate the toughness enhancement and the effect of stochastic variations in brick strength in representative bio-composites. Our simulations indicate that the hierarchical structure of bio-composites not only plays a key role in toughness optimization, but also reduces the sensitivity of the structure to biomineral imperfections. This work demonstrates a simple and efficient simulation platform for designing novel biomimetic materials.