Nonhomogeneous Material Properties Research Articles

A Domain-Independent Interaction Integral for Dynamic Fracture in Nonhomogeneous Magneto-Electro-Elastic Materials

In practice, magneto-electro-elastic (MEE) equipment are inevitably subjected to impact loading during service, resulting in fracture failure of the devices. This paper proposes a dynamic domain-independence interaction integral (DII-integral) to solve the dynamic intensity factors (IFs) at the crack tip of MEE materials. Here, this is a new expression of the interaction integral (I-integral) for dynamic crack study of nonhomogeneous MEE media, and we further extend it to MEE materials containing complex interfaces. Domain-independent property of the I-integral is theoretically demonstrated and excellent numerical results (relative deviation<1%) obtained from the various integration domains again verify the domain-independence for nonhomogeneous and multi-interface material properties without considering the material continuity requirements. Through the combination of the DII-integral and extended finite element method (XFEM), a good agreement is observed by comparing with the published data subjected to different MEE impact loadings. Typical examples reveal that the magnitude of dynamic mode-I stress intensity factor (SIF), dynamic electric displacement intensity factor (EDIF) and dynamic magnetic induction intensity factor (MIIF) decrease with the increase of the poling angle of the MEE material, while the dynamic mode-II SIF shows the opposite change. The peaks of dynamic EDIF and MIIF are more sensitive to the distance between the parallel cracks than the SIF, and all IFs are smaller than single crack due to the shielding effect. Finally, the significant difference on the dynamic IFs is observed with various material discontinuities.

FEniCS implementation of the Virtual Fields Method (VFM) for nonhomogeneous hyperelastic identification

It is of great significance to identify the nonhomogeneous distribution of material properties in human tissues for different clinical and medical applications. This leads to the requirement of solving an inverse problem in elasticity. The Virtual Fields Method (VFM) is a rather recent inverse method with remarkable computational efficiency compared with the optimization-based methods. In this study, we aim to identify nonhomogeneous hyperelastic material properties using the VFM. We propose two novel algorithms, RE-VFM and NO-VFM. In RE-VFM, the solid is partitioned in different regions and the elastic properties of each region are determined. In NO-VFM, the distribution of elastic properties is completely reconstructed through the inverse problem without partitioning the solid. As the VFM requires to use virtual fields, we proposed an efficient way to construct them and implemented the approach in the FEniCS package. We validated the proposed methods on several examples, including a bilayer structure, a lamina cribosa (LC) model and a cube model embedded with a spherical inclusion. The numerical examples illustrate the feasibility of both RE-VFM and NO-VFM. Notably, the spatial variations of the Young's modulus distribution can be recovered accurately within only 5 iterations. The obtained results reveal the potential of the proposed methods for future clinical applications such as estimating the risk of vision loss related to glaucoma and detecting tumors.

Deep learning in frequency domain for inverse identification of nonhomogeneous material properties

The inverse identification of nonhomogeneous material properties from measured displacement/strain fields, especially when noise exists, is crucial for both engineering and material science. The conventional physics-based solutions either require time-consuming iterative calculations, or are sensitive to noise. While the new machine learning methods either need excess data for high-dimensional matchups, or mainly apply to case-by-case analyses with informed physics. In this paper, to solve the complex matchup between the measured displacement/strain fields and the randomly distributed modulus field rapidly and robustly, a novel method of deep learning in frequency domain is proposed, with discrete cosine transform (DCT) to achieve frequency domain transformation as well as dimensionality reduction and convolutional neural network (CNN) to implement learning in frequency domain. Results show that our method not only has high prediction accuracy on zero-noise samples (with L1-error of 4.249%) but also presents great robustness to noise (with L1-error of 5.085% on large-noise samples). Besides, by our method, only one-time training on a dataset with mixed noise is basically enough to deal with arbitrary levels of noise (with L1-errors below 5.202%), improving the efficiency significantly in practical applications. Moreover, our method can be directly transferred to neighbor sampling spaces with different sampling points, showing a great generalization. The study provides a powerful approach for inverse identification of material properties and promises for wide applications such as real-time elastography and high-throughput non-destructive evaluation techniques.

Fully Coupled Vibrations of Cable-Harnessed Beams with a Non-Periodic Wrapping Pattern

Power- and signal- cable attachments have a significant impact on the vibrations of space structures. Recent works show the importance of having an analytical model to gain physical insight into the influence of cabling on the dynamics of host structures. The models in the literature focus mainly on pure bending vibrations and ignore the effect of coupling between different coordinates. Recently, the authors demonstrated the importance of modeling the coupling effects in cable-harnessed (CH) beams with straight and periodic wrapping patterns. In real-life situations, the cable attachment patterns are mostly non-periodic, and the cables are also attached to host structures that consist of a combination of several harness elements of same (homogenous) or different (non-homogenous) material properties. Hence, the fully coupled vibration model developed in this article is the first to analyze the vibrations of homogenous and non-homogenous CH beams with non-periodic wrapping patterns. The Frequency Response Functions (FRFs) of the developed model are compared with experiment FRFs in the case of the homogenous non-periodic wrapping pattern. The study shows that the coupling effects are pronounced in non-periodic wrapped CH beams, and the advantage of developing the coupled model over the decoupled model is shown through experimental validation.

Time Period of Thermal-Induced Vibration of Skew Plate with Two-Dimensional Circular Thickness

In this study, two-dimensional circular thickness effects on time period of skew plate made with nonhomogeneous material properties are computed in variable temperature environment. Clamped C , simply supported S , and free-edge F conditions are the various combinations considered in this work. One-dimensional circular density and two-dimensional parabolic temperature variation on the plate are considered due to the nonhomogeneous nature of the material and variable temperature environment. Rayleigh–Ritz method is used to compute the time period for the first two modes of vibration. A convergence study of frequency modes of skew plate on various edge conditions has also been performed. The present work has been compared with the frequency modes of parallelogram and rectangle plate on various edge conditions. The objective of the study is to present a numerical data of time period as well as to show how one can control the frequencies of plate by taking an appropriate variation in plate parameters.

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.

Interaction integral method for crack-tip intensity factor evaluations of magneto-electro-elastic materials with residual strain

Magneto-electro-elastic (MEE) materials generally consist of piezoelectric and piezomagnetic constituent phases so that the residual strain often occurs during the manufacturing process. The distributed residual strain and complex interfaces between constituents cause a great challenge to the fracture analysis of MEE materials. Considering the effect of the residual strain, this paper develops an interaction integral (I-integral) method for the extraction of the fracture parameters of an impermeable crack in nonhomogeneous MEE materials. The I-integral method has several merits: 1) it can decouple stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF); 2) the I-integral is independent of the size of the integration domain, even when the integration domain contains nonhomogeneous and discontinuous material properties. Combined with the extended finite element method (XFEM), the I-integral is adopted to investigate several crack problems subjected to homogeneous, nonhomogeneous and discontinuous residual strains. The I-integral values calculated by different integration domains agree well with each other (relative deviation < 1%) for nonhomogeneous and discontinuous MEE properties. Then, single-crack and multi-crack problems are studied to show the influences of residual strain distributions on the intensity factors. Homogeneous residual strain affects the SIFs dramatically but has a slight influence on the EDIF and MIIF. By comparison, nonhomogeneous residual strains affect the SIFs, EDIF and MIIF significantly. In addition, the residual strain gradients affect the variations of intensity factors with crack location extremely.

Thermoelastic response of a nonhomogeneous elliptic plate in the framework of fractional order theory

This article aims to examine the thermoelastic problem for a medium with nonhomogeneous material properties within a finite-length elliptical plate. An analytical method is established under the assumption that the thermomechanical material inhomogeneity obeys the product form of power and exponential functions through the plate's thickness. At the same time, calorific capacity and Poisson's ratio are assumed to be constant. The fractional time derivative heat conduction differential equation with an internal heat source is solved using integral transformation technique in terms of generalized Laguerre polynomial, and Mathieu function in the Laplace domain. The Gaver–Stehfest approach is used to invert Laplace domain outcomes. The convergence of infinite series solutions has been discussed. In the specific case, the elliptic region that degenerates into the problem of the circular area by adding limiting conditions is also considered. As a numerical example, the temperature, displacement and thermal stress distributions are calculated, taking into account influence of inhomogeneity, and the numerical results are graphically demonstrated.

Sustainable Architecture Creating Arches Using a Bamboo Grid Shell Structure: Numerical Analysis and Design

Bamboo is known as a sustainable alternative for green building design, and it has been tied culturally to some regions around the world. However, bamboo’s structural strength for different design scenarios needs more investigation before it is widely adopted. Timber and bamboo have similar mechanical properties, but the latter is a repaid, renewable, sustainable, disaster-resilient system and is versatile, which has more advantages for construction purposes. Natural bamboo and its derivatives have been considered as a high-demand green and environmentally responsible alternative construction material, and this interest is increasing globally. Because of the rapid growth rate and large developing area, it is more useable than the limited timber resource. However, natural bamboo has an anisotropic and nonhomogeneous material property, which varies in multiple directions. There is limited engineering data and investigation of bamboo material and its use in and impact on construction. In this study, three different bamboo models were analyzed by Strand7; each of them had different features in structure and in the major construction material. A new model was proposed by improving the three given structures and was maximized in the mechanical capacity. Some design guidelines were proposed based on the analysis and comparison of different bamboo structures. The model will replace natural bamboo with bamboo scrimber, which is an engineered bamboo derivative that has more uniform material properties.

Precise residual stress profile in ion-exchanged silicate glass by modified contour method

In this work, the contour method is modified and used to precisely estimate the residual stress profiles in ion-exchanged aluminosilicate glasses, including the effects of non-homogeneous material properties. The method requires the precise measurements of the fracture surface’s topography and exploits fractographic features to mate and accurately determine the out-of-plane deformation after fracture. Residual stresses are computed from the out-of-plane deformation through a weight function approach rather than by numerical methods, as weight functions can easily account for concentration-dependent material properties. The stress profiles in aluminosilicate glass with three different molten salt soaking times were calculated and compared with the approximate profiles measured by optical waveguide. It was found that the optical waveguide’s estimations under-predicts the surface stress by as much as 10–30 %, likely due to the assumption that the stress optic coefficient is unaffected by the K+↔Na+ exchange.

The Effect of Porosity on Elastic Stability of Toroidal Shell Segments Made of Saturated Porous Functionally Graded Materials.

This research deals with the stability analysis of shallow segments of the toroidal shell made of saturated porous functionally graded (FG) material. The nonhomogeneous material properties of porous shell are assumed to be functionally graded as a function of the thickness and porosity parameters. The porous toroidal shell segments with positive and negative Gaussian curvatures and nonuniform distributed porosity are considered. The nonlinear equilibrium equations of the porous shell are derived via the total potential energy of the system. The governing equations are obtained on the basis of classical thin shell theory and the assumptions of Biot's poroelasticity theory. The equations are a set of the coupled partial differential equations. The analytical method including the Airy stress function is used to solve the stability equations of porous shell under mechanical loads in three cases. Porous toroidal shell segments subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. Closed-form solutions are expressed for the elastic buckling behavior of the convex and concave porous toroidal shell segments. The effects of porosity distribution and geometrical parameters of the shell on the critical buckling loads of porous toroidal shell segments are studied.

Design and numerical investigation of an adaptive intramedullary nail with a novel interlocking mechanism

Abstract Malalignment is a common complication in the treatment of distal fractures of the tibia. Numerous efforts have been made to reduce the malalignment ratio. However, the reported cases with this disorder are still high. This study aimed at investigating an adaptive design of an intramedullary nail with a novel interlocking mechanism (AINIM), as an alternative for the customary nailing, in reducing malalignment ratio. A verified finite element model was employed to compare the performance of AINIM with the customary nail. The finite element model of the tibia follows the exact shape of the medullary canal, and nonhomogeneous material properties were assigned to the bone from bone ash density. It was assumed that the nails were implanted and interlocked in the tibia according to surgical protocols, and physiological-like loading was applied to finite element models. The results of this study showed that AINIM reduces the mean shear interfragmentary strains by about 30%, and the axial interfragmentary strain by 55%, also it increases the uniformity in the interfragmentary movements, compared to the customary nail. It was also found that AINIM caused a reduction of the stress on the nail by 60%, and an increase of 25% on the bone, compared to the customary nail. Moreover, average compressive principal strains in the tibia fixed by AINIM increased by 40% from 485 to 678 με, compared to the tibia fixed by the customary nailing method. The results of this work also showed that AINIM causes an increase in the contact area with the intramedullary canal, particularly at the fracture site, and it also escalates the magnitude of contact pressure. Results of this work indicate that, from the biomechanical standpoint, the adaptive nail, i.e. AINIM, with an innovative interlocking mechanism, compared to the customary nailing, can lessen intra- and post-operative malalignment occurrence, and it also mitigates the side effects of stress shielding, and thus better conserves neighboring bone density in a long period.

A meshless regularized local boundary integral equation method and the selection of weight function and geometrical parameters

The local boundary integral equation (LBIE) method is a pure meshless integral method proposed by Atluri and co-workers. Since the fundamental solutions are used as the test functions, the sub-domain can be very small and the LBIE method is a promising method for solving problems of elasticity with nonhomogeneous material properties. In this paper, the subtraction technique is introduced to remove the strong singularity that results in a regularized local boundary integral equation (RLBIE) method. The formula is rather straightforward and more accurate than other treatment techniques. Compared with other meshless methods, more geometrical parameters need to be assigned in the method. If the parameters change a little bit, the solutions obtained from the method may fluctuate quickly and uncertainly. Therefore the effects of weight function, support domain, sub-domain, and monomial basis are investigated and how to select the parameters is discussed. The appropriate parameters can be determined and the meshless RLBIE method is an accurate and robust method.

Boundary element formulation of axisymmetric problems in vertically non-homogeneous solids subject to normal traction

This paper presents an efficient and accurate boundary element method (BEM) for the elastic analysis of axisymmetric problems in vertically non-homogeneous solids without or with cavity. This BEM uses the fundamental solution of the elastic field in a multilayered elastic solid induced by the body force uniformly concentrated at a circular ring. This solution is also called Yue's solution. The effective integration methods are used for dealing with the integrals in the discretized boundary integral equations. The discretization of the boundary surface uses one-dimensional boundary elements. It also adopts the infinite boundary element to take into account the influence of a far-field region on the boundary surface. Numerical verifications of displacements and stresses for three benchmark problems are conducted, which gives the excellent agreement with previously published results. Case studies are presented to numerically illustrate the influences of both vertically non-homogeneous elastic material properties and spherical cavity on the elastic fields induced by uniform tractions on the boundary surface. These numerical results show that this new BEM is a fast and simple numerical algorithm for accurately computing the axisymmetric elastic fields in vertically non-homogeneous solids with or without cavity induced by normal tractions.

A Fat boundary-type method for localized nonhomogeneous material problems

Problems with localized nonhomogeneous material properties arise frequently in many applications and are a well-known source of difficulty in numerical simulations. In certain applications (including additive manufacturing), the physics of the problem may be considerably more complicated in relatively small portions of the domain, requiring a significantly finer local mesh compared to elsewhere in the domain. This can make the use of a conforming, highly non-uniform mesh numerically unfeasible. In fact, such meshes may be challenging to generate (particularly for regions with complex boundaries) and more difficult to precondition. The problem becomes even more prohibitive when the region requiring a finer-level mesh changes in time, requiring the introduction of refinement and derefinement techniques. To address the aforementioned challenges, we employ a technique related to the Fat boundary method as a possible alternative. We analyze the proposed methodology from a mathematical point of view and validate our findings with a series of two-dimensional numerical tests.

Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.

A Novel Method of Residual Strain Analysis Via a Non-Bonded Interface Technique in Combination With the Circular Grid Analysis Method

AbstractA novel non-bonded interface technique (NBIT) is used to analyze internal residual strain by combining a pre-split sample of AISI 4340 steel with the circular grid residual strain analysis technique. NBIT is compared with an implicit non-linear finite element (FE) model using LS-DYNA. A split FE model was compared with a quarter FE model to determine the split interface that causes an average difference of 9.0% on the residual von Mises strain field from a 588.6 N indentation. The homogeneous FE quarter model was then compared with the experimental split model using 588.6, 981.0, and 1471.5 N indentation forces. An average displacement difference of 3.92 µm was found when comparing the experimental split and FE homogeneous samples from a 588.6 N indentation. The internal residual major and minor principal strains from the split experimental sample and homogeneous FE model were compared for each indentation force. The minor principal strain results show the 588.6, 981.0, and 1471.5 N indentation forces resulted in a difference between the experimental split and homogeneous FE model of 28.5%, 34.8%, and 26.0%, respectively. The difference between the comparisons was explained by the inability of the FE model to simulate local non-homogeneous material properties such as grain composition and orientation whereas NBIT does. NBIT can be used for micro- or macro-scale residual strain analysis as the spatial resolution is highly adjustable.

2D Hybrid Steady-State Magnetic Field Model for Linear Induction Motors

This paper presents a 2D hybrid steady-state magnetic field model, capable of accurately modeling the electromagnetic behavior in a linear induction motor, including primary slotting, finite yoke length, and longitudinal end-effects by primary motion. This model integrates a complex harmonic modeling technique with a discretized magnetic equivalent circuit model. The Fourier model is applied to regions with homogeneous material properties, e.g., air regions and the track of the motor, while the magnetic equivalent circuit (MEC) approach is used for the regions containing non-homogeneous material properties, e.g., the primary of the linear induction motor (LIM). By only meshing the domains containing highly-permeable materials, the computational effort is reduced in comparison with the finite element method (FEM). The model is applied to a double-layer single-sided LIM, and the resulting thrust and normal forces show an excellent agreement with respect to finite element analysis and measurement data.

An improved local boundary integral equation method implemented by the transformed MLS approximation with the delta property

The meshless local boundary integral equation (LBIE) method is a promising method for solving problems of elasticity with nonhomogeneous material properties. However, the shape functions of the LBIE method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property. To impose boundary conditions, numerical integrations of LBIE method are carried out under consideration of boundary conditions and the fictitious nodal values would be solved out. In this paper, the transformed MLS shape functions can be obtained, satisfying the Kronecker delta property and requiring no singular weight function. And then the potential and flux in 2D potential problems would be regarded as independent variables each other. The numerical integrations of LBIEs can be processed without regard to boundary conditions and the coefficient matrix of LBIE method would be irrelated to boundary conditions. Finally boundary conditions would be directly imposed and the unknown nodal potentials or the unknown fluxes on the boundary nodes would be solved out. Three numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the proposed method with the traditional LBIE method shows the feasibility of the proposed method.

Investigating the flue-wall deformation effects on performance characteristics of an open-top aluminum anode baking furnace

The carbon anode baking is typically the bottleneck in the production of anodes for the aluminum industry. The challenge is to produce high quality baked anodes while keeping the energy consumption, environmental emissions, and cost to a minimum. Anode baking homogeneity is an important consideration in the design and operation of the anode baking furnaces. The flue-walls, into which the firing takes place, are the heart of the furnace. The flue-walls must be well designed and well maintained in order to be able to regulate them properly. However, during the service life of a baking furnace, flue-walls deform which affect the baking uniformity and consequently result in over-consumption energy and reduction in carbon anode quality. Studying the effects of flue-wall deformation by plant tests is highly challenging and expensive. Hence, this study aims at investigating this phenomenon by developing a three-dimensional (3D) model which includes many physical phenomena and parameters that play vital roles in the baking process. It was observed that indeed the flue-wall deflection has a significant impact on heat transfer characteristics of the anode baking process and furnace energy consumption. In specific, the flue-wall deformation results in an increase in temperature gradients within the anode pack such that the differences between anode pack minimum, average and maximum temperatures lead to overbaking or underbaking of anodes. In fact, this non-uniform baking gives rise to the evolution of non-homogeneous carbon anodes material properties, which is the main reason for excess energy consumption and various instabilities in the aluminum reduction cell. The insights obtained in the present study can be employed in modifying the furnace geometrical and operational parameters with the deformed flue-walls.