Infinite Solids Research Articles

A semi-analytical approach for thermoelastic wave propagation in infinite solids subject to linear heat supply using two-phase lag theory

A semi-analytical approach for thermoelastic wave propagation in infinite solids subject to linear heat supply using two-phase lag theory

Elastic multipole method for describing deformation of infinite two-dimensional solids with circular inclusions.

Elastic materials with holes and inclusions are important in a large variety of contexts ranging from construction material to biological membranes. More recently, they have also been exploited in mechanical metamaterials, where the geometry of highly deformable structures is responsible for their unusual properties, such as negative Poisson's ratio, mechanical cloaking, and tunable phononic band gaps. Understanding how such structures deform in response to applied external loads is thus crucial for designing novel mechanical metamaterials. Here we present a method for predicting the linear response of infinite 2D solid structures with circular holes and inclusions by employing analogies with electrostatics. Just like an external electric field induces polarization (dipoles, quadrupoles, and other multipoles) of conductive and dielectric objects, external stress induces elastic multipoles inside holes and inclusions. Stresses generated by these induced elastic multipoles then lead to interactions between holes and inclusions, which induce additional polarization and thus additional deformation of holes and inclusions. We present a method that expands the induced polarization in a series of elastic multipoles, which systematically takes into account the interactions of inclusions and holes with the external stress field and also between them. The results of our method show good agreement with both linear finite element simulations and experiments.

Numerical analysis of wave mixing in 3‐D infinite elastic solids with a spherical damage

AbstractIn this paper the 3‐D wave propagation in an infinite elastic solid with a spherical damage is numerically simulated by a mapped staggered Chebyshev pseudo‐spectral collocation method. In the numerical simulation process, the so‐called Convolutional‐Perfectly‐Matched‐Layers (CPML) are used to model the absorbing boundaries and the wave excitations are specified inside the corresponding physical domain. Furthermore, to consider different damage models the classical nonlinear elastic and non‐classical hysteretic material laws are used. The main objective of this study is to evaluate the influences of the particular wave modes and the mixing of the incident waves on the generated nonlinear scattered wave field. To analyze the specific scattered wave fields around the spherical damage region the computed time‐domain signals are transformed to the frequency‐domain.

Quantifying uncertainties in contact mechanics of rough surfaces using the multilevel Monte Carlo method

We quantify the effect of uncertainties on quantities of interest related to contact mechanics of rough surfaces. Specifically, we consider the problem of frictionless non adhesive normal contact between two semi infinite linear elastic solids subject to uncertainties. These uncertainties may for example originate from an incomplete surface description. To account for surface uncertainties, we model a rough surface as a suitable Gaussian random field whose covariance function encodes the surface’s roughness, which is experimentally measurable. Then, we introduce the multilevel Monte Carlo method which is a computationally efficient sampling method for the computation of the expectation and higher statistical moments of uncertain system output’s, such as those derived from contact simulations. In particular, we consider two different quantities of interest, namely the contact area and the number of contact clusters, and show via numerical experiments that the multilevel Monte Carlo method offers significant computational gains compared to an approximation via a classic Monte Carlo sampling.

Theory of dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings

This article presents a general theory on dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings. First, a novel extended point-force Green's function is developed that not only guarantees the numerical stability but also admits a direct interpretation of the image method in multilayered solids. Then, based on the new Green's function, the classical theory of dislocation loops in pure-elastic infinite solids is systematically generalized by accounting for both the surface/interface conditions and the magneto-electro-elastic couplings. Two major contributions are made in the generalization: (i) the extended displacement and stress fields induced by an arbitrary dislocation loop are derived and expressed concisely as a simple line integral along the closed dislocation curve; (ii) the interaction energy between two arbitrary dislocation loops is also derived and expressed compactly as double line integrals along the two closed dislocation curves. Based on the new line-integral expressions for multilayered solids, special configurations such as straight dislocation segments and elliptic dislocation loops are analytically studied in detail; and more strikingly, elegant line-integral expressions for dislocation loops in bi-materials or in a semi-infinite solid are obtained by simple reduction. All the expressions derived in this article are compared with available solutions in existing literature and the correctness of them is fully verified. As novel applications, the present theory is utilized to calculate the self-energy of interfacial elliptic dislocation loops efficiently, which clearly shows that both the surface/interface conditions and the multi-field couplings have a considerable influence on the self-energy of interfacial dislocation loops. The line-integral based formulae developed here for dislocation-loop problems will not only enrich the theory of magneto-electro-elasticity, but also have potential applications in three-dimensional discrete dislocation dynamics simulations associated with multilayered thin-film structures.

Anion-induced structural varieties of CuII coordination complexes with 3-phenyl-4-(3-pyridyl)-5-(4-pyridyl)-1,2,4-triazole

Two CuII coordination complexes {[Cu(L430)(H2O)2](NO3)2}n (I) and {[Cu(L430)2(H2O)2](ClO4)2(H2O)0.3}n (II) have been synthesized based on 3-phenyl-4-(3-pyridyl)-5-(4-pyridyl)-1,2,4-triazole (L430) in the presence of different anions (CIF file CCDC nos. 1489348 (I) and 1489349 (II)). Notably, complex I shows a 2D coordination network while polymer II represents a 1D box-like structure. Further, extended supramolecular architectures are constructed via secondary interactions in both structures. The structural distinction on both compounds fully demonstrates the role of lattice anions NO3– or ClO4– in the assembly of infinite coordination solids. In addition, complex II exhibits some certain efficiency for dichromate (Cr2O72−) capture in the water system.

EHD Effects in Lubricated Journal Bearing

This paper presents a numerical analysis of the influence of deformation of infinite parallel cylindrical solids in partial journal bearing on the oil film characteristics. The stationary elastohydrodynamic EHD problems for three design models of bearings are considered: (1) The bearing in which the basic contribution to the elastic displacement of the surface brings the thin elastic liner; (2) The elastic cylinder and the elastic bushing which is modeled by the elastic space with a cylindrical cut; (3) The elastic cylinder and the elastic bushing in the presence of the thin elastic liner with a small module of elasticity. It is shown that when the minimum film thickness is fixed and deformations of the elastic solids increase, then the load capacity increases, reaches a maximum, and then decreases. The deformation of solids can raise load capacity many times over. When the deformation of solids increases from zero, the pressure distribution changes from the distribution of pressure in the case of rigid bodies to the distribution of pressure which takes place with the dry contact of elastic bodies.

Analysis of Nonclassical Fracture Problems for Prestressed Bodies with Interacting Cracks

We consider two types of nonclassical fracture mechanisms, namely, the fracture of cracked bodies with initial (residual) stresses acting along the crack planes and the fracture of solids under compression along parallel cracks. To investigate nonaxisymmetric and axisymmetric problems for infinite solids containing two parallel coaxial cracks or a periodic set of coaxial parallel cracks, we use a combined analytic-numerical method within the framework of the three-dimensional linearized mechanics of solids. The analysis involves the representation of stresses and displacements in the linearized theory via harmonic potential functions. With the use of the integral Fourier–Hankel transformations, the problems are reduced to the solution of Fredholm integral equations of the second kind. This approach allows us to investigate problems in a unified general form for compressible and noncompressible homogeneous isotropic or transversely isotropic elastic bodies with an arbitrary structure of the elastic potential, and the material specification of the model is carried out only in the stage of numerical analysis of the resolving equations obtained in the general form. The effects of initial stresses on the stress intensity factors are analyzed for highly elastic materials and layered composites (modeled as transversely isotropic elastic bodies). The “resonance-like” effects are revealed when compressive initial stresses reach the values corresponding to the local loss of stability of the material in the vicinity of cracks, which, according to the indicated combined method, allows one to determine the critical (limiting) load parameters under the conditions of compression of the body along the cracks. The conclusions concerning the dependences of the stress intensity factors and critical (limiting) parameters of compression on the geometric parameters of the problems are formulated as well as concerning the dependences on physical and mechanical characteristics of the materials.

A physical model and analysis for whisker growth caused by chemical intermetallic reaction

Abstract The formation of intermetallic compound Cu 6 Sn 5 gives rise to the internal stress in the lead-free coating, which causes the growth of Sn whiskers. This phenomenon is characterized with the expansion of inclusion in a plate perfectly bonded between two infinite solids. Based on the grain boundary diffusion mechanism, a model is established to evaluate the growth rate of Sn whiskers. The results show that the growth rate of whisker varies with the relative site between whisker and inclusion. When the distance between the whisker and inclusion exceeds a critical value, negative growth rate will appear, and it approaches zero as the distance increases. They explain some phenomena observed in experiments.

Carbon mass balance and microbial ecology in a laboratory scale reactor achieving simultaneous sludge reduction and nutrient removal

Solids reduction in activated sludge processes (ASP) at source using process manipulation has been researched widely over the last two-decades. However, the absence of nutrient removal component, lack of understanding on the organic carbon, and limited information on key microbial community in solids minimizing ASP preclude the widespread acceptance of sludge minimizing processes. In this manuscript, we report simultaneous solids reduction through anaerobiosis along with nitrogen and phosphorus removals. The manuscript also reports carbon mass balance using stable isotope of carbon, microbial ecology of nitrifiers and polyphosphate accumulating organisms (PAOs). Two laboratory scale reactors were operated in anaerobic-aerobic-anoxic (A(2)O) mode. One reactor was run in the standard mode (hereafter called the control-SBR) simulating conventional A(2)O type of activated sludge process and the second reactor was run in the sludge minimizing mode (called the modified-SBR). Unlike other research efforts where the sludge minimizing reactor was maintained at nearly infinite solids retention time (SRT). To sustain the efficient nutrient removal, the modified-SBR in this research was operated at a very small solids yield rather than at infinite SRT. Both reactors showed consistent NH3-N, phosphorus and COD removals over a period of 263 days. Both reactors also showed active denitrification during the anoxic phase even if there was no organic carbon source available during this phase, suggesting the presence of denitrifying PAOs (DNPAOs). The observed solids yield in the modified-SBR was 60% less than the observed solids yield in the control-SBR. Specific oxygen uptake rate (SOUR) for the modified-SBR was almost 44% more than the control-SBR under identical feeding conditions, but was nearly the same for both reactors under fasting conditions. The modified-SBR showed greater diversity of ammonia oxidizing bacteria and PAOs compared to the control-SBR. The diversity of PAOs in the modified-SBR was even more interesting in which case novel clades of Candidatus Accumulibacter phosphatis (CAP), an uncultured but widely found PAOs, were found.

Extended displacement discontinuity method for nonlinear analysis of penny-shaped cracks in three-dimensional piezoelectric media

Abstract The polarization saturation (PS) model and the dielectric breakdown (DB) model are both used, under the electrically impermeable crack assumption, to analyze penny-shaped cracks in the isotropic plane of three-dimensional (3D) infinite piezoelectric solids. Using the extended displacement discontinuity integral equation method, we obtained analytical solutions for the size of the electric yielding zone, the extended displacement discontinuities, the extended field intensity factor and the J-integral. Integrating the Green function for the point extended displacement discontinuity provided constant element fundamental solutions. These solutions correspond to an annular crack element applied with uniformly distributed extended displacement discontinuities in the transversely isotropic plane of a 3D piezoelectric medium. Using the obtained Green functions, the extended displacement discontinuity boundary element method was developed to analyze the PS model and DB model for penny-shaped cracks. The numerical method was validated by the analytical solution. Both the analytical results and numerical results show that the PS and the DB models give equivalent solutions for nonlinear fracture analysis of 3D piezoelectric materials, even though they are based on two physically different grounds.

A meshless Galerkin method with moving least square approximations for infinite elastic solids

Combining moving least square approximations and boundary integral equations, a meshless Galerkin method, which is the Galerkin boundary node method (GBNM), for two- and three-dimensional infinite elastic solid mechanics problems with traction boundary conditions is discussed. In this numerical method, the resulting formulation inherits the symmetry and positive definiteness of variational problems, and boundary conditions can be applied directly and easily. A rigorous error analysis and convergence study for both displacement and stress is presented in Sobolev spaces. The capability of this method is illustrated and assessed by some numerical examples.

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

Abstract This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.

Boundary integral equations and the boundary element method for fracture mechanics analysis in 2D anisotropic thermoelasticity

Abstract This paper develops the Somigliana type boundary integral equations for fracture of anisotropic thermoelastic solids using the Stroh formalism and the theory of analytic functions. In the absence of body forces and internal heat sources, obtained integral equations contain only curvilinear integrals over the solid’s boundary and crack faces. Thus, the volume integration is eliminated and also there is no need to evaluate integrals over the contours in the mapped temperature domain as it was done before. In addition to finite solids, the case of an infinite anisotropic medium with a remote thermal load is also studied. The dual boundary element method for fracture of anisotropic thermoelastic solids is developed based on the obtained boundary integral equations. Presented numerical examples show the validity and efficiency of the obtained equations in the analysis of both finite and infinite solids with cracks.

The influences of non-linear electrical, magnetic and mechanical boundary conditions on the dynamic intensity factors of magnetoelectroelastic solids

Abstract In this paper advanced transient dynamic crack analysis in two-dimensional (2D), homogeneous and linear magnetoelectroelastic solids by taking non-linear electrical, magnetic and mechanical crack-face boundary conditions into account is presented. Stationary cracks in infinite and finite magnetoelectroelastic solids under different impact loadings are investigated. Thus, a time-domain boundary element method (TDBEM) is developed for this purpose. The TDBEM uses the Galerkin-method for the spatial discretization, while the collocation method is implemented for the temporal discretization. An explicit time-stepping scheme is used to compute the unknown boundary data numerically. Iterative solution algorithms are developed to compute the non-linear semi-permeable electrical and magnetic crack-face boundary conditions. An additional iteration scheme is implemented for crack-face contact analysis at time-steps when a physically unacceptable crack-face intersection occurs. Numerical examples are presented and discussed to investigate the influences of the electrical, magnetic and mechanical crack-face boundary conditions on the dynamic field intensity factors.

Effective Contact Temperature of Solid with Different Material Surface

The estimation of thermal sensation at the instant when a certain object is touched is becoming an important consideration in material evaluation. In a previous paper, we showed that the effective contact temperature of the object –when covered with a thin surface of a different material– could be implied from the thermal contact resistance of the nonsteady heat conduction between half infinite solids. However, when the thickness of the surface material of the contact object becomes the same depth as the thermoreceptor in the skin, the approximation by the thermal contact resistance no longer applies, and, therefore, deriving a strict analytical solution becomes difficult. We calculated the temperature distributions in the material for various material surfaces and the skin using a numerical method. We examined the influence on the effective contact temperature of the thickness and the heat properties of each surface material. As a result, deflection of effective contact temperature from the temperature of a material is determined by the temperature difference between the skin and the surface material/object, the heat properties of each, the thickness of the surface material, and the depth of thermoreceptor in the skin. This can be expressed by the relation of the nondimensional parameter function. When the thickness of surface material is thick enough, effective contact temperature is determined only by the heat property of the surface material, and it is unrelated to the thickness of the surface material and the thermal property of the object under it. The influences of the thickness of the surface material and the thermal property of the object on effective contact temperature appear when the thickness of surface material becomes very thin, and increases rapidly with a decrease in the thickness of the surface material. The criterion between thermally thick and/or thin material is quantitatively presented by the numerical simulation.

A BEM for transient thermoelastic fracture analysis of FGMs

AbstractA boundary element method for the transient thermoelastic fracture analysis in isotropic, continuously non‐homogeneous and linear elastic functionally graded materials subjected to a thermal shock is presented. The material parameters are assumed to be continuous functions of the Cartesian coordinates. Laplace‐domain fundamental solutions of linear coupled thermoelasticity for infinite, isotropic, homogeneous and linear elastic solids are applied to derive the boundary‐domain integral equation formulation. The numerical implementation is performed by using a collocation method for the spatial discretization. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transient dynamic crack analysis in linear magnetoelectroelastic solids by a hypersingular time-domain BEM

Abstract This paper presents a hypersingular time-domain boundary element method (TDBEM) for transient dynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic and linear magnetoelectroelastic solids. Stationary cracks in infinite and finite solids under impact loading are investigated. A combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction boundary integral equations is used in the present analysis. The Galerkin-method is applied for spatial discretization, while a collocation method is implemented for the temporal discretization. The time-domain fundamental solutions for linear magnetoelectroelastic solids are derived. Both temporal and spatial integrations can be carried out analytically. The line integrals over the unit circle arising in the time-domain fundamental solutions are computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. To ensure a direct and an accurate computation of the dynamic intensity factors (IFs) from the generalized CODs, special crack-tip elements are implemented. Several numerical examples are shown to confirm the accuracy and the efficiency of the present hypersingular time-domain BEM.

A performance evaluation of three membrane bioreactor systems: aerobic, anaerobic, and attached-growth

Water sustainability is essential for meeting human needs for drinking water and sanitation in both developing and developed countries. Reuse, decentralization, and low energy consumption are key objectives to achieve sustainability in wastewater treatment. Consideration of these objectives has led to the development of new and tailored technologies in order to balance societal needs with the protection of natural systems. Membrane bioreactors (MBRs) are one such technology. In this investigation, a comparison of MBR performance is presented. Laboratory-scale submerged aerobic MBR (AMBR), anaerobic MBR (AnMBR), and attached-growth aerobic MBR (AtMBR) systems were evaluated for treating domestic wastewater under the same operating conditions. Long-term chemical oxygen demand (COD) and total organic carbon (TOC) monitoring showed greater than 80% removal in the three systems. The AnMBR system required three months of acclimation prior to steady operation, compared to one month for the aerobic systems. The AnMBR system exhibited a constant mixed liquor suspended solids concentration at an infinite solids retention time (i.e. no solids wasting), while the aerobic MBR systems produced approximately 0.25 g of biomass per gram of COD removed. This suggests a more economical solids management associated with the AnMBR system. Critical flux experiments were performed to evaluate fouling potential of the MBR systems. Results showed similar critical flux values between the AMBR and the AnMBR systems, while the AtMBR system showed relatively higher critical flux value. This result suggests a positive role of the attached-growth media in controlling membrane fouling in MBR systems.

Dynamic crack analysis in piezoelectric solids with non-linear electrical and mechanical boundary conditions by a time-domain BEM

Abstract This paper presents advanced transient dynamic crack analysis in two-dimensional (2D), homogeneous and linear piezoelectric solids using non-linear mechanical and electrical crack-face boundary conditions. Stationary cracks in infinite and finite piezoelectric solids subjected to impact loadings are considered. For this purpose a time-domain boundary element method (TDBEM) is developed. A Galerkin-method is implemented for the spatial discretization, while a collocation method is applied for the temporal discretization. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. An iterative solution algorithm is developed to consider the non-linear semi-permeable electrical crack-face boundary conditions. Furthermore, an additional iteration scheme for crack-face contact analysis is implemented at time-steps when a physically meaningless crack-face intersection occurs. Several numerical examples are presented and discussed to show the effects of the electrical crack-face boundary conditions on the dynamic intensity factors.