Infinite Body Research Articles

Modified fractional thermoelasticity model with multi-relaxation times of higher order: application to spherical cavity exposed to a harmonic varying heat

Several researchers have introduced different models of fractional order thermoelasticity theory depending on fractional calculus. In this research, a new modified model of generalized thermoelasticity with multi-relaxation times is derived based on fractional calculus and Taylor series expansion of time-fractional order. The models of Green and Lindsay two relaxation times [1] and coupled thermoelasticity [2] as well as fractional thermoelasticity with two relaxation times proposed by Hamza et al. [3] follow as limiting cases. The model is then adopted to study thermoelastic interactions in an infinite body with a spherical cavity whose boundary is subjected to harmonic varying heat and traction free. The basic equations governing the problem were solved using Laplace transformations. To find the analytical formulas of the physical quantities and to find inverse Laplace transforms, a numerical method was used. Detailed analysis of the effects of the fractional order parameter and the angular frequency of thermal vibration on the absolute temperature, stresses and displacement are studied.

Study of stress–strain state of elastic body with hyperbolic notch

The paper considers elastic stress distributions in infinite space with hyperbolic notch when normal or tangential stresses are given on the boundary of notch. The work considers plane deformation. So, exact (analytical) solution of two-dimensional boundary value problems of elasticity in the domain with hyperbolic boundary in the elliptic coordinate system is constructed using the method of separation of variables. The stress–strain state of a homogeneous isotropic infinite body with a hyperbolic cut is studied when there are non-homogeneous (nonzero) boundary conditions given on the hyperbolic cut. Finally, the numerical simulation is performed to the stress and displacement distributions over a finite size volume surrounding the notch and relevant graphs for the numerical results of some test problems are presented.

Transient response analysis of a spherical shell embedded in an infinite thermoelastic medium based on a memory-dependent generalized thermoelasticity

In the context of a memory-dependent generalized thermoelasticity, the thermal-induced transient response in an infinite elastic body containing a spherical shell is investigated. A thermal shock is applied on the inner surface of the spherical shell. The infinite body and the spherical shell are assumed to be isotropic but two dissimilar materials. By using an analytical technique based on the Laplace transform along with its numerical inversion, the governing equations of the problem are solved and the non-dimensional physical quantities in the two materials, i.e., temperature, displacement and stress, are obtained and illustrated graphically respectively. In simulation, the accuracy of memory-dependent derivative (MDD) is verified by degrading the present model into L-S model to compare the results obtained from the cases with interfacial thermal resistance and without interfacial thermal resistance. In addition, the effects of the different kernel functions as well as the ratios of the two materials, including the ratios of the density, the thermal-conductivity and the time-delay, on the distributions of the considered variables are obtained and demonstrated graphically respectively.

Mathematical Modelling of Stationary Thermoelastic State for a Plate with Periodic System of Inclusions and Cracks

Abstract Two-dimensional stationary problem of heat conduction and thermoelasticity for infinite elastic body containing periodic system of inclusions and cracks is considered. Solution of the problem is constructed using the method of singular integral equations (SIEs). The numerical solution of the system integral equations are obtained by the method of mechanical quadrature for a plate heated by a heat flow, containing periodic system elliptic inclusions and thermally insulated cracks. There are obtained graphic dependences of stress intensity factors (SIFs), which characterise the distribution of intensity of stresses at the tops of a crack, depending on the length of crack, elastic and thermoelastic characteristics inclusion, relative position of crack and inclusion.

Method of automorphic functions for an inverse problem of antiplane elasticity

A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a ($3n-4$) parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

Many physical equations have the form $$\mathbf{J }(\mathbf{x })=\mathbf{L }(\mathbf{x })\mathbf{E }(\mathbf{x })-\mathbf{h }(\mathbf{x })$$ with source $$\mathbf{h }(\mathbf{x })$$ and fields $$\mathbf{E }$$ and $$\mathbf{J }$$ satisfying differential constraints, symbolized by $$\mathbf{E }\in \mathcal E$$ , $$\mathbf{J }\in \mathcal J$$ where $$\mathcal E$$ , $$\mathcal J$$ are orthogonal spaces. We show that if $$\mathbf{L }(\mathbf{x })$$ takes values in certain nonlinear manifolds $$\mathcal M$$ , and coercivity and boundedness conditions hold, then the infinite body Green’s function (fundamental solution) satisfies exact identities. The theory also links Green’s functions of different problems. The analysis is based on the theory of exact relations for composites, but without assumptions about the length scales of variations in $$\mathbf{L }(\mathbf{x })$$ , and more general equations, such as for waves in lossy media, are allowed. For bodies $$\Omega $$ , inside which $$\mathbf{L }(\mathbf{x })\in \mathcal{M}$$ , the “Dirichlet-to-Neumann map” giving the response also satisfies exact relations. These boundary field equalities generalize the notion of conservation laws: the field inside $$\Omega $$ satisfies certain constraints that leave a wide choice in these fields, but which give identities satisfied by the boundary fields, and moreover provide constraints on the fields inside the body. A consequence is the following: if a matrix-valued field $$\mathbf{Q }(\mathbf{x })$$ with divergence-free columns takes values within $$\Omega $$ in a set $$\mathcal B$$ (independent of $$\mathbf{x }$$ ) that lies on a nonlinear manifold, we find conditions on the manifold, and on $$\mathcal B$$ , that with appropriate conditions on the boundary fluxes $$\mathbf{q }(\mathbf{x })=\mathbf{n }(\mathbf{x })\cdot \mathbf{Q }(\mathbf{x })$$ (where $$\mathbf{n }(\mathbf{x })$$ is the outward normal to $$\partial \Omega $$ ) force $$\mathbf{Q }(\mathbf{x })$$ within $$\Omega $$ to take values in a subspace $$\mathcal D$$ . This forces $$\mathbf{q }(\mathbf{x })$$ to take values in $$\mathbf{n }(\mathbf{x })\cdot \mathcal D$$ . We find there are additional divergence-free fields inside $$\Omega $$ that in turn generate additional boundary field equalities. Consequently, there exist partial null Lagrangians, functionals $$F(\mathbf{w },\nabla \mathbf{w })$$ of a vector potential $$\mathbf{w }$$ and its gradient that act as null Lagrangians when $$\nabla \mathbf{w }$$ is constrained for $$\mathbf{x }\in \Omega $$ to take values in certain sets $$\mathcal A$$ , of appropriate nonlinear manifolds, and when $$\mathbf{w }$$ satisfies appropriate boundary conditions. The extension to certain nonlinear minimization problems is also sketched.

Computational study on thermal energy around diamond shaped cylinder at varying inlet turbulent intensity

Transport phenomena over a diamond shaped cylinder (DC) at varying inlet turbulent intensity is reported in the work. An infinite diamond cylinder bluff body with hydraulic diameter D is investigated numerically. The turbulent SST model is engaged for simulation. The simulation is accomplished with the aim to achieve an understanding of physical behavior of thermo-hydraulic flow over diamond cylinder at 15D spacing from the inlet. The computational analysis is done to cover cross flow encompassing laminar, transition, quasi turbulent and turbulent flow regime in the range of Reynolds number upto 1,00,000 and inlet intensities from 5% to 20%. The computational fluid dynamics results are confirmed with correlations which show reasonable agreements. Moreover, the diamond cylinder provides better performance than circular cylinder. The effects of Nusselt number are esteemed and the proposed work computes the influence of inlet turbulence intensity on augmenting heat transfer from the diamond cylinder.

Stress distribution caused by co-phase locally spatially curved layers in an infinite elastic body under bi-axial compression

In the present paper, the stress distribution is studied in an infinite elastic body, reinforced by an arbitrary number of non-intersecting co-phase locally spatially curved filler layers under bi-axial compression is studied. It is assumed that this system is loaded at infinity with uniformly distributed normal forces with intensity p1(p3) acting in the direction which is parallel to the layers? location planes. It is required to determine the self-equilibrated stresses within, caused by the spa?tially local curving of the layers. The corresponding boundary and contact value problem is formulated within the scope of geometrically non-linear exact 3-D equations of the theory of elasticity by utilizing of the piece-wise homogeneous body model. The solution the formulated problem is represented with the series form of the small parameter which characterizes the degree of the aforementioned local curving. The boundary-value problems for the zeroth and the first approximations of these series are determined with the use of the exponential double Fourier transform. The original of the sought values is determined numerically. Consequently, in the present investigation, the effect of the local curving on the considered interface stress distribution is taken into account within the framework of the geometrical non-linear statement. The numerical results related to the considered interface stress distribution and to the influence of the problem parameters on this distribution are given and discussed.

Analytical model for estimation of energy release rate at mode I crack tip in bi-material of identical steels joined by an over-matched weld interlayer

Structure fabricated by welding identical steels or a fractured steel component repaired by steel weld filler of different strength leads to the presence of two interfaces with strength mismatch across them. Mode I crack in such a bi-material upon growing across the weld interlayer experiences the effect of the interfaces that manifests by way of change in crack tip energy release rate from the far field or remote value. The paper presents a strain based analytical model for estimation of the effect of interface over the crack tip. Crack tip energy release rate is determined at all the positions of the crack under SSY or K dominant conditions. The weld interlayer is assumed to be homogeneous and over-matched which possesses higher strength than that of the constituent steels. The modulus of elasticity is same throughout the domain. Edge crack of various discrete lengths is investigated in infinite body under uniform monotonic tension of 150 MPa in plane stress situation for demonstration of the model behavior.

On internal stability loss of a row unidirected periodically located fibers in the visco-elastic matrix

In the present paper, the microbuckling or internal stability loss in the viscoelastic composites containing unidirected fibers under compression along the fibers is studied by use of piecewise homogeneous body model. In this model, it is used the 3-D geometrically non-linear exact equations of viscoelasticity theory. The composite material was considered as an infinite viscoelastic body with a row unidirected periodically located elastic fibers that have an initial infinitesimal imperfection. When the initial imperfection starts to increase and becomes indefinitely, this is taken as a stability loss criterion and co-phase microbuckling mode out of plane are taken into account. The numerical results about the influence of the interaction between the fibers on the values of the critical time are obtained and presented.

On the Application of the Adomian’s Decomposition Method to a Generalized Thermoelastic Infinite Medium with a Spherical Cavity in the Framework Three Different Models

A mathematical model is elaborated for a thermoelastic infinite body with a spherical cavity. A generalized set of governing equations is formulated in the context of three different models of thermoelasticity: the Biot model... | Find, read and cite all the research you need on Tech Science Press

Application of the internal and external Williams functions to plane elasticity problem for A Mode I crack

The main idea of this research is in applying Williams functions to calculate stress intensity factors (SIF) in case of 2D elastic bodies with cracks. Firstly, we employ the concept of Williams functions converging on infinity which can be used along withtraditional functions to analyse infinite bodies and to study partially loaded crack boundaries. We show the convergence of SIF for the case of a strip and for the bodies with circular boundary depending on the number of Williams functions and on the number ofintervals on the boundary in case of numerical integration. Secondly, we complement the standard set of equations with the global equilibrium conditions of the elastic body to improve the accuracy of the calculation. We compared the computed stress with the onedefined on the boundary.

Two-temperature generalized thermoelasticity with fractional order strain of an infinite body with a spherical cavity

Two-temperature generalized thermoelasticity with fractional order strain of an infinite body with a spherical cavity

Uniformly moving screw dislocation in strain gradient elasticity

In the present paper, Mindlin's strain gradient theory of elasticity (form II) is utilized to determine the deformation and strain fields due to a straight screw dislocation that moves uniformly at a subsonic velocity in an infinite isotropic body. The theory employed herein, in addition to the effect of strain gradient, is capable of accounting for that of acceleration gradient through the incorporation of the micro-inertia term into the analysis of the problem. Integral representations are obtained for the displacement and strain fields, and it is shown that the utilization of the strain gradient theory for the continuum description of a moving screw dislocation leads to discarding the discontinuity of the induced displacement field, consequently resulting in the removal of the classical singularities of the associated strain and stress fields.

Plasticity retards the formation of creases

When an elastomer is compressed, its surface forms creases at a critical strain about 0.35. When a plastically deformable material is compressed, however, its surface remains smooth at much larger strains. As the smooth surface folds locally into a crease, the material around the crease loads and unloads. We show that the hysteresis of plasticity retards the formation of the crease. For a crease growing in an infinite body, the stress field is self-similar as the crease grows, and the critical strain ec is the applied strain needed to maintain the self-similar growth. We simulate the formation of creases using an elastic-plastic model with linear hardening, and characterize the degree of plasticity of the model by the ratio of the tangent modulus to elastic modulus, Et/E. A small value of Et/E leads to a large critical strain for the onset of creases. We further show experimentally that creases can form at a strain of 0.49 for high-density polyethylene (HDPE) with Et/E ∼ 0.025, but cannot form for metals (aluminum, copper, and stainless steel) with Et/E ∼ 0.001.

Analysis of thermal effect around an underground storage cavern with a combined three-dimensional indirect boundary element method

This paper combines a displacement discontinuity method (DDM) for elasticity and an indirect boundary element method (IBEM) for heat conduction to investigate three dimensional thermal displacements and stresses, such as those around an underground storage cavern. The thermal displacements and stresses due to temperature change are expressed in terms of a thermal displacement potential function. For a constant point heat source, we present a derivation for solution of the thermal displacement potential function and the solution is the same as found in literature. A hybrid scheme with semi-analytical integration and a direct numerical integration are used to integrate the thermal displacement potential function due to a constant heat source on a triangular region. The hybrid scheme is used to overcome a difficulty arising with the semi-analytical integration for points whose projection on the integration plane is close to the vertices of the triangle. The combined IBEM with the hybrid integration scheme is first verified with the analytical solution for an infinite elastic body with a spherical cavity. The results agree well, although differences are observed in the early stages of simulation. Then the method is used to simulate the deformation around a liquefied natural gas underground storage cavern.

Spherical inclusion with time-harmonic eigenfields in strain gradient elasticity considering the effect of micro inertia

In the present paper, the complete form of Toupin–Mindlin strain gradient theory of elasticity is utilized to determine the elastic field of a spherical inclusion undergoing a time-harmonic eigenfield in an infinite isotropic body. The theory adopted herein involves three characteristic lengths, two ones of which are present therein due to the incorporation of strain gradients in the strain energy density function of the material under consideration, and the presence of the third one arises from taking into account the role of velocity gradients in its kinetic energy density function, reflecting the effect of micro inertia. After the derivation of an expression for the associated Green’s function of the problem, closed-form solutions are obtained for the interior and exterior elastic fields of a spherical inclusion subjected to a time-harmonic eigenfield whose spatial distribution is expressible as a function of the radial distance of points from the center of the inclusion.

Applicability of the classical fracture mechanics criteria to predict the crack propagation path in rock under compression

A higher order displacement discontinuity method (HDDM) with quadratic elements is used in this study to evaluate the validity of classical fracture criteria in order to predict crack propagation path under mixed mode loading I–II in the case of a crack in an infinite body. Crack propagation path is predicted here by defining three stress states, uniaxial tensile stress (UTS), uniaxial compressive stress (UCS) and biaxial compressive stress (BCS). Comparing with experimental results, it was found that by increasing the crack inclination angle, the length of crack propagation path increases and the slope of this path decreases. For BCS state, G-criterion almost loses the validity and can be replaced with σ-criterion. S-criterion is strongly dependent on Poisson’s ratio and it is recommended to study fracture under brittle and ductile behaviour. Due to the complexity of crack tip element equations, the effect of this element on the results has been studied in detail and it was found that by considering the mid-point of the last element as a reference point to evaluate displacement discontinuities (Ds and Dn), there is no need to use special crack tip element. This procedure facilitates the modelling and shortens the solution.

Regional and residual gravity anomaly separation using the singular spectrum analysis-based low pass filtering: a case study from Nagpur, Maharashtra, India

We present here a singular spectrum analysis (SSA)-based low pass filtering algorithm for regional and residual gravity anomaly separation. The data adaptive decomposition in SSA frequency filtering algorithm facilitates reduction of the artefacts in the filtering of the non-linear and non-stationary gravity data. Initially, the method was tested on synthetic gravity data representing combined response of regional and residual components derived from the geological structures like infinite horizontal layers, intrusive dykes, deep-seated faults and volcanic intrusive bodies and compared with fast Fourier transform (FFT) wavenumber and wavelet-based filtering techniques. The results show that SSA-based filtered output exhibits a better match with pure synthetic data than the output generated from the FFT and wavelet filtering methods. The underlying method was then applied to two parallel gravity profiles of real data from the Umred coalfield, Nagpur, Maharashtra, India. Further, the SSA-based filtered regional anomaly was modelled using Geosoft GM-SYS software to construct the model of crustal structure of the study region. In essence, the modelling results suggest the following: (i) a basin-type graben structure with variable trap and sedimentary thicknesses and (ii) deep seated faults on either sides of the basin. These results correlate fairly well with known regional geology attesting the authenticity of the regional models generated from the two gravity profiles, which also agree well with each other. We, therefore, conclude that the SSA-based filtering technique is robust for regional gravity anomaly separation and could be effectively exploited for filtering other geophysical data.

An embedded couple stress micro-/nano-obstacle with micro-inertia incident upon by SH-waves

An elliptic micro-/nano-obstacle bonded to an infinite body incident upon by SH-waves, where both domains are couple stress media with micro-inertia, is of major concern. The formulation of this problem in the mathematical framework of couple stress elasticity with micro-inertia leads to angular and radial Mathieu differential equations which are solved analytically. These equations carry two characteristic lengths which are peculiar to the discrete nature of each domain enabling the capture of size effect, dispersion phenomenon, as well as the enhancement of the accuracy of the results. For verification, the ratio of the semi-axis of the elliptic obstacle is set equal to 1, and the result associated with the circular fiber which was previously considered by Shodja et al. (Int J Solids Struct 58:73–90, 2015) is recovered. The distribution of the induced stresses along an elliptic obstacle boundary, depending on its major-to-minor semi-axis ratio (e.g., 2), can significantly differ from that pertinent to a circular one.