Linear and non-linear elastic and plastic strains in a plate with a large hole loaded axially in its plane

Strain analysis of the ligament of a plate with a large hole indicates that both geometric and material non-linearity may take place. Geometric non-linearity may occur even when the material is strained elastically. This may explain the disagreement in the results obtained recently, for the stress concentration factors, by several investigators. The experimental analyses were conducted on an aluminum plate, using electrical resistance strain gages, and on a polyurethane rubber sheet using photography. The strain concentration factor was found to vary from 1 to 2 depending on the level of deformation.     

In plane loaded masonry walls: DEM and FEM/DEM models. A critical review

This work is dedicated to the assessment of the nonlinear behaviour of masonry panels with regular texture and subject to in-plane loads, by means of numerical pushover analysis and an analytical homogenized model. Two numerical models are considered and adopted for performing a set of numerical tests: a discrete model developed by authors and a discrete/finite element model frequently adopted in rock mechanics field and effectively extended to masonry structures. In both models the hypotheses of rigid blocks and elastic–plastic joints following a Mohr–Coulomb yield criterion are adopted. The aim of this work is twofold: (1) a comparison and a calibration of the numerical models, evaluating their effectiveness in determining ultimate loads and collapse mechanisms of masonry panels, by assuming a nonlinear homogenized model for regular masonry as reference solution; (2) the evaluation of sensitivity of masonry behaviour and numerical models to panel dimension ratio and to varying masonry texture. In a first case study, sliding collapse mechanisms changing to overturning collapse mechanisms for increasing panel and block height-to-width ratio are obtained and the results given by the numerical models turn out to be in good agreement. Furthermore, a second case study, dedicated to square panels supported at base ends and vertically loaded, shows different ‘arch mechanisms’ depending on block height-to-width ratio.     

Some problems in plane elastic dielectrics

Representations of Galerkin type are obtained for the displacement vector, polarization vector and the potential fields in the static plane theory of elastic dielectrics using the method of associated matrices. Fundamental matrix solutions of an infinite elastic dielectric plane subjected to a concentrated body force, electric force and charge density are derived from the singular solutions of harmonic, biharmonic and Helmholtz equations. Using boundary operatorsY, Z, M, the fundamental matrix solutions, and Betti's formulae, a matrix (x, y) is constructed and an integral representation for (u ₁,u ₂,P ₁,P ₂, ) is obtained. Discontinuity theorems are stated for the double layer potential andQ operator of the single layer potential. By means of these theorems, the solutions of interior and exterior boundary value problems are reduced to the solution of a system of five singular integral equations. The index of one of the systems is shown to be zero and it is concluded that Fredholm theorems and its alternatives hold.

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Zusammenfassung Durch Anwendung der Assoziativ-Matrizen-Methode werden Galerkinische Darstellungen für den Verschiebungsvektor, den Polarisierungsvektor und die potentiellen Prüffelder einer statischen ebenen Theorie des elastischen Dielektrikums entwickelt. Von den singulären Lösungen der harmonischen, biharmonischen und Helmholtz-Gleichungen werden grundlegende Matrizenlösungen für das unendliche, ebene und elastische Dielektrikum, das durch konzentrierte Raumkräfte, elektrische Kräfte und Ladungsdichte beansprucht wird, abgeleitet. Unter Verwendung der Randwert-OperatorenY, Z, M, der grundlegenden Matrizen-Lösungen und Betti's Formel, wird eine Spezial-Matrize (x, y) konstruiert und eine Integraldarstellung für (u ₁,u ₂,P ₁,P ₂, ) erhalten. Unstetigkeitssätze werden für das Doppelschicht-Potential und denQ-Operator eines einschichtigen Potentials angeführt. Durch Anwendung dieser Sätze werden die Lösungen der inneren und äusseren Grenzwertprobleme zur Lösung eines Systems von fünf singulären Integralgleichungen reduziert. Der Index eines der Systeme wird als Null bewiesen und es wird die Schlussfolgerung gezogen, dass der Fredholmsche Satz und seine Alternativen fur diese Theorie anwendbar sind.

     

Transport equations for elastic and other waves in random media

We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.     

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.     

Coordinates of principal stresses for elastic plane problem

I.IntroductionInelasticmechanics,thereisakindofproblemsthatcouldbesolveddirectlybyequilibriumequations,i.e.,whenal1oftheboundaryconditionsaretheknownstressesorforcessuchasthestressfunction.Becausestressfunctionsmustsatisfyharmonicequationorbi-harmonicequa…     

Saint-Venants principle and the plane elastic wedge

The stress field due to self-equilibrating loading on the inner or outer arc of a plane strain elastic wedge sector is affected by two agencies: a geometric effect of increasing or decreasing area, and decay as anticipated by Saint-Venants principle (SVP) . When the load is applied to the inner arc the two effects act in concert ; however, when the load is applied to the outer arc the two effects act in opposition and for a wedge angle in excess of the half-space, 2α > π, for the symmetric case, and for 2α > 1.43π for the asymmetric case, the geometric effect is dominant over Saint-Venant decay and stress level increases as one moves away from the outer arc, confirming the inapplicability of SVP. This is additional to previously reported difficulties at these angle when a self-equilibrated load on the inner arc decays at the same rate as does a concentrated moment, and is explained in terms of the interaction of a near-field geometric effect and a far-field stress interference effect at a traction-free edge. For wedge angle 2α = 2π the unique Modes I and II inverse square root stress singularities at the crack tip, which are at the heart of Linear Elastic Fracture Mechanics (LEFM) , can be attributed to this inapplicability for just one symmetric and one asymmetric eigenmode.     

Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies＊

4-node, 8-node and 8(4)-node quadrilateral plane isoparametric elements are used for the solution of boundary value problems in linear isotropic Cosserat elasticity. The patch test is applied to validate the finite elements. Engineering problems of stress concentration around a circular hole in plane strain condition and mechanical behaviors of heterogeneous materials with rigid inclusions and pores are computed to test the accuracy and capability of these three types of finite elements.The project supported by the National Natural Science Foundation of China (10225212, 50178016, 10421002) and the Program for Changjiang Scholars and Innovative Research Team in University of China The English text was polished by Keren Wang.     

Fundamental-solution-based finite element model for plane orthotropic elastic bodies

A new hybrid finite element formulation is presented for solving two-dimensional orthotropic elasticity problems. A linear combination of fundamental solutions is used to approximate the intra-element displacement fields and conventional shape functions are employed to construct elementary boundary fields, which are independent of the intra-element fields. To establish a linkage between the two independent fields and produce the final displacement-force equations, a hybrid variational functional containing integrals along the elemental boundary only is developed. Results are presented for four numerical examples including a cantilever plate, a square plate under uniform tension, a plate with a circular hole, and a plate with a central crack, respectively, and are assessed by comparing them with solutions from ABAQUS and other available results.     

A modulus gradient model for an axially loaded inhomogeneous elastic rod

A new gradient elasticity formulation is proposed for a one-dimensional linear elastic inhomogeneous rod. In the new formulation, similar to the differential relation between the local strain and the gradient enhanced strain in the classical models of gradient elasticity, a differential relation is proposed for the Young’s modulus. Analytical and finite element solutions of the proposed formulation are derived. Results of the proposed model are compared with a classical model of gradient elasticity for a model problem of carbon nanotube reinforced polymer composite.     

Canonical ensemble for static elastic structures with random microstructures

A constant-(N,V,T,θ) ensemble is proposed to describe the elastostatics of random solid structures. Within the harmonic approximation, the energy of such a solid structure is the sum of a thermal and a strain component without mutual interaction. Systems in this ensemble thus draw energies from two separate baths: one thermal and the other mechanical. A mechanical entropy and an effective temperature (θ) can then be defined on the same rigorous basis as the thermal entropy and the Kelvin temperature (T). This ensemble approach can be used to calculate the properties of solid structures sharing similar microstructural randomness.     

二维非局部线弹性平面问题的辛分析

将二维非局部线弹性理论引入到Hamilton体系下,基于变分原理推导得出了二维线弹性理论的对偶方程和相应的边界条件.在分析验证对偶方程的准确性的基础上,该套方法被应用于二维弹性平面波问题的求解.将精细积分与扩展的W-W算法相结合在Hamilton体系下建立了求解平面Rayleigh波的数值算法.从推导到计算的保辛性确保了辛体系非局部理论与算法的准确性.通过对不同算例的数值计算,分析和对比了非局部理论方法与传统局部理论方法的差别,并进一步指出了该套算法的适用性和优势所在.     

Complementary energy variational approach for plane elastic problems with singularities

Presented is the numerical analysis of plane elastic problems involving stress concentrations and/or singularities using a physically meaningful complementary energy variational approach. The continuum body is modeled by a non-conventional truss structure. Stress distributions in laminated composite bodies and orthotropic sheets with a through crack are obtained. The present results are compared with the analytical solutions for different numerical methods.     

Residual stresses in random elastic composites: nonlocal micromechanics-based models and first estimates of the representative volume element size

Random elastic composites with residual stresses are examined in this paper with the aim of understanding how the prestress may influence the overall mechanical properties of the composite. A fully non-local effective response is found in perfect analogy with the un-prestressed case examined in (Drugan and Willis, J. Mech. Phys. Solids 44(4):497–524, 1996). The second gradient approximation is considered and the impact of the residual stresses on the estimate of the RVE size is studied whenever the local response is used to describe the mechanical properties of the heterogeneous medium. To this aim, total and incremental formulations are worked out in this paper and the influence of both uniform and spatially varying prestresses are studied. Among other results, it is shown how rapid oscillations of relatively “small” residual stresses in most cases may result in the impossibility of describing the overall behavior of the composite with a local constitutive equation. On the other hand, prestresses with relatively high amplitudes and slow spatial oscillations may even reduce the RVE size required for approximating the mechanical properties of un-prestressed heterogeneous media with a local constitutive equation.     

Bounds and estimates for elastic constants of random polycrystals of laminates

In order to obtain formulas providing estimates for elastic constants of random polycrystals of laminates, some known rigorous bounds of Peselnick, Meister, and Watt are first simplified. Then, some new self-consistent estimates are formulated based on the resulting analytical structure of these bounds. A numerical study is made, assuming first that the internal structure (i.e., the laminated grain structure) is not known, and then that it is known. The purpose of this aspect of the study is to attempt to quantify the differences in the predictions of properties of the same system being modelled when such internal structure of the composite medium and spatial correlation information is and is not available.     

Analogues relating the plane and bending problems for a bi-material elastic composite

Southwell's analogues for an elastically homogeneous plate, are shown to be applicable to a bi-material elastic composite.