Elasto-plastic cracked plates in plane strain

The plane strain behavior of an clasto-plastic cracked plate in tension is examined at two levels of work-hardening. The results are compared to similar data for the case of plane stress

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Zusammenfassung Das Flächenanspannungsverhalten einer elastisch-plastisch gespaltenen Platte unter Spannung wurde in zwei Verfertigungsvorgängen untersucht. Die Ergebnisse warden mit ähnlichen Daten im Bereich von Flächenanspannung verglichen.

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Résumé Le comportement en état plan de déformation d'une tô1e comportant une fissure et soumise à tension dans le domaine é1asto-plastique est examiné pour deux niveaux de taux d'écrouissage. Les résultats sont comparés à ceux que l'on obtient, dans des conditions similaires, en état plan de tension.

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This work was supported by the National Aeronautics and Space Administration Research Grant NGR-39-002-023.     

Generalised plane strain problem for a circular inclusion in anisotropic elasticity

This paper is concerned with a generalised plane deformation problem in the linear theory of anisotropic elasticity. As is well known, the generalised plane deformation is the deformation of a body of infinite length bounded by a cylindrical surface, when all the stress and strain components exist but they are functions of two co-ordinates x₁, and x₂ only. It may be shown that if u₃ = 0, it is impossible to satisfy all the three equations of equilibrium of anisotropic elastic body. One has to choose u₃ as a non-zero function of x₁, x₂ for satisfying equations of equilibrium. In isotropic elasticity, u₃ = 0, makes the third equation of equilibrium identically equal to zero.The problem in this paper concerns an elastic circular cylindrical inclusion embedded in a matrix of different anisotropic material. The matrix and the inclusion are perfectly bonded at the interface. Each of the two materials possesses anisotropy of a general form with all the 21 elastic constants. The matrix is subjected to a uniform stress at infinity. The equations of elasticity theory demand that the rotation component ω₃ must also be prescribed at infinity. The complex variable technique is used and exact analytical expressions are derived for the elastic field in both the regions.     

Finite plane strain deformations of an infinite compressible nonlinear elastic body containing either a cylindrical elliptical void or a rigid cylindrical elliptical inclusion

Summary We study the title nonlinear problem for a rubberlike material by using the domain perturbation method. The perturbation is deviation from unity of the ratio of the minor to the major axes of the ellipse. For the body containing an ellipsoidal void, two different loadings are considered; one in which the void surface is taken to be traction free and a uniform compressive load is applied at infinity, and the other in which a uniform pressure is applied to the void surface and null tractions at infinity. For the case of a rigid inclusion, uniform normal tensile tractions are applied at infinity. It is shown that a slight deviation from the circular shape of the cavity or the inclusion has a noticeable effect on the maximum stresses induced in the nonlinear elastic body.     

A strong solution for a generalized plane strain unsymmetrical piezoelectric bi-layer laminates

Interfacial stresses, electric fields, and electric displacements of a piezoelectric unsymmetrical bi-layer orthotropic laminate are presented. A state space equation for orthotropic piezoelectric material is derived from three-dimensional piezoelectric elasticity directly. With the application of the transfer matrix and recursive solution approach, a strong solution for the unsymmetrical piezoelectric generalized plane strain bi-layer laminated structure is sought after considering all elastic and piezoelectric constants of materials. Electromechanical boundary layer effect is identified quantitatively at free edges. To facilitate the discussion on the results, the corresponding calculations from finite element models are compared with those of the strong solution.     

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

Elasto-plastic strain analysis by a semi-analytical method

The aim of this paper is to develop a simulation model of large deformation problems following a semi-analytical method, incorporating the complications of geometric and material non-linearity in the formulation. The solution algorithm is based on the method of energy principle in structural mechanics, as applicable for conservative systems. A one-dimensional solid circular bar problem has been solved in post-elastic range assuming linear elastic, linear strain hardening material behaviour. Type of loading includes uniform uniaxial loading and gravity loading due to body force, whereas the geometry of the bar is considered to be non-uniformly taper. Results are validated successfully with benchmark solution and some new results have also been reported. The location of initiation of elasto-plastic front and its growth are found to be functions of geometry of the bar and loading conditions. Some indicative results have been presented for static and dynamic problems and the solution methodology developed for one-dimension has been extended to the elasto-plastic analysis of two-dimensional strain field problems of a rotating disk.     

Dynamic shear band development in plane strain compression of a viscoplastic body containing a rigid inclusion

Summary We study the plane strain thermomechanical deformations of a viscoplastic body containing a rigid non-heat-conducting ellipsoidal inclusion at the center. Two different problems, one in which the major axis of the inclusion is parallel to the axis of compression and the other in which it is perpendicular to the loading axis are considered. In each case the deformations are presumed to be symmetric about the two centroidal axes and consequently deformations of a quarter of the block are analyzed. The material of the block is assumed to exhibit strain-rate hardening, but thermal softening. The applied load is such as to cause deformations of the block at an overall strain-rate of 5000 sec^–1. The rigid inclusion simulates the presence of second phase particles such as oxides or carbides in a steel and acts as a nucleus for the shear band.It is found that a shear band initiates near the tip of the inclusion and propagates along a line inclined at 45° to the horizontal axis. At a nominal strain of 0.25, the peak temperature rise near the tip of the vertically aligned inclusion equals 75% of that for the horizontally placed inclusion. The precipitous drop in the effective stress near the inclusion tip is followed somewhat later by a rapid rise in the maximum principal logarithmic strain there.     

The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain problem

The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain (CPS) problem is investigated in this paper. By employing the solutions of doubly quasi-periodic and doubly-periodic Riemann boundary value problems for analytic functions we obtain the general solution in closed form. And approximate analytical expression of the stress intensity factors, which are identical with the known results, are obtained when there are no holes or gaskets in the periodic parallelogram and e ₃=0, F _k=0, T _k=0, k=1,2, with constant load applied at the edges of the crack.     

Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder

By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow cylinder with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are finally presented.     

Thermal mismatch stress of a cylindrical inclusion in a cubic crystal

Thermal mismatch stress of a [0 0 1] cylindrical inclusion in a cubic crystal is theoretically investigated by using complex variable formulation of anisotropic elasticity. The closed-form expressions for the interface stress in terms of material constants are derived. Both the stresses in the inclusion and in the matrix are determined. Some numerical results are provided to illustrate the general features of the stress distribution. The stress intensity factors associated with radial cracks emanating from the inclusion are also determined. Based on these results brief discussions on the possible damage modes of the composite due to thermal mismatch are given.     

Elasto-plastic dynamic analysis of plane frames and deep arches

The dynamic response of elasto-plastic frames and arches is investigated using a discrete system approach. The governing equations of motion are formulated through the virtual work principle and supplemented by the compatibility conditions established through the conjugate segment analogy. Time marching is carried out through direct time integration process using backward differences. The method, requiring small core storage and short computer time, can be easily implemented on any personal computer.     

Plane deformation of a body with rigid cylindrical inclusion

We study the problem of plane deformation of an infinite elastic body with thin rigid cylindrical inclusion with oval cross section. The body is loaded by biaxial uniform tensile forces at infinity. The solution of the problem is reduced to two singular integral equations with Cauchy kernels for the jumps of normal and tangential stresses on the surface of the inclusion. The solutions of these equations are obtained in the closed analytic form and, used to deduce the formulas for the concentration of stresses near the inclusion, for stresses inside the inclusion, and for the angle of rotation of the inclusion as a rigid body. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Ukrainian State University of Forestry Engineering, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 87–92, November–December, 1996.     

Boundary-value problem of heat conduction for a layer with foreign cylindrical inclusion

Using generalized functions, we obtain a heat-conduction equation with discontinuous and singular coefficients for an isotropic layer with heat-generating foreign cylindrical inclusion. Using a piecewise-linear approximation of temperature on the surfaces of the inclusion and the Hankel integral transformation, we construct an analytic solution of the boundary-value problem of heat conduction with heat emission. A numerical analysis of the considered system is carried out.     

An approximate solution for a plane strain hydraulic fracture that accounts for fracture toughness,fluid viscosity,and leak-off

The goal of this paper is to develop an approximate solution for a propagating plane strain hydraulic fracture, whose behavior is determined by a combined interplay of fluid viscosity, fracture toughness, and fluid leak-off. The approximation is constructed by assuming that the fracture behavior is primarily determined by the three-process (viscosity, toughness, and leak-off) multiscale tip asymptotics and the global fluid volume balance. First, the limiting regimes of propagation of the solution are considered, that can be reduced to an explicit form. Thereafter, applicability regions of the limiting solutions are investigated and transitions from one limiting solution to another are analyzed. To quantify the error of the constructed approximate solution, its predictions are compared to a reference numerical solution. Results indicate that the approximation is able to predict hydraulic fracture parameters for all limiting and transition regimes with an error of under one percent. Consequently, this development can be used to obtain a rapid solution for a plane strain hydraulic fracture with leak-off, which can be used for quick estimations of fracture geometry or as a reference solution to evaluate accuracy of more advanced hydraulic fracture simulators.     

Closed-form solution for a coated circular inclusion under uniaxial tension

A coated circular inclusion embedded in an infinite matrix is analyzed in the framework of two-dimensional isotropic linear elasticity. A closed-form solution is obtained for the case of far-field uniaxial tension using Muskelishvilis complex potential method. The solutions for the stress and strain distributions for all three regions, that is, matrix, coating, and inclusion, were obtained for various coating-to-matrix shear modulus ratios, while keeping the fiber and matrix shear moduli the same. Test cases for an inclusion without the coating and hollow inclusion were also studied. The energy release rate was evaluated using the path-independent M-integral, which is used to calculate the energy release rate for the self-similar expansion of defects surrounded by the closed contour of the integral. The results for the stress and strain concentrations along with the energy release rate due to this material inhomogeneity were analyzed to yield a better understanding of the mechanics of materials with circular inclusions. This can be helpful in designing intelligent composite structures with embedded optical fiber sensors.     

An axisymmetric external crack problem for an infinite medium with a cylindrical inclusion

Summary This paper deals with the determination of stresses in an infinite medium containing an external crack surrounding a cylindrical inclusion. The two media are assumed to be homogeneous, isotropic and elastic but with different elastic constants. The continuity of stresses and displacements is assumed at the common cylindrical surface due to perfect bonding. The problem is reduced to the solution of a Fredholm integral equation of the second kind. A closed-form expression is obtained for the stress-intensity factor. The integral equation is solved numerically and the results are used to obtain the numerical values of the stress-intensity factor which are displayed graphically.The authors thank the National Research Council of Canada for supporting this research through NRC Grant No. A-4177.