Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots

This paper deals with an elastic orthotropic inhomogeneity problem due to non-uniform eigenstrains. The specific form of the distribution of eigenstrains is assumed to be a linear function in Cartesian coordinates of the points of the inhomogeneity. Based on the polynomial conservation theorem, the induced stress field inside the inhomogeneity which is also linear, is determined by the evaluation of 10 unknown real coefficients. These coefficients are derived analytically based on the principle of minimum potential energy of the elastic inhomogeneity/matrix system together with the complex function method and conformal transformation. The resulting stress field in the inhomogeneity is verified using the continuity conditions for the normal and shear stresses on the boundary. In addition, the present analytic solution can be reduced to known results for the case of uniform eigenstrain.     

Geometrical shape of in-plane inclusion characterized by polynomial internal stress field under uniform eigenstrains

The internal stress field of an inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains is studied. The study is restricted to the inclusion shapes defined by the polynomial mapping functions mapping the exterior of the inclusion onto the exterior of a unit circle. The inclusion shapes, giving a polynomial internal stress field, are determined for three types of inclusions, i.e., an inhomogeneous inclusion with an elastic modulus different from the surrounding matrix, an inhomogeneous inclusion with the same shear modulus but a different Poisson’s ratio from the surrounding matrix, and a homogeneous inclusion with the same elastic modulus as the surrounding matrix. Examples are presented, and several specific conclusions are achieved for the relation between the degree of the polynomial internal stress field and the degree of the mapping function defining the inclusion shape.     

Acoustoelastic determination of stress in slightly orthotropic materials

A method is proposed to determine stresses in acoustoelasticity by making use of orthotropic stress-acoustic relations and the equations of equilibrium. It is derived theoretically that shear stress is determined ny ultrasonic data ofB and ?, which denote a magnitude of acoustic birefringence and its principal direction, respectively. Other stress components are obtained by numerical integration of the equilibrium equation with the shear stress thus determined. Experiments were carried out to show the validity and usefulment of the method. This method was applied to the measurement of stress field on a plate with a circular hole subjected to axial tension. Ultrasonic measurements were made by a Y-cut quartz transducer with 5-MHz fundamental frequency. The specimen was cut out from 1100 aluminum plate of 4-mm thickness, which shows a slight orthotropy due to roll working. The values ofB and ? were measured in both stressed and unstressed state. Then, stress distributions were determined by the method proposed here, and are compared with the known theoretical distributions.     

正交异性材料平面裂纹尖端应力场

根据正交各向异性材料力学性能确定出了用应力函数表示的弹性力学基本方程,利用坐标变换和复变函数方法求解了正交异性材料平面裂纹体的应力边值问题。借鉴一般断裂力学解法构造了I型和II型裂纹问题的应力函数,推导出了正交各向异性板裂纹尖端区的奇异应力场。通过数值计算说明了裂纹尖端应力表达式的正确性,验证了裂尖前沿应力变化规律,即σx与材料特征参数h2成正比,而σy和τxy不随材料特性变化。     

Direction dependent orthotropic model for Mullins materials

The motivating key for this work was the absence of a phenomenological model that can reasonably predict a variety of non-proportional experimental data on the anisotropic Mullins effect for different types of rubber-like materials. Hence, in this paper, we propose a purely phenomenological direction dependent orthotropic model that can describe the anisotropic Mullins behaviour with permanent set and, has orthotropic invariants that have a clear physical interpretation. The formulation is based on an orthotropic principal axis theory recently developed for nonlinear elastic problems. A damage function and a direction dependent damage parameter are introduced in the formulation to facilitate the analysis of anisotropic stress softening in rubber-like materials. A direction dependent free energy function, written explicitly in terms of principal stretches, is postulated. The proposed theory is able to predict and compares well with experimental data available in the literature for different types of rubberlike materials.     

Torsion of cylindrically orthotropic elastic circular bars with radial inhomogeneity: some exact solutions and end effects

Torsion of elastic circular bars of radially inhomogeneous, cylindrically orthotropic materials is studied with emphasis on the end effects. To examine the conjecture of Saint-Venant’s torsion, we consider torsion of circular bars with one end fixed and the other end free on which tractions that results in a pure torque are prescribed arbitrarily over the free end surface. Exact solutions that satisfy the prescribed boundary conditions point by point over the entire boundary surfaces are derived in a unified manner for cylindrically orthotropic bars with or without radial inhomogeneity and for their counterparts of Saint-Venant’s torsion. Stress diffusion due to the end effect is examined in the light of the exact solutions.     

Residual-stress measurement in orthotropic materials using the hole-drilling method

The hole-drilling method is used here to measure residual stresses in an orthotropic material. An existing stress-calculation method adapted from the isotropic case is shown not to be valid for orthotropic materials. A new stress-calculation method is described, based on the analytical solution for the displacement field around a hole in a stressed orthotropic plate. The validity of this method is assessed through a series of experimental measurements. A table of elastic compliances is provided for practical residual-stress measurements in a wide range of orthotropic materials.     

Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroeiastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity , totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.     

Plane-stress crack-tip stress fields for orthotropic perfectly-plastic materials

Under the hypothesis that the stress components of crack-tip fields are only thefunctions ofθ,the differential equations of plane-stress crack-tip stress fields fororthotropic perfectly-plastic materials are obtained by using Hill’s yield condition andequilibrium equations.By combining the general analytical expression with the numericalmethod the crack-tip stress fields for orthotropic perfectly-plastic materials for plane stressare presented.     

Surface instability of orthotropic laminated materials under compression

The piecewise-homogeneous material model and the three-dimensional linearized theory of stability with the assumption of small subcritical strains are used to study the surface buckling of orthotropic and transtropic laminates. A plane problem is formulated, and characteristic equations are derived. A solution is found for a specific transtropic material with different orientations of the isotropy axis __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 64–72, July 2006.     

Symmetry relations for orthotropic and transversely isotropic materials

Archive for Rational Mechanics and Analysis -     

Designing orthotropic materials for negative or zero compressibility

There has been considerable interest in materials exhibiting negative or zero compressibility. Such materials are desirable for various applications. A number of models or mechanisms have been proposed to characterize the unusual phenomena of negative linear compressibility (NLC) and negative area compressibility (NAC) in natural or synthetic systems. In this paper we propose a general design technique for finding metamaterials with negative or zero compressibility by using a topology optimization approach. Based on the bi-directional evolutionary structural optimization (BESO) method, we establish a systematic computational procedure and present a series of designs of orthotropic materials with various magnitudes of negative compressibility, or with zero compressibility, in one or two directions. A physical prototype of one of such metamaterials is fabricated using a 3D printer and tested in the laboratory under either unidirectional loading or triaxial compression. The experimental results compare well with the numerical predictions. This research has demonstrated the feasibility of designing and fabricating metamaterials with negative or zero compressibility and paved the way towards their practical applications.     

Interface crack problem in layered orthotropic materials under thermo-mechanical loading

In the present paper, the behavior of an interface crack for a homogeneous orthotropic strip sandwiched between two different functionally graded orthotropic materials subjected to thermal and mechanical loading is considered. It is assumed that interface crack is partly insulated, and the temperature drop across the crack surfaces is the result of the thermal resistance due to the heat conduction through the crack region. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The complicated mixed boundary problems of equations of heat conduction and elasticity are converted analytically into singular integral equations, which are solved numerically. The main objective of the paper is to study the effects of material nonhomogeneity parameters and the dimensionless thermal resistance on the thermal stress intensity factors for the purpose of gaining better understanding of the thermal behavior of graded layer.     

Buckling of an orthotropic graded coating with an embedded crack bonded to a homogeneous substrate

To simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an orthotropic stress–strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the critical buckling strain. The Finite Element Method was additionally used to model the crack problem. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions, coating thicknesses and different orthotropic FGMs.