An extension of generalized plane stress for problems with out-of-plane restraint

The standard concept of generalized plane stress is extended to obtain a new mathematical model for studying the effect of local out-of-plane displacement restraint on the in-plane stresses and displacements in thin plates. It is pointed out how this model may be used by the photoelastician, whose otherwise plane-stress experiment introduces an unavoidable out-of-plane restraint condition in the model, to obtain some estimate of the deviation to be expected between the results of his experiment and the actual plane-stress solution of the problem. In this way, the model may be applied to aid in the interpretation of a large class of two-dimensional photoelastic analyses involving the determination of stresses near rigid inclusions and rigid boundaries. The extended model is then applied to the problem of an annular disk subjected to thermal shrinkage and completely restrained at its outer boundary. In view of the simplicity of the model, the predicted radial and circumferential stress distributions agree remarkably well with existing photoelastic data. In contrast, results obtained from standard generalized plane-stress theory, which cannot account for the out-of-plane displacement restraint at the outer boundary, show substantial deviation from experimental values, especially near the restrained boundary.     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part II

Summary From the continuum mechanics points of view, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity in a continued work (Part III, forthcoming).The supports from the Alexander von Humboldt Foundation, Germany and the Research Foundation of Tsinghua University, P. R. China are acknowledged by the first author.     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part I

Summary From the continuum mechanical point of view, most of practical fibre-reinforced composites may be considered to be transversely isotropic, orthotropic, tetratropic, hexatropic, or octotropic according to material symmetries. Engineering applications of fibre-reinforced materials have already been well established. Based on tensor function representations, the formulation of constitutive equations for fibre-reinforced composites constitutes a consistent mathematical basis for modelling the mechanical behaviour of these oriented materials. Although we have good knowledge of transversely isotropic and orthotropic tensor function representations, only integrity bases for tetratropic and hexatropic polynomials could be learnt up to now in very limited cases. In this paper, we present the complete and irreducible representations for tetratropic, hexatropic and octotropic scalr-, symmetric tensor-, skew-symmetric tensor- and vector-valued functions (not only polynomials) of any finite number of symmetric tensors, skew-symmetric tensors and vectors in plane problems; a generalization of these results is also given. By use of these representations we shall perform the formulation of constitutive equations of fibre-reinforced composites in plane problems of elasticity, heat conduction, asymmetric elasticity, plasticity and continuum damage mechanics in a continued paper (Part II).

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Formulierung von Stoffgleichungen für faserverstärkte Verbundwerkstoffe bei ebenen Problemen: Teil I

Übersicht Im Hinblick auf Materialsymmetrien kann man für den praktischen Einsatz transversal-isotrope, orthotrope, tetratrope, hexatrope oder auch oktotrope faserverstärkte Verbundwerkstoffe unterscheiden. Für Anwendungen im Ingenieurbereich sind sie von größter Bedeutung. Die Darstellungstheorie tensorwertiger Funktionen bildet die Grundlage zur Formulierung von Stoffgleichungen anisotroper Stoffe, zu denen auch die oben erwähnten Verbundwerkstoffe gehören. Tensorfunktionen speziell für transversal-isotrope und orthotrope Stoffe sind hinreichend bekannt. Hingegen fehlen entsprechende Untersuchungen zum tetratropen und hexatropen Verhalten. Daher sollen im folgenden vollständige und irreduzible Darstellungen für derartige Materialorientierungen entwickelt werden. Dazu müssen vektor- und tensorwertige Funktionen mit unterschiedlichen Tensor- und Vektorargumenten in Betracht gezogen werden. Eine weiterführende Untersuchung (Teil II), in der u. a. auch anisotrope Werkstoffschädigungen berücksichtigt werden, ist in Vorbereitung.

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Paper presented at the XVIIIth ICTAM Congress in Haifa, Israel, August 1992.     

Plane strain problems in second-order elasticity theory

The equations of second-order elasticity are developed in polar coordinates R, θ for plane strain deformations of incompressible isotropic elastic materials. By considering a ‘displacement function’ the second-order problem is reduced to the solution of an equation of the form ⁴ψ = g(R, Θ) where ² is Laplace's differential operator and g(R, Θ) depends only on the first-order solution. The displacement function technique is then applied to obtain a second-order solution to the problem of an elastic body contained between two concentric rigid circular boundaries, when the outer boundary is held fixed and the inner is subjected to a rigid body translation.     

Singular boundary method for solving plane strain elastostatic problems

This study documents the first attempt to apply the singular boundary method (SBM), a novel boundary only collocation method, to two-dimensional (2D) elasticity problems. Unlike the method of fundamental solutions (MFS), the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest. Three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the MFS, boundary element method (BEM), and finite element method (FEM).     

Analytical solutions of mixed mode plane strain crack problems in elastic perfectly plastic materials

In the present paper analytical solutions concerning the stress state at the tip of a crack in an elastic-perfectly plastic body, subjected to mixed mode loadings under plane strain conditions, are presented. Analytical solutions of the nonlinear ordinary differential equations are obtained and the dominant singularity is completely determined with the aid of suitable boundary conditions. The obtained results are in perfect agreement with those given by other investigators, both analytical and numerical. The novel aspect here is the methodology used for the solution, as well as the direct determination of the plastic zones. As a consequence, the resulting analytical solutions cover many more problems in the mathematical theory of plasticity compared to similar existing methods and they may be proved of importance in various applications.     

磁弹性广义平面问题的分解形式

通过将分解形式从弹性梁板推广到磁弹性广义平面问题,得到了磁弹性板的分解定理,表明表面不受外力的板内的应力状态可以分解为四部分:平面应力状态、剪切状态、Papkovich-Fadle(P-F)状态、磁应力状态。通过引入并证明了四个引理,简明直接地给出了分解定理的一个严格数学证明。此证明不依赖于双调和函数的P-F本征函数展开,且在证明过程中只应用了一些基本的数学方法,更易于理解。     

A simplified nonlinear theory of the generalized plane strain

A finite deformation theory of plane strain is formulated for transversely isotropic, homogeneous bodies with nonlinear stress-strain law. A new set of simplified field equations, which is valid in the case of some deviations from Hooke's law, is derived systematically with the help of the method of order estimation. For illustration purposes, a circular hole in a body under generalized plane strain is considered, together with the solution of an example problem by perturbation techniques.     

Plane strain slip line theory for anisotropic rigid/plastic materials

A slip line theory governing states of incipient plane flow at the yield point load is developed for anisotropic rigid/plastic materials which exhibit a reduced yield criterion, governing states of plane flow, that depends only on the deviatoric parts of the in-plane stress tensor. It is shown that every homogeneous, incompressible material which complies with the principle of maximum plastic work, but which is of otherwise arbitrary anisotropy, is of this class. The stress equilibrium requirements are seen to take a remarkably simple form expressing the constancy of the quantities mean in-plane normal stress plus or minus arc length around the governing yield contour in a Mohr stress plane along members of the two slip line families. Further, this generalization of the Hencky equations is valid for every material of the considered class. Some special features of yield contours containing corners and flat segments are discussed, and velocity equations are given for materials complying with the maximum work inequality. The theory is applied to obtain the solution for indentation of an arbitrarily anisotropic half-space with a flat-ended, rigid, frictionless punch. A simple, universal formula, involving only geometrical dimensions of the governing yield contour, is derived for the yield point indentation pressure.     

Influence of damage on properties of strain localization in geomaterials at plane stress and plane strain

Summary By regarding geomaterials under loading as a mixture of intact and damaged parts, we investigate the influence of damage on the properties of strain localization in elastoplastic geomaterials at plane stress and plane strain. Conditions for the onset of strain localization including the effects of damage are derived for the cases of plane strain and plane stress. Discussed are the inclination of the localized band and the hardening modulus corresponding to the onset of strain localization. It is shown that the properties of the strain localization are dependent on the damage and the capacity of bearing hydrostatic pressure by the damaged part, and that damage may induce an earlier onset of strain localization and lead to instability of a geomaterial.accepted for publication 11 March 2004     

Bifurcation conditions for ideal fibre-reinforced materials

A bifurcation of an equilibrium state for ideal fibre-reinforced material is discussed. It is assumed that the material is elastic, locally transversely isotropic, incompressible and inextensible in the direction of fibres. On a finite state of strain an arbitrary field of small displacements is superposed and a set of governing equations for the perturbed state is derived.As an example a stability problem of a rectangular block. Objected to a finite, homogeneous deformation is considered. A discussion of the results is focused on the influence on the stability of the pressure applied in the direction of fibres.Due to the assumption of inextensibility this pressure has no influence on the state of strain, but it is shown that it may cause a loss of stability.     

含主应力轴旋转的土体平面应变问题弹塑性数值模拟

在含主应力轴旋转的土体本构关系研究的基础上,通过含主应力轴旋转的土体平面应变问题的弹塑性数值模拟结果分析,总结主应力轴旋转对土体应力分布与应形影响的规律,得出需要考虑主应力轴旋转的条件及影响的相对大小。     

The elastostatic plane strain mode I crack tip stress and displacement fields in a generalized linear neo-Hookean elastomer

A general asymptotic plane strain crack tip stress field is constructed for linear versions of neo-Hookean materials, which spans a wide variety of special cases including incompressible Mooney elastomers, the compressible Blatz–Ko elastomer, several cases of the Ogden constitutive law and a new result for a compressible linear neo-Hookean material. The nominal stress field has dominant terms that have a square root singularity with respect to the distance of material points from the crack tip in the undeformed reference configuration. At second order, there is a uniform tension parallel to the crack. The associated displacement field in plane strain at leading order has dependence proportional to the square root of the same coordinate. The relationship between the amplitude of the crack tip singularity (a stress intensity factor) and the plane strain energy release rate is outlined for the general linear material, with simplified relationships presented for notable special cases.