Showing papers on "Orthotropic material published in 1990"

Singularities, interfaces and cracks in dissimilar anisotropic media

Abstract: For a non-pathological bimaterial in which an interface crack displays no oscillatory behaviour, it is observed that, apart possibly from the stress intensity factors, the structure of the near-tip field in each of the two blocks is independent of the elastic moduli of the other block. Collinear interface cracks are analysed under this non-oscillatory condition, and a simple rule is formulated that allows one to construct the complete solutions from mode III solutions in an isotropic, homogeneous medium. The general interfacial crack-tip field is found to consist of a two-dimensional oscillatory singularity and a one-dimensional square root singularity. A complex and a real stress intensity factors are proposed to scale the two singularities respectively. Owing to anisotropy, a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields, however defined, does not recover the classical stress intensity factors as the bimaterial degenerates to be non-pathological. Collinear crack problems are also formulated in this context, and a strikingly simple mathematical structure is identified. Interactive solutions for singularity-interface and singularity interface-crack are obtained. The general results are specialized to decoupled antiplane and in-plane deformations. For this important case, it is found that if a material pair is non-pathological for one set of relative orientations of the interface and the two solids, it is non-pathological for any set of orientations. For bonded orthotropic materials, an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized. A complex-variable representation is presented for a class of degenerate orthotropic materials. Throughout the paper, the equivalence of the Lekhnitskii and Stroh formalisms is emphasized. The article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids.

Constitutive modelling of orthotropic plasticity in sheet metals

Abstract: T he classical quadratic yield criterion for orthotropic metals is known not to be sufficiently flexible in practice. By the simple expedient of admitting non-integer exponents, however, an improved criterion was devised for sheet with in-plane isotropy (so-called normal anisotropy). On the other hand, an acceptable proposal has not been forthcoming for sheet with in-plane anisotropy (so-called planar anisotropy). It is suggested here that improvement should be sought by incorporating a compatible dependence on orientation in a homogeneous yield function of arbitrary degree. In so doing, the practicalities of forming technology are respected by keeping the number of arbitrary parameters as small as possible. A new criterion is constructed along these lines and its implications are explored in detail. Additionally, a simple means of representing anisotropic yield criteria of any kind is presented with supporting general theorems.

Delamination Specimens for Orthotropic Materials

Abstract: A semi-infinite crack in an infinite strip of orthotropic material is analyzed. Analytic expressions for mixed-mode stress intensity factors are derived with only parameter undetermined, which is then extracted from numerical solutions to integral equations. The results are relatively simple and complete, and provide the flexibility to simulate a wide range of practical problems, such as fracture specimens and edge delamination phenomena of woods and fiber-reinforced composites. As an illustration, specimens with transverse splitting from notches are analyzed based on the general solution. The validity of using solutions for an isotropic material to calibrate some testing geometries of orthotropic materials is discussed.

Nonlinear analysis in soil mechanics : theory and implementation

Abstract: Part I. FUNDAMENTALS. 1. Introduction. Characteristics of soil behavior.Idealizations and material modeling. Historical review of plasticity in soil mechanics. Nonlinear stress analyses in geotechnical engineering. Need, objectives and scope. References. 2. Basic Concept of Continuum Mechanics. Introduction. Notations. Stresses in three dimensions. Definitions and notations. Cauchy's formulas, index notation, and summation convention. Principal axes of stresses. Deviatoric stress. Geometrical representation of stresses. Strains in three dimensions. Definitions and notations. Deviatoric strain. Octahedral strains and principal shear strains. Equations of solid mechanics. Equations of equilibrium (or motion). Geometric (compatibility) conditions. Constitutive relations. Summary. References. Part II. MATERIAL MODELING-BASIC CONCEPTS. 3. Elasticity and Modeling . Introduction. Elastic models in geotechnical engineering. Linear elastic model (generalized Hooke's law). Cauchy elastic model. Hyperelastic model. Hypoelastic model. Uniqueness, stability, normality, and convexity for elastic materials. Uniqueness. Drucker's stability postulate. Existence of W and v. Restrictions - normality and convexity. Linear elastic stress-strain relations. Generalized Hooke's law. A plane of symmetry. Two planes of symmetry (orthotropic symmetry). Transverse and cubic isotropies. Full isotropy. Isotropic linear elastic stress-strain relations. Tensor forms. Three-dimensional matrix forms. Plane stress case. Plane strain case. Axisymmetric case. Isotropic nonlinear elastic stress-strain relations based on total formulation. Nonlinear elastic model with secant moduli. Cauchy elastic model. Hyperelastic (green) model. Isotropic nonlinear elastic stress-strain relations based on incremental formulation. Nonlinear elastic model with secant muduli. Cauchy elastic model. Hyerelastic model. Hypoelastic model. Summary. References. 4. Perfect Plasticity and Modeling. Introduction. Deformation theory. An illustrative example. Variable moduli models. Flow theory. Yield criteria. Flow rule. Basic requirements. Perfect plasticity models. Tresca and von Mises models. Coulomb model. Drucker-Prager model. Prandtl-Reuss stress-strain relations. Generalized stress-strain relations. Stiffness formulation. General description. Stiffness coefficients. Summary. References. 5. Hardening Plasticity and Modeling. Introduction. Flow theory. Loading function. Hardening rule. Flow rule. Drucker's postulate. Hardening plasticity models. Lade-Duncan model. Lade model. Nested yield surface models. Generalized multi-surface models. Bounding surface models. Prandtl-Reuss stress-strain relations. Prandtl-Reuss equations. Matrix form of Prandtl-Reuss equations. Generalized stress-strain relations. Incremental stress-strain relations. Isotropic hardening. Kinematic hardening. Mixed hardening. Stiffness formulation. General description. Stiffness coefficients. Summary. References. PART III.

Studies in anisotropic plasticity with reference to the Hill criterion

Abstract: Algorithms based upon the notion of return mapping have been developed for the Hill yield function of anisotropic plasticity. The relative accuracy of two algorithms is assessed by means of iso-error maps. The choice of the algorithm turns out to be much more critical for the orthotropic Hill criterion than for the underlying isotropic von Mises plasticity model. A tangent operator that is consistent with the developed integration algorithm is formulated and its efficiency is assessed compared with the classical continuum tangent operator. The model has been applied to three shell/plate structures.

Bounds on elastic energy in solids of orthotropic materials

Abstract: Elastic energy is the measure for the overall stiffness/flexibility of a solid. For orthotropic materials, a combined finite element/optimization procedure is described, by which the orientation field of the most stiff/flexible solid can be determined. A study of these bounds for possible materials is performed and more general insight obtained. Recent analytical results from sensitivity analysis are applied and the paper also adds to theoretical aspects, such as proving coinciding directions for principal strains and principal stresses, even with different (but optimal) principal directions of the material.

Effective Elastic Moduli of Ribbon-Reinforced Composites

Abstract: Based on the Eshelby-Mori-Tanaka theory the nine effective elastic constants of an orthotropic composite reinforced with monotonically aligned elliptic cylinders, and the five elastic moduli of a transversely isotropic composite reinforced with two-dimensional randomly-oriented elliptic cylinders, are derived. When the aspect ratio approaches zero, the elliptic cylinders exist as thin ribbons, and these moduli are given in very simple, explicit forms as a function of volume fraction. These results are in direct contrast to those of circular fibers

Two-Dimensional Stress-Strain Relationship for Canine Pericardium

Abstract: Two-dimensional pseudoelastic mechanical properties of the canine pericardium were investigated in vitro. The pericardium was assumed to be orthotropic. The material symmetry axis was determined a priori and aligned with the stretching axis. Various biaxial stretching tests were then performed and a set of data covering a wide range of strains was constructed. This complete data set was fitted to a new exponential type constitutive model, and a set of true material constants was determined for each specimen. Using the constitutive model and the true material constants, the results from constant lateral force tests and constant lateral displacement tests were predicted and compared with experiment.

Analysis of Simply-Supported Orthotropic Cylindrical Shells Subject to Lateral Impact Loads

Abstract: The analysis is based on an expansion of the loads, displacements, and rotations in a double Fourier series which satisfies the end boundary conditions of simple support. By neglecting in-plane and rotary inertia the problem becomes a second-order ordinary differential equation in time for the Fourier coefficients of the radial deflection

Reinforced Concrete Membrane Element Formulations

Abstract: Finite‐element formulations are presented for the analysis of reinforced concrete membrane structures. Cracked reinforced concrete is treated as an orthotropic material based on a smeared, rotating crack model. Secant‐stiffness moduli are defined for concrete and reinforcement, and these are used in the development of linear displacement rectangular and triangular membrane finite elements. Procedures are discussed by which these elements can then be incorporated into a nonlinear analysis algorithm. Extensions to the formulations are also described that permit the consideration of prestrain effects in the component materials such as prestressing of the reinforcement, shrinkage of the concrete, or thermal expansion. The constitutive relations currently utilized are those of the modified compression field theory, although the element formulations are sufficiently generic to easily accommodate other constitutive models. A numerical example is provided to illustrate the simplicity of the calculation procedure ...

Predictor-corrector procedures for stress and free vibration analysis of multilayered composite plates and shells

Abstract: A study is made of two predictor-corrector procedures for the accurate determination of the global, as well as detailed, static and vibrational response characteristics of plates and shells. Both procedures use first-order shear deformation theory in the predictor phase, but differ in the elements of the computational model being adjusted in the corrector phase. The first procedure calculates a posteriori estimates of the composite correction factors and uses them to adjust the transverse shear stiffnesses of the plate (or shell). The second procedure calculates a posteriori the functional dependence of the displacement components on the thickness coordinate. The corrected quantities are then used in conjunction with the three-dimensional equations to obtain better estimates for the different response quantities. Extensive numerical results are presented showing the effects of variation in the geometric and lamination parameters for antisymmetrically laminated anisotropic plates, and simply supported multilayered orthotropic cylinders, on the accuracy of the linear static and free vibrational responses obtained by the predictor-corrector procedures. Comparison is also made with the solutions obtained by other computational models based on two-dimensional shear deformation theories. For each problem the standard of comparison is taken to be the analytic three-dimensional elasticity solution. The numerical examples clearly demonstrate the accuracy and effectiveness of the predictor-corrector procedures.

Relations between elastic constants Cij and texture parameters for hexagonal materials

Abstract: Ultrasonic techniques have recently been applied to the texture characterization in polycrystalline aggregates of hexagonal crystals The basis of this application lies in the relations between the elastic constants Cij of the aggregates, which can be inferred from ultrasonic wave velocity measurements, and the orientation distribution coefficients This communication presents such relations for aggregates which possess orthotropic material symmetry and hexagonal crystal symmetry for Voigt, Reuss, and Hill averaging methods in a unified and concise representation

Interface cracks in anisotropic bimaterials

Abstract: It is known that the displacement at the interface crack surface in a bimaterial under a two-dimensional deformation may be oscillatory. The existence of oscillation depends on the material properties and the orientations of the two materials on both sides of the crack-interface. We show in this paper that the existence of oscillation depends on the material properties only and is otherwise independent of the individual orientation of the two materials. This means that. if the crack surface displacement is oscillatory for one choice of orientations of the two materials, no other orientations obtained by rotating about the x3-axis can alleviate the oscillation. Conversely, if the crack surface displacement is not oscillatory for one choice of orientations of the two materials, no other orientations can generate oscillation. An exception is the Type B bimaterials defined in the paper for which the crack surface displacement is always oscillatory except for a particular Type B bimaterial at a particular choice of relative orientation. For bimaterials which consist of two different orthotropic materials, it is shown that whether the crack surface displacement is oscillatory or not depends on whether C 11 C 22 +C 12 of the two materials are different or not.

BEM for general anisotropic 2D elasticity using particular integrals

Abstract: In the context of the direct boundary-element method (DBEM) for most general two-dimensional elastostatics, new particular integrals are presented in this paper to account for inertial and centrifugal body forces, as well as for those cases of three-dimensional behaviour where the assumption of plane strain (or stress) may remain approximately valid despite the presence of non-zero strain (or stress) in the out-ofplane direction. These integrals have been implemented in a general-purpose multi-region boundary-element code (namely, GPBEST) and tested via examples.

Thermal buckling of laminated composite shells

Abstract: The linear buckling analysis of laminated composite cylindrical and conical shells under thermal load using the finite element method is reported here. Critical temperatures are presented for various cases of cross-ply and angle-ply laminated shells. The effects of radius/thickness ratio, number of layers, ratio of coefficients of thermal expansion, and the angle of fiber orientation have been studied. The results indicate that the buckling behavior of laminated shell under thermal load is different from that of mechanically loaded shell with respect to the angle of fiber orientation. Content H IGH-speed aerospace vehicles consisting of thin-shell elements are subjected to aerodynamic heating. This induces a temperature distribution over the surface and thermal gradient through the thickness of the shell. The compressive stresses, which in these circumstances develop, may cause buckling. Recently fiber-reinforced, laminated composites have begun to be used extensively in aerospace vehicle construction due to the high specific properties of the composites. In view of the above, the thermal buckling analysis of laminated composite shell assumes importance. Thermal buckling of isotropic cylindrical and conical shells have been reviewed by Bushnell. 1 Chang and Card2 investigated thermal buckling of stiffened, orthotropic, multilayered cylindrical shells. The governing equations obtained through the minimization of the total potential energy were solved by the finite difference technique with a few cases of practical problems. Nevertheless, this technique cannot be extended to other complex geometries and loading conditions. In this paper the Semiloof shell element formulated by Irons,3 which was adapted by the authors for thermal stress analyses of laminated plates and shells,4 is being extended to thermal buckling problems. The derivation of governing equation is a standard procedure, which uses the principle of minimum total potential energy. The characteristic finite element equilibrium equation thus obtained is [Ks] [q] = [F] where [Ks] is the structural stiffness matrix, [q} is the nodal displacement vector and [F] is the consistent nodal load vector. In order to establish the critical buckling state corresponding to the neutral equilibrium condition, the second variation of the total potential must be equated to zero, which gives rise to the condition, | [Ks] + \[Kg] j = 0, where [Kg] is the geometric stiffness matrix and X is the eigenvalue. The computer program (COMSAP) developed based on this formulation can handle general temperature variations, lamination parameters, and various boundary conditions. The material properties considered in the analysis of laminated shells are En/En = 10, Glt/Ett = 0.5,

On Isotropic Laminate Configurations

Abstract: A particular class of laminated composites is presented where the stiffness characteristics are isotropic with respect to both in-plane and out-of-plane stiffnesses. The paper presents a determining method for isotropic and orthotropic laminate configurations and shows examples of isotropic laminates consisting of 40 plies. Experimental verifica tion by a tension test and a four-point flexure test has been made on a 40-ply isotropic lam inate. Experimental results of elastic properties show almost isotropic properties.

The Initial Phase of Transient Thermal Stresses Due to General Boundary Thermal Loads in Orthotropic Hollow Cylinders

Abstract: The stresses and displacements are obtained using the Hankel asymptotic expansions for the Bessel functions of the first and second kind. The material properties are assumed to be independent of temperature. A constant applied temperature at the one surface and convection into a medium at a different temperature at the other surface is studied