Orthotropic Axes Research Articles

Modeling the Rotation of Orthotropic Axes of Sheet Metals Subjected to Off-Axis Uniaxial Tension

A simplified version of a newly developed anisotropic plasticity theory is presented to describe the anisotropic flow behavior of orthotropic polycrystalline sheet metals under uniaxial tension. The theory is formulated in terms of the intrinsic variables of principal stresses and a loading orientation angle and its uniaxial tension version requires a non-quadratic stress exponent and up to five anisotropic material functions of the loading orientation angle to specify a flow condition, a flow rule for plastic strain rates, a flow rule for macroscopic plastic spin, and an evolution law of isotropic hardening. In this investigation, the proper analytical form and the associated parameter identification of the anisotropic material functions defining the flow rule of macroscopic plastic spin are discussed for sheet metals with persistent but rotated orthotropic symmetry axes under off-axis uniaxial tension. It is shown that the proposed flow rule of macroscopic plastic spin can successfully model the experimental data on the rotation of orthotropic symmetry axes in the three sheet metals reported, respectively, by Boehler et al. (Boehler and Koss, 1991, Advances in Continuum Mechanics, O. Bruller et al., eds., Springer, Heidelberg, pp. 143–158; Losilla, Boehler, and Zheng, 2000, Acta Mech. 144, pp. 169–183); Kim and Yin (1997, J. Mech. Phys. Solids 45, pp. 841–851); and Bunge and Nielsen (1997 Int. J. Plasticity 13, pp. 435–446).

Wrinkling of anisotropic metal sheets under deep-drawing: analytical and numerical study

The onset of wrinkling in sheet metal forming is investigated using an analytical approach and finite element (FE) simulations. In both cases the yield criterion proposed by Ferron et al. [Int. J. Plast. 10 (1994) 431] for orthotropic sheets is employed. The analytical approach is developed on the basis of the bifurcation criterion developed by Hutchinson [J.W. Hutchinson, Plastic buckling, in: C.-S. Yih (Ed.), Advances in Applied Mechanics, vol. 14, 1974, pp. 67–144] for thin and shallow shells submitted to a biaxial plane stress loading. Wrinkling limit curves obtained with the bifurcation analysis are drawn for different orientations of the orthotropic axes with respect to the principal stress axes, assuming also different orientations with respect to the geometric axes of principal curvatures. The numerical simulations are performed with the FE code ABAQUS/Explicit. Experimental results of deep-drawing processes taken from the literature are also analysed to check the reliability of the wrinkling predictions obtained with the bifurcation analysis and with the FE simulations. Both analytical and numerical predictions compare reasonably well with experiments.

Influence of anisotropy on the limit load of bi-material welded cracked joints subject to tension

A plane-strain upper bound limit load solution for bi-material welded joints subject tension is found. It is assumed that each material obeys Hill's orthotropic yield criterion and one of the principal axes of orthotropy coincides with the tensile direction. A crack of arbitrary length is located at the bi-material interface. The solution is based on a simple discontinuous kinematically admissible velocity field and is an extension of the corresponding solution for the specimen made of isotropic materials. The main purpose of the paper is to demonstrate that the influence of anisotropy on the magnitude of limit loads may be much more significant than other effects.

Numerical methods for porous metals with deformation-induced anisotropy

A constitutive model for porous metals subjected to general three-dimensional finite deformations is presented. The model takes into account the evolution of porosity and the development of anisotropy due to changes in the shape and the orientation of the voids during deformation. Initially, the pores are assumed to be ellipsoids distributed randomly in an elastic–plastic matrix (metal). This includes also the special case in which the initial shape of the voids is spherical and the material is initially isotropic. Under finite plastic deformation, the voids are assumed to remain ellipsoids but to change their volume, shape and orientation. At every material point, a “representative” ellipsoid is considered and the homogenized continuum is assumed to be locally orthotropic, with the local axes of orthotropy coinciding with the principal axes of the representative local ellipsoid. The orientation of the principal axes is defined by the unit vectors n (1) , n (2) , n (3)= n (1)× n (2) and the corresponding lengths are 2 a 1, 2 a 2 and 2 a 3. The basic “internal variables” characterizing the state of the microstructure at every point in the homogenized continuum are given by the local equivalent plastic strain ϵ ̄ p in the metal matrix, the local void volume fraction (or porosity) f, the two aspect ratios of the local representative ellipsoid ( w 1= a 3/ a 1, w 2= a 3/ a 2) and the orientation of the principal axes of the ellipsoid ( n (1), n (2), n (3)) . A methodology for the numerical integration of the elastoplastic constitutive model is developed. The problems of uniaxial tension, simple shear, plastic flow localization and necking in plane strain tension, and ductile fracture initiation at the tip of a blunt crack are analyzed in detail; comparisons with the isotropic Gurson model are made.

Rules of crown development in the clonal shrub Vaccinium hirtum in a low-light understory: a quantitative analysis of architecture

The rules of crown development in the clonal shrub Vaccinium hirtum Thunb. in a low-light understory were identified by architectural analysis, and the structure and dynamics of current-year shoots were quantified. Development started from an initial orthotropic axis, which forked into plagiotropic axes; consequently, arched stems were formed. Subsequently, a new orthotropic shoot arose from the dormant meristem on the stem. The process from orthotrophy to plagiotrophy was then repeated. Ramets developed vertically as a result of the repeated formation of such orthotropic shoots and reached a maximum height of about 2 m. These processes were mainly characterized by the sequential change from orthotrophy to plagiotrophy in stem orientation and by a sequential reduction in shoot growth, in which a long shoot forked into shorter shoots with increasing allocation to leaves relative to stems. These rules may be adaptive for efficient light capture under conditions of low light availability, in terms of a low degree of self-shading and low cost of supporting tissue. Simultaneously, these rules worked to restrict the developmental potential of crowns, including the lateral expansion of crowns and the longevity of branch systems. This may be associated with a shrub-specific, throwaway design for stems that are expendable in high-stress environments.Key words: crown architecture, branching, leaf display, architectural unit, reiteration.

Cracking behavior of RC shear walls subject to cyclic loadings

This paper presents a numerical model for simulating the nonlinear response of reinforced concrete (RC) shear walls subject to cyclic loadings. The material behavior of cracked concrete is described by an orthotropic constitutive relation with tension-stiffening and compression softening effects defining equivalent uniaxial stress-strain relation in the axes of orthotropy. Especially in making analytical predictions for inelastic behaviors of RC walls under reversed cyclic loading, some influencing factors inducing the material nonlinearities have been considered. A simple hysteretic stress-strain relation of concrete, which crosses the tension-compression region, is defined. Modification of the hysteretic stress-strain relation of steel is also introduced to reflect a pinching effect depending on the shear span ratio and to represent an average stress distribution in a cracked RC element, respectively. To assess the applicability of the constitutive model for RC element, analytical results are compared with idealized shear panel and shear wall test results under monotonic and cyclic shear loadings.

Cracking behavior of RC shear walls subject to cyclic loadings

This paper presents a numerical model for simulating the nonlinear response of reinforced concrete (RC) shear walls subject to cyclic loadings. The material behavior of cracked concrete is described by an orthotropic constitutive relation with tension-stiffening and compression softening effects defining equivalent uniaxial stress-strain relation in the axes of orthotropy. Especially in making analytical predictions for inelastic behaviors of RC walls under reversed cyclic loading, some influencing factors inducing the material nonlinearities have been considered. A simple hysteretic stress-strain relation of concrete, which crosses the tension-compression region, is defined. Modification of the hysteretic stress-strain relation of steel is also introduced to reflect a pinching effect depending on the shear span ratio and to represent an average stress distribution in a cracked RC element, respectively. To assess the applicability of the constitutive model for RC element, analytical results are compared with idealized shear panel and shear wall test results under monotonic and cyclic shear loadings.

Influence of anisotropy on a limit load of weld strength overmatched middle cracked tension specimens

ABSTRACT A plane‐strain upper bound limit load solution for weld strength overmatched middle cracked tension specimens (M(T) specimens), is found. It is assumed that the weld material is isotropic, but the base material is orthotropic and its axes of orthotropy are straight and parallel to the axes of symmetry of the specimen. A quadratic orthotropic yield criterion is adopted. The solution is based on a simple discontinuous kinematically admissible velocity field and is an extension of the corresponding solution for the specimen made of isotropic materials. These two solutions are compared to demonstrate the influence of anisotropy on the magnitude of the limit load.

Description of anisotropy of textured metals using macro and micro models

In the paper description of anisotropy of textured metals using macro and micro models is compared. In the present macro-approach the texture anisotropy is described by introducing a second order microstructure tensor whose principal directions specify the orthotropy axes. A scalar orientation parameter is introduced in order to specify the relative orientation of the generalized traction vector with respect to anisotropy axes. The yield stress is assumed to depend on the parameter η with specific forms introduced in order to provide quantitative description of yield stress variation with loading orientation. The phenomenological approach is next confronted with the microstructural approach based on the analysis of crystallographic slip and induced lattice reorientation in a representative grain aggregate. The resulting yield surface for polycrystalline aggregate is then compared with the macroscopic description based on the orientational variation of the yield stress. The proposed macro description seems much simpler from micro-approach and could prove convenient in the analysis of technological processes for textured materials.

Petiole twisting in the crowns of Psychotria liminesis: implications for light interception and daily carbon gain.

We used Y-plant, a computer-based model of crown architecture, to examine the implications of leaf reorientation resulting from petiole bending in Psychotria limonensis (Rubiaceae) seedlings. During this reorientation process, bending of the petioles of lower leaves that are potentially self-shaded by the upper leaves rotates the lamina around the stem's orthotropic axis so that self-shading is reduced. Simulations of daily light capture and assimilation revealed a 66% increase in daily C gain due to reorientation of the leaves as compared to simulations where the leaves remained in their characteristic opposite decussate pattern set by the phyllotaxy. This was due to enhanced carbon (C) gain of the lower leaves because of the reduction of shading by upper developing leaves in the same vertical plane. The light signal for this movement was experimentally examined by placing leaf-shaped filters above already fully expanded leaves and following the resulting shade-avoiding movements. The filters were either neutral density shade cloth that reduced the photon flux density (PFD) but did not alter the red to far red ratio (R:FR) or a film that reduced the PFD equivalently but also reduced the R:FR. Leaf reorientation was much more rapid and complete under the low R:FR as compared to the high R:FR indicating involvement of a phytochrome photosensory system that detected the presence of a shading leaf. Plants in gaps were found to lack a reorientation response indicating that the reorientation is specific to the shaded understory environment.

Limit analysis of orthotropic plates

The limit analysis problem for plates in bending is considered. The failure criterion for the material is assumed as orthotropic, with possible non-symmetric strength properties. According to Kirchhoff’s hypothesis, the plate is conceived as a superposition of layers, individually in plane stress situation, and continuity is enforced by means of a kinematic assumption. By exploiting previous results, recently established by the authors, the expression of the dissipation power per unit plate area is defined on this basis and the kinematic (upper bound) theorem of limit analysis is cast in a form suitable for numerical solutions. To this purpose, efficient algorithms successfully employed in the isotropic case can be used with minor modifications. The effectiveness of the procedure is demonstrated by solving some homogeneous plate examples. Results permit the assessment of the influence of different aspects, such as the ratio between strengths along the orthotropy directions, the tensile to compressive strength differential and the inclination of the orthotropy axes with respect to the sides. The effects of in-plane edge constraints are also discussed and it appears that they are emphasized considerably by anisotropy. Even if referred to specific cases, some conclusions can be regarded as fairly general.

Complete In-plane Elastic Characterisation under Tensile Tests of Angle-Ply Laminates Composed of Polymer-Matrix Layers

In this paper we present a new strategy to completely characterise the in-plane elastic properties of a large range of angle-ply laminates using only unidirectional tests. We consider laminates having the same number of identical plies in the α and – α directions. This new method uses some preceding results found by Verchery for orthotropic laminates, namely the conditions of existence of a specific direction ω, in which the shear-extension coupling is null. The characterisation of the laminate is then made using the results of three tensile tests: two in the orthotropy axes, and the third one in the ω direction, in order to have always a pure one-dimensional state of stress. We show that for the most common unidirectional fibre-reinforced materials, the angle ω is, in most cases, close to the α direction of the fibres. This result permits a complete experimental characterisation of the laminate, which then does not need any a priori knowledge of the elastic properties of the elementary layer. In addition, it provides a simple method to verify the predictions of the laminate behaviour obtained by the Classical Laminated Plate Theory (CLPT) when the elementary layer is completely known. The paper ends with numerical examples and with the results of some experimental tests.

Response of Cross-Ply Composite to Off-Axis Loading

Polymer composites are known to exhibit nonlinear stress–strain response due to nonlinearly elastic or plastic deformation of the matrix and damage accumulation. Mechanistic modeling of material response explicitly accounting for these interacting factors often leads to complex theories. Plasticity theory formalism provides an alternative for nonlinear deformation description of composite material. We examine the applicability of an orthotropic plasticity model, developed by Sun et al. for unidirectionally reinforced composite, to composite laminate. The response of a symmetric and balanced cross-ply glass/epoxy laminate is studied under uniaxial tensile loading at different angles to the material orthotropy axis. It is found that the associated flow rule and a quadratic approximation of the orthotropic potential function provide satisfactory description for the nonlinear strain component under monotonic loading for 15–45° off-axis angle range. The nonlinearity in on-axis loading (modulus degradation)is well described by stiffness reduction due to the cracks in transverse plies. Meanwhile the change of elastic modulus due to intralaminar cracking can be neglected in off-axis loading, but the possible intensification of nonlinear deformation in off-axis loading caused by the presence of intralaminar cracks agrees well with orthotropic potential formalism.

Quantifying the Plastic Anisotropy for Particulate Metal Matrix Composites

The failure of orthotropic materials, traditionally described by Hill's criterion, in terms of the anisotropy ratio r, i.e. the ratio of transverse to through thickness increments of logarithmic strain, is studied in the present work. In spite of the critical role of r for the description of the failure of such materials, its quantification is not yet standardized. Based on results obtained from long series of experiments r is here quantified, for the case of a modern particulate composite material, extensively used in aerospace applications. The variation of r versus the level of the plastic strain is determined and it is concluded that its values depend both on the orientation of the specimen and load with respect to the initial orthotropy axes as well as on the plastic strain induced in case the specimen and load are not oriented along one of the axes of orthotropy.

Solution of plane elasticity problems for orthotropic bodies with cracks by the displacement discontinuity method

The force problem of fracture mechanics for orthotropic body is solved by boundary element method. The equations of plane strain state are used. The key point of this paper is obtaining the influence functions for finite segment with constant displacement discontinuities in the main axes of orthotropy. They have been deduced by means of two-dimensional Fourier transform and theory of generalized functions. The straight crack under the action of uniform tension in the infinity is considered by using obtained influence functions. The calculations were carried out for isotropic and two orthotropic materials.

Desarrollo arquitectónico de tres especies de Acacia

Existen pocos estudios sobre el desarrollo vegetativo de árboles de la familia de las leguminosas. Para poder interpretar la evolución de diferentes patrones de desarrollo se requiere el estudio de especies taxonómicamente cercanas. Se analizaron tres especies de Acacia (A. collinsii, A. cedilloi y A. dolichostachya), que a simple vista crecen según los modelos arquitectónicos de Champagnat y Troll, para contestar en un futuro preguntas sobre la relación evolutiva entre estos modelos. Se describe el desarrollo arquitectónico de estas especies. Se encontró que su patrón corresponde a combinaciones de modelos arquitectónicos, y que los ejes mixtos plagiotropos en A. collinsii son reemplazados por ejes ortótropos durante el desarrollo del árbol. Esta observación abre una discusión sobre el origen evolutivo del eje mixto.

COMPARISON OF COMPUTER-AIDED INTEGRATION TECHNIQUES WHEN OBTAINING INITIAL DISPLACEMENTS MATRIX FOR GEOMETRICALLY NON-LINEAR LAMINATED FINITE ELEMENT /KOMPIUTERINIŲ INTEGRAVIMO TECHNOLOGIJŲ PALYGINIMAS SUDARANT GEOMETRIŠKAI NETIESINIO SLUOKSNIUOTOJO BAIGTINIO ELEMENTO PRADINIŲ POSLINKIŲ MATRICĄ

Geometrical non-linearity of the laminated element has not been realized so far in the widely known commercial finite element method packages such as ABAQUS, ALGOR, ANSYS, COSMOS although researches in that field are actively carried out. On the other hand, there is a lot of problems where large displacements and deformations must be dealt with to obtain a precise decision. A wide range of composite orthotopic materials is used in constructions and other fields of technology. Various numerical methods were implemented to handle laminated plates and shells, however most of them are intended for application only with particular types of the structures. The author's aim is to develop a geometrically nonlinear finite element that could be effectively used for analysis of various laminated slabs regardless of their shape, thickness of laminae, properties of materials, direction of orthotropy axes, way of loading and boundary conditions. Obtaining and handling the element's initial displacement matrix used in the iteration process is a highly complicated issue requiring significant amount of computer resources to be involved. One of the most important aims of the research is to develop an element which could be used not only in an expensive multiprocessor mainframes, but also in an usual personal computer. For the structure, a sophisticated finite element TRIPLT having 50 degrees of freedom is used. The geometrical matrix for this element is obtained involving L-coordinates' array while displacements and rotations in the middle of the element are expressed through the nodal displacements (rotations), their derivatives, and displacements (rotations) in the central point. Linear and non-linear components for the geometrical matrix are shown in Eqs 2 and 5. The behaviour of a geometrical non-linear finite elements structure is described by Eq 8. The tangent stiffness matrix consists of the conventional linear elastic stiffness matrix, initial stress matrix and initial displacements matrix which is obtained by Eq 10, using both analytical and/or numerical integrating. The analytical integrating involves expanding of the appropriate expressions into basic matrices (Eqs 11, 12) and using formula 15. The initial displacement matrix in term of constitutive matrix's elements and the basic matrices is shown in Eqs 13 and 14. Numerical integrating is conducted by two methods: those using Hammer and Gauss-Radau weight coefficients. Numerical approach is applied both to the basic matrices and factorised expressions of submatrices involving intermediate arrays and matrices (Eqs 23, 24). Two ways of obtaining the intermediate arrays and matrices are discussed. Because of high complexity of the procedures involved the computer algebra system Mathematica was used for the integrating and recording FORTRAN codes. Comparison of the effectiveness of all the procedures is presented in a table. The investigation results show that the initial displacement matrix obtained by means of numerical integration involves a small amount of arithmetic operations to be handled with a usual personal computer.

Material database optimization using Monte Carlo method in engineering application

Ceramic materials were simulated numerically. The practical goal of the research was to assess numerically the average residual stress value in such materials. The scientific goal of the work was to establish a basic analytical apparatus. One of the main aspects was to create material database, enabling simulation of polycrystalline structure with random orientations of thermal expansion orthotropy axes. Monte Carlo method was adopted for this purpose. The calculation results were analyzed with the use of statistical method, which is analogous to experimental measure analysis. However, the developed methods can be readily used to analyze both ceramics and ceramic composites. Currently, the methods can take advantage of more precise geometrical description to achieve more accurate models reflecting real internal structure of the material.

Limit analysis of anisotropic structures based on the kinematic theorem

Abstract The paper considers perfectly plastic materials with a yield condition of the form Φ σ =F ij σ ij +F ijkl σ ij σ kl ⩽1 corresponding to a second order truncation of the tensor polynomial expression proposed by Tsai and Wu for failure criteria. Such an expression is often employed for materials exhibiting particular forms of anisotropic failure properties, including orthotropic ones, and accounts for non-symmetric strengths. The limit analysis problem is considered next. The formulation based on the kinematic theorem, reducing to the search of the constrained minimum of a convex functional, was successfully employed in the isotropic case for numerical solutions and can be extended to the present context without modifications, provided that the expression for the dissipation power as an explicit function of strain rates is available. For the material considered, this expression is established in this paper. The result is specialized to plane stress orthotropy and an example is worked out. Although extremely simple, it permits the assessment of the influence of the ratio between tensile and compressive strength and of the inclination of the orthotropy axes with respect to loading directions.

Torsion of thin-walled composite beams with midplane symmetry

The optimization of thin-walled beams can lead to the design of structures whose walls have significant differences in stiffness. The stress field resulting from a twisting moment is not correctly modeled by classical analytical theories, so numerical modeling is therefore essential. The analytical theory presented in this study provides us with a simple way of determining stress and stiffness in this type of structure, without having to resort to complex methods of calculation. The walls are made of isotropic or composite materials with midplane symmetry and one orthotropic axis is the longitudinal axis. The section may be open or closed and the constrained warping effect can be taken into account. The results obtained correlate accurately with those obtained from 3D FE modeling.