Principal Coordinate System Research Articles

A progressive damage simulation algorithm for GFRP composites under cyclic loading. Part I: Material constitutive model

A life prediction algorithm and its implementation for a thick-shell finite element formulation for GFRP composites under constant or variable amplitude loading is introduced in this work. It is a distributed damage model in the sense that constitutive material response is defined in terms of meso-mechanics for the unidirectional ply. The algorithm modules for non-linear material behaviour, pseudo-static loading–unloading–reloading response, Constant Life Diagrams and strength and stiffness degradation due to cyclic loading were implemented on a robust and comprehensive experimental database for a unidirectional glass/epoxy ply. The model, based on property definition in the principal coordinate system of the constitutive ply, can be used, besides life prediction, to assess strength and stiffness of any multidirectional laminate after arbitrary, constant or variable amplitude multi-axial cyclic loading. Numerical predictions were corroborated satisfactorily by test data from constant amplitude fatigue of glass/epoxy laminates of various stacking sequences.

A measure of the equivalent shear stress amplitude from a prismatic hull in the principal coordinate system

A measure is proposed in this paper in order to assess the equivalent shear stress amplitude for high-cycle multiaxial tests. It is obtained from a prismatic hull oriented along the principal load axes while it encloses the stress path. Harmonic and non-harmonic load examples are presented for illustrating this definition. Moreover, a comparison with various definitions is presented from two same amplitude stress loads. On these examples, the definitions based on the elliptic and prismatic hulls are too safe and this could have a significant impact on the cost of design.

Shear Deformation Effect in Flexural-torsional Vibrations of Composite Beams by Boundary Element Method (BEM)

In this paper a boundary element method (BEM) is developed for the general flexural-torsional vibration problem of Timoshenko beams of arbitrarily shaped composite cross-section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a non-symmetric cross-section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM-based method. Both free and forced vibrations are examined. Several beams are analyzed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.

Shear deformation effect in flexural-torsional buckling analysis of beams of arbitrary cross section by BEM

In this paper a boundary element method is developed for the general flexural-torsional buckling analysis of Timoshenko beams of arbitrarily shaped cross section. The beam is subjected to a compressive centrally applied concentrated axial load together with arbitrarily axial, transverse and torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-walled theory and the significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.

Mechanical behavior of glass/epoxy tubes under combined static loading. Part I: Experimental

A series of biaxial static tests of E-glass/epoxy tubular specimens [±45] 2, subjected to combined torsion and tension/compression were performed to simulate complex stress states encountered in a wind turbine rotor blade. The failure locus in the effective axial-shear stress plane was derived experimentally while in-plane strain tensor components were measured in the tube outer surface. By means of shell theory and strain measurements in the surface of the specimen, the in-plane shear response of the outer ply was obtained, revealing dependence each time to the combined tube loading. The correlation established between the ratio of transverse normal and in-plane shear stress in the principal coordinate ply system and the elastic shear modulus, suggested a strong dependence, warning on the implications for design and certification procedures.

A homogenisation-based continuum damage mechanics model for cyclic damage in 3D composites

AbstractThis paper develops a 3D homogenisation based continuum damage mechanics (HCDM) model for fibre-reinforced composites undergoing micromechanical damage under cyclic loading. Micromechanical damage in a representative volume element (RVE) of the material occurs by fibre-matrix interfacial debonding, which is incorporated in the model through a hysteretic bilinear cohesive zone model. The proposed model expresses a damage evolution surface in the strain space in the principal damage co-ordinate system or PDCS. PDCS enables the model to account for the effect of non-proportional load history. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterised as a macroscopic internal variable. Cyclic damage parameters are introduced in the monotonic HCDM model to describe the material degradation due to fatigue. Three dimensional damage in composites is accounted for through functional forms of the fourth order damage tensor in terms of components of macroscopic strain and elastic stiffness tensor. The HCDM model parameters are calibrated from homogenisation of micromechanical solutions of the RVE for a few representative cyclic strain histories. The proposed model is validated by comparing results of the HCDM model with pure micromechanical analysis results followed by homogenisation. Finally, the potential of cyclic HCDM model as a design tool is demonstrated through macro-micro analysis of cyclic damage progression in composite structures.

Homogenization-based continuum plasticity-damage model for ductile failure of materials containing heterogeneities

This paper develops an accurate and computationally efficient homogenization-based continuum plasticity-damage (HCPD) model for macroscopic analysis of ductile failure in porous ductile materials containing brittle inclusions. Example of these materials are cast alloys such as aluminum and metal matrix composites. The overall framework of the HCPD model follows the structure of the anisotropic Gurson–Tvergaard–Needleman (GTN) type elasto-plasticity model for porous ductile materials. The HCPD model is assumed to be orthotropic in an evolving material principal coordinate system throughout the deformation history. The GTN model parameters are calibrated from homogenization of evolving variables in representative volume elements (RVE) of the microstructure containing inclusions and voids. Micromechanical analyses for this purpose are conducted by the locally enriched Voronoi cell finite element model (LE-VCFEM) [Hu, C., Ghosh, S., 2008. Locally enhanced Voronoi cell finite element model (LE-VCFEM) for simulating evolving fracture in ductile microstructures containing inclusions. Int. J. Numer. Methods Eng. 76(12), 1955–1992]. The model also introduces a novel void nucleation criterion from micromechanical damage evolution due to combined inclusion and matrix cracking. The paper discusses methods for estimating RVE length scales in microstructures with non-uniform dispersions, as well as macroscopic characteristic length scales for non-local constitutive models. Comparison of results from the anisotropic HCPD model with homogenized micromechanics shows excellent agreement. The HCPD model has a huge efficiency advantage over micromechanics models. Hence, it is a very effective tool in predicting macroscopic damage in structures with direct reference to microstructural composition.

RECONSTRUCTION PERMITTIVITY TENSOR AND PRINCIPAL AXIS FOR UNIAXIAL MEDIUM IN MICROWAVE BAND

The relationship of permittivity tensor of anisotropic medium in principal coordinate system and laboratory coordinate system is given. The characteristic of permittivity tensor of uniaxial anisotropic medium in the laboratory coordinate system is discussed. The transverse permittivity of an anisotropic plate is reconstructed in laboratory coordinate system based on the resonance and polarization characteristics of backscattering radar cross section (RCS) in wide band. Then, a new scheme of reconstructing the principal axis direction for a uniaxial sample plate is proposed, subject to the principal axis is unknown. The backscattering characteristics of a sample plate are discussed when the electromagnetic (EM) wave of different polarization is incident perpendicular to the sample plate. Three sample plates, which are cut perpendicularly to the x � , y � ,a nd zaxes in the laboratory coordinate system, are required. A numerical reconstruction example is given to demonstrate the availability of presented scheme.

Damage Evolution in Composites with a Homogenization-based Continuum Damage Mechanics Model

This paper develops a 3D homogenization-based continuum damage mechanics (HCDM) model for fiber reinforced composites undergoing micromechanical damage. Micromechanical damage in the representative volume element (RVE) is explicitly incorporated in the form of fiber—matrix interfacial debonding. The model uses the evolving principal damage coordinate system as its reference in order to represent the anisotropic coefficients. This is necessary for retaining accuracy with nonproportional loading. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterized as macroscopic internal variable. Damage in 3D composites is accounted for through functional forms of the fourth order damage tensor in terms of macroscopic strain components. The HCDM model parameters are calibrated by using homogenized micromechanical (HMM) solutions for the RVE for a few strain histories. The proposed model is validated by comparing the CDM results with HMM response of single and multiple fiber RVEs subjected to arbitrary loading history. Finally the HCDM model is incorporated in a macroscopic finite element code to conduct damage analysis in a structure. The effect of different microstructures on the macroscopic damage progression is examined through this study.

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.

Homogenization based continuum damage mechanics model for monotonic and cyclic damage evolution in 3D composites

This paper develops a 3D homogenization based continuum damage mechanics (HCDM) model for fiber reinforced composites undergoing micromechanical damage under monotonic and cyclic loading. Micromechanical damage in a representative volume element (RVE) of the material occurs by fiber-matrix interfacial debonding, which is incorporated in the model through a hysteretic bilinear cohesive zone model. The proposed model expresses a damage evolution surface in the strain space in the principal damage coordinate system or PDCS. PDCS enables the model to account for the effect of non-proportional load history. The loading/unloading criterion during cyclic loading is based on the scalar product of the strain increment and the normal to the damage surface in strain space. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterized as a macroscopic internal variable. Three dimensional damage in composites is accounted for through functional forms of the fourth order damage tensor in terms of components of macroscopic strain and elastic stiffness tensors. The HCDM model parameters are calibrated from homogenization of micromechanical solutions of the RVE for a few representative strain histories. The proposed model is validated by comparing results of the HCDM model with pure micromechanical analysis results followed by homogenization. Finally, the potential of HCDM model as a design tool is demonstrated through macro-micro analysis of monotonic and cyclic damage progression in composite structures.

Flexural-torsional buckling analysis of composite beams by BEM including shear deformation effect

In this paper, a boundary element method is developed for the general flexural-torsional linear buckling analysis of Timoshenko beams of arbitrarily shaped composite cross-section. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The beam is subjected to a compressive centrally applied load together with arbitrarily axial, transverse and/or torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the method can treat composite beams of both thin and thick walled cross-sections taking into account the warping along the thickness of the walls, while the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross-section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the analog equation method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency. The significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.

Homogenization Based 3D Continuum Damage Mechanics Model for Composites Undergoing Microstructural Debonding

This paper develops a microscopic homogenization based continuum damage mechanics (HCDM) model framework for fiber reinforced composites undergoing interfacial debonding. It is an advancement over the 2D HCDM model developed by Raghavan and Ghosh (2005, “A Continuum Damage Mechanics Model for Unidirectional Composites Undergoing Interfacial Debonding,” Mech. Mater., 37(9), pp. 955–979), which does not yield accurate results for nonproportional loading histories. The present paper overcomes this shortcoming through the introduction of a principal damage coordinate system (PDCS) in the HCDM representation, which evolves with loading history. The material behavior is represented as a continuum constitutive law involving a fourth order orthotropic tensor with stiffness characterized as a macroscopic internal variable. The current work also extends the model of Raghavan and Ghosh to incorporate damage in 3D composites through functional forms of the fourth order damage tensor in terms of macroscopic strain components. The model is calibrated by homogenizing the micromechanical response of the representative volume element (RVE) for a few strain histories. This parametric representation can significantly enhance the computational efficiency of the model by avoiding the cumbersome strain space interpolations. The proposed model is validated by comparing the CDM results with homogenized micromechanical response of single and multiple fiber RVEs subjected to arbitrary loading history.

Identification of principal screws of two- and three-screw systems

The current paper proposes a unified analytical methodology to identify the principal screws of two- and three-screw systems. Based on the definition of the pitch of a screw, it first obtains an identical homogeneous quadric equation. According to functional analysis theory, it is known that the partial derivatives of an identical quadric equation with respect to its variables must be zero. Therefore, the paper deduces a set of linear homogeneous equations that are made up of the partial derivatives of the quadric equation. With the existing criteria of non-zero solutions for homogeneous linear algebra equations, it ultimately obtains the formulas of the principal pitches and the associated principal screws of the system. The most outstanding contribution of this methodology is that it proposes a unified analytical approach to identify the principal pitches and the principal coordinate systems of the second-order and the third-order screw systems. This should be a new contribution to the screw theory and will boost its applications to the kinematics analysis of robots and spatial mechanisms.

Upscaling of elastic properties of anisotropic sedimentary rocks

SUMMARY In this paper, the term ‘upscaling’ means the theoretical prediction of rock's elastic properties at lower frequency (seismic or cross-well data) using higher frequency logging data on sonic velocities (VP, VS1 and VS2), porosity and density. The mineral composition and water saturation derived from other logs are used. Due to the special treatment of sonic logging data provided by the dipole sonic probe, all the sonic velocities are obtained in the principal coordinate system of the rock's stiffness tensor. The upscaling procedure includes two steps. The first step involves the solution of an inverse problem on reconstruction of the parameters of the rock's microstructure from the logging data. The inversion is based on the effective medium theory. As a result of the inverse problem solution, the effective stiffness tensor is found for depths at which the sonic wave velocities are measured. At the second step, the velocities of waves at given lower frequencies are calculated as propagating in a layered medium. The number of layers in the medium depends on the given frequency and logging step. Each layer of the medium has the stiffness tensor found at the first step. This upscaling procedure has been applied to a wellbore for which the dipole sonic data are available. The rocks penetrated by the well are shales. In general, the resulting medium exhibits orthorhombic symmetry at sonic frequency. This symmetry results from the preferential orientation of clay platelets and grain-related cracks and vertical cracks. The existence of the latter is indicated by the dipole sonic tool. Depending on the microstructure parameters (orientation of clay platelets and cracks, pore/crack connectivity and shale mineralogical composition), the shales, at lower frequency, have either transversely isotropic symmetry (with the vertical axis of symmetry, a.k.a. VTI) or orthorhombic symmetry.

Modeling Cross Anisotropy in Granular Materials

A constitutive model has been developed to capture the behavior of cross-anisotropic frictional materials. The elastoplastic, single hardening model for isotropic materials serves as the basic framework. Based on the experimental results of cross-anisotropic sands in isotropic compression tests, the principal stress coordinate system is rotated such that the model operates isotropically within the rotated framework. Experimental plastic work contours on the octahedral plane are plotted for a series of true triaxial tests on dense Santa Monica Beach sand to study the effects of cross anisotropy on the evolution of yield surfaces. The amount of rotation of the yield and plastic potential surfaces decreases to zero (isotropic state) with loading. The model is constructed for cases where the principal stress and material symmetry axes are collinear and no significant rotation of principal stresses occur. The model incorporates fourteen parameters that can be determined from simple experiments, such as isotropic compression, drained triaxial compression, and triaxial extension tests. A series of true triaxial and isotropic compression tests on dense Santa Monica Beach sand are used as a basis for verification of the capabilities of the proposed model.

A new free‐space method for measurement of electromagnetic parameters of biaxial materials at microwave frequencies

AbstractA free‐space method for the measurement of biaxial material is proposed in this paper. Four complex transverse constitutive parameters are directly computed from the reflection and transmission coefficients of a planar sample in free space for the normally incident plane wave with two polarizations. However, because of the extreme insensibility of the reflection and transmission coefficients versus the longitudinal constitutive parameters for thin planar biaxial materials, the correct reconstruction of longitudinal parameters cannot be accomplished by the free‐space method. Considering that the principal coordinate system (PCS) to characterize biaxial material possibly does not coincide with the measurement coordinate system (MCS), an approach to determine PCS is also presented to ensure practicability of this method. A measurement system has been set up to perform measurements of transverse parameters of biaxial material and the validity of this method is experimentally verified. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 72–78, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20905

Dipole moments of HDO in highly excited vibrational states measured by Stark induced photofragment quantum beat spectroscopy

We report here a measurement of electric dipole moments in highly vibrationally excited HDO molecules. We use photofragment yield detected quantum beat spectroscopy to determine electric field induced splittings of the J=1 rotational levels of HDO excited with 4, 5, and 8 quanta of vibration in the OH stretching mode. The splittings allow us to deduce mua and mub, the projections of dipole moment onto the molecular rotation inertial axes. We compare the measured HDO dipole moment components with the results of quantitative calculations based on Morse oscillator wave functions and an ab initio dipole moment surface. The vibrational dependence of the dipole moment components reflect both structural and electronic changes in HDO upon vibrational excitation; principally the vibrational dependence of the O-H bond length and bond angle, and the resulting change in orientation of the principal inertial coordinate system. The dipole moment data also provide a sensitive test of theoretical dipole moment and potential energy surfaces, particularly for molecular configurations far from equilibrium.

Analysis of geometrically nonlinear anisotropic membranes: theory and verification

A new anisotropic constitutive model is introduced to simulate the geometrically nonlinear dynamic behavior of general anisotropic membranes experiencing large deformations. This model relates the second Piola–Kirchhoff stress with the Green–Lagrange strain in a 3-D convected curvilinear coordinate frame. The construction of the anisotropic constitutive law from the predefined principal material coordinate system to the final convected curvilinear coordinate system using coordinate transformations is presented. The proposed theory is implemented into a finite element code and several numerical examples are given for validation. An application problem is described to demonstrate the importance of the new model in real applications.

Special solutions of Kirchhoff equations and their Lyapunov stability

The special solutions of the Kirchhoff equations, which are those rela tive to fixed coordinate system, principal coordinate system of a cross section of the rod, and Frenet coordinate system of the central line of the rod, respect ively, are derived in this paper. Lyapunov stability of these solutions is disc us sed by use of theory on the first-approximation stability, and at the same time stability area in parameter's plane is given.