Showing papers on "Orthotropic material published in 2007"

Finite element analysis of composite materials

Abstract: MECHANICS OF ORTHOTROPIC MATERIALS Material Coordinate System Displacements Strain Stress Contracted Notation Equilibrium and Virtual Work Boundary Conditions Continuity Conditions Compatibility Coordinate Transformations Transformation of Constitutive Equations 3D Constitutive Equations Engineering Constants From 3D to Plane Stress Equations Apparent Laminate Properties Suggested Problems References INTRODUCTION TO THE FINITE ELEMENT METHOD Basic FEM Procedure General FEM Procedure FE Analysis with CAE Systems Suggested Problems References ELASTICITY AND STRENGTH OF LAMINATES Kinematics of Shells FE Analysis of Laminates Failure Criteria Suggested Problems References BUCKLING Bifurcation Methods Continuation Methods Suggested Problems References FREE EDGE STRESSES Poisson's Mismatch Coefficient of Mutual Influence Suggested Problems References COMPUTATIONAL MICROMECHANICS Analytical Homogenization Numerical Homogenization Local-Global Analysis Laminated RVE Suggested Problems References VISCOELASTICITY Viscoelastic Models Boltzmann Superposition Correspondence Principle Frequency Domain Spectrum Representation Micromechanics of Viscoelastic Composites Macro-Mechanics of Viscoelastic Composites FEA of Viscoelastic Composites Suggested Problems References DAMAGE MECHANICS One-Dimensional Damage Mechanics Multi-Dimensional Damage and Effective Spaces Thermodynamics Formulation Kinetic Law in Three-Dimensional Space Damage and Plasticity Suggested Problems References A DAMAGE MODEL FOR FIBER REINFORCED COMPOSITES Theoretical Formulation Numerical Implementation Model Identification Laminate Damage References Bibliography DELAMINATIONS Two-Dimensional Delamination Delamination in Composite Plates Suggested Problems References APPENDIX A: TENSOR ALGEBRA Principal Directions of Stress and Strain Tensor Symmetry Matrix Representation of a Tensor Inner-Product Tensor Multiplication Tensor Inversion Tensor Differentiation APPENDIX B: STRAIN CONCENTRATION TENSORS APPENDIX C: SECOND ORDER DIAGONAL DAMAGE MODELS Effective and Damaged Spaces Thermodynamic Force Y Damage Surface Unrecoverable-Strain Surface APPENDIX D: NUMERICAL INVERSE LAPLACE TRANSFORM APPENDIX E: INTRODUCTION TO THE SOFTWARE INTERFACE ANSYS BMI3 References INDEX

A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues

Abstract: In this paper a generalized anisotropic hyperelastic constitutive model for fiber-reinforced materials is proposed. Collagen fiber alignment in biological tissues is taken into account by means of structural tensors, where orthotropic and transversely isotropic material symmetries appear as special cases. The model is capable to describe the anisotropic stress response of soft tissues at large strains and is applied for example to different types of arteries. The proposed strain energy function is polyconvex and coercive. This guarantees the existence of a global minimizer of the total elastic energy, which is important in the context of a boundary value problem.

‘Morphing’ bistable orthotropic elliptical shallow shells

Abstract: This study is concerned with the equilibrium shapes of orthotropic, elliptical plates and shells deforming elastically without initial stresses. The aim is to explore potential bistable configurations and their dependencies on material parameters and initial shape for elucidating novel morphing structures. A strain energy formulation gives way to a compact set of governing equations of deformation, which can be solved in closed form for some isotropic and orthotropic conditions. It is shown that bistability depends on the change in Gaussian curvature of the shell, in particular, for initially untwisted shells, isotropy precludes bistability, where there is negative initial Gaussian curvature, but orthotropic materials yield bistability irrespective of the sign of the initial Gaussian curvature. This improved range of performance stems from increasing the independent shear modulus, which imparts sufficient torsional rigidity to stabilize against perturbations in the deformed state. It is also shown that the range of bistable configurations for initially twisted shells generally diminishes as the degree of twist increases.

A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials

Abstract: In this paper, a cohesive grain boundary integral formulation is proposed, for simulating intergranular microfracture evolution in polycrystalline brittle materials. Artificially generated polycrystalline microstructures are discretized using the proposed anisotropic boundary element method, considering the random location, morphology and material orientation of each grain. Each grain is assumed as a single crystal with general elastic orthotropic mechanical behaviour. Crack initiation and propagation along the grain boundaries interfaces are modelled using a linear cohesive law, considering mixed mode failure conditions. Furthermore, a non-linear frictional contact analysis is performed over cracked grain interfaces to encounter cases where crack surfaces come into contact, slide or separate. The effect of randomly located pre-existing flaws on the overall behaviour and microcracking evolution of a polycrystalline material is also investigated for different Weibull moduli. The stochastic effects of each grain morphology-orientation, internal friction and randomly distributed pre-existing flaws, under different loading conditions, are studied probabilistically by simulating various randomly generated microstructures. Copyright © 2006 John Wiley & Sons, Ltd.

A continuum model for remodeling in living structures

Abstract: A new remodeling theory accounting for mechanically driven collagen fiber reorientation in cardiovascular tissues is proposed. The constitutive equations for the living tissues are motivated by phenomenologically based microstructural considerations on the collagen fiber level. Homogenization from this molecular microscale to the macroscale of the cardiovascular tissue is performed via the concept of chain network models. In contrast to purely invariant-based macroscopic approaches, the present approach is thus governed by a limited set of physically motivated material parameters. Its particular feature is the underlying orthotropic unit cell which inherently incorporates transverse isotropy and standard isotropy as special cases. To account for mechanically induced remodeling, the unit cell dimensions are postulated to change gradually in response to mechanical loading. From an algorithmic point of view, rather than updating vector-valued microstructural directions, as in previously suggested models, we update the scalar-valued dimensions of this orthotropic unit cell with respect to the positive eigenvalues of a tensorial driving force. This update is straightforward, experiences no singularities and leads to a stable and robust remodeling algorithm. Embedded in a finite element framework, the algorithm is applied to simulate the uniaxial loading of a cylindrical tendon and the complex multiaxial loading situation in a model artery. After investigating different material and spatial stress and strain measures as potential driving forces, we conclude that the Cauchy stress, i.e., the true stress acting on the deformed configuration, seems to be a reasonable candidate to drive the remodeling process.

Micromechanical modeling of solid-type and plate-type deformation patterns within softwood materials. A review and an improved approach

Abstract: Wood exhibits a highly diverse microstructure. It appears as a solid-type composite material at a length scale of some micrometers, while it resembles an assembly of plate-like elements arranged in a honeycomb fashion at the length scale of some hundreds of micrometers. These structural features result in different load-carrying mechanisms at different observation scales and under different loading conditions. In this paper, we elucidate the main load-carrying mechanisms by means of a micromechanical model for softwood materials. Representing remarkable progress with respect to earlier models reported in the literature, this model is valid across various species. The model is based on tissue-independent stiffness properties of cellulose, lignin, hemicellulose, and water obtained from direct testing and lattice-dynamics analyses. Sample-specific characteristics are considered in terms of porosity and the contents of cellulose, lignin, hemicelluloses and water, which are obtained from mass density measurements, environmental scanning micrographs, analytical chemistry, and NMR spectroscopy. The model comprises three homogenization steps, two based on continuum micromechanics and one on the unit cell method. The latter represents plate-like bending and shear of the cell walls due to transverse shear loading and axial straining in the tangential stem direction. Accurate representation of these deformation modes results in accurate (orthotropic) stiffness estimates across a variety of softwood species. These stiffness predictions deviate, on average, by less than 10% from corresponding experimental results obtained from ultrasonic or quasi-static testing. Thus, the proposed model can reliably predict microscopic and macroscopic mechanical properties from internal structure and composition, and is therefore expected to significantly support wood production technology (such as drying techniques) and mechanical analyses of timber structures.

A fibre reorientation model for orthotropic multiplicative growth. Configurational driving stresses, kinematics-based reorientation, and algorithmic aspects.

Abstract: The main goal of this contribution consists in the development of a remodelling framework for orthotropic continua whereby the underlying symmetry group is incorporated via two fibre families. Special emphasis is placed on the modelling of biological tissues at finite deformations. Besides the incorporation of a referential mass source, anisotropic growth is addressed by means of a multiplicative decomposition of the overall deformation gradient into an elastic and a growth distortion. Projected quantities of a configurational growth stress tensor are advocated as driving forces for time-dependent saturation–type evolution of the principal values of the growth distortion. Moreover, the reorientation of both fibre families, which directly affects the strain energy as well as the growth distortion itself, is guided by analyzing critical energy points. In particular, a time-dependent formulation is developed which aligns the fibre directions according to the principal stretch directions. Finally, the proposed framework is embedded into a finite element context so that representative numerical examples, examining growth and resorption in volume and density together with fibre reorientation, close this study.

Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures

Abstract: An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases.

Identification of the Orthotropic Elastic Stiffnesses of Composites with the Virtual Fields Method: Sensitivity Study and Experimental Validation

Abstract: This paper presents the use of a sensitivity study to find the best combinations of free length and fibre angle in an unnotched Iosipescu test processed with the virtual fields method. The sensitivity to noise coefficients arising from the special virtual fields procedure are used to build up a cost function. This function is aimed at balancing out the coefficients of the different orthotropic stiffnesses so that the same confidence level can be reached on all these parameters. Then, experimental validation was performed using a speckled interferometry (ESPI) system. Full-field strains were measured and stiffnesses identified and compared between the usual 0°, 30 mm configuration and an improved 25°, 40 mm configuration. The outcome of the optimisation was confirmed by testing the same specimen several times and comparing scatter between the two configurations. This is a first promising result on the way to the design of a new test for orthotropic stiffness identification on a single specimen from full-field measurements.

Analysis of an Orthotropic Deck Stiffened with a Cement-Based Overlay

Abstract: Over the past years, with increasing traffic volumes and higher wheel loads, fatigue damage in steel parts of typical orthotropic steel bridge decks has been experienced on heavily trafficked routes. A demand exists to find a durable system to increase the fatigue safety of orthotropic steel bridge decks. A solution might be to enhance the stiffness of the traditional orthotropic bridge deck by using a cement-based overlay. In this paper, an orthotropic steel bridge deck stiffened with a cement-based overlay is analyzed. The analysis is based on nonlinear fracture mechanics, and utilizes the finite-element method. The stiffness of the steel deck reinforced with an overlay depends highly on the composite action. The composite action is closely related to cracking of the overlay and interfacial cracking between the overlay and underlying steel plate (debonding). As an example, a real size structure, the Faro bridges located in Denmark, are analyzed. The steel box girders of the Faro bridges spans 80 m, and have a depth of 3.5 m, and a width of 19.5 m. The focus of the present study is the top part of the steel box girders, which is constructed as an orthotropic deck plate. Numerous factors can influence the cracking behavior of the cement-based overlay system. Both mechanical and environmental loading have to be considered, and effects such as shrinkage, temperature gradients, and traffic loading are taken into account. The performance of four overlay materials are investigated in terms of crack widths. Furthermore, the analysis shows that debonding is initiated for a certain crack width in the overlay. The load level where cracking and debonding is initiated depends on the stress-crack opening relationship of the material.

Novel experimental approach for longitudinal-radial stiffness characterisation of clear wood by a single test

Abstract: Experimental results obtained from maritime pine (Pinus pinaster Ait.) wood are presented for the characterisation of all LRs(1,2) orthotropic stiffness parameters of clear wood specimens by a single test. The approach relies on application of the virtual field method (VFM) to a rectangular specimen loaded in the Iosipescu fixture. The displacement field over the gauge surface of the specimen is measured by the grid method. Two configurations are investigated: (1) with grain aligned along the specimen length (08 configuration) and (2) with grain at 458. For the 08 configuration, only the parameters Q11 and Q66 are correctly identified, with coefficients of variation of the same order of magnitude as those obtained from reference tensile and shear tests. Better identification is obtained for the 458 configuration, for which only the parameter Q12 exhibits large scatter. This improvement results from a more balanced influence of all stiffness parameters on the response of the 458 specimen. However, all stiffness parameters identified were systematically underestimated by approximately 30% in comparison to reference values. This deviation is due to the vertical spatial variation of the mechanical properties of wood within the stem. Literature data confirm this interpretation.

Cross-Laminated Timber Plates: Evaluation and Verification of Homogenized Elastic Properties

Abstract: Cross-laminated solid wood panels are used in timber structures as load bearing plates and shear panels. Since timber is a relatively soft construction material, the design of such structures is driven by serviceability criteria. Therefore, accurate elastic properties are required. In this paper a fully automated procedure to determine global elastic properties of full-scale cross-laminated wood panels is developed. Experimental modal analysis is used to determine resonance frequencies and mode shapes of rectangular wooden specimens. An analytical model based on Reddy's higher order plate theory is applied to calculate natural frequencies and mode shapes numerically. Corresponding frequencies are allocated using the modal assurance criterion. All three shear moduli and the two in-plane stiffness moduli are identified successfully by minimizing the difference between measured and estimated resonance frequencies in a total least squares sense. By comparing resonance frequencies and additionally by a static bending experiment, it is shown that the global mechanical behavior of the specimen is accurately described using an orthotropic, homogenized, linear elastic material behavior.

Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

Abstract: We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.

The structural efficiency of orthotropic stalks, stems and tubes

Abstract: An optimised structure is one which uses the smallest quantity of the best material to perform its function, with adequate safety factor or margin for error. Structural optimisation occurs not only in mechanical engineering, but also in nature: plants with hollow stems or stalks gain a height advantage, and are thus more efficient, by approaching the optimum shape. Here we consider the optimisation of orthotropic tubes, typifying, in a mechanical sense, stalk and stem. The stiffness and strength of orthotropic tubes of initially circular section are reviewed, and diagrams are proposed which allow the optimum section shape to be selected.