Orthotropic Axes Research Articles

NONLINEAR REFINEMENT OF THE DEFORMATION MODEL OF ORTHOTROPIC MATERIALS, THE RIGIDITY OF WHICH DEPENDS ON THE TYPE OF STRESS STATE

As a result of numerous experimental and theoretical studies, it has been established that when deforming both traditional and new structural materials, which are polymers, composites and synthetic structures used in construction, mechanical engineering, and power engineering devices, complicated mechanical properties manifest themselves.Most of these materials have an orthotropic structure complicated by the dependence of deformation and strength characteristics on the type of stress state, which can be interpreted as deformation anisotropy. These properties contradict the generally accepted theories of deformation. Therefore, a number of models have been specially developed for such materials over the past 56 years, taking into account the complicated properties of materials. However, all of them have disadvantages and certain contradictions with the fundamental rules for constructing equations of state. The previous works of the authors of the presented studies establish general approaches to the construction of energetically nonlinear deformation models of composite materials with recommendations for calculating their constants based on a wide range of experiments.It turned out that the set of necessary experiments should include experiments on complex stress states, most of which are technically unrealizable. In another work of the authors in 2021, using the tensor space of normalized stresses, the deformation potential was formulated in a quasi-linear form, constructed in the main axes of orthotropy of materials, for determining constants, which is sufficient for the data of the simplest experiments. Along with the obvious advantages of the introduced potential, it has one drawback, which is to replace real nonlinear diagrams with direct rays with minimal error. Despite the unconditional adequacy of the quasi-linear potential, the use of this level of approximations leads to quantitative errors.Therefore, simplified nonlinear equations of state for composite materials are proposed here, the simplest experiments are sufficient to determine the material functions of which. These equations are based on the general laws of mechanics, on the basis of which the constants of material polynomials for a carbon-graphite composite are calculated, taking into account the Drucker constraints.

Research of orthotropic composites failure taking into account their structural stochasticity

The object of the study is the construction of the reliability assessing algorithm for the orthotropic composite plate, taking into account the stochasticity of its structure under the conditions of plane deformation. The plate consists of a matrix and reinforcement elements. The main orthotropic directions of the material coincide with the directions of the loading. The conducted studies are based on the failure criteria expressed through the components of macro stresses. The hypothesis of the weakest link is used, which for the case of the statistical theory of strength sounds like this: the ultimate (failure) loading for an orthotropic composite plate is equal to the ultimate loading for its weakest element. Defects-cracks are characterized by independent random variables – the half-length and the orientation angle between the defect line and the axis of orthotropy with a higher modulus of elasticity. The proposed model of orthotropic composite material corresponds to known experimental studies epoxy phenolic fiberglass on the cord glass fiber. The distribution probabilities density of defect orientation takes into account their predominant orientation in the direction of reinforcement. On the basis of the obtained composite failure loading integral probability distribution function, the construction and study of the dependence of the plate failure probability diagrams for different number of cracks, structural inhomogeneity and type of applied loading was carried out. Complex application of the composite materials fracture mechanics deterministic solution and the methods of probability theory and mathematical statistics allows for a more adequate assessment of the composite materials reliability, taking into account the stochasticity of their structure. The main content of this work is the construction and analysis of dependence of stochastically defective reinforced composite materials failure probability diagrams. The obtained results make it possible to evaluate the reliability of orthotropic stochastically defective materials under conditions of plane deformation. The algorithm of a compatible combination of defectiveness and randomness of the orthotropic composite material structure makes it possible to qualitatively investigate its failure under various types of applied loading.

Study of the laminas’ stress-strain state for cross-ply composites with the [± 45°]2s lay-up: tensile tests

In this paper we have analysed the relations that allows one to determine the components of strains and stresses in the orthotropic axes of layered fiber reinforced composite materials in tensile and compression experiments for test specimens with the [±45°]2s lay-up. In particular, such relations were compiled on the assumption that in a cross-ply fiber composite two adjacent layers with the [±φ] lay-ups can be considered as one symmetrically reinforced layer with orthotropic properties. To establish the degree of accuracy of these relationships and assumptions, numerical experiments were carried out to determine the parameters of the stress-strain state of specimens consisting of two and four monolayers of a unidirectional fibrous composite material with the [±45°]2s lay-up. The analysis has been performed in the ANSYS finite element analysis program system in a linear formulation of two- and three-dimensional problems. It is shown that in the central zone of the specimen, the theoretical-experimental methodology based on the above mentioned relations has a sufficient degree of accuracy. Based on the analysis of results of three-dimensional problems, the formation of a linear torque in the specimen, which causes the twisting of some zones of the specimen along its length, has been revealed.

An enriched improved complex variable element-free Galerkin method for efficient fracture analysis of orthotropic materials

The fracture problem of orthotropic materials has been widely concerned by researchers in the fields of mechanics and engineering. The meshless method using nodes to discrete the problem domain is an effective way to analyze cracked structures. In this paper, an enriched improved complex variable moving least squares (EICVMLS) approximation, in which a group of intrinsic enrichments is introduced into the conventional ICVMLS adopting a well-defined weighted complex variable error norm, is proposed for capturing the crack-tip fields in orthotropic media accurately. Then, an enriched improved complex variable element-free Galerkin (EICVEFG) method is developed to simulate fracture behavior in orthotropic media. The visibility criterion is used to model the discontinuity behavior around cracks and the M−integral is considered to evaluate stress intensity factor (SIF). The stress field and the displacement field at the crack tip, the SIF for different crack inclinations, and different directions of orthotropy axes are investigated to verify the validity of the proposed method. In addition, the effects of linear basis functions and enriched basis functions on the results are compared in this study. The numerical results reveal that the established EICVEFG approach can achieve higher accuracy than the extended finite element method (XFEM) and traditional meshless methods such as the element-free Galerkin (EFG) method and the meshless finite volume (MFV) method. Similarly, the EICVEFG method has been proved to offer a higher convergence rate by parameter studies (including dimensionless factor dmax, integral size, and the number of integral cells in M−integral).

Modeling tension-compression asymmetry and failure anisotropy in bending operations of a magnesium alloy

The deformation behavior of the magnesium sheet alloy AZ31 at room temperature is investigated using different standard and non-standard mechanical tests as well as respective finite-element simulations. In order to address the bendability, three-point bending tests were performed for different orientations of the sample with respect to the sheets axes of orthotropy. The elastic-plastic deformation was modeled using a two-yield-surface model, which is able to describe the evolving strength differential effect as well as the anisotropy of the material. For the prediction of material failure, a numerically efficient damage criterion based on transformation of strain rates is calibrated and applied. It is shown that this model is able to predict the observed edge failure of specimens with good accuracy.

The effect of material orientation on void growth

In this work, we have brought to light the effect of material orientation on void growth. For that purpose, we have performed finite element calculations using a cubic unit-cell model with a spherical void at its center and subjected to periodic boundary conditions. The behavior of the material is described with an elastic isotropic, plastic orthotropic constitutive model with yielding defined by Yld2004-18p criterion (Barlat et al., 2005). We have used the multi-point constraint subroutine developed by Dakshinamurthy et al. (2021) to enforce constant values of macroscopic stress triaxiality T and Lode parameter L in calculations that have been carried out for different stress states resulting from the combination of T=0.33, 1 and 2, with L=−1, 0 and 1 (axisymmetric tension, generalized shear and axisymmetric compression, respectively). Firstly, we have performed numerical simulations in which the loading directions are collinear with the orthotropy axes of the material, so that the principal directions of macroscopic stress and strain are parallel. Investigation of the cases for which the minor loading axis coincides either with the rolling, the transverse or the normal direction, has shown that the initially spherical void turns into an ellipsoid whose rate of growth and eccentricity depend on both stress state and material orientation. A key result is that for specific material orientations the anisotropy switches the effect of Lode parameter on void growth, reversing the trends obtained for isotropic von Mises materials. Secondly, we have carried out calculations using a novel strategy which consists of including angular misalignments within the range 0∘≤θ≤90∘, so that one loading direction is parallel to one of the symmetry axes of the material, and θ is the angle formed between the other two loading directions and the second and third orthotropy axes. In fact, to the authors’ knowledge, these are the first unit-cell calculations ever reported in which the material is modeled using a macroscopic anisotropic yield function with prescribed misalignment between loading and material axes and, at the same time, the macroscopic stress triaxiality and the Lode parameter are controlled to be constant during loading. The finite element calculations have shown that the misalignment between loading and material axes makes the void and the faces of the unit-cell to rotate and twist during loading. Moreover, the main contribution of this work is the identification of an intermediate value of the angle θ for which the growth rate of the void reaches an extreme value (minimum or maximum), so that the numerical results indicate that material orientation and angular misalignment can be strategically exploited to control void growth, and thus promote or delay localization and fracture of anisotropic metal products. The conclusions of this research have been shown to be valid for three different materials (aluminum alloys 2090-T3, 6111-T4 and 6013) and selected comparisons have also been performed using two additional yield criteria (CPB06ex2 and Yld2011-27p).

Triaxial variation of the modulus of elasticity in the thermo-elastic range of six tropical wood species

ABSTRACT This work highlights the effect of temperature on the modulus of elasticity following the orthotropic axis of six species of tropical woods: assamela, bibolo, doussié, iroko, movingui, and sapelli of scientific names Afzelia, Pericopsiselata, Distemonanthus benthamianus, Entandrophragma cylindricum, Miliciaexcelsa, and Lovoatrichilioides, respectively. The four-point bending and compression tests were conducted to determine the failure loads, which can help further tests performed in the elastic domain. In order to determine the Young's modulus in the thermoselastic field, the above tests were carried out at different levels of loads and temperature; the samples of 20 × 20 × 340 mm dimensions for bending tests and 20 × 20 × 60 mm dimensions for compression tests according to current standards. The strains were read by means of two strain gauges, connected to the strain gauge bridge. Test results under the different constant thermo-hydric conditions showed an increase in the longitudinal, radial and tangential MOE of all six species as the temperature increased from 22°C to 40°C, and a regression above 40°C. In terms of lost in elastic modulus, due to the temperature, the tests showed the following classification: . Those results may be crucial while selecting the wood species for a typical use in a range of temperature.

Thermal diffusivity of polymeric composite cured in a magnetic field

Curing of the composite in a constant magnetic field of low intensity (Н∼104 A/m) leads to the anisotropic spatial distribution of the filler particles - carbonyl nickel powder. Thermal diffusivity of the composite with orthotropic structure was measured using "pulsed heat source" method. Increase of thermal diffusivity value was found in the direction of the orthotropic axis.

Variational-asymptotic homogenization of thermoelastic periodic materials with thermal relaxation

A multi-scale variational-asymptotic homogenization method for periodic microstructured materials in presence of thermoelasticity with periodic spatially dependent one relaxation time is introduced. The asymptotic expansions of the micro-displacement and the micro-temperature fields are rewritten on the transformed Laplace space and expressed as power series of the microstructural length scale, leading to a set of recursive differential problems over the periodic unit cell. The solution of such cell problems leads to the perturbation functions. Up-scaling and down-scaling relations are then defined, and the latter allow expressing the microscopic fields in terms of the macroscopic ones and their gradients. The variational-asymptotic scheme to establish an equivalence between the equations at macro-scale and micro-scale is developed. Average field equations of infinite order are also derived. The efficiency of the proposed technique was tested in relation to a bi-dimensional orthotropic layered body with orthotropy axis parallel to the direction of the layers, where the mechanical and temperature constitutive properties were well established. The dispersion curves of the homogenized medium, truncated at the first order are compared with the dispersion curves of the heterogeneous continuum obtained by the Floquet-Bloch theory. The results obtained with the two different approaches show a very good agreement.

Numerical Investigation of an Orthotropic Plate with Interactions of Crack, Inclusions and Voids under Uniaxial Tensile Loading by XFEM

This work is focused to investigate the effect of various discontinuities like cracks, inclusions and voids for an orthotropic plate, to evaluate the normalized mixed-mode stress intensity factors (NMMSIFs) by implementing the extended finite element method (XFEM) under uniaxial tensile loading though considering the various numerical examples. The NMMSIFs are investigated with the interaction of crack, single- and multi-inclusions/voids for an orthotropic plate. The effect of NMMSIFs is analyzed for an orthotropic plate with several orthotropy axis orientations by changing the position of single- and multi-inclusions/voids while aligned, above and away with respect to an edge crack of the plate and for the both side inclusions/voids aligned the center crack. It is also investigated for the effect of various shapes of inclusions/voids for an edge crack orthotropic plate under uniaxial tensile loading using XFEM.

Ellipsoidal inhomogeneity in elliptically orthotropic elastic solid

An unbounded elliptically orthotropic elastic solid containing a single anisotropic ellipsoidal inhomogeneity is considered. The problem statement assumes arbitrary orientation of ellipsoid with respect to the orthotropy axes and non-uniform far loading. By applying the appropriate affine transformations, the boundary value problem is reduced to that one with the isotropic constituents. The complete displacement solution to the transformed problem is obtained using the Papkovich-Neuber representation with the scalar potentials written in terms of ellipsoidal solid harmonics. By accurate fulfilling the interface conditions, the boundary value problem is reduced to the system of the linear algebraic equations for the constants entering the potentials. For the Eshelby problem (uniform remotely applied loading and perfect interface), an explicit analytical solution is written in terms of ellipsoidal harmonics. This result yields solution for the elliptical crack problem in the remarkably simple way. The numerical examples illustrate an effect of the matrix anisotropy on the stress concentration and stress intensity factors. An application of the obtained solution to the micromechanics is discussed. The effective elastic stiffness tensor of a particulate composite with elliptically orthotropic matrix is evaluated using Maxwell homogenization scheme.

Modelling-assisted description of anisotropic edge failure in magnesium sheet alloy under mixed-mode loading

An uncoupled fracture criterion based on a simple damage indicator computed using a mean-field crystal plasticity framework is proposed and applied to predict failure of an AZ31 sheet originating from the edges. The damage indicator quantifies the contribution of strain components along the axes of orthotropy leading to material failure. The model is calibrated by uniaxial tension tests. The damage indicator is validated for various mixed-mode deformation histories realized by modified Arcan tests in various loading configurations. The loading history of respective fracture sites obtained from DIC analyses is directly employed to a visco-plastic self-consistent crystal plasticity model to obtain the stress responses. The results indicate that the damage indicator requires – beside the strain history – an input of stress triaxiality, by which an improved predictive accuracy can be achieved. This effect is quantified for various loading scenarios, in which cracks are initiated near or at the edge of the sample.

Deformation patterns and their stability in finitely strained circular cell honeycombs

The mechanics of cellular honeycombs—part of the rapidly growing field of architected materials—in addition to its importance for engineering applications has a great theoretical interest due to the complex bifurcation mechanisms leading to failure in these nonlinear structures of high initial symmetry.Of particular interest to this work are the deformation patterns and their stability of finitely strained circular cell honeycomb. Given the high degree of symmetry of these structures, the introduction of numerical imperfections is inadequate for the study of their behavior past the onset of first bifurcation. Thus, we further develop and explain a group-theoretic approach to investigate their deformation patterns, a consistent and general methodology that systematically finds bifurcated equilibrium orbits and their stability. We consider two different geometric arrangements, hexagonal and square, biaxial compression along loading paths, either aligned or at an angle with respect to the axes of orthotropy, and different constitutive laws for the cell walls which can undergo arbitrarily large rotations, as required by the finite macroscopic strains applied.We find that the first bifurcation in biaxially loaded hexagonal honeycombs of infinite extent always corresponds to a local mode, which is then followed to find the deformation pattern and its stability. Depending on load path orientation, these first bifurcations can be simple, double or even triple. All bifurcated orbits found are unstable and have a maximum load close to their point of emergence. In contrast, the corresponding instability in square honeycombs always corresponds to a global mode and hence the deformation pattern will depend on specimen size and boundary conditions.

LOCALIZATION ANALYSIS OF AN ORTHOTROPIC MULTI-SURFACE PLASTICITY MODEL UNDER UNIAXIAL STRESS

Numerical simulations of masonry structures are often based on continuum macromodelling approaches that need constitutive laws able to phenomenologically reproduce the behavior of the material. To capture the deformation process up to failure, appropriate softening laws are needed to take into account the contraction of the yield stress domain caused by cracking and crushing. It is well known that softening may lead to localization of inelastic strain. This paper focuses on localization analysis of an orthotropic macro-scale model in the framework of multi-surface plasticity, which describes the in-plane behavior of masonry structures. Preliminary results reported in this short paper are limited to uniaxial stress states. Analytical localization conditions are first derived for uniaxial stress states with principal axes aligned with the material axes of orthotropy. Then, localization analysis is extended to an arbitrary angle between the principal stress axes and the axes of orthotropy.

Ontogenesis and architectural analysis of Myricaria bracteata Royal in East Kazakhstan

The article presents research of Myricaria bracteata shrub ontogenesis and architecture under East Kazakhstan conditions. The study revealed that the specimen ontogenesis is complete and includes two phases of development: a primary shoot and a primary bush. The adult speciment structure is built due to consistent development of perennial orthotropic/skewed orthotropic skeletal axes was forming from the resting buds. Annually dying lateral vegetative and generative shoots are formed of branching resting buds located on these axes. The architectural unit of M. bracteata consists of 5 categories of axes: 1 – perennial orthotropic, 2 – perennial lateral skewed orthotropic, 3 – annual lateral generative bracteose, 4 – annual lateral generative frondose, 5 – annual lateral vegetative ones.

Experimental and Numerical Bendability Analysis of a 3rd Generation Magnesium Alloy

Motivated by the demand for lightweight materials, magnesium alloys in use for structural components in the automotive and aircraft industry are getting attention. In the current contribution, the bendability of the innovative magnesium alloy “E-form” AZ31 at room temperature is investigated by means of 3-point bending tests and respective numerical simulations. The bending tests were conducted for different orientations of the sample with respect to the sheets axes of orthotropy. The force-displacement records confirmed the plastic anisotropy of the material evidenced in tensile testing. The limits of bendability were given by cracking of the samples, which was found to be strongly direction-dependent. A correlation between bendability and Lankford coefficient was explored. The elastic-plastic deformation was modelled using finite elements and a two-yield-surface model, which is able to describe the evolving strength differential effect and the anisotropy. Approaches based on 1D, 2D and 3D discretisations were compared. Zones with tensile and compressive stresses were identified by microstructural changes and mapped with its numerically obtained counterparts.

Technique for determining the orthotropic properties of the bone organ according to computer tomography

The actual task of modern biomechanics of bone tissue is to determine the elastic properties. Nowadays computed tomography coupled with computer technology allows the use of numerical experiments to determine the mechanical properties of biological objects. This paper presents such a technique. The technique is based on the hypothesis that the anisotropy of bone tissue is determined by the uneven distribution of pore volume. In the proposed approach, the organ under study is proposed to be divided into a set of representative volumes. A set of numerical experiments is carried out for each volume, which makes it possible to determine the components of the elastic tensor. Then, the axes of orthotropy and elastic constants are determined. The described technique is applied for diaphysis of the femoral bone rat. It is shown that the resulting elastic constants remain constant in the organ, and the body is transversally isotropic.

Orthotropic-equivalent strain measures and their application to the elastic response of porcine skin

A Green-elastic response function based on a non-algebraic invariant representation for the in-plane deformation of tissue (porcine skin) is developed. It retains orthotropy and simulates the characteristic bimodal behavior (a very low modulus regime followed by a rapidly stiffening response) with experimentally determinable and meaningful parameters. This is achieved by using a recently developed upper triangular (or QR decomposition) of the deformation gradient using the orthotropy axes and defining physically meaningful ‘equivalent strains’ that are non-algebraic functions of the usual invariants. We demonstrate that the strain energy density can be written as a function of one or more such ‘equivalent strains’ and can be used to model the orthotropic response of such bimodal materials. The model is used to simulate the orthotropic unload-reload response of porcine-skin when loaded in different directions with results that compare well with experiments based on a single orthotropic-equivalent strain measure.

Characterization of the orthotropic viscoelastic tensor of composites using the Ultrasonic Polar Scan

An enhanced two-stage inversion scheme for the determination of the viscoelastic properties of orthotropic plates (e.g. fiber reinforced polymers) based on the pulsed ultrasonic polar scan (P-UPS) is presented. A forward numerical model for the P-UPS is combined with an efficient optimizer routine (particle swarm optimizer, PSO) to inversely extract the complex orthotropic stiffness tensor, as well as the orientation of the main axes of orthotropy. The developed inversion technique is validated on synthetic data, yielding accurate results with small deviations: elasticity tensor <0.5%, viscosity tensor <2% and main axes of orthotropy <0.03°. The inversion procedure has been further applied to analyze actual experimental P-UPS recordings on an aluminum plate as well as a unidirectional [0]8 carbon/epoxy laminate. The obtained inversion results yield complex stiffness parameters which are in close agreement to reported values in literature, and confirms the potential of the UPS technique and its inversion optimization routine as an enhanced NDT technique to quickly and accurately obtain characteristic material properties of composites.

A para-universal relation for orthotropic materials

A universal relation is an algebraic relation between stress and strain that holds for any material within a certain class, irrespective of the exact form of the material response function and parameter values. Classical universal relations, such as Rivlin’s famous relation for simple shear, apply to stress components produced by one and the same deformation. We present a family of relations that connect stress components under different deformations, which we call para-universal relations to highlight this difference. The proposed para-universal relations hold for any orthotropic material whose response function is additively decomposed into terms, each of which possesses a symmetry with respect to one of the axes of orthotropy. Using basic properties of the permutation group S3, we demonstrate that such an additive decomposition implies the proposed identities. The established para-universal relations hold for an arbitrary local deformation and, like classical universal relations, are linked to material symmetry and apply to a wide class of materials. Since the proposed para-universal relations do not hold for all orthotropic material models, they present a convenient way to test for the suitability of additively split strain-energy functions, which are often used to model the nonlinearly elastic response of soft tissues. Such a test can be performed on collected experimental data prior to choosing an exact form of the response function and fitting its parameters. We use published experimental data for human myocardium and also synthetic data to illustrate this.