Theory Of Orthotropic Elasticity Research Articles

Modified state space method for long-term behavior of viscoelastic laminated angle-ply plate

This work proposes a modified state space method (SSM) to develop the analytical solution for viscoelastic laminated angle-ply plates subjected to long-term loading situations. Within the analytical model, the plate’s stresses and displacements are governed by orthotropic elasticity theory combined with Boltzmann superposition. The convolution within the governing equation is resolved through the application of Laplace transform. The modified SSM is proposed to provide a general solution for each lamina by extracting the complex variable of the Laplace transform from the state space equation. This modification addresses a limitation of the traditional SSM where the exponential function of the state space matrix is computationally infeasible to calculate. The analytical solution of the laminated angle-ply plate is ultimately derived using the transfer matrix method followed by the inverse Laplace transform. Through comparative analysis, it is observed that the proposed solution exhibits greater accuracy than simplified solutions and higher efficiency compared to the finite element method. In addition, the work explores the bending creep and recovery behaviors of viscoelastic laminated angle-ply plate under various parameters.

Modified state space method for bending creep and recovery behaviors of viscoelastic orthotropic laminated plate

A modified state space method (SSM) is proposed first to solve the analytical solution of viscoelastic orthotropic laminated (VOL) plate under time-varying load. This modification extends the applicability of traditional SSM from elastic materials to viscoelastic materials by resolving a limitation of the traditional SSM, where computing the exponential function of the state space matrix is infeasible. In the analytical model, the long-term stresses and displacements are established based on the orthotropic elasticity theory combined with the Boltzmann superposition. The convolution within the governing equation is resolved through Laplace transform. By using the proposed modified SSM, the general solution for each viscoelastic lamina is derived out, by separating the complex variable of Laplace transform out from the state space matrix. The analytical solution is ultimately derived using series expansion, transfer matrix method and inverse Laplace transform. The proposed model can provide long-term creep stress distribution in the VOL plate by creep deflection data from experiment. The comparison with existing results indicates the proposed solution exhibits greater accuracy than simplified solutions and higher efficiency compared to the finite element method. At last, the bending creep and recovery behaviors of the VOL plate are investigated in detail.

On Regularization of Singular Solutions of Orthotropic Elasticity Theory

On Regularization of Singular Solutions of Orthotropic Elasticity Theory

Oblique surface cracking and crack closure in an orthotropic medium under contact loading

This article presents analytical methods for analysis of an inclined surface crack in an orthotropic medium subjected to contact loading. Governing partial differential equations are derived in accordance with plane theory of orthotropic elasticity. Contact and crack problems are formulated separately utilizing Fourier transformation techniques. The contact problem consists of a rigid flat punch pressing upon an orthotropic half-plane, and is reduced to a singular integral equation of the second kind. The surface crack is assumed to be tilted at an arbitrary angle with respect to the normal of the half-plane surface. Three different crack configurations are considered, which are completely closed, partially-closed, and fully-open cracks. Frictional contact between the crack faces is implemented in the cases of completely closed, and partially-closed cracks. A single singular integral equation is derived for the completely closed crack, whereas each of the problems of partially- and fully-open cracks requires derivation of a set of two singular integral equations. The contact problem and each of the crack problems are solved numerically by means of the expansion-collocation technique. A flowchart is developed and implemented to detect the type of crack closure configuration under a given contact loading. The parametric analyses carried out illustrate the influences of factors such as crack inclination angle, crack face and contact surface friction coefficients, material orthotropy, and relative punch location on crack face contact stresses, degree of crack closure, and mode I and II stress intensity factors.

Investigation of elastic constants and ultimate strengths of Korean pine from compression and tension tests

Korean pine (Pinus koraiensis) is a wood species recently adopted in China for the rehabilitation of traditional timber buildings. This paper investigates its mechanical properties with laboratory tests on typical specimens to obtain the moduli of elasticity, Poisson’s ratios, shear moduli, coefficients of mutual influence, crushing strengths and tension strengths in various directions. Highly different failure patterns of specimens in compression test were observed when loaded in different directions relative to the grain while only brittle failure mode was observed for tension specimens. The measured parameters of Korean pine were compared with those obtained from theory of orthotropic elasticity, the empirical formula and the Norris failure criterion, good agreements were reached for all examinations in general which indicate it is admissible to treat Korean pine as ideal orthotropic material.

Dependence of Poisson’s ratio and Young’s modulus on microfibril angle (MFA) in wood

AbstractMicrofibril angle (MFA) is a major structural variable that describes the fine structure of the cell wall in wood. In this study, the relationships between the MFA of the S2 layer and the Poisson’s ratios and Young’s moduli (modulus of elasticity, MOE) of five wood species (agathis, larch, Japanese cedar, Japanese cypress and ginkgo) were determined by analyzing both their normal and compression woods. It was found that both the longitudinal MOE (MOEL) and MOE of the cell-wall substance (MOEW) decreased with increasing MFA, while the peaks values of Poisson’s ratio (νLT) were obtained at MFAs of ≈25°. In particular, at MFAs lower than 25°, theνLTincreased with increasing MFA, and the opposite relationship was observed at MFA values exceeding 25°. This trend is in good agreement with the estimates obtained based on the theory of orthotropic elasticity with the underlying assumption that the orthotropic elasticity of materials is MFA-dependent. Hence, the MFA parameter incorporated into the orthotropic elasticity theory is useful for determination of the Poisson’s ratio.

Surface cracking in an orthotropic medium subjected to frictional contact

This article presents an analytical method capable of resolving the coupled problem of surface cracking in an orthotropic elastic medium subjected to frictional contact by a rigid flat punch. Reciprocal influences between the surface crack and the flat punch are accounted for by establishing a fully coupled formulation. Governing partial differential equations involving the displacement components are derived in accordance with plane theory of orthotropic elasticity. General solutions corresponding to mode I and II crack problems and contact problem are obtained employing Fourier transformation techniques. These separate solutions are then reconciled; and three coupled singular integral equations are developed by applying crack surface and contact zone conditions. Singular integral equations are solved numerically through an expansion–collocation method in which the primary unknowns are expanded into series in terms of Jacobi polynomials. Comparisons to the results available in the literature for certain special cases do verify the proposed procedures. Further numerical results are presented to be able to demonstrate the influences of material orthotropy, coefficient of friction, and geometric parameters upon the mixed-mode stress intensity factors and the contact stress.

Time dependence of Poisson’s effect in wood IV: influence of grain angle

Off-axis tensile creep tests were conducted on woods taken from Japanese cypress and Kalopanax by changing the angle of load to the grain direction in the longitudinal–tangential (LT) plane. The dependence of the Poisson’s ratio and trend of the viscoelastic Poisson’s ratio on grain angle were investigated. The Poisson’s ratios were found to reach their extrema when the grain angle was around 30°. Moreover, the Poisson’s ratio in the LT plane was observed to be negative when the grain angle was in the range of 15°–45°. Comparing the experimental results with theoretical values obtained from the theory of orthotropic elasticity, it was revealed that, although the Poisson’s ratio reached an extremum in both cases, the specific values did not match, especially when the angle was between 15° and 45°. Furthermore, the temporal variation of the viscoelastic Poisson’s ratio was found to depend on the grain angle and the measurement plane. It also appeared to be affected by the Poisson’s ratio, showing an increasing tendency above a specific Poisson’s ratio (Japanese cypress: 0.196, Kalopanax: 0.102) and a decreasing tendency below it, regardless of the grain angle and measurement plane. Additionally, the increment in the viscoelastic Poisson’s ratio after 24 h of creep was observed to reach its extremum when the grain angle was around 30°. Finally, by improving the six-element Frandsen–Muszynski viscoelastic model, which simultaneously considers the longitudinal and transverse strains, an eight-element model was presented, and the trend of the viscoelastic Poisson’s ratio was well reproduced by this model.

Paleotectonic residual strain energy in Southwest China

Based on the orthotropic elastic theory of rock masses, the X-ray method was used to measure the distribution of macro-residual strain energy density along a depth profile, using core samples taken from 47 large-aperture deep boreholes in four regions of Southwest China: the Longmenshan, Anninghe, Honghe, and Xianshuihe fault zones. Then, the vertical gradients of the macro-residual strain energy density and the macro-residual strain energy contained in high-energy cuboid block segments along each fault zone were determined. The results demonstrate that the macro-residual strain energy stored at shallow levels in the rock mass in these fault zones may be partly responsible for generating many large earthquakes and may explain why the large earthquakes in this region are typically shallow.

On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework

The mechanical testing of anisotropic nonlinearly elastic solids is a topic of considerable and increasing interest. The results of such testing are important, in particular, for the characterization of the material properties and the development of constitutive laws that can be used for predictive purposes. However, the literature on this topic in the context of soft tissue biomechanics, in particular, includes some papers that are misleading since they contain errors and false statements. Claims that planar biaxial testing can fully characterize the three-dimensional anisotropic elastic properties of soft tissues are incorrect. There is therefore a need to clarify the extent to which biaxial testing can be used for determining the elastic properties of these materials. In this paper this is explained on the basis of the equations of finite deformation transversely isotropic elasticity, and general planar anisotropic elasticity. It is shown that it is theoretically impossible to fully characterize the properties of anisotropic elastic materials using such tests unless some assumption is made that enables a suitable subclass of models to be preselected. Moreover, it is shown that certain assumptions underlying the analysis of planar biaxial tests are inconsistent with the classical linear theory of orthotropic elasticity. Possible sets of independent tests required for full material characterization are then enumerated.

Dynamic properties of three types of wood-based composites

This study used a vibration test method to show that grain angles of face veneer have substantial effects on sound velocities and dynamic Young’s moduli of three types of wood-based composites. The sound velocity at 0° grain angle of face veneer was the highest, and it decreased with increasing grain angle in the range of 0° to 90°. This tendency was similar to that for dynamic Young’s modulus. The relationship between the grain angle of face veneer and the sound velocity of three types of wood-based composites can be expressed in the form of Hankinson’s equation or a second-order parabolic equation. This study also showed that the application of orthotropic elasticity theory was valid for the three types of wood-based composites. The relationship between the grain angle of the face veneer and the Young’s modulus of three types of wood-based composites can be expressed in the form of the Jenkin equation, Hankinson’s equation, or a second-order parabolic equation. Rule of mixture can also be used to predict the Young’s modulus of wood-based composite from the Young’s moduli of the two elements.

Exact Solutions of Cross-Ply Laminates with Bonding Imperfections

State-space approach is developed to analyze the bending and free vibration of a simply supported, cross-ply laminated rectangular plate featuring interlaminar bonding imperfections, for which a general linear spring layer model is adopted. The analysis is directly based on the three-dimensional theory of orthotropic elasticity and is completely exact. Numerical comparison is made, showing that although the plate theory developed in the literature behaves well for moderately thick perfect laminates it can become inaccurate when bonding imperfections are present. The special problem of the laminate in cylindrical bending is also considered, and the validity of the assumption of cylindrical bending is investigated through numerical examples.

An Analytic Method for Interlaminar Stress in A Laminated Cylindrical Shell

The present article studies the problem of interlaminar stress in a cross-ply laminated cylindrical shell under radial pressure. The elasticity solution of the problem can be obtained by making use of orthotropic elastic theory. At first, an analytic solution for a separate cross-ply cylindrical shell is woked out by means of orthotropic elastic theory. Then, boundary conditions at internal-external surfaces and continuity conditions at the interface are used to determine the unknown constants involved in this solution. Thus, an exact solution for interlaminar stresses in a laminated cylindrical shell is obtained. Finally, the distributions of the interlaminar stresses in laminated shells with various stacking sequences under radial pressure are given and the length of the bending boundary layer-giving effect on interlaminar stresses in a laminated cylindrical shell is discussed.

Compressive behaviour of softwood under uniaxial loading at different orientations to the grain

The compressive behaviour of spruce wood under uniaxial loading is studied at different orientations with regard to the longitudinal and radial direction. The dependence of the Young modulus, Poisson ratio and crushing strength on the loading angle with respect to the longitudinal direction is shown and described by a simple theory of orthotropic elasticity and a Tsai–Hill like strength criterion. The deformation and failure behaviour was also strongly influenced by the loading orientation. The different deformation and failure mechanisms found ranged from buckling of the elongated cells at loading in the longitudinal direction followed by final failure due to longitudinal cracks to shear deformation and failure at annual ring borders at loading angles of 20° and 45° due to the abrupt density change at the ring border and to the plastic yielding and collapsing followed by densification for loading in the radial direction.

Stress Intensity Factors for Interface Crack between Dissimilar Orthotropic Materials. The Case Where Principal Axes Are Not Aligned with the Interface.

In this study, stress and displacement fields at the interface crack tip between dissimilar orthotropic materials are presented for the case where the principal axes of orthotropic materials are not aligned with the interface, based on the orthotropic elasticity theory. The correct form of the definition of stress intensity factors for the interface crack is clarified. This definition is consistent with that of orthotropic material and also of dissimilar isotropic materials. The extrapolation method to determine the stress intensity factors by numerical analysis is proposed based on this definition. Some interface cracks in a finite plate of dissimilar orthotropic materials are analyzed by BEM and the stress intensity factors are presented.

On the relationship between the microstructure of bone and its mechanical stiffness

A recent study of bone structure shows that the plate-shaped carbonate apatite crystals in individual lamellae are arranged in layers across the lamellae, and that the orientation of these layers are different in alternate lamellae. Based on these findings, a new micromechanical model for the Young's modulus of bone is proposed, which accounts for the anisotropy and geometrical characteristics of the material. The model incorporates the platelet-like geometry of the basic reinforcing unit, the presence of alternating thin and thick lamellae, and the orientations of the crystal platelets in the lamellae. The thin and thick lamellae are modeled as orthotropic composite layers made up of thin rectangular apatite platelets within a collagen matrix, and classical orthotropic elasticity theory is used to calculate the Young's modulus of the lamellae. Bone is viewed as an assembly of such orthotropic lamellae bent into cylindrical structures, and having a constant, alternating angle between successive lamellae. The micromechanical model employs a modified rule-of-mixtures to account for the two types of lamellae. The model provides a curve similar to the published experimental data on the angular dependence of Young's modulus, including a local maximum at an angle between 0 and 90°. A rigorous testing of the model awaits additional experimental data.

Further Studies on Elastic Properties of Douglas Fir

The objective of this study was to examine the elastic behavior of Douglas fir in the radial‐longitudinal plane. Parallel‐ and perpendicular‐to‐grain tensile test results were used to establish linear elastic behavior. Off‐axis tensile test results were used to determine modulus of elasticity for grain angles of 15°, 30°, 45°, and 60° and the radial‐longitudinal shear modulus. The results of this study indicated that the material behaved in a linear elastic manner shown by symmetry in the compliance matrix. The measured off‐axis modulus of elasticity was found to be similar to that predicted by the transformation laws. It was concluded that the behavior of the material studied could be adequately described by linear orthotropic elastic theory in the radial‐longitudinal plane. Longitudinal and radial modulus of elasticity, and Poisson's ratios, νRL and νRL, were found to be within the same range as those previously reported. However, the radial‐longitudinal shear modulus was found to be from 10 to 30% lower than previously reported values. The difference in shear modulus was primarily due to the different testing methods used.

An improved boundary force method for analysing cracked anisotropic materials

In this paper, the Boundary Force Method (BFM), a form of an indirect boundary element method, is used to analyse composite laminates with cracks. The BFM uses the orthotropic elasticity solution for a concentrated horizontal and vertical force and a moment applied at a point in a cracked, infinite sheet as the fundamental solution. The necessary stress functions for this fundamental solution were formulated using the complex variable theory of orthotropic elasticity. The current method is an improvement over a previous method that used only forces and no moment. The improved method was verified by comparing it to accepted solutions for a finite-width, center-crack specimen subjected to uniaxial tension. Four graphite/epoxy laminates were used: [0/±45/90] s , [0], [±45] s , and [±30] s . The BFM results agreed well with accepted solutions. Convergence studies showed that with the addition of the moment in the fundamental solution, the number of boundary elements required for a converged solution was significantly reduced. Parametric studies were done for two configurations for which no orthotropic solutions are currently available: a single edge crack and an inclined single edge crack.

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich‐Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two‐dimensional orthotropic thick plate theories with stress or mixed edge data.

Analysis of a shear fracture in unidirectional fiber-reinforced composites. II (For the case that the displacement on boundaries is constrained in the direction normal to the shear fracture)

A semi-infinite shear fracture initiated from notch roots or other stress concentrators in undidrectional fiber-reinforced composites is considered on the basis of the two-dimensional theory of orthotropic elasticity. In a previous paper we have dealt with the stress intensity factors at the tip of a semi-infinite shear fracture in an infinite strip with stress-free-boundaries under tensile loading. As a result we have pointer out that the well-known method of energy balance is inappropriate for estimating stress intensity factors. The present paper continues from the previous paper and deals with the stress intensity factors for the case where the displacement on boundaries is constrained in the direction normal to the shear fracture. In contrast to the result of the previous paper, the energy balance method gives a good approximation to the stress intensity factors for both the opening and in-plane-shear modes.