Orthotropic Solids Research Articles

On the Transition from a Model to a Real Solid in the Course of Photoelastic Simulation of Problems of the Mechanics of Orthotropic Solids

We propose a method for the recalculation of dynamic stresses from a photoelastic model to a real solid in plane problems of the mechanics of orthotropic bodies. To estimate the accuracy of the method, we recalculate stresses for a problem with known theoretical solution.

Combined Isotropic-Kinematic Hardening Laws with Anisotropic Back-stress Evolution for Orthotropic Fiber-Reinforced Composites

In order to describe the Bauschinger and transient behavior of orthotropic fiber-reinforced composite solids, a combined isotropic-kinematic hardening law based on the non-linear kinematic hardening rule was considered here, in particular, based on the Chaboche type law. In this modified constitutive law, the anisotropic evolution of the back-stress was properly accounted for. Also, to represent the orthotropy of composite materials, Hill's 1948 quadratic yield function and the orthotropic elasticity constitutive equations were utilized. Furthermore, the numerical formulation to update the stresses was also developed based on the incremental deformation theory for the boundary value problems. Numerical examples confirmed that the new law based on the anisotropic evolution of the back-stress complies well with the constitutive behavior of highly anisotropic materials such as fiber-reinforced composites.

Optimal experiment design for the identification of thermo-physical properties of orthotropic solids

This numerical study deals with the identification of several thermal properties of orthotropic solids by using an optimal experimental design. The system under investigation is a cylindrical vertical sample, which is submitted to an irradiation flux on the upper face on a central circular area. The temperature response, during and after the irradiation, is measured at the opposite face. In this article, the system is particularly developed to the identification of the Fourier number including the axial diffusivity, the ratio of radial to axial conductivities, a Biot number and a heat flux term from one temperature evolution versus time. The temperature sensor position is adjusted simultaneously with the identification of the above four parameters.

Analysis of Two-Dimensional Nonlinear Transient Heat Conduction in Anisotropic Solids by Boundary Element Method

Liao and Chwang proposed a new boundary element method using homotopy for nonlinear transient heat conduction problems in orthotropic solids. This study is concerned with an application of this approach to anisotropic solids. Mathematical formulations of the boundary element method are presented in detail for two-dimensional nonlinear transient heat conduction. Computation is started from the initial stage subject to a given initial condition, and advanced by an iterative procedure until the root mean square (RMS) of the residual is a minimum or the rate of change in RMS is less than a tolerance. The proposed solution procedure is applied to some typical examples, and the usefulness of the proposed BEM is demonstrated through discussions of the results obtained. Emphasis is placed on comparison between the present scheme and isotropic solid scheme of the Liao-Chwang method.

Effects of the Heating Process and Body Dimensions on the Estimation of the Thermal Conductivity Components of Orthotropic Solids

In this article, we design the experiment for estimating the thermal conductivity components of an orthotropic medium, with respect to the solid dimensions and to the magnitude of the fluxes that are used to heat the solid. The D-optimum approach is used in the experimental design. By using a transformation that maps the heat conduction equation for an orthotropic medium into the heat conduction equation for an isotropic medium, we show that thermal conductivity components with identical relative accuracies can be estimated by appropriately selecting the experimental variables under picture. The Levenberg-Marquardt method is applied for the solution of the present parameter estimation problem by using simulated temperature measurements.

Parameter estimation of orthotropic solids with uncertainty in the sensor position: Use of Levenberg-Marquardt and conjugate gradient methods

This numerical study deals with the estimation of radial and axial thermal diffusivities of orthotropic solids by two different parameter estimation methods. The system under investigation is a cylindrical vertical sample, which is submitted to an irradiation flux on the upper face on a central circular area. The temperature response, during and after the irradiation, is measured at the opposite face. In this paper, the system is particularly designed for the identification of the Fourier number including the axial diffusivity, the ratio of radial to axial thermal conductivities, the Biot number, and a heat flux term from one temperature evolution versus time. The temperature sensor position is adjusted simultaneously with the identification of the above four parameters.

A continuum damage model for linear viscoelastic composite materials

This paper presents a constitutive model for linear viscoelastic orthotropic solids containing a fixed level of distributed cracks. The model is formulated in a continuum damage mechanics framework using internal variables taken as second rank tensors. Use is made of the correspondence principle for linear viscoelastic solids to define a pseudo strain energy function in the Laplace domain. This function is then expressed as a polynomial in transformed strain and tensorial damage variables using the integrity bases restricted by the initial orthotropic symmetry of the material. The constitutive relationships derived in the Laplace domain are then converted to the time domain by using the inverse Laplace transform. The model is applied to the specific case of cross-ply laminates with transverse matrix cracks. The material coefficient functions appearing in the model are determined by a numerical (finite element) method for one cross-ply laminate configuration at one damage level. Predictions of the viscoelastic response are then made for the same laminate at other damage levels and for other cross-ply laminate configurations at different damage levels. These predictions agree well with independently determined time variations of properties by an analytic method (Kumar and Talreja, 2001, Linear viscoelastic behavior of matrix cracked cross-ply laminates. Mechanics of Materials 33 (3), 139–154) as well as with the numerically calculated values. Extension of the model to incorporate effects of transient temperature, physical aging and moisture is outlined.

A 2-D Time-Domain BIEM for Dynamic Analysis of Cracked Orthotropic Solids

A 2-D Time-Domain BIEM for Dynamic Analysis of Cracked Orthotropic Solids

Crack initiation behaviour of orthotropic solids as predicted by the strain energy density theory

The strain energy density theory is applied to determine crack initiation in orthotropic solids. An approach is used to derive the complex variable expressions of the elastic fields. The formulation is used to solve the boundary value problem by superimposing of Mode-I and Mode-II crack problems. It is shown that asymptotic expressions of the crack tip stress and displacement fields are affected by non-singular terms related to biaxial loading.

BEM Analysis of Two-Dimensional Elastodynamic Problems of Anisotropic Solids

A direct BEM (boundary element method) formulation and its numerical implementation for steady-state elastodynamic analysis of two-dimensional problems involving orthotropic and nonorthotropic solids of arbitrary shape is presented. The present formulation is developed by incorporating isoparametric quadratic boundary elements, a rigorous self-adapting numerical integration scheme, and the Green's functions of Wang and Rajapakse for two-dimensional orthotropic solids. The full-space as well as the half-space Green's functions have been implemented. The formulation is capable of treating multidomain problems by substructuring and satisfying the equilibrium and displacement compatibility at the interfaces. Thus, problems related to the layered media and soil-structure interaction can all be analyzed. The accuracy and reliability of the present BEM analysis are demonstrated through comparison of results obtained for a few example problems with published solutions. Finally, the applicability of this methodolo...

The nonaxisymmetric thermostressed state of laminated isotropic and rectilinearly orthotropic solids of revolution

A technique for determination of the nonaxisymmetric elastoplastic stress-strain state of laminated isotropic and rectilinearly orthotropic solids of revolution under nonisothermal loading is described. The semianalytic method of finite elements is used together with the method of successive approximations. The deformation of isotropic materials is described by the equations of the theory of deformation along trajectories of small curvature, and the deformation of orthotropic materials is described by the equations of the theory of elasticity. The thermostressed state of a three-layered solid of revolution of complex form made from isotropic and cylindrically and rectilinearly orthotropic materials subject to nonaxisymmetric heating is studied.

The nonaxisymmetric elastoplastic state of isotropic and cylindrically orthotropic solids of revolution under nonisothermal loading

A technique for numerical analysis of the nonaxisymmetric elastoplastic stress-strain state of laminated isotropic and cylindrically orthotropic bodies of revolution under nonisothermal loading with allowance for its history is stated. The strain of isotropic materials is described by the equations of the theory of strain along small-curvature trajectories, while Hooke's law is used to describe the strain of orthotropic materials. The necessity of accounting for the loading history is shown by an example of a nonaxisymmetrically heated sandwich solid of revolution.

On radiation-free transonic motion of cracks and dislocations

Eshelby has shown that a glide dislocation can move without radiation of energy at 2 of the shear wave speed. It is also known that the same velocity plays a special role in shear crack propagation. This result has not received wide attention in the past due to lack of experiments and numerical simulations of transonic defects. Recent experiments on transonic shear fracture and molecular dynamics simulations of dislocation motion have stimulated renewed interest in the behavior of cracks and dislocations beyond the subsonic regime. We attempt to provide a unified treatment of transonic cracks and dislocations by elaborating on the fundamental result of Eshelby. We develop a unified treatment of radiation-free transonic motion of both cracks and dislocations. We use Stroh’s method to generalize the Eshelby theorem to orthotropic and anisotropic elastic solids. In the case of orthotropic solids, we provide a proof of existence of the radiation-free speed. In the case of general anisotropic solids, there are three wave speeds c 3< c 2< c 1 in any given crystal orientation at which a moving defect is considered. In the first transonic regime c 3< v< c 2, we show that there always exists a radiation-free state for any given velocity v of a moving defect. In the second transonic regime c 2< v< c 1, the existence of radiation-free states appears to depend on both the symmetry properties of the material and the defect orientation. Examples of existence in the second transonic regime include a crack propagating in an isotropic solid and a crack propagating along a plane of symmetry in an orthotropic solid.

Laser ultrasonic investigation of the elastic properties of unidirectional graphite-epoxy composites

An ultrasonic method is proposed for the determination of the elastic constants of orthotropic solids with a laser source of ultrasound and wide-band registration of acoustic pulses. The propagation of acoustic transients in unidirectional fiber-reinforced graphite-epoxy composites is investigated. The experimental data show that the model of orthotropic solids is valid for the description of the mechanical properties of these materials. The absence of frequenc dispersion of the phase velocity in the spectral range of 1–15 MHz for every direction of propagation of ultrasound waves in the composite was confirmed. A complete set of elastic constants of unidirectional fiber-reinforced graphite-epoxy composites is calculated.

Methods for calculating stress intensity factors for interfacial cracks between two orthotropic solids

Stress intensity factors for interfacial cracks between two dissimilar orthotropic materials were considered. Due to the oscillatory characteristics of stresses and displacements near the crack tip, individual strain energy release rates no longer exist. Instead, the individual strain energy release rates corresponding to a finite crack extension were obtained in terms of the stress intensity factors and the assumed crack extension Δa. The finite element methods in conjunction with the crack closure technique were used to calculate these finite extension strain energy release rates from which accurate stress intensity factors were obtained. An alternative method based on crack surface displacement ratio was also discussed. Non-oscillatory (Δa-independent) Mode I and Mode II “strain energy release rates” were also proposed to provide an alternate measure of fracture mode mixity or to be used as a fracture criterion for interfacial cracks.

Microfracturing Patterns And Effective Elastic Properties Of Solids With Interacting Holes

We analyze some basic aspects of micromechanics of solids with multiple holes and microcracks. Two groups of properties are addressed: 1. Stress concentrations and microfracturing patterns in 2-D isotropic elastic solids with interacting elliptical holes. We examine various physical effects produced by interactions and the impact of hole eccentricities on these effects. Patterns of stress concentrations in such mixtures imply certain microfracturing patterns in materials with multiple defects. 2. Effective elastic properties of the isotropic and orthotropic solids with arbitrarily (randomly or non-randomly) oriented holes of diverse shapes. We describe the potential-based procedure to derive effective moduli of porous materials. The example of one family of parallel elliptical holes in the orthotropic matrix is considered in detail. 1 Interaction of holes in elastic solids To study interactions of elliptical holes (of diverse eccentricities) in a 2-D elastic solid, we employ the Neumann-Schwartz alternating method, or stress feedbacks. It appears to be more efficient and accurate (particularly, for closely spaced strongly interacting holes) than series Transactions on Engineering Sciences vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

On the asymmetric yield surface of plastically orthotropic materials: A phenomenological study

Hill's (1948) yield criterion for plastically orthotropic solids is extended to include materials whose tensile and compressive yield stresses, obtained from uniaxial loading tests, are different. First, we review and discuss the classic yield criteria in plasticity theory including plastically isotropic and anisotropic materials. Then, we present a yield criterion for plastic anisotropic solids whose initial yield surface may not be symmetric about the origin in stress space. The new criterion is based on the criterion given by Hill ( Proc. Roy. Soc. London A, 1948, 193, 281) for anisotropic solids and the criterion proposed by Drucker and Prager ( Q. Appl. Math., 1952, 10, 157) for soil whose whose tensile and compressive strengths are far apart. The coefficients that appear in the new criterion can be determined from uniaxial tension and compression, and simple shearing tests. The formulae for the determination of those coefficients are given. Specifically, the new criterion is applied to the yielding of thin sheets where a plane stress condition prevails. Finally, the prediction of the criterion is compared with the experimental data obtained by Lee and Backofen ( Trans. Metall. Soc. AIME, 1966, 236, 1966) on titanium alloy sheets.

Bifurcation of Orthotropic Solids

We consider a large deformation plane-strain problem involving a compressible orthotropic solid subjected to uniaxial compressive loading along one of the principle directions which is aligned with the boundary of a half-space. An exact solution for the displacement field is obtained and a condition for the smallest compressive load corresponding to the onset of a surface instability is determined. It is shown that when the compression occurs along the stiffest direction this condition is expressible in terms of a cubic polynomial, and that the corresponding critical load is lower than the well-known estimate which determines the critical load to be equal to the inplane shear modulus.

Fast transient dynamic plane stress analysis of orthotropic Hill-type solids at finite elastoplastic strains

The objective of this work is the geometrically nonlinear plane stress analysis of orthotropic Hill type elastoplastic solids under short term dynamic loading conditions. Thereby, the underlying motivation is the necessity to model the anisotropic behaviour of metal specimens which is due to the manufacturing procedure in terms of an orthotropic yield condition. To this end, the classical von Mises condition in the framework of multiplicative elastoplasticity is substituted by a yield criterion which invokes a second order anisotropy tensor acting on the deviatoric stresses in the plastic intermediate configuration. A reparametrization of this model reveals its relation to the classical criterion originally proposed by Hill in the geometrically linear setting. Moreover an element technology is outlined recovering the plane stress response of arbitrary 3D constitutive models without any plane stress specific modifications in the large strain regime. The intriguing influence of certain types of orthotropy on the failure patterns of thin metal sheets under plane stress uniaxial extension is investigated numerically. Thereby, an explicit time stepping procedure designed to incorporate the developed plane stress element is employed to trace the short term response predictions of the investigated specimens.

A boundary element method based on time-stepping approximation for transient heat conduction in anisotropic solids

The time-stepping boundary element method has been so far applied by the authors to transient heat conduction in isotropic solids as well as in orthotropic solids. In this paper, attempt is made to extend the method to 2-D transient heat conduction in arbitrarily anisotropic solids. The resulting boundary integral equation is discretized by means of the boundary element with quadratic interpolation. The final system of equations thus obtained is solved by advancing the time step from the given initial state to the final state. Through numerical compuation of a few examples the potential usefulness of the proposed method is demonstrated.