Effect Of Material Inhomogeneity Research Articles

Mode-I pressurized axisymmetric penny-shaped crack in graded interfacial zone with variable modulus and Poisson’s ratio

This paper presents an analytical treatment to the fracture problem of a Mode-I pressurized penny-shaped crack in a graded interfacial zone or FGMs interlayer with variable shear modulus and Poisson’s ratio. An efficient multilayered model is proposed to address the general form of material inhomogeneity. The mixed boundary value problem is reduced to a Fredhom integral equation of the second kind. The solution procedure is also extended to address the Leonov-Panasyuk-Dugdale plastic crack. Analytical solutions in explicit form are derived for the full stress fields at the crack plane, Mode-I stress intensity factor (SIF), crack opening displacement (COD), T-Stress and plastic zone in terms of the solution of the integral equations. Numerical results of SIF, elastic stress fields, COD, T-Stress and length of plastic zone are presented to quantitively explore the effects of material inhomogeneity on the fracture response. It is demonstrated that the graded Poisson’s ratio has non-negligible influences on COD and T-Stress under certain condition. The present solutions can be used to advanced fracture analysis of pressurized crack in engineering composite material.

Forced vibrations of functionally graded annular and circular plates by domain‐boundary element method

AbstractAxi‐symmetric dynamic response of functionally graded circular and annular Mindlin plates with through‐the‐thickness variations of physical properties is investigated by a new domain‐boundary element formulation. Three governing partial differential equations of motion of the inhomogeneous plate are converted to integral equations by utilizing the static fundamental solutions of the displacement components. These integral equations are then spatially discretized by dividing the entire domain into a number of cells and incorporating suitable shape functions to approximate the variation of unknown parameters across the cells. The resulting set of ordinary differential equations in time are then solved by the Houbolt method. To verify the accuracy of the developed procedure, dynamic responses generated for a homogeneous plate are compared to those calculated through an analytical solution. Different loading conditions such as step, harmonic, and impulsive loadings are considered for generation of numerical results of dynamic responses of functionally graded circular and annular plates. Convergence characteristics and effects of material inhomogeneity, and geometric parameters are extensively investigated. It has been shown that the presented method is a fast and accurate and technique in elastodynamic analysis of functionally graded plates.

Effect of material inhomogeneity on characteristics of a functionally graded hollow cylinder

In this paper, we present a statement of the axisymmetric problem on steady-state oscillations of an inhomogeneous elastic cylinder on the basis of the general relations of the linear theory of elasticity. The material parameters and density are considered to be variable along the radial coordinate. The oscillations are caused by a distributed load applied to the outer part of the cylinder boundary. We reduce the direct problem on determination of the radial and longitudinal displacement components to the numerical solution of a number of canonical systems of differential equations with variable coefficients. On the basis of the solution obtained, we carry out the computational experiments to analyze the influence of the variable Lamé parameters on the displacement field components, the stress tensor components, the amplitude-frequency characteristics and the values of resonant frequencies.

LARGE DEFLECTION OF CANTILEVER FUNCTIONALLY GRADED SANDWICH BEAM UNDER END FORCES BASED ON A TOTAL LAGRANGE FORMULATION

A two-node beam element for large deflection analysis of cantilever functionally graded sandwich (FGSW) beams subjected to end forces is formulated in the context of total Lagrange formulation. The beams consist of three layers, a homogeneous core and two functionally graded layers with material properties varying in the thickness direction by a power gradation law. Linear functions are adopted to interpolate the displacement field and reduced integral technique is applied to evaluate the element formulation. Newton-Raphson based iterative algorithm is employed in combination with arc-length control method to compute equilibrium paths of the beams. Numerical investigations are given for the beam under a transverse point load and a moment to show the accuracy of the element and to illustrate the effects of material inhomogeneity and the layer thickness ratio on the large deflection behavior of the FGSW beams.

LARGE DEFLECTION OF CANTILEVER FUNCTIONALLY GRADED SANDWICH BEAM UNDER END FORCES BASED ON A TOTAL LAGRANGE FORMULATION

A two-node beam element for large deflection analysis of cantilever functionally graded sandwich (FGSW) beams subjected to end forces is formulated in the context of total Lagrange formulation. The beams consist of three layers, a homogeneous core and two functionally graded layers with material properties varying in the thickness direction by a power gradation law. Linear functions are adopted to interpolate the displacement field and reduced integral technique is applied to evaluate the element formulation. Newton-Raphson based iterative algorithm is employed in combination with arc-length control method to compute equilibrium paths of the beams. Numerical investigations are given for the beam under a transverse point load and a moment to show the accuracy of the element and to illustrate the effects of material inhomogeneity and the layer thickness ratio on the large deflection behavior of the FGSW beams.

Finite element model based on an efficient layerwise theory for dynamics and active vibration control of smart functionally graded beams

In this work, we present a two-noded efficient finite element (FE) model incorporating the layer-wise mechanics for the dynamics and active vibration control of smart functionally graded (FG) beams. The material properties in the FG beam are assumed to vary smoothly in the thickness direction according to power law variation. The effective properties are computed using Mori-Tanaka homogenization scheme. Electric potential profile in the electroded piezoelectric layers/patches is assumed quadratic across its thickness. The equations of motion are derived using extended Hamilton’s principle. Due to the complex algebraic expressions involved in the effective properties of FG system, the inertia and stiffness parameters are computed numerically using six point Gauss integration method. To fulfill the convergence requirements for weak integration of various energy terms in the variational formulation, the transverse displacement is interpolated with Hermite interpolation function possessing C1-continuity while the inplane displacement, shear rotation and quadratic component of the electric potential are interpolated with linear Lagrangian functions of C0-continuity. The equipotential condition of the electroded piezoelectric sensors and actuators is conveniently modelled using electric node concept. The control system is designed for constant gain velocity feedback (CGVF), and optimal LQR and LQG control strategies for a reduced order model using state space approach. The control performance is studied for single-input-single-output (SISO) and multi-input-multi-output (MIMO) configurations under various excitations. The effect of material inhomogeneity on stability/instability of the closed-loop response in the CGVF control has been discussed.

On subsurface initiated failures in marine bevel gears

In this study, the microstructure of three large bevel gears with evident tooth flank fractures is investigated to define the effect of material inhomogeneities on the probability of subsurface initiated fatigue in case hardened structural steels. The observed micro-segregation structure in the core and case/core interface of the gear teeth features a bainitic and martensitic matrix with large variations in the local hardness. Applying an orthogonal shear stress based subsurface fatigue model, alongside a hardness map obtained through multiple hardness profile measurements, an improved method to predict tooth flank fracture is proposed.

Longitudinal fracture of inhomogeneous beams with material non-linearity: An analytical investigation

This paper describes an analytical approach for analysing of longitudinal fracture in beam structures which exhibit smooth material inhomogeneity along the height and length. A solution to the strain energy release rate is derived for a general loading scheme assuming non-linear mechanical behaviour of the material. The Ramberg-Osgood stress-strain relation is used to model the material non-linearity. It is assumed that the three material properties which are involved in the Ramberg-Osgood stress-strain relation vary continuously along the height and length of the beam (exponential laws are applied for describing the distribution of these properties along the height and length of the beam). The solution derived is valid for a longitudinal crack located arbitrary along the beam height. It is shown that the approach developed can be applied for evaluating the effects of material inhomogeneity along the height and length of the beam, non-linear mechanical behaviour of the material, crack location along the beam height and the crack length on the longitudinal fracture.

Application of the material inhomogeneity effect for the improvement of fracture toughness of a brittle polymer

In a multilayered structure with a crack, a spatial change in the mechanical properties of the material strongly influences the crack driving force. This material inhomogeneity effect can be utilized to improve the fracture toughness of a given structure by inserting thin, soft interlayers into the material. The effectiveness of this procedure has been demonstrated on high-strength materials, such as metallic alloys and ceramics. It is shown in this article that the material inhomogeneity effect can be also successfully applied to polymers and that it is possible to predict the improvement in fracture toughness by a numerical analysis. First, a numerical case study based on the configurational force concept is performed on a brittle polymer matrix with interlayers made of materials with different strength and Young’s modulus. After selecting the most appropriate interlayer material, a composite is fabricated, which contains a single interlayer. Fracture toughness experiments show approximately 7 times higher fracture toughness for the composite in comparison to the homogeneous matrix material. Numerical fracture mechanics tests are performed on homogeneous and composite material using the cohesive zone model for crack growth simulation. A procedure to calibrate the cohesive zone parameters is worked out, which is relatively easy for the homogeneous material, but more sophisticated for the composite material. The numerical analysis provides a tool for predicting the fracture toughness of multilayered polymer composites.

Development of Damage-Tolerant and Fracture-Resistant Materials by Utilizing the Material Inhomogeneity Effect

AbstractThe improvement of fracture strength by insertion of thin, soft interlayers is a strategy observed in biological materials such as deep-see sponges. The basic mechanism is a reduction of the crack driving force due to the spatial variation of yield strength and/or Young's modulus. The application of this “material inhomogeneity effect” is demonstrated in this paper. The effectiveness of various interlayer configurations is investigated by numerical simulations under application of the configurational force concept. Laminated composites, made of high-strength tool steels as matrix materials and low-strength deep-drawing steel as interlayer material, were manufactured by hot press bonding. The number of interlayers and the interlayer thickness were varied. Fracture mechanics experiments show crack arrest in the first interlayer and significant improvements in fracture toughness, even without the occurrence of other toughening mechanisms, such as interface delamination. The application of the material inhomogeneity effect for different types of matrix materials is discussed.

Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

• A set of analytical Green's functions for a cross anisotropic inhomogeneous tri-material solid is given. • The response of a graded solid sandwiched between two homogenous transversely isotropic layers is studied numerically. • The analysis of an finite bi-material layer has been given in terms of Green's functions and analyzed numerically. • The results may be used as benchmarks for numerical studies or developing numerical methods like boundary element methods . This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.

Longitudinal Fracture Analysis of Nonlinear Elastic Circular Shafts Loaded in Torsion

The longitudinal fracture of circular shafts loaded in torsion is analytically investigated. It is assumed that a longitudinal crack with a circular crack front is arbitrarily located to the radial direction of the shaft cross section. The shafts are made of nonlinear elastic material, which exhibits continuous inhomogeneity in the radial direction. The material nonlinearity is described by the Ramberg–Osgood stress–strain relation. The longitudinal fracture is studied in terms of the strain energy release rate. The developed analysis is applied to evaluate the longitudinal fracture behavior of a clamped nonlinear elastic inhomogeneous shaft loaded in torsion by a torque applied to a free end of internal crack arm. For the method verification, the strain energy release rate is also assessed by analyzing the energy balance in the clamped shaft. A parametric study is performed to assess the effects of material inhomogeneity, its nonlinearity, and crack orientation to the shaft radial direction on the longitudinal fracture behavior of the clamped shaft.

Longitudinal cracks in non-linear elastic beams exhibiting material inhomogeneity

Longitudinal fracture behavior of non-linear elastic beam configurations is studied in terms of the strain energy release rate. It is assumed that the beams exhibit continuous material inhomogeneity along the width as well as along the height of the cross-section. The Ramberg-Osgood stress-strain relation is used for describing the non-linear mechanical behavior of the inhomogeneous material. A solution to strain energy release rate is derived that holds for inhomogeneous beams of arbitrary cross-section under combination of axial force and bending moments. Besides, the solution may be applied at any law of continuous distribution of the modulus of elasticity in the beam cross-section. The longitudinal crack may be located arbitrary along the beam height. The solution is used to investigate a longitudinal crack in a beam configuration of rectangular cross-section under four-point bending. The crack is located symmetrically with respect to the beam mid-span. It is assumed that the modulus of elasticity varies continuously according a cosine law in the beam cross-section. The longitudinal fracture behavior of the inhomogeneous beam is studied also by applying the J-integral approach for verification of the non-linear solution to the strain energy release rate derived in the present paper. Effects of material inhomogeneity, crack location along the beam height and non-linear mechanical behavior of the material on the longitudinal fracture behavior are evaluated. Thus, the solution derived in the present paper can be used in engineering design of inhomogeneous non-linear elastic structural members to assess the influence of various material and geometrical parameters on longitudinal fracture.

Crack arrest in thin metallic film stacks due to material- and residual stress inhomogeneities

Miniaturized materials, in general, exhibit higher strength compared to their bulk counterparts. As a consequence, their resistance to fracture is often compromised. However, the effect of material inhomogeneities can be used to significantly improve the fracture toughness of thin film components. In this work, the material inhomogeneity effect on the crack driving force, caused by material property and residual stress variations in thin tungsten and copper stacks, is numerically investigated. To this purpose, a finite element analysis is performed using the concept of configurational forces. In this way, we are able to distinguish between the various inhomogeneity effects and draw conclusions about the effective crack driving force. It is demonstrated that the material inhomogeneity effect is not solely determined by the material property variations at the interfaces, since an important contribution emerges due to a smooth residual stress gradient within the layers. The possibility to separate the different effects represents an opportunity for cost efficient design of future reliable thin film microelectronic components.

On the stability of internally pressurized thick-walled spherical and cylindrical shells made of functionally graded incompressible hyperelastic material

Abstract In this paper, stability analysis of thick-walled spherical and cylindrical shells made of functionally graded incompressible hyperelastic material subjected to internal pressure is presented. Instability point happens in the inflation of above mentioned shells and in this paper effect of material inhomogeneity and shell thickness has been investigated. Extended Ogden strain energy function with variable material parameter is used to model the material behavior. To model inhomogeneity, we assume that material parameter varies by a power law function in the radial direction and inhomogeneity factor is a power in the power law function. Analytical method is used to find the internal pressure versus hoop extension ratio relations in explicit form for both of cylindrical and spherical shells and the non-monotonic behavior of the inflation curves is studied. Following this, profile of inflation pressure versus hoop stretch is presented and effect of the inhomogeneity and shell thickness in the onset of instability is studied. The obtained results show that the material inhomogeneity parameter and shell thickness have a significant influence on the stability of above mentioned shells. Thus with selecting a proper material inhomogeneity parameter and shell thickness, engineers can design a specific FGM hollow cylinder that can meet some special requirements.

Finite element analysis of fretting contact for nonhomogenous materials

Fretting problem arises in the case of relatively small sliding motion between contacting surfaces. Fatigue life of the components that are in contact with each other, especially in rotorcraft may be significantly reduced due to fretting. The purpose of this study is to investigate material inhomogeneity near the contact region on the fretting problem in a cylindrical on flat contact configuration. A finite element (FE) model was constructed by using commercial finite element package ABAQUSTMto study partial sliding and stress concentrations. In order to investigate the effect of material inhomogeneity, the fretting contact is analyzed by introducing voids near the contact region. The void size and an array of voids is introduced into the substrate. The results are compared in terms of pressure, shear traction, tangential stress magnitudes and relative slip between the contacting materials.

Plane strain cylindrical indentation of functionally graded half-plane with exponentially varying shear modulus in the presence of residual surface tension

A functionally graded half-plane with shear modulus varying exponentially along the direction normal to the surface and surface effects accounted through Gurtin–Murdoch model, indented by long rigid smooth cylindrical indenter is solved to understand the effect of material inhomogeneity and surface effects on indentation response. The Green’s function relating surface load to surface displacement under plane strain condition is obtained semi-analytically through the combination of Airy stress function approach and Fourier transforms and utilized to solve the contact problem. The solution is used to study the effect of inhomogeneity through grading parameter and surface effects through residual surface tension based intrinsic length scale on the contact pressure, contact size and in-plane normal stress on the surface responsible for cracks during indentation.

Elastoplastic analysis of FGM plate with a central cutout of various shapes under thermomechanical loading

ABSTRACTThe present work aims to study the elastoplastic buckling, postbuckling, and failure behavior of perforated Ni/Al2O3 functionally graded material (FGM) plate with various shaped cutouts (i.e., circular, square, diamond, and elliptical) of various sizes under thermomechanical loading conditions using finite element method (FEM). The nonlinear FEM formulation is based on the first-order shear deformation theory and von Kármán’s nonlinear kinematics in which the material nonlinearity is incorporated. The nonlinear temperature-dependent thermoelastic material properties of FGM plate are varied in the thickness direction by controlling the volume fraction of the constituent materials (i.e., ceramic and metal) as per a power law, and Mori–Tanaka homogenization scheme is applied to evaluate the properties at a particular thickness coordinate of FGM. In accordance with the Tamura–Tomota–Ozawa model (TTO model), the ceramic phase of FGM is considered to be elastic, whereas the metal phase is assumed to be elastoplastic. Further, the elastoplastic analysis of FGM is assumed to follow J2 plasticity with isotropic hardening. After validating the present formulation with the results available in the literature, various numerical studies are conducted to examine the effects of material inhomogeneity, thermal loading, cutout shape, and size on the elastoplastic buckling, postbuckling, and failure behavior of perforated FGM plate. It is observed that for smaller cutout sizes, the FGM plate with square shape cutout possesses maximum value of ultimate failure load; however, for larger cutout size, the FGM plate with diamond cutout depicts highest ultimate failure load. Furthermore, for all cutout shapes, the ultimate failure load of FGM plate decreases with an increase in cutout size. It is also revealed that irrespective of shape and size of cutout, the material plastic flow has considerable effect on postbuckling path of FGM plate, and under thermomechanical loading conditions, the FGM plate shows destabilizing response after the point of maximum postbuckling strength.

Improving strength and toughness of materials by utilizing spatial variations of the yield stress

The introduction of thin interlayers with low yield stress can greatly improve the strength and the fracture toughness of inherently brittle materials. The reason is that the spatial yield stress variation affects the crack driving force, which strongly decreases when the crack tip is located in the interlayer region, near the boundary to the hard matrix material. This can lead to crack arrest. The material inhomogeneity effect appears without previous delamination of the interlayer. The decisive parameters influencing the effect are the interlayer spacing (the wavelength of the yield stress variation), the interlayer thickness and the yield stress ratio between interlayer and matrix. Based on numerical simulations with the configurational forces concept, it is demonstrated how the architectural parameters of the multilayer must be chosen in order to enhance the fracture stress and the fracture toughness of the material. An iterative procedure is proposed to find the optimum configuration. It is found that the optimum wavelength is inversely proportional to the square of the applied stress. A similar relation is given for composites with spatial variations in Young's modulus.

In-plane material inhomogeneity of functionally graded plates: A higher-order shear deformation plate isogeometric analysis

Abstract Numerical investigation of buckling and free vibration of functionally graded plates considering in-plane material inhomogeneity is presented in this paper. A novel and effective approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) is developed for fulfilling that purpose, analyzing the critical buckling parameter and natural eigenvalues of functionally graded plates involving the in-plane material inhomogeneity. The HSDT allows us to account for shear deformation effect without requiring any shear correction factors. Non-uniform rational B-spline is used as basis functions, resulting in both exact geometric representation and high order approximations, enabling to easily achieve the C 1 -continuity requirement of the HSDT without any additional variables. The material properties of functionally graded plates are assumed to vary along in-plane direction. Numerical examples are presented and discussed, in which the effects of material inhomogeneity, length to thickness ratio and boundary conditions on natural frequencies and critical buckling loads are investigated.