A micromechanical model for predicting thermal properties and thermo-viscoelastic responses of functionally graded materials

This study introduces a micromechanical model for predicting effective thermo-viscoelastic behaviors of a functionally graded material (FGM). The studied FGM consists of two constituents with varying compositions through the thickness. The microstructure of the FGM is idealized as solid spherical particles spatially distributed in a homogeneous matrix. The mechanical properties of each constituent can vary with temperature and time, while the thermal properties are allowed to change with temperature. The FGM model includes a transition zone where the inclusion and matrix constituents are not well defined. At the transition zone, an interchange between the two constituents as inclusion and matrix takes place such that the maximum inclusion volume contents before and after the transition zone are less than 50%. A micromechanical model is used to determine through-thickness effective thermal conductivity, coefficient of thermal expansion, and time-dependent compliance/stiffness of the FGM. The material properties at the transition zone are assumed to vary linearly between the two properties at the bounds of the transition zone. The micromechanical model is designed to be compatible with finite element (FE) scheme and used to analyze heat conduction and thermo-viscoelastic responses of FGMs. Available experimental data and analytical solutions in the literature are used to verify the thermo-mechanical properties of FGMs. The effects of time and temperature dependent constituent properties on the overall temperature, stress, and displacement fields in the FGM are also examined.     

Dynamic fracture parameters and constraint effects in functionally graded syntactic epoxy foams

Functionally graded syntactic foam sheets are developed by dispersing microballoons in epoxy for studying dynamic fracture behavior under low velocity impact loading. The volume fraction of microballoons is graded linearly over the width of the sheets. The mode-I crack initiation and growth behaviors are studied using reflection coherent gradient sensing technique and high-speed photography in samples with crack on the compliant and stiff sides and oriented along the compositional gradient. Crack growth along the gradient in each case shows sudden acceleration followed by steady state growth and deceleration during the window of observation. In both cases, the crack accelerations are similar while crack decelerations show differences. The dynamic stress intensity factor history prior to crack initiation in each case shows a rapid increase at different rates with the crack on the compliant side of the graded sheet experiencing higher rate of loading relative to the one with the crack on the stiffer side. Post-crack initiation stress intensity factor histories suggest increasing fracture toughness with crack growth in the graded sample with the crack on the compliant side while a decreasing trend is seen when the crack is on the stiffer side.Optical measurements are supplemented by finite element simulations for studying crack tip constraint effects on fracture behavior of graded foam sheets. Computed plane strain constraints in graded configurations are essentially identical to the homogeneous counterpart and the computed stress intensity factors obtained from plane stress elasto-dynamic analyses of the graded foams correlate well with the experimental measurements prior to crack initiation. The computed T-stress histories however, show an earlier loss of negative crack tip constraint in case of the graded foam sample with a crack on the compliant side. This correlates well with the higher crack tip loading rate and earlier crack initiation suggesting a possible role of in-plane constraint on fracture of graded foam. The coincidence of the time rate of change of in-plane constraint parameter becoming stationary close to experimentally observed crack initiation times are noted.     

功能梯度材料的平面断裂力学分析

针对材料参数在厚度方向可能按任意连续变化的梯度材料，给出了一个新的分层模型，利用该模型求解了面内加载下梯度界面层和涂层中的界面裂纹问题，借助Fburier积分技术和传递矩阵方法，将该问题化为一个Cauchy型奇异积分方程，通过数值求解，得到感兴趣的应力强度因子，对不同形式的杨氏模量和泊松比，计算了界面裂纹应力强度因子，结果表明泊松比的变化形式对应力强度因子影响不大，可当作常数处理，而杨氏模量的影响则很大。     

Interaction integrals for fracture analysis of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials. Conservation integrals of J-type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

Anti-plane fracture of a functionally graded material strip

This paper considers the anti-plane (or mode III) crack problem in a functionally graded material strip. The shear modulus of the strip is considered for a class of functional forms for which the equilibrium equation has an analytical solution. The problem is solved by means of singular integral equation technique. Both a single crack and a series of collinear cracks are studied. The results are tabulated and plotted to show the effect of the material nonhomogeneity and crack location on the stress intensity factors.     

A novel technique for the fabrication of laboratory scale model functionally graded materials

In this work, the authors describe the design, fabrication and testing of model functionally graded materials (FGMs). The inhomogeneous property variations were generated by altering material properties through selective ultraviolet (UV) irradiation. Poly(ethylene co-carbon monoxide) (ECO) was chosen to make the FGMs because of its rapid degradation under UV light. Irradiated ECO becomes stiffer, stronger and more brittle with increasing irradiation time. Through a series of tension tests, the authors characterized the mechanical behavior of the specific ECO used as a function of UV exposure time. Furthermore, by controlling exposure time, specimens with continuously and discretely varying mechanical properties were produced. The resulting graded materials exhibited a Young's modulus that varied from about 160 MPa to 250 MPa and a strain to failure that varied from about 900 percent to 10 percent over the width in a 150 mm wide specimen. Microhardness measurements were used to determine the differences between discretely and continuously varying mechanical properties.     

Hybrid graded element model for transient heat conduction in functionally graded materials

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest’s algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.     

功能梯度材料构件三维分析的细观元模型

提出一种新颖的功能梯度构件分析的细观元法,给出了方法模型、基本算式及特点与功能。细观元法对构件的常规有限单元内部设置密集细观单元以反映材料特性梯度变化,又通过协调条件将各细观元结点自由度转换为同一常规有限元自由度,再上机计算。这种细观元法既能充分反映材料功能梯度及组分变化特性,而其计算单元与自由度又与常规有限元一样,是一种针对功能梯度构件分析的有效数值方法。算例表明了细观元法对不同情况下功能梯度构件分析的适应性与精度。     

Mechanical behavior of sandwich composite beams made of foams and functionally graded materials

We investigate sandwich composite beams using a direct approach which models slender bodies as deformable curves endowed with a certain microstructure. We derive general formulas for the effective stiffness coefficients of composite elastic beams made of several non-homogeneous materials. A special attention is given to sandwich beams with foam core, which are made of functionally graded or piecewise homogeneous materials. In the case of small deformations, the theoretical predictions are compared with experimental measurements for the three-point bending of sandwich beams, showing a very good agreement. For functionally graded sandwich columns we obtain the analytical solutions of bending, torsion and extension problems and compare them with numerical results computed by the finite element method.     

Computation of thermal fracture parameters for orthotropic functionally graded materials using Jk-integral

This article introduces a computational method based on the J_k-integral for mixed-mode fracture analysis of orthotropic functionally graded materials (FGMs) that are subjected to thermal stresses. The generalized definition of the J_k-integral is recast into a domain independent form composed of line and area integrals by utilizing the constitutive relations of plane orthotropic thermoelasticity. Implementation of the domain independent J_k-integral is realized through a numerical procedure developed by means of the finite element method. The outlined computational approach enables the evaluation of the modes I and II stress intensity factors, the energy release rate, and the T-stress. The developed technique is validated numerically by considering two different problems, the first of which is the problem of an embedded crack in an orthotropic FGM layer subjected to steady-state thermal stresses; and the second one is that of periodic cracks under transient thermal loading. Comparisons of the mixed-mode stress intensity factors evaluated by the J_k-integral based method to those calculated through the displacement correlation technique (DCT) and to those available in the literature point out that, the proposed form of the J_k-integral possesses the required domain independence and leads to numerical results of high accuracy. Further results are presented to illustrate the influences of the geometric and material constants on the thermal fracture parameters.     

On a micromechanical fracture model for cracked reinforced composites

The fracture process of reinforced composite materials is examined. In the outer region of the crack tip anisotropic continuum mechanics is employed, while for the crack tip region a heterogeneous micromechanical model is proposed. A solution is obtained using combined boundary layer — non-linear finite elements.     

Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Dynamic anti-plane fracture problem of an exponentially graded linear magnetoelectroelastic plane with a finite impermeable crack subjected to time-harmonic SH-waves is solved. Directions of wave propagation and material inhomogeneity are chosen in an arbitrary way. The fundamental solution for the coupled system of partial differential equations with variable coefficients is derived in a closed form by the hybrid usage of both an appropriate algebraic transformation for the displacement vector and the Radon transform. The formulated boundary-value problem is solved by a nonhypersingular traction boundary integral equation method (BIEM). The collocation method and parabolic approximation for the unknown generalized crack opening displacements are used for the numerical solution of the posed problem. Quarter point elements placed next to the crack-tips ensure properly modeling the singular behavior of the field variables around the crack tip. Fracture parameters as stress intensity factor, electric field intensity factor and magnetic field intensity factor are computed. Intensive simulations reveal the sensitivity of the generalized intensity factors (GIF) at the crack-tips to the material inhomogeneity, characteristics of the incident wave, coupling effects, wave-material and wave-crack interaction phenomena.     

Thermally induced fracture of a smart functionally graded composite structure

Discussed is the fracture behavior of a cracked smart actuator on a substrate under thermal load. The actuator is made of piezoelectric material with functionally graded material (FGM) properties. Integral transform method is used to reduce the problem to the solution of a set of singular integral equations and is solved numerically. This paper is completed by including graphical plots of the thermal flow, stress and electric displacement intensity factors around the crack for different crack positions and material gradients. Directions of crack initiation are also predicted by using the energy density criterion.     

A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity

We formulate a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical implementation. We show that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. In particular, we derive optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely, the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains and to the strain-gradient elasticity regularization. We show how the critical energy-release rate of specific materials can be determined from test data. Finally, we demonstrate the scope and fidelity of the model by means of an example of application, namely, Taylor-impact experiments of polyurea 1000 rods.     

Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic plate containing an internal or an edge crack parallel to the graded direction is investigated. The crack is assumed to be magneto-electrically impermeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Field intensity factors and energy release rate are derived, analyzed and partially calculated numerically. The effects of material graded index, loading combination parameter (including size and direction) and geometry criterion of the plate on the dynamic energy release rate are shown graphically. Numerical results indicate that increasing the graded index can all retard the crack extension, and that both the applied magnetic field loadings and electric field loadings play a dominant role in the dynamic fracture behaviors of crack tips.     

Discontinuous crack-bridging model for fracture toughness analysis of nacre

Studying the structure–property relation of biological materials can not only provide insight into the physical mechanisms underlying their superior properties and functions but also benefit the design and fabrication of advanced biomimetic materials. In this paper, we present a microstructure-based fracture mechanics model to investigate the toughening effect due to the crack-bridging mechanism of platelets. Our theoretical analysis demonstrates the crucial contribution of this mechanism to the high toughness of nacre. It is found that the fracture toughness of nacre exhibits distinct dependence on the sizes of platelets, and the optimized ranges for the thickness and length of platelets required to achieve higher fracture toughness are given. In addition, the effects of such factors as the mechanical properties of the organic phase (or interfaces), the effective elastic modulus of nacre, and the stacking pattern of platelets are also examined. Finally, some guidelines for the biomimetic design of novel materials are proposed based on our theoretical analysis.     

Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions

Thermoelastic behavior of functionally graded particulate materials is investigated with a micromechanical approach. Based on a special representative volume element constructed to represent the graded microstructure of a macroscopic material point, the relation between the averaged strains of the particle and matrix phases is derived with pair-wise particle interactions, and a set of governing equations for the thermoelastic behavior of functionally graded materials is presented. The effective coefficient of thermal expansion at a material point is solved through the overall averaged strain of two phases induced by temperature change under the stress-free condition, and is shown to exhibit a weak anisotropy due to the particle interactions within the graded microstructure. When the material gradient is eliminated, the proposed model predicts the effective coefficient of thermal expansion for uniform composites as expected. If the particle interactions are disregarded, the proposed model recovers the Kerner model. The proposed semi-analytical scheme is consistent and general, and can handle any thermal loading variation. As examples, the thermal stress distributions of graded thermal barrier coatings are solved for two types of thermal loading: uniform temperature change and steady-state heat conduction in the gradation direction.     

A concurrent micromechanical model for predicting nonlinear viscoelastic responses of composites reinforced with solid spherical particles

A concurrent micromechanical model for predicting nonlinear viscoelastic responses of particle reinforced polymers is developed. Particles are in the form of solid spheres having micro-scale diameters. The composite microstructures are idealized by periodically distributed cubic particles in a matrix medium. Each particle is assumed to be fully surrounded by polymeric matrix such that contact between particles can be avoided. A representative volume element (RVE) is then defined by a single particle embedded in the cubic matrix. A spatial periodicity boundary condition is imposed to the RVE. One eighth unit-cell model with four particle and polymer subcells is generated due to the three-plane symmetry of the RVE. The solid spherical particle is modeled as a linear elastic material. The polymeric matrix follows nonlinear viscoelastic behaviors of thermorheologically simple materials. The homogenized micromechanical relation is developed in terms of the average strains and stresses in the subcells and traction continuity and displacement compatibility at the subcells’ interfaces are imposed. A stress–strain correction scheme is also formulated to satisfy the linearized micromechanical and the nonlinear constitutive relations. The micromechanical model provides three-dimensional (3D) effective properties of homogeneous composite responses, while recognizing microstructural geometries and in situ material properties of the heterogeneous medium. The micromechanical formulation is designed to be compatible with general displacement based finite element (FE) analyses. Experimental data and analytical micromechanical models available in the literature are used to verify the capability of the above micromechanical model for predicting the overall composite behaviors. The proposed micromodel is also examined in terms of computational efficiency and accuracy.