Marek-Jerzy Pindera

Higher-Order Theory for Functionally Graded Materials

This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composites functionally graded microstructure. The generalization outlined herein involves extension of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions. Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.

Higher-order theory for periodic multiphase materials with inelastic phases

Abstract An extension of a recently-developed linear thermoelastic theory for multiphase periodic materials is presented which admits inelastic behavior of the constituent phases. The extended theory is capable of accurately estimating both the effective inelastic response of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the materials periodic microstructure. The models analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite-element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading. The models predictive accuracy in generating both the effective inelastic stress-strain response and the local stress and inelastic strain fields is demonstrated by comparison with the results of an analytical inelastic solution for the axisymmetric and axial shear response of a unidirectional composite based on the concentric cylinder model and with finite-element results for transverse loading.

An efficient implementation of the generalized method of cells for unidirectional, multi-phased composites with complex microstructures

An efficient implementation of the generalized method of cells micromechanics model is presented that allows analysis of periodic unidirectional composites characterized by repeated unit cells containing thousands of subcells. The original formulation, given in terms of Hills strain concentration matrices that relate average subcell strains to the macroscopic stains, is reformulated in terms of the interfacial subcell tractions as the basic unknowns. This is accomplished by expressing the displacement continuity equations in terms of the stresses and then imposing the traction continuity conditions directly. The result is a mixed formulation wherein the unknown interfacial subcell traction components are related to the macroscopic strain components. Because the stress field throughout the repeating unit cell is piece-wise uniform, the imposition of traction continuity conditions directly in the displacement continuity equations, expressed in terms of stresses, substantially reduces the number of unknown subcell traction (and stress) components, and thus the size of the system of equations that must be solved. Further reduction in the size of the system of continuity equations is obtained by separating the normal and shear traction equations in those instances where the individual subcells are, at most, orthotropic. Comparison of execution times obtained with the original and reformulated versions of the generalized method of cells demonstrates the new versions efficiency. As demonstrated through examples, the reformulated version facilitates previously unattainable detailed analysis of the impact of fiber cross-section geometry and arrangement on the response of multi-phased unidirectional composites.

Shear characterization of unidirectional composites with the off-axis tension test

The influence of end constraints on accurate determination of the intralaminar shear modulusG12 from an off-axis tension test is examined both analytically and experimentally. The Pagano-Halpin model is employed to illustrate that, when the effect of end constraints is properly considered, the exact expression forG12 is obtained. When the effect of end constraints is neglected, expressions for the apparent shear modulusG12* and apparent Youngs modulusExx* are obtained. Numerical comparison for various off-axis configurations and aspect ratios is carried out using typical material properties for graphite/polyimide unidirectional composites. It is demonstrated that the end-constraint effect influences accurate determination ofG12 more adversely than it affects the laminate Youngs modulusExx in the low off-axis range. Experimental results obtained from off-axis tests on unidirectional Gr/Pi specimens confirm the above. Based on the presented analytical and experimental evidence, the 45-deg off-axis coupon is proposed for the determination of the intralaminar shear modulusG12.

Thermoelastic theory for the response of materials functionally graded in two directions

A recently developed micromechanical theory for the thermoelastic response of functionally graded composites with nonuniform fiber spacing in the through-thickness direction is further extended to enable analysis of material architectures characterized by arbitrarily nonuniform fiber spacing in two directions. In contrast to currently employed micromechanical approaches applied to functionally graded materials, which decouple the local and global effects by assuming the existence of a representative volume element at every point within the composite, the new theory explicitly couples the local and global effects. The analytical development is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense. Results are presented that illustrate the capability of the derived theory to capture local stress gradients at the free edge of a laminated composite plate due to the application of a uniform temperature change. It is further shown that it is possible to reduce the magnitude of these stress concentrations by a proper management of the microstructure of the composite plies near the free edge. Thus by an appropriate tailoring of the microstructure it is possible to reduce or prevent the likelihood of delamination at free edges of standard composite laminates.

Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials

A new micromechanics model is presented which is capable of accurately estimating both the effective elastic constants of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the materials periodic microstructure. The models analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading which, in turn, can be incorporated into a structural analysis computer code. The models predictive accuracy is demonstrated by comparison with reported results of detailed finite element analyses of periodic composites as well as with the classical elasticity solution for an inclusion in an infinite matrix.

Response of functionally graded composites to thermal gradients

Abstract A new micromechanical theory is presented for the response of functionally graded metal-matrix composites subjected to thermal gradients. In contrast to existing micromechanical theories that utilize standard homogenization schemes in the course of calculating microscopic and macroscopic field quantities, in the present approach the actual microstructural details are explicitly coupled with the macrostructure of the composite. The theory is particularly well-suited for predicting the response of thin-walled metal-matrix composites with a finite number of large-diameter fibers in the thickness direction subjected to thermal gradients. Standard homogenization techniques which decouple micromechanical and macromechanical analyses may not produce reliable results for such configurations. Examples presented illustrate the usefulness of the outlined approach in generating favorable stress distributions in the presence of thermal gradients by appropriately grading the internal microstructural details of the composite.

A Methodology for Accurate Shear Characterization of Unidirectional Composites

A combined experimental-analytical methodology is presented for accurate determina tion of the intralaminar shear modulus G 12 of unidirectional composites using the off-axis tension test and/or the Iosipescu test. It is demonstrated that consistent values of G 12 can be obtained with the two methods provided that: (1) specimen geometry is optimized for the off-axis test, (2) correction factors are employed to account for the shear stress nonuni formity in the test section of Iosipescu specimens. The problem of measuring the shear strength with these specimens is discussed. The 45-deg off-axis tensile coupon is recom mended for determination of the shear modulus. The 0-deg Iosipescu specimen is recom mended for determining an upper lower bound on the shear strength.

Elastic response of metal matrix composites with tailored microstructures to thermal gradients

Abstract A new micromechanical theory is presented for the response of heterogeneous metal matrix composites subjected to thermal gradients. In contrast to existing micromechanical theories that utilize classical homogenization schemes in the course of calculating microscopic and macroscopic field quantities, in the present approach the actual microstructural details are explicitly coupled with the macrostructure of the composite. Examples are offered that illustrate limitations of the classical homogenization approach in predicting the response of thin-walled metal matrix composites with large-diameter fibers to thermal gradients. These examples include composites with a finite number of fibers in the thickness direction that may be uniformly or nonuniformly spaced, thus admitting so-called functionally gradient composites. The results illustrate that the classical approach of decoupling micromechanical and macromechanical analyses in the presence of a finite number of large-diameter fibers, finite dimensions of the composite, and temperature gradient may lead to serious errors in the calculation of both macroscopic and microscopic field quantities. The usefulness of the new outlined approach in generating favorable stress distributions in the presence of thermal gradients by appropriately tailoring the internal microstructural details of the composite is also demonstrated.

Local/global stiffness matrix formulation for composite materials and structures

Abstract An efficient algorithm is outlined for solving boundary-value problems involving laminated composite materials and structures that require satisfaction of both continuity of tractions and displacements along common interfaces. The method is based on the systematic construction of a global stiffness matrix for the entire laminated structure in terms of local stiffness matrices of the individual layers. The local stiffness matrix relates the traction components at the upper and lower (or inner and outer) surface of a given layer to the corresponding displacements. The assembly of local stiffness matrices into a global stiffness matrix is carried out by enforcing continuity conditions along the interfaces which, in effect, leads to reformulation of the problem in terms of interfacial displacements as the basic unknown variables. This, in turn, results in the elimination of certain redundant continuity conditions and thus reduction in the number of simultaneous algebraic equations that need to be solved. An additional advantage of the local/global stiffness matrix formulation is the ease with which certain mixed boundary-value problems can be reduced to singular integral equations of the Fredholm type for the determination of unknown quantities such as the contact pressure in the case of contact problems and the crack-opening displacement in the case of interfacial crack problems.