Micromechanical analysis of fatigue cracks in cord–rubber composites

Analyses of two different types of cracks due to fatigue of cord–rubber composites is carried out using micromechanical two-dimensional (2D) and three-dimensional (3D) finite element analysis. The fracture parameter, tearing energy (TE)/J-integral that characterizes the severity of crack tip stresses in rubber composites, is computed from the finite element results of stresses and strains. The results obtained are validated with existing analytical methods in the literature. Numerical results of J-integral values are presented for two crack types, and crack sizes under transverse strain and shear strain loading conditions. The results presented illustrate that crack type, loading, and crack size have a strong effect on the values of J-integral. The results of the J-integral should help our understanding in estimating the severity of local failures in cord–rubber composites.     

Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor.     

The role of flaw geometry in thin film delamination from two-dimensional interface flaws along free edges

Delamination from planar interface edge flaws between a thin film and a semi-infinite substrate is examined to determine the roles of flaw width and depth relative to the film thickness. The flaws have curved and straight sections, and the crack front intersects the free edge at a right angle. Three-dimensional finite element models are used to extract local energy release rates and mode-mixity angles along the entire crack front. This paper focuses the crack dimensions required to reach steady state, wherein energy release rates are independent of flaw dimensions along the entire crack front. Results indicate that moderate elastic mismatch, although affecting mode mixity, plays a small role in determining the crack aspect ratios required to reach steady state. For wide cracks, the energy release rate for crack advance into the film interior approaches the plane-strain steady-state value when the half-width of the crack is approximately four times its depth (for cracks whose depths is several times the film thickness). For narrow cracks, the energy release rate near the free edge is significantly greater than the plane-strain steady-state result, and reaches a steady state when the depth approximately 10 times its width (for widths several time the film thickness). The results imply that delamination from wide cracks is reasonably accurately predicted via plane-strain analyses. Conversely, two-dimensional models are incapable of accurately predicting delamination from narrow cracks, which have a tendency to widen into flaws with more balanced aspect ratios (i.e. without growth in the depth direction).     

Finite element modeling of plasticity-induced crack closure with emphasis on geometry and mesh refinement effects

Two-dimensional, elastic-perfectly plastic finite element analyses of middle-crack tension (MT) and compact tension (CT) geometries were conducted to study fatigue crack closure and to calculate the crack-opening values under plane-strain and plane-stress conditions. The behaviors of the CT and MT geometries were compared. The loading was selected to give the same maximum stress intensity factor in both geometries, and thus approximately similar initial forward plastic zone sizes. Mesh refinement studies were performed on both geometries with various element types. For the CT geometry, negligible crack-opening loads under plane-strain conditions were observed. In contrast, for the MT specimen, the plane-strain crack-opening stresses were found to be significantly larger. This difference was shown to be a consequence of in-plane constraint. Under plane-stress conditions, it was found that the in-plane constraint has negligible effect, such that the opening values are approximately the same for both the CT and MT specimens.     

Strain energy release rates for a straight-fronted edge crack in a circular bar subject to bending

The strain energy release rate for a straight-fronted edge crack in a bar of circular cross section subjected to pure bending is determined. The cracked bar is modelled with two-dimensional plane-stress finite elements and strain energy release rates, determined from this model, are shown to be in close agreement with existing results for a bar subjected to three-point bending in which strain energy release rates were determined by measuring the compliance of the bar experimentally. The strain energy release rates for a crack in the circular cross section bar are found to be lower than those in a rectangular cross section bar having the same relative crack length and subjected to the same bending moment. Previously determined results for uniform tension are superimposed to obtain strain energy release rates for a circular cross section bar which is subjected simultaneously to a tensile load and a bending moment.     

A generalized plane-strain theory for transversally isotropic plates

Summary.  By adopting Kane and Mindlin's assumption that the through-the-thickness extensional strain is uniform through the plate thickness, a generalized plane-strain theory is developed for transversely isotropic plates. The three-dimensional governing equations are successfully reduced to two coupled equations in the two-dimensional space. With the new theory, explicit solutions of the three-dimensional stresses, especially the through-the-thickness component, around a circular hole and a V-shaped notch in a transversely isotropic plate of arbitrary thickness are derived. The analytical solutions are verified by comparing with finite element results. Received September 5, 2002 Published online: March 20, 2003     

Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory

A two-dimensional finite element model is presented to perform the linear static analysis of laminated orthotropic composite plates based on a refined higher order shear deformation theory. The theory accounts for parabolic distributions of transverse shear stresses and requires no shear correction factors. A finite element program is developed using serendipity element with seven degrees of freedom per node. The present solutions are compared with those obtained using three-dimensional elasticity theory and those obtained by other researchers. The theory accurately predicts displacements and transverse shear stresses compared to previously developed theories for thick plates and are very close to three-dimensional elasticity solutions. The effects of transverse shear deformation, material anisotropy, aspect ratio, fiber orientation and lamination sequence on transverse shear stresses are investigated. The error in values of transverse shear stresses decreases as the number of lamina increases, for a plate of same thickness. An increase in degree of anisotropy results in lower values of deflection in the plate. For cross-ply plate an increase in anisotropy results in an increase in effective stress whereas for angle-ply plate the effect is almost negligible. Through thickness variation of transverse shear stresses are independent of anisotropy. The maximum effective stress increases exponentially at lower values of anisotropy and reaches to an asymptotic value at higher values. The stacking sequence has a significant effect on the transverse deflections and shear stress. Rectangular plates experience less effective, in-plane and transverse shear stresses compared to square plates.     

Mixed-mode analysis of superimposed thermo-elastic effects in fiber-reinforced composites with embedded interface delaminations

This paper deals with the 3D finite element analysis of superimposed thermo-elastic effect on embedded interfacial delamination crack growth characteristics in fiber-reinforced laminated composites. Interlaminar fracture at the delamination front is found to be a mixed-mode phenomenon due to the anisotropy and heterogeneity of thermo-physical properties of composite materials. This leads to the requirement of finite element evaluation of energy release rates, based on the principles of linear elastic fracture mechanics. The strain energy release rate components along the delamination front due to a uniform temperature drop, during the manufacturing stages of composite laminates, to room temperature and subsequent mechanical loading is obtained by superimposing their respective effects based on the assumptions of linear elasticity. Numerical calculations are carried out for multi-layered cross-ply and angle-ply composite laminates and energy release rate plots demonstrate large asymmetries along the delamination front due to the interaction of residual stresses and superimposed transverse loading.     

Micromechanics of rubber–toughened polymers

A new micromechanical model is provided to account for the full interaction between rubber particles in toughened polymers. Three-dimensional large deformation elastic–plastic finite element analysis is carried out to obtain the local stress and strain fields and then a homogenization method is adopted to obtain the effective stress–strain relation. The dependence of the local stress and strain distributions and effective stress–strain relation on phase morphology and mechanical properties of rubber particles is examined under various transverse constraints. The profile for the effective yield surface is obtained at four different particle volume fractions. It is shown that stress triaxiality affects significantly the effective yield stress and the local stress concentrations. Rubber cavitation and matrix shear yielding are two coupled toughening mechanisms; which one occurs first depends on the properties of rubber particles and matrix and the imposed triaxiality. Rubber cavitation plays an important role in the toughening process under high tensile triaxial stresses. Axisymmetric modelling may underestimate, and two-dimensional plane-strain modelling may overestimate, the inter-particle interaction compared with three-dimensional modelling.     

A shell/3D modeling technique for the analysis of delaminated composite laminates

A shell/3D modeling technique was developed for which a local three-dimensional solid finite element model is used only in the immediate vicinity of the delamination front. The goal was to combine the accuracy of the full three-dimensional solution with the computational efficiency of a plate or shell finite element model. Multi-point constraints provided a kinematically compatible interface between the local three-dimensional model and the global structural model which has been meshed with plate or shell finite elements. Double cantilever beam (DCB), end notched flexure (ENF), and single leg bending (SLB) specimens were modeled using the shell/3D technique to study the feasibility for pure mode I (DCB), mode II (ENF) and mixed mode I/II (SLB) cases. Mixed mode strain energy release rate distributions were computed across the width of the specimens using the virtual crack closure technique. Specimens with a unidirectional layup and with a multidirectional layup where the delamination is located between two non-zero degree plies were simulated. For a local three-dimensional model, extending to a minimum of about three specimen thicknesses on either side of the delamination front, the results were in good agreement with mixed mode strain energy release rates obtained from computations where the entire specimen had been modeled with solid elements. For large built-up composite structures modeled with plate elements, the shell/3D modeling technique offers a great potential for reducing the model size, since only a relatively small section in the vicinity of the delamination front needs to be modeled with solid elements.     

Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels

The present paper shows a comparison between classical two-dimensional (2D) and three-dimensional (3D) finite elements (FEs), classical and refined 2D generalized differential quadrature (GDQ) methods and an exact three-dimensional solution. A free vibration analysis of one-layered and multilayered isotropic, composite and sandwich cylindrical and spherical shell panels is made. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the FE and GDQ methods (numerical solutions) and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 3D and 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D and 3D FE solutions, classical and refined 2D GDQ models and 3D exact solutions depend on several parameters. These include the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.     

A quadrilateral hybrid-Trefftz plate bending element for the inclusion of warping based on a three-dimensional plate formulation

A plate formulation, for the inclusion of warping and transverse shear deformations, is considered. From a complete thick and thin plate formulation, which was derived without ad hoc assumptions from the three-dimensional equations of elasticity for isotropic materials, the bending solution, involving powers of the thickness co-ordinate z, is used for constructing a quadrilateral finite plate bending element. The constructed element trial functions, for the displacements and stresses, satisfy, a priori, the three-dimensional Navier equations and equilibrium equations, respectively. For the coupling of the elements, independently assumed functions on the boundary are used. High accuracy for both displacements and stresses (including transverse shear stresses) can be achieved with rather coarse meshes for thick and thin plates.     

SiC/Ti-6Al-4V复合材料在横向拉伸载荷下界面失效行为的数值模拟

采用有限元模拟了SiC/Ti-6Al-4V复合材料冷却过程和横向拉伸试验过程, 横向拉伸试样采用十字形试样。分别建立了平面应力和轴对称有限元模型, 采用平面应力有限元模型计算环绕纤维圆周的界面微区应力分布, 预测界面失效机制。采用轴对称有限元模型分析复合材料界面脱粘过程以及残余应力对界面径向应力分布的影响。结果表明: 对于SiC/Ti-6Al-4V复合材料十字形试样,在横向拉伸载荷下的界面失效由径向应力导致,界面失效模式为法向失效, 剪切失效模式未发生; 十字形试样在横向拉伸载荷下界面初始脱粘位置处于界面中间; 随横向拉伸应力增加, 十字形试样的界面脱粘对称向两边扩展; 界面径向应力随残余应力降低而升高。     

Effect of fiber array irregularities on microscopic interfacial normal stress states of transversely loaded UD-CFRP from viewpoint of failure initiation

A detailed investigation has been carried out to determine the effect of local fiber array irregularities and controlling fiber distribution parameters on microscopic interfacial normal stress states for transversely-loaded unidirectional carbon fiber (CF)/epoxy composites. Linear elastic finite element analyses were carried out for two-dimensional image-based models composed of about 70 fibers. The relationship between the geometrical distribution of two adjacent fibers and the interfacial normal stresses (INSs) is investigated for all fibers in different image-based models. Three boundary conditions for loading were selected: Case A involved cooling from the curing temperature (the difference in temperature was ?155 K); Case B involved transverse loading of 75 MPa chosen as an example of macroscopic transverse fracture strength; and Case C involved both cooling from the curing temperature and transverse loading of 75 MPa. High compressive INSs due to the difference in the coefficients of thermal expansion are observed at the location of the shortest interfiber distance for Case A (cooling). High tensile INSs are observed at the location of the shortest interfiber distance and where the fiber alignment angle to the loading direction is small for Case B (loading). For Case C (cooling and loading), the high thermal residual compressive INSs and the high mechanical tensile INSs compensate each other, and the INSs at a short interfiber distance are much lower than those for Case B. These results clearly indicate the importance of the contribution of the thermal residual stresses to the transverse tensile failure initiation of CF/epoxy laminates.     

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

Abstract

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C⁰-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.     

A C1 finite element for flexural and torsional analysis of rectangular piezoelectric laminated/sandwich composite beams

This work deals with the development of a new C¹ finite element for analysing the bending and torsional behaviour of rectangular piezoelectric laminated/sandwich composite beams. The formulation includes transverse shear, warping due to torsion, and elastic–electric coupling effects. It also accounts for the inter-layer continuity condition at the interfaces between layers, and the boundary conditions at the upper and lower surfaces of the beam. The shear strain is represented by a cosine function of a higher order in nature and thus avoiding shear correction factors. The warping function obtained from a three-dimensional elasticity solution is incorporated in the present model. An exact integration is employed in evaluating various energy terms due to the application of field consistency approach while interpolating the transverse shear and torsional strains. The variation of the electric potential through the thickness is taken care of in the formulation based on the observation of three-dimensional solution. The performance of the laminated piezoelectric element is tested comparing with analytical results as well as with the reference solutions evaluated using three-dimensional finite element procedure. A detailed study is conducted to highlight the influence of length-to-thickness ratio on the displacements, stresses and electric potential field of piezoelectric laminated beam structures subjected to flexural and torsional loadings. Copyright © 2004 John Wiley & Sons, Ltd.     

Application of BEM to generalized plane problems for anisotropic elastic materials in presence of contact

It is in many cases very instructive and useful to have the possibility of treating three-dimensional problems by means of two-dimensional models. It always implies a reduction in computing cost which is particularly significant in presence of non-linearities, derived for instance from the presence of contact between the solids involved in the problem. The term generalized plane problem is adopted for a three-dimensional problem in a homogeneous linear elastic cylindrical body where strains and stresses are the same in all transversal sections. This concept covers many practical cases (for instance in the field of composites), a particular situation called generalized plane strain (strains, stresses and displacements are the same in all transversal sections) being the most frequently analyzed. In this paper, a new formulation is developed in a systematic way to solve generalized plane problems for anisotropic materials, with possible friction contact zones, as two-dimensional problems. The numerical solution of these problems is formulated by means of the boundary element method. An explicit expression of a new particular solution of the problem associated to constant body forces is introduced and applied to avoid domain integrations. Some numerical results are presented to show the performance and advantages of the formulation developed.     

Mixed-mode delamination fracture in laminated composites

Quasi-static fracture of laminated composite plates with circular or elliptic delaminations due to transverse shear stresses is considered. The system is modeled as two laminated plates bonded everywhere except over the area of the disbond. A laminated plate theory with the effects of surface tractions and shear deformations is used for each plate and the problem is reduced to a set of integral equations. Because of the nature of the model, stress singularities along the disbond periphery appear as concentrated line forces, which are different from those obtained from three-dimensional elasticity analysis. Physically meaningful quantities such as strain energy release rates for mode I, II and III types of deformation are evaluated. Failure loads for catastrophic fracture are computed based on firstly a point criterion and secondly a physically realistic growth pattern. Results are compared with test data for disbonds located in the midplane. For such disbonds the mode I component of energy release rate is absent and, therefore, fracture is due to mode II and III interactions.     

Elastic-plastic analysis of stress state and spread of plastic zone around single edge-cracked plate by finite element method

A two-dimensional finite element model for estimating elastic-plastic displacements and stresses near a crack for the elastic-plastic situation of loadings is presented. Singularity effects near the crack tip are solved by introducing 12-node cubic isoparametric elements in the crack tip region and collapsing them into triangular elements. The proposed finite element model is used to determine the spread of the plastic zone and mouth opening displacements of an edge-cracked structural steel plate. The spread of the plastic zone is obtained by successive increments of applied nominal tensile stress transverse to the crack length. The mouth opening displacement values obtained by this model are also compared with those measured experimentally.     

Three-dimensional finite element analysis of fracture modes for the pull-off test of a thin film from a stiff substrate

A three-dimensional geometrically nonlinear finite element analysis model is presented to study the interfacial delamination for the pull-off test of a thin film strip debonded from a stiff substrate. The strain energy release rates of all three modes (mode I, mode II, and mode III) along the debond front are considered and calculated to investigate the mixed fracture modes for the entire deformation regime from bending plate to stretching membrane. These results indicate that the individual strain energy release rates and the total energy release rate vary along the width of the debond front and strong three-dimensional edge effects exist near the free edges of the film. Interestingly, residual stress also plays an important role in controlling mixed fracture modes and the variation of the energy release rates. Finally, the three-dimensional finite element model is compared with an analytical solution developed earlier. The three-dimensional finite element model is found to provide additional insights for interfacial delamination for the pull-off test.