BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient 3D‐finite element formulation for thin mechanical and piezoelectric structures

It is the aim to compute efficiently the deformations of the mechanical structures that are excited or damped by piezoelectric actuators. They are treated as 3D structures to have much flexibility. Special design of the finite element concept is required since the structures are thin walled and locking effects have to be avoided. Although the computations are performed in the framework of 3D elasticity, we use ideas from modern plate elements and mixed methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Anti‐symmetrical curved composite laminate subject to delamination induced by thermal cycling

Composite structures usually undergo to temperature variations in aircraft during landing/taking off and when cruising at high altitude. Under these conditions and in combination with curved structures, they can generate severe thermal stresses that induce delaminations. Considering the importance of studying delamination in these conditions, this research imposed an anti‐symmetrical laminate to cyclic temperature variations of 130 °C and ?70 °C with the objective of inducing varied curvatures and, consequently, crack growth. Different from standardized test procedures, this test setup elastically deformed coupons without external forces and forward experimentally and numerically evaluated the strain energy release rate (SERR) during crack propagation. This procedure enabled the assessment of delamination rate (da/dN) as a function of maximum SERR. The experimental results were compared with numerical results obtained by ABAQUS Finite Element code. Despite large scatter in experimental results, a reasonable correlation between experimental and numerical results was obtained in terms of crack growth rate (da/dN) as a function of the maximum SERR.     

Modeling of the electromechanical coupling between crack propagation and piezoelectric behavior in active layers of smart devices

In this article, a new computational approach is investigated to predict the crack propagation inside some smart structures equipped with surface-bonded piezoelectric layers; meanwhile, the electromechanical coupling is exploited. The current industrial need of analyzing rather irregular geometries motivates resorting to a numerical approach. Therefore, the finite element method (FEM) is used to predict both the fracture behavior and the coupled response of piezoelectric material through a suitable interoperation of the two computational environments, available in some commercial code like the ABAQUS^©. The two solutions are then coupled by means of a subroutine, in this case operated through the ISIGHT^© tool. After a preliminary analysis and a validation, results of some numerical simulations are shown to highlight some significant peculiarities of the coupled behavior of the piezoelectric material.     

Unified correlation of in‐plane and out‐of‐plane constraints with fracture toughness

In this paper, the specimens with different geometries and loading configurations were used to study the unified correlation of in‐plane and out‐of‐plane constraints with fracture toughness by using numerical simulation method. The results show that the unified constraint parameter A_p which was defined on the basis of the areas surrounded by the equivalent plastic strain isolines ahead of crack tip can characterise both in‐plane and out‐of‐plane constraints induced by different specimen geometries and loading configurations. A sole linear relation between the normalised fracture toughness J_IC/J_ref and was obtained. The J_IC/J_ref ‐ line is a unified correlation line of in‐plane and out‐of‐plane constraints with fracture toughness of a material, and the constraint dependent fracture toughness of a material can be determined from the unified correlation line. The results also demonstrate that the out‐of‐plane constraint effect is related to the in‐plane constraint effect, and there exists interaction between them. The higher in‐plane constraint strengthens the out‐of‐plane constraint effect, whereas the lower in‐plane constraint is not sensitive to the out‐of‐plane constraint effect.     

A 3D beam‐to‐beam contact finite element for coupled electric–mechanical fields

In this paper the formulation of an electric–mechanical beam‐to‐beam contact element is presented. Beams with circular cross‐sections are assumed to get in contact in a point‐wise manner and with clean metallic surfaces. The voltage distribution is influenced by the contact mechanics, since the current flow is constricted to small contacting spots. Therefore, the solution is governed by the contacting areas and hence by the contact forces. As a consequence the problem is semi‐coupled with the mechanical field influencing the electric one. The electric–mechanical contact constraints are enforced with the penalty method within the finite element technique. The virtual work equations for the mechanical and electric fields are written and consistently linearized to achieve a good level of computational efficiency with the finite element method. The set of equations is solved with a monolithic approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Three‐dimensional analyses of in‐plane and out‐of‐plane crack‐tip constraint characterization for fracture specimens

Three‐dimensional elastic–plastic finite element analyses have been conducted for 21 experimental specimens with different in‐plane and out‐of‐plane constraints in the literature. The distributions of five constraint parameters (namely T‐stress, Q, h, T_z and A_p) along crack fronts (specimen thickness) for the specimens were calculated. The capability and applicability of the parameters for characterizing in‐plane and out‐of‐plane crack‐tip constraints and establishing unified correlation with fracture toughness of a steel were investigated. The results show that the four constraint parameters (T‐stress, Q, h and T_z) based on crack‐tip stress fields are only sensitive to in‐plane or out‐of‐plane constraints. Therefore, the monotonic unified correlation curves with fracture toughness (toughness loci) cannot obtained by using them. The parameter A_p based on crack‐tip equivalent plastic strain is sensitive to both in‐plane and out‐of‐plane constraints, and may effectively characterize both of them. The monotonic unified correlation curves with fracture toughness can be obtained by using A_p. In structural integrity assessments, the correlation curves may be used in the failure assessment diagram (FAD) methodology for incorporating both in‐plane and out‐of‐plane constraint effects in structures for improving accuracy.     

Unified characterisation of in‐plane and out‐of‐plane constraint based on crack‐tip equivalent plastic strain

Constraint can be divided into two conditions of in‐plane and out‐of‐plane, and each of them has its own parameter to characterize. However, in most cases, there exists a compound change of both in‐plane and out‐of‐plane constraint in structures, a unified measure that can reflect both of them is needed. In this paper, the finite element method (FEM) was used to calculate the equivalent plastic strain (ɛ_p) distribution ahead of crack tips for specimens with different in‐plane and out‐of‐plane constraints, and the FEM simulations based on Gurson–Tvergaard–Needleman (GTN) damage model and a small number of tests were used to obtain fracture toughness for the specimens with different constraints. Unified measure and characterisation parameter of in‐plane and out‐of‐plane constraints based on crack‐tip equivalent plastic strain has been investigated. The results show that the area A_PEEQ surrounded by the ɛ_p isoline ahead of crack tips can characterize both in‐plane and out‐of‐plane constraints. Based on the area A_PEEQ, a unified constraint characterisation parameter A_p was defined. It was found that there exists a sole linear relation between the normalised fracture toughness J_IC/J_ref and regardless of the in‐plane constraint, out‐of‐plane constraint and the selection of the ɛ_p isolines. The unified J_IC/J_ref−reference line can be used to determine constraint‐dependent fracture toughness of materials. The FEM simulations with the GTN damage model (local approach) can be used in obtaining the unified J_IC/J_ref−reference line for materials with ductile fracture.     

The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive finite element computation of dielectric and mechanical intensity factors in piezoelectrics with impermeable cracks

The paper deals with the application of an adaptive, hierarchic‐iterative finite element technique to solve two‐dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for monolithic thermo‐hydro‐mechanical simulations at finite strain

An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo‐hydro‐mechanical behavior of porous media undergoing large deformation. We first formulate a finite‐deformation thermo‐hydro‐mechanics field theory for non‐isothermal porous media. Projection‐based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two‐fold inf‐sup condition of the equal‐order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non‐isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo‐hydro‐mechanical model. Copyright © 2015 John Wiley & Sons, Ltd.     

Out‐of‐plane singular stress fields in V‐notched plates and welded lap joints induced by in‐plane shear load conditions

When a crack or sharp notch is subjected to antisymmetric plane loading the Poisson's effect leads to the generation of a coupled out‐of‐plane singular mode. The latter was known to exist for problems with cracks for a long period of time; meanwhile this mode was largely ignored in theoretical studies of V‐shaped notches subjected to in‐plane loading as well as in practical fracture problems associated with such geometries. Only recently a characteristic equation describing the strength of the singularity of this mode was derived within the first order plate theory. Preliminary numerical investigations confirmed that a highly localized out‐of‐plane singular state linked to the transverse shear stress components does exist in the close vicinity of the notch tip with the singular behaviour as theoretically predicted. However, until now it is unclear how significant this mode is and whether it has to be taken into consideration in the stress analysis of engineering structures. This paper is aimed to discuss important features of this recently identified singular mode, out‐of‐plane singular mode, conduct a comprehensive three‐dimensional numerical study of a typical problem of a welded lap joint to investigate the contribution of this mode into the overall stress state in the close vicinity of the notch tip and discuss the implementation of these new results to the failure and integrity assessment of plate structures with sharp notches.     

Interlaminar Fracture Toughness of Carbon Fibre‐Reinforced Polymer Laminates With Nano‐ and Micro‐Fillers

Abstract: Vapour growth carbon nanofibres (CNF) and lead zirconate titanate (PZT) piezoelectric particles were added in the matrix of carbon fibre‐reinforced polymer laminates. The fracture toughness of the modified composites was measured under mode I and mode II loading and compared with plain epoxy carbon fibre‐reinforced composites. The mode I fracture toughness of the composites improved with the incorporation of the carbon nanofibres and deteriorated with the incorporation of PZT piezoelectric particles. When both fillers were added in the composite matrix, the mode I fracture toughness improved but to a lesser extend. The mode II fracture toughness of the modified composites was improved in all three cases. The aforementioned behaviour was attributed to competing fracture mechanisms instigated by the different fillers, and backed by fractographic evidence from the failed composite coupons; during the tests, the acoustic emission activity of the coupons was monitored and classified in three major energy absorbing mechanisms which were attributed to the failure of distinct composite phases.     

Non‐oscillatory and non‐singular asymptotic solutions to stress fields at interface cracks

This work concerns the complex oscillatory singularities revealed in Williams's asymptotic solutions to stress fields around arbitrary interface cracks, which are the foundation of phenomenological interface fracture mechanics. First, we highlight the fatal discrepancy between the asymptotic stress fields for cracks in a homogeneous material obtained by assigning an identical material on both regions embracing an interface crack, and the solutions directly derived from cracks in a single material. Next, following a brief introduction to Williams's formulation process, we adopt the method of repeatedly eliminating variables instead of solving the determinant equation for the coefficient matrix to reformulate the asymptotic analysis of stress fields at arbitrary interface cracks. The resultant stresses get rid of oscillatory character. Further, under two specific loading conditions, namely, remotely uniaxial tension or shear, non‐oscillatory and non‐singular asymptotic solutions to stress fields around interface cracks are obtained.     

Load increment procedure for post‐buckling analysis of laminated plates under in‐plane loads by finite strip method

A load increment procedure has been presented to integrate with the finite strip method for the post‐buckling analysis of laminated plates when subjected to uniform end shortening. In‐plane loads are introduced to reflect the end shortening effect. The Newton–Raphson procedure is implemented to attend a solution that satisfies the equilibrium condition and at the same time meets the loading requirements. Error associated with loading condition is minimised by adjusting the load factor to preserve the rate of convergence. The enhanced capability can be easily incorporated into the context of both classical and shear deformation plate theories. A range of application has been described. Convergence test and numerical results are presented for isotropic plate and laminates with general lay‐up arrangement. Copyright © 2000 John Wiley & Sons, Ltd.     

Estimation of high‐temperature fracture parameters for small punch specimen with a surface crack

An outline of a newly proposed methodology for evaluating creep crack growth (CCG) parameters using cracked small‐punch (SP) specimens is explained. Three‐dimensional finite element analyses were performed to calculate the stress intensity factor along the crack front for a surface crack formed at the centre of a SP specimen. Effects of crack ratio, (a/t); crack aspect ratio, (a/c); and thickness of the specimen, (t), on the fracture parameters were studied. It was observed that the minimum variation of K‐value along the crack front can be achieved when a/c was 0.50 except the location very near the intersection of the crack and free surface. This condition is similar to the case of constant K‐values along the crack front of the conventional compact tension specimen. Thus, it can be argued that the SP specimen with a surface crack is a suitable specimen geometry for CCG testing. The proposed CCG test method was found to be practically applicable for the crack geometry of 0.10 to 0.30 of a/t with constant aspect ratio of 0.50. An estimation of the K and C_t‐parameter under the small scale creep condition was derived. Future work for further development of the suggested CCG testing is discussed.     

Reissner mixed variational theorem-based finite cylindrical layer methods for the static analysis of functionally graded piezoelectric circular hollow cylinders under electro-mechanical loads

A unified formulation of Reissner's mixed variational theorem-based finite cylindrical layer methods is developed for the static analysis of simply-supported, multilayered functionally graded piezoelectric material (FGPM) circular hollow cylinders. The material properties of the cylinder are assumed to obey an exponent-law exponentially varying through the thickness coordinate of this. The trigonometric functions and Lagrange polynomials are used to interpolate the in-surface and thickness variations of the primary variables of each individual layer, respectively. The coupled electro-elastic effects on the static behaviors of multilayered FGPM cylinders are closely examined.