Final Report Summary - INTERCRACKS (Unsolved problems in fracture mechanics of heterogeneous materials)
According to the work plan, the emphasis of the research was given on the analytical modelling of interfacial cracks in heterogeneous materials. This included models of elastic media with cracks at the interface between dissimilar materials, accounting for the interactions with inclusions, microcracks, voids and other heterogeneities. The effect of the microstructure on the crack stability and crack propagation was also investigated using both micropolar and discrete lattice formulations.
The main research objectives of the project were:
(1) the evaluation of transmission conditions for perfect and imperfect interfaces;
(2) the solution and analysis of the asymptotic behaviour of the physical solution;
(3) the evaluation of the weight functions;
(4) the stability and sensitivity analysis of the crack propagation.
Main contributions of the research project:
Significant results were obtained as related to the objectives of the project. These are detailed for each task in the following:
(1) Evaluation of transmission conditions for perfect and imperfect interfaces
Solutions for the perturbation problem of a mode III interfacial crack have been obtained in paper (1). The perturbation is of geometrical type and can be both perturbation of the crack faces and perturbation of the interface, which can deviate from the initial straight line configuration. Asymptotic formulae are derived for the first-order perturbation of the stress intensity factor and numerical results are presented for different geometrical perturbations of a half-plane interfacial crack in an infinite bimaterial structure. The problem of a crack lying at the interface between dissimilar materials with microstructure undergoing antiplane deformations was studied in paper (3). The micropolar behaviour of the materials is described by the theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the two materials. We perform an asymptotic analysis to investigate the behaviour of the solution near the crack tip. It turns out that the stress singularity at the crack tip is strongly influenced by the microstructural parameters and it may or may not show oscillatory behaviour depending on the ratio between the characteristic lengths.
The effects of imperfection of the interface have been considered in paper (18). Transmission conditions and corresponding weight functions taking into account the imperfection have been derived using the Wiener-Hopf technique and then used to analyse the interaction between a macrocrack at the imperfect interface and an elastic inclusion.
(2) Solution and analysis of the asymptotic behaviour of the physical solution
The effects of the microstructure of heterogeneous media on the asymptotic fields near crack tips have been analysed and models of couple stress elastic materials with cracks have been developed. The modelling includes both static and dynamic cases.
The interaction of an interfacial crack with small impurities has been analysed in paper (2) on the basis of an asymptotic formula derived by the authors. The interaction between the main crack and the defects (e.g. small cracks or inclusions) is described asymptotically by analysing the dipole fields and the corresponding dipole matrices of the defects in question. Shielding and amplification effects of the defects on the propagation of the main crack along the interface are investigated. Numerical computations based on the explicit analytical formulae show potential applications in the design of composite and fiber reinforced materials.
The paper (7) is concerned with the problem of a semi-infinite crack steadily propagating in an elastic solid with microstructures subject to loading applied on the crack surfaces. The loading is moving with the same constant velocity as that of the crack tip. The material behaviour is described by the indeterminate theory of couple stress elasticity developed by Koiter. The analysis confirms and extends earlier results on the static case by including the effects of crack velocity and rotational inertia. By adopting the criterion of maximum total shear stress, we discuss the effects of microstructural parameters on the stability of crack propagation.
The problem of a rectilinear semi-illimitate crack in an elastic solid with microstructures subject to dynamic loadings applied on the crack surfaces has been investigated in paper (16). The material behaviour is described by the indeterminate theory of couple stress elasticity.
(3) Evaluation of the weight functions
New integral identities for interfacial cracks in two-dimensional (2D) and 3D elasticity have been derived for the first time in analytical closed form. This is a crucial step for the implementation of numerical schemes for the efficient calculation of asymptotic fields and stress intensity factors for heterogeneous bodies containing an interfacial crack. These results have been extended also to anisotropic materials.
The paper (5) is concerned with the problem of a semi-infinite crack at the interface between two dissimilar elastic half-spaces, loaded by a general asymmetrical system of forces distributed along the crack faces. On the basis of the weight function approach and the fundamental reciprocal identity (Betti formula), we formulate the elasticity problem in terms of singular integral equations relating the applied loading and the resulting crack opening. Such formulation is fundamental in the theory of elasticity and extensively used to solve several problems in linear elastic fracture mechanics (for instance various classic crack problems in homogeneous and heterogeneous media).
The paper (8) extends the integral identities to the case of interfacial cracks between dissimilar anisotropic elastic materials. Recently derived symmetric and skew-symmetric weight functions matrices are introduced for both plane strain and antiplane shear cracks, and used together with the fundamental reciprocal identity in order to derive the required singular integral formulation.
The notion of skew-symmetric weight functions for solving interfacial crack problems has been introduced in Piccolroaz et al. (2009) and Piccolroaz et al. (2010), where the 2D skew-symmetric weight functions are presented and used to solve various perturbation problems in two dimensions. In paper (17), we extend the notion to the 3D case, by constructing the 3D skew-symmetric weight functions for semi-infinite interfacial cracks in a bi-material space.
(4) Stability and sensitivity analysis of the crack propagation
Models for the propagation of interfacial cracks in heterogeneous media containing small defects and impurities have been developed on the basis of the weight function and dipole matrix approach. These models show that it is possible to design and tune composite structures such that interfacial cracks can easily propagate along the interface, or, vice versa, are arrested at a fixed position.
The paper (4) addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as microcracks, rigid line inclusions and voids. We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate effective tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects. The analytical results are used to analyse the shielding and amplification effects of various types of defects in different configurations.
The paper (6) considers the quasi-static propagation of a Mode III crack along the interface in a bimaterial plane containing a finite array of small line defects (microcracks and rigid line inclusions). The microdefects are arranged to form a channel around the interface that can facilitate (or prevent) the crack propagation. Numerical examples are provided showing potential applications of the proposed approach in the analysis of failure of composite materials. Extension to the case of infinite number of defects is discussed.
The solution of the scalar problem of a semi-infinite crack propagating at the interface between dissimilar square lattices has been obtained in the paper (19). The fault propagates along the interface generating waves. The paper presents dispersion diagrams analysing possible waves propagating in such structure. A matrix equation of the Wiener-Hopf type is also derived and solved along the crack faces in order to compute the stress intensity factor. The crack stability is analysed on the basis of the energy release rate evaluated for different contrasts in mass and stiffness of the two lattices.
Project website: http://fp7.imaps.aber.ac.uk/intercracks.html(opens in new window)
The main research objectives of the project were:
(1) the evaluation of transmission conditions for perfect and imperfect interfaces;
(2) the solution and analysis of the asymptotic behaviour of the physical solution;
(3) the evaluation of the weight functions;
(4) the stability and sensitivity analysis of the crack propagation.
Main contributions of the research project:
Significant results were obtained as related to the objectives of the project. These are detailed for each task in the following:
(1) Evaluation of transmission conditions for perfect and imperfect interfaces
Solutions for the perturbation problem of a mode III interfacial crack have been obtained in paper (1). The perturbation is of geometrical type and can be both perturbation of the crack faces and perturbation of the interface, which can deviate from the initial straight line configuration. Asymptotic formulae are derived for the first-order perturbation of the stress intensity factor and numerical results are presented for different geometrical perturbations of a half-plane interfacial crack in an infinite bimaterial structure. The problem of a crack lying at the interface between dissimilar materials with microstructure undergoing antiplane deformations was studied in paper (3). The micropolar behaviour of the materials is described by the theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the two materials. We perform an asymptotic analysis to investigate the behaviour of the solution near the crack tip. It turns out that the stress singularity at the crack tip is strongly influenced by the microstructural parameters and it may or may not show oscillatory behaviour depending on the ratio between the characteristic lengths.
The effects of imperfection of the interface have been considered in paper (18). Transmission conditions and corresponding weight functions taking into account the imperfection have been derived using the Wiener-Hopf technique and then used to analyse the interaction between a macrocrack at the imperfect interface and an elastic inclusion.
(2) Solution and analysis of the asymptotic behaviour of the physical solution
The effects of the microstructure of heterogeneous media on the asymptotic fields near crack tips have been analysed and models of couple stress elastic materials with cracks have been developed. The modelling includes both static and dynamic cases.
The interaction of an interfacial crack with small impurities has been analysed in paper (2) on the basis of an asymptotic formula derived by the authors. The interaction between the main crack and the defects (e.g. small cracks or inclusions) is described asymptotically by analysing the dipole fields and the corresponding dipole matrices of the defects in question. Shielding and amplification effects of the defects on the propagation of the main crack along the interface are investigated. Numerical computations based on the explicit analytical formulae show potential applications in the design of composite and fiber reinforced materials.
The paper (7) is concerned with the problem of a semi-infinite crack steadily propagating in an elastic solid with microstructures subject to loading applied on the crack surfaces. The loading is moving with the same constant velocity as that of the crack tip. The material behaviour is described by the indeterminate theory of couple stress elasticity developed by Koiter. The analysis confirms and extends earlier results on the static case by including the effects of crack velocity and rotational inertia. By adopting the criterion of maximum total shear stress, we discuss the effects of microstructural parameters on the stability of crack propagation.
The problem of a rectilinear semi-illimitate crack in an elastic solid with microstructures subject to dynamic loadings applied on the crack surfaces has been investigated in paper (16). The material behaviour is described by the indeterminate theory of couple stress elasticity.
(3) Evaluation of the weight functions
New integral identities for interfacial cracks in two-dimensional (2D) and 3D elasticity have been derived for the first time in analytical closed form. This is a crucial step for the implementation of numerical schemes for the efficient calculation of asymptotic fields and stress intensity factors for heterogeneous bodies containing an interfacial crack. These results have been extended also to anisotropic materials.
The paper (5) is concerned with the problem of a semi-infinite crack at the interface between two dissimilar elastic half-spaces, loaded by a general asymmetrical system of forces distributed along the crack faces. On the basis of the weight function approach and the fundamental reciprocal identity (Betti formula), we formulate the elasticity problem in terms of singular integral equations relating the applied loading and the resulting crack opening. Such formulation is fundamental in the theory of elasticity and extensively used to solve several problems in linear elastic fracture mechanics (for instance various classic crack problems in homogeneous and heterogeneous media).
The paper (8) extends the integral identities to the case of interfacial cracks between dissimilar anisotropic elastic materials. Recently derived symmetric and skew-symmetric weight functions matrices are introduced for both plane strain and antiplane shear cracks, and used together with the fundamental reciprocal identity in order to derive the required singular integral formulation.
The notion of skew-symmetric weight functions for solving interfacial crack problems has been introduced in Piccolroaz et al. (2009) and Piccolroaz et al. (2010), where the 2D skew-symmetric weight functions are presented and used to solve various perturbation problems in two dimensions. In paper (17), we extend the notion to the 3D case, by constructing the 3D skew-symmetric weight functions for semi-infinite interfacial cracks in a bi-material space.
(4) Stability and sensitivity analysis of the crack propagation
Models for the propagation of interfacial cracks in heterogeneous media containing small defects and impurities have been developed on the basis of the weight function and dipole matrix approach. These models show that it is possible to design and tune composite structures such that interfacial cracks can easily propagate along the interface, or, vice versa, are arrested at a fixed position.
The paper (4) addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as microcracks, rigid line inclusions and voids. We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate effective tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects. The analytical results are used to analyse the shielding and amplification effects of various types of defects in different configurations.
The paper (6) considers the quasi-static propagation of a Mode III crack along the interface in a bimaterial plane containing a finite array of small line defects (microcracks and rigid line inclusions). The microdefects are arranged to form a channel around the interface that can facilitate (or prevent) the crack propagation. Numerical examples are provided showing potential applications of the proposed approach in the analysis of failure of composite materials. Extension to the case of infinite number of defects is discussed.
The solution of the scalar problem of a semi-infinite crack propagating at the interface between dissimilar square lattices has been obtained in the paper (19). The fault propagates along the interface generating waves. The paper presents dispersion diagrams analysing possible waves propagating in such structure. A matrix equation of the Wiener-Hopf type is also derived and solved along the crack faces in order to compute the stress intensity factor. The crack stability is analysed on the basis of the energy release rate evaluated for different contrasts in mass and stiffness of the two lattices.
Project website: http://fp7.imaps.aber.ac.uk/intercracks.html(opens in new window)
