V. Mantič

A new formula for the C-matrix in the Somigliana identity

By making use of a convenient decomposition of the fundamental tractions, a new formula for the C-matrix in the Somigliana identity for a three- or two-dimensional elastic isotropic body is derived. This kind of formula is more advantageous for analytical and computational C-matrix evaluations than the currently well-known formula. A general closed analytical formula of the C-matrix for the case of any finite number of tangent planes to the boundary of the body at a non-smooth boundary point, presented in the final section of this paper, demonstrates the usefulness of the new formula.

Kinking of Transversal Interface Cracks Between Fiber and Matrix

Under loads normal to the direction of the fibers, composites suffer failures that are known as matrix or interfiber failures, typically involving interface cracks between matrix and fibers, the coalescence of which originates macrocracks in the composite. The purpose of this paper is to develop a micromechanical model, using the boundary element method, to generate information aiming to explain and support the mechanism of appearance and propagation of the damage. To this end, a single fiber surrounded by the matrix and with a partial debonding is studied. It has been found that under uniaxial loading transversal to the fibers direction the most significant phenomena appear for semidebonding angles in the interval between 60 deg and 70 deg. After this interval the growth of the crack along the interface is stable (energy release rate (ERR) decreasing) in pure Mode II, whereas it is plausibly unstable in mixed mode (dominated by Mode I for semidebondings smaller than 30 deg) until it reaches the interval. At this interval the direction of maximum circumferential stress at the neighborhood of the crack tip is approximately normal to the applied load. If a crack corresponding to a debonding in this interval leaves the interface and penetrates into the matrix then: (a) the growth through the matrix is unstable in pure Mode I; (b) the value of the ERR reaches a maximum (in comparison with other debonding angles); and (c) the ERR is greater than that released if the crack continued growing along the interface. All this suggests that it is in this interval of semidebondings (60-70 deg) that conditions are most appropriate for an interface crack to kink. Experiments developed by the authors show an excellent agreement between the predictions generated in this paper and the evolution of the damage in an actual composite.

Singularity analysis of anisotropic multimaterial corners

Singular stress states induced at the tip of linear elastic multimaterial corners are characterized in terms of the order of stress singularities and angular variation of stresses and displacements. Linear elastic materials of an arbitrary nature are considered, namely anisotropic, orthotropic, transversely isotropic, isotropic, etc. Thus, in terms of Stroh formalism of anisotropic elasticity, the scope of the present work includes mathematically non-degenerate and degenerate materials. Multimaterial corners composed of materials of different nature are typically present at any metal-composite, or composite-composite adhesive joint. Several works are available in the literature dealing with a singularity analysis of multimaterial corners but involving (in the vast majority) only materials of the same nature (e.g., either isotropic or orthotropic). Although many different corner configurations have been studied in literature, with almost any kind of boundary conditions, there is an obvious lack of a general procedure for the singularity characterization of multimaterial corners without any limitation in the nature of the materials. With the procedure developed here, and implemented in a computer code, multimaterial corners, with no limitation in the nature of the materials and any homogeneous orthogonal boundary conditions, could be analyzed. As a particular case, stress singularity orders in corners involving extraordinary degenerate materials are, to the authors’ knowledge, presented for the first time. The present work is based on an original idea by Ting (1997) in which an efficient procedure for a singularity analysis of anisotropic non-degenerate multimaterial corners is introduced by means of the use of a transfer matrix.

Bem solution of two-dimensional contact problems by weak application of contact conditions with non-conforming discretizations

Abstract Non-conforming discretizations of the surfaces involved in the contact problem are sometimes required, due to the geometry and/or the load. A strong application of the contact conditions directly relating variables (displacements and tractions) of the nodes of the discretizations, a general approach called by the authors node-to-point contact scheme, may lead to unsatisfactory results. In this paper, a weak application of the contact conditions, by means of the principle of virtual work, is developed for boundary integral equations. The formulation is presented for two-dimensional problems without or with friction, using the Coulomb model. The modelization is made using the Boundary Element Method and the problem is solved with an incremental procedure based on a displacement scaling approach. The solution scheme proposed is applicable to any contact problem (with small or large displacements) and is validated in this paper by applying it to receding, conforming and advancing contact problems, the jumps in the contact stresses that appeared in node-to-point contact schemes, not having been found in the problems tested.

Existence and evaluation of the two free terms in the hypersingular boundary integral equation of potential theory

Hypersingular boundary integral equations (HBIE) have been studied very intensively in recent years especially because of their application for precise computations of potential gradient and stresses on the boundary. One free term which in general does not vanish for non-smooth boundary points with adjacent curved boundary parts has been omitted in previous formulations of HBIE. This paper demonstrates the presence of this new free term in 2D and 3D HBIE of potential gradient. Simple formulae for coefficients of both this new free term related to potential value and that related to potential gradient value, suitable for evaluation in the boundary element method, are derived in detail.

ON THE REMOVAL OF RIGID BODY MOTIONS IN THE SOLUTION OF ELASTOSTATIC PROBLEMS BY DIRECT BEM

A theoretical and numerical study of the removal of rigid body motions in the solution of the boundary form of Somigliana identity and of the corresponding discretized linear system of the direct BEM is presented. This study is based on the Fredholm theory of linear operators and mechanical aspects of the problem. Various methods suitable for implementation in BEM codes are analyzed and relations between apparently different methods are shown. The relation between global equilibrium conditions and solvability of the discretized linear system of the direct BEM is discussed.

Stress singularities in 2D orthotropic corners

A new technique for analysis of two-dimensional linear elastostatic solutions with stress singularities at orthotropic corners is developed. An explicit and general representation of the associated eigenequation given as ‘zero determinant condition’ of a matrix with half dimension in comparison with the current approach is derived by application of Stroh relations of anisotropic elasticity. The technique is directly applicable to anisotropic corners. Analytical formulae for stress singularity exponents, roots of the associated eigenequations, at orthotropic half-plane and semiinfinite crack problems for all combinations of basic homogeneous boundary conditions, including slip with friction, are derived. it is noteworthy that the singularity exponents are invariant with respect to the relative orientation of boundary edges and orthotropic material for nine of these combinations. Numerical analysis of singularity exponents for some configurations typical in the modelization of material tests of fiber-matrix composite materials is presented.

A linear elastic-brittle interface model: application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads

A new linear elastic and perfectly brittle interface model for mixed mode is presented and analysed. In this model, the interface is represented by a continuous distribution of springs which simulates the presence of a thin elastic layer. The constitutive law for the continuous distribution of normal and tangential initially-linear-elastic springs takes into account possible frictionless elastic contact between adherents once a portion of the interface is broken. A perfectly brittle failure criterion is employed for the springs, which enables the study of crack onset and propagation. This interface failure criterion takes into account the variation of the interface fracture toughness with the fracture mode mixity. A unified way to represent several phenomenological both energy and stress based failure criteria is introduced. A proof relating the energy release rate and tractions at an interface point (not necessarily a crack tip point) is introduced for this interface model by adapting Irwin’s crack closure technique for the first time. The main advantages of the present interface model are its simplicity, robustness and computational efficiency, even in the presence of snap-back and snap-through instabilities, when the so-called sequentially linear (elastic) analysis is applied. This model is applied here in order to study crack onset and propagation at the fibre-matrix interface in a composite under tensile/compressive remote biaxial transverse loads. Firstly, this model is used to obtain analytical predictions about interface crack onset, while investigating a single fibre embedded in a matrix which is subjected to uniform remote transverse loads. Then, numerical results provided by a 2D boundary element analysis show that a fibre-matrix interface failure is initiated by the onset of a finite debond in the neighbourhood of the interface point where the failure criterion is first reached (under increasing proportional load); this debond further propagates along the interface in mixed mode or even, in some configurations, with the crack tip under compression. The analytical predictions of the debond onset position and associated critical load are used for several parametric studies of the influence of load biaxiality, fracture-mode sensitivity and brittleness number, and for checking the computational procedure implemented.

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.

Local-solution approach to quasistatic rate-independent mixed-mode delamination

The quasistatic rate-independent evolution of a delamination at small strains in the so-called mixed mode, i.e.~distinguishing opening (Mode I) from shearing (Mode II) is rigorously analyzed in the context of a concept of stress-driven local solutions. The model has separately convex stored energy and is associative, namely the 1-homogeneous potential of dissipative force driving the delamination depends only on rates of internal parameters. An efficient fractional-step-type semi-implicit discretisation in time is shown to converge to specific, stress-driven like) local solutions that may approximately obey the maximum-dissipation principle. Making still a spatial discretisation, this convergence as well as relevancy of such solution concept are demonstrated on a nontrivial 2-dimensional example.