Influence of micro-cracking and contact on the effective properties of composite materials

In the present work a novel micro-mechanical approach to analyze the influence of micro-crack evolution and contact on the effective properties of elastic composite materials is proposed, based on homogenization techniques, interface models and fracture mechanics concepts. By means of the finite element method, enhanced non-linear macroscopic constitutive laws are developed by taking into account changes in micro-structural configuration associated with the growth of micro-cracks and with contact between crack faces. Numerical simulations are carried out for the cases of a porous composite with edge cracks and of a debonded fibre reinforced composite, loaded along extension/compression uniaxial macro-strain paths. Micro-crack propagation is modelled by using an original methodology based on the J-integral technique in conjunction with an interface model taking into account the unilateral contact of crack faces. In the context of a micro-to-macro transition obtained by controlling the macro-deformation of the micro-structure, the effects of adopting three types of boundary conditions on the macroscopic constitutive law, namely linear deformation, uniform tractions and periodic deformations and anti-periodic tractions, are studied. As a consequence, the proposed method can be applied to a large class of problems including periodic, locally periodic and irregular composite materials. Micro-crack and contact evolution result in a progressive loss of stiffness and can lead to failure for homogeneous macro-deformations associated with unstable crack propagation.     

The comparison between three different elements within a center crack plate considering elastic-plastic action

Stress near a crack tip in plasticity was analyzed using three different finite element modelings; a constant strain triangle, eight-noded quadrilaterial, and a crack tip singularity element all considering viscoplasticity. The specimen under consideration was a center cracked plate made from IN-100, a nickel-base superalloy containing a half-crack length equal to 0.1367 in. (3.472 mm). An elastic solution was formulated in conjunction with two different loadings to generate plasticity. Fine mesh and coarse mesh solutions for the higher order elements were generated and compared considering equal number of degrees of freedom in two specific regions referred to as the near field and the far field regions.

The authors determined that the elements whose elastic solutions conformed best to linear elastic fracture mechanics predictions were the constant strain triangle and the eight-noded quadrilateral in a fine mesh. The crack tip element did not perform as well as expected. For the plastic analysis, the constant strain triangle exhibited the largest plastic region. The eight-noded isoparametric element came within 15% of the stress levels generated from the constant strain triangle. The stress singularity that is characteristic of the crack tip element forced that element to behave unnaturally stiff immediately adjacent to the crack tip.

Because it is not as stiff as either the crack tip element or the eight-noded element, the constant strain triangle offered the most promising solutions as verified through experimentation. It was therefore determined that the constant strain triangle offered the best solution to elastic-plastic finite element problems for the center cracked plate.     

The use of optimization techniques in the analysis of cracked members by the finite element displacement and stress methods

Mathematical programming is applied to the two-dimensional stationary crack problem of a body composed of nonlinear elastic incompressible material. Fully admissible displacement as well as stress formulations are used to discretize the problem. Crack tip singularity is introduced in the displacement formulation by enriched elements for plane stress and, in certain cases, by superposition for plane strain. Pointwise incompressibility is obtained through constrained displacement functions. For three crack geometries Rice's J integral is evaluated by the energy difference method for different values of the hardening index. The numerical results, which are also applicable to secondary creep problems, appear to suggest a bounding character.     

Mixed mode fracture analysis of rectilinear anisotropic plates using singular boundary elements

This paper reports a multidomain boundary element formulation for direct calculation of stress intensity factors in rectilinear anisotropic plates subjected to arbitrary in-plane loading. The √r displacement and 1/√r traction behaviour near a crack tip is correctly represented in crack elements and transition elements. The use of these singular boundary elements is investigated for mode I and mixed mode crack problems.     

T-stress,mixed-mode stress intensity factors,and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method

For linear elastic functionally graded materials (FGMs), the fracture parameters describing the crack tip fields include not only stress intensity factors (SIFs) but also T-stress (nonsingular stress). These two fracture parameters are important for determining the crack initiation angle under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). In this paper, the mixed-mode SIFs and T-stress are evaluated by means of the interaction integral, in the form of an equivalent domain integral, in combination with the finite element method. In order to predict the crack initiation angle in brittle FGMs, this paper makes use of a fracture criterion which incorporates the T-stress effect. This type of criterion involves the mixed-mode SIFs, the T-stress, and a physical length scale r_c (representative of the fracture process zone size). Various types of material gradations are considered such as continuum models (e.g. exponentially graded material) and micromechanics models (e.g. self-consistent model). Several examples are given to show the accuracy and efficiency of the interaction integral scheme for evaluating mixed-mode SIFs, T-stress, and crack initiation angle. The techniques developed provide a basic framework for quasi-static crack propagation in FGMs.     

Elastoplastic analysis of tubular joints of offshore platforms

The subject of this paper is the analysis and design of complex tubular joints, eventually including internal and external gussets and stiffener rings, corresponding to fixed and mobile offshore structures.Two different joints are analysed. In the first case an X joint is studied for elastic and elastoplastic behaviour, loading up to collapse in order to determine ultimate strength and safety factor. Finite elements which can reproduce the elastic and plastic singularities of the stress and the strain fields in the crack tip, are then used for the analysis of a T joint. Both direction and rate are considered in the crack propagation, and an elastoplastic analysis is carried out, to determine the crack opening displacement (COD).Finally, the consideration of fatigue effects in tubular joints is discussed, and techniques for evaluating fatigue life are outlined.     

Recent advances in the DtN FE Method

Summary The Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fields of application. Its name comes from the fact that it involves the nonlocal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems governed by the Helmholtz equation or by the equations of time-harmonic elasticity. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the author's group and by others. This article is a state-of-the-art review of the method. In particular, it concentrates on two major recent advances: (a) the extension of the DtN finite element method tononlinear elliptic and hyperbolic problems; (b) procedures forlocalizing the nonlocal DtN map, which lead to a family of finite element schemes with local artificial boundary conditions. Possible future research directions and additional extensions are also discussed.     

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen-Loève (K-L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K-L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.     

Topology optimization considering fracture mechanics behaviors at specified locations

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.     

Mixed finite element formulations of strain-gradient elasticity problems

Theories on intrinsic or material length scales find applications in the modeling of size-dependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which, in this case, depends not only on the strain tensor but also on gradients of the rotation and strain tensors. In the present paper, the strain-gradient elasticity theories developed by Mindlin and co-workers in the 1960s are treated in detail. In such theories, when the problem is formulated in terms of displacements, the governing partial differential equation is of fourth order. If traditional finite elements are used for the numerical solution of such problems, then C¹ displacement continuity is required. An alternative “mixed” finite element formulation is developed, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an“integral-sense”. A variational formulation is developed which can be used for both linear and non-linear strain-gradient elasticity theories. The resulting finite elements require only C⁰ continuity and are simple to formulate. The proposed technique is applied to a number of problems and comparisons with available exact solutions are made.     

Micro- and macro-failure models of heterogeneous media with micro-structure

In this paper, the effectiveness of homogenization techniques for media with micro-structure subject to large deformations has been studied by comparing their micro- and macro-failure mechanisms. The material has been studied by considering its representative volume element (RVE) which entails all the geometric and constitutive information of the micro-structure. First, the formulation of the elastostatic problem governing the non-linear (large deformation and non-linear elastic) behavior of the structure of the RVE is presented. The RVE is subject to loading paths that produce uniform macroscopic strains. In this way it has been possible to use an homogenization procedure in order to simulate the overall behavior of the material, i.e. its constitutive tensor, at each point of the equilibrium path. Then, a macro failure surface has been defined as the locus of the points, in the macroscopic stretches space, corresponding to the loss of positivity of the macroscopic fourth order constitutive tensor in terms of the Biot stress [Encyclopedia of Physics, vol. 3(3), Springer-Verlag, Berlin]. Further, a micro-failure surface is defined as the locus of the points, in the overall stretches space, corresponding to the first critical point detected along the equilibrium path which can be characterized by an eigenmode compatible with the boundary conditions. Finally, a representative volume element, schematized by plane rods with strongly non-linear elastic constitutive behavior, is considered and the corresponding micro- and macro-failure surfaces are obtained in order to validate the proposed methodology.     

A high-order absorbing boundary condition for 2D time-harmonic elastodynamic scattering problems

In this paper, we are concerned with the construction of a new high-order Absorbing Boundary Condition (ABC) for 2D-elastic scattering problems. It is defined by an approximate local Dirichlet-to-Neumann (DtN) map. First, we explain the derivation of this approximation. Next, a detailed analytical study in terms of Hankel functions in the circular case is addressed. The new ABC is compared with the standard low-order Lysmer–Kuhlemeyer ABC. Finally, its accuracy and efficiency are investigated for various numerical examples, particularly at high frequencies.     

Finite element mesh for a complete solution of a problem with a singularity

It is well known that the stress field at the tip of a crack in an elastic body exhibits a singularity. In this paper, a complete elastic solution for a center cracked plate loaded under uniform tension is obtained by a finite element plane-stress analysis using constant strain elements. No a priori assumptions about the form of the stress singularity are required. The numerical results are compared with the exact analytic solution. It is shown that a particular mesh arrangement in which the size of the elements decreases in a geometric series as the crack tip is approached yields stress and strain fields which are accurate over the entire plate, even at distances very close to the crack tip. The effects of changes in mesh arrangements on the accuracy of the solution are considered. The computations are carried out on the CRAY-1 computer and the advantages of vectorization are discussed for this problem.     

Back analysis for rock model surrounding underground roadways in coal mine based on black hole algorithm

The huge plastic deformation is the characteristic of the underground roadways in coal mine. Therefore, to compute the stability of underground roadways, a elastic–plastic constitutive model of surrounding rock must be obtained. Many elastic–plastic constitutive models for rock mass have been proposed. In this study, a generalized constitutive law for an elastic–plastic constitutive model is applied. Using this generalized constitutive law, the problem of model identification is transformed to a problem of parameter back analysis, which is a typical and complicated optimization. To improve the efficiency of the traditional optimization method, a black hole algorithm is applied in this study. Combining the generalized constitutive law for an elastic–plastic constitutive model and black hole algorithm, a new back analysis method for model identification of rocks surrounding underground roadways in coal mine is proposed. Using this new method, the elastic–plastic constitutive models for two underground roadways in Huainan coal mine has been back-calculated. The results are compared with those of traditional genetic algorithm, fast genetic algorithm and immune continuous ant colony algorithm, that proposed in previous studies. The results show that the new model back analysis algorithm can significantly improve the computation efficiency and the computation effect, and is a very good method for back analysis the rock model surrounding underground roadways in coal mine.     

Nonlinear finite element analysis of reinforced concrete plates and shells under monotonic loading

Plane stress constitutive models are proposed for the nonlinear finite element analysis of reinforced concrete structures under monotonic loading. An elastic strain hardening plastic stress-strain relationship with a nonassociated flow rule is used to model concrete in the compression dominating region and an elastic brittle fracture behavior is assumed for concrete in the tension dominating area. After cracking takes place, the smeared cracked approach together with the rotating crack concept is employed. The steel is modeled by an idealized bilinear curve identical in tension and compressions. Via a layered approach, these material models are further extended to model the flexural behavior of reinforced concrete plates and shells. These material models have been tested against experimental data and good agreement has been obtained.     

The application of moving finite elements for the study of crack propagation in linear elastic solids

Mode I crack propagation in linear elastic solids is studied by a convecting finite element mesh. The finite element formulation is briefly described. Two different approaches for the determination of the stress intensity factor K₁ have been investigated, one using the COD relation to K₁ and one using the dynamic G-integral. Different test problems, including stationary cracks, steady-state crack propagation and transient crack propagation, have been thoroughly examined.     

Planar crack occupying a finite doubly connected region inside an infinite elastic solid

A method is proposed for the approximate solution of the problem of an embedded pressurized planar crack occupying a finite doubly connected region inside an infinite elastic solid. The formulation of the problem produces a system of two integral equations for determining the unknown normal stresses on the plane of the crack outside the crack region, which can be solved using numerical procedures. The proposed method has been applied to obtain the opening mode stress intensity factors K_I, along the boundary lines of an annular crack subjected to a uniform internal pressure.