J. Alfaiate

On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture

In this paper, strong discontinuities embedded in finite elements are used to model discrete cracking in quasi-brittle materials. Special attention is paid to (i) the constitutive models used to describe the localized behaviour of the discontinuities, (ii) the enforcement of the continuity of the crack path and (iii) mixed-mode crack propagation. Different constitutive relations are adopted to describe the localized behaviour of the discontinuities, namely two damage laws and one plasticity law. A numerical algorithm is introduced to enforce the continuity of the crack path. In the examples studied, an objective dissipation of energy with respect to the mesh is found. Examples of mode-I and mixed-mode crack propagation are presented, namely a double notch tensile test and a single-edge notched beam subjected to shear. In the former case different crack patterns are obtained depending on the notch offset; in the latter case special emphasis is given to the effect of shear on the global structural response. In particular, both the peak load and the softening response of the structure are related to the amount of shear tractions allowed to develop between crack faces. The results obtained are compared to experimental results. As a general conclusion, it is found that crack path continuity allows for the development of crack patterns similar to those found in experiments, even when reasonably coarse meshes are used.

Non-homogeneous displacement jumps in strong embedded discontinuities

In this paper, strong discontinuities are embedded in finite elements to describe fracture in quasi-brittle materials. A new numerical formulation is introduced in which the displacement jumps do not need to be homogeneous within each finite element. Both the crack path and the displacement jumps are continuous across element boundaries. This formulation is compared with the discrete approach, in which interface elements are inserted to model the discontinuities, as well as with other embedded discontinuity approaches and with the partition of unity method. Numerical results have been obtained with relatively coarse meshes, which compare well with experimental results and with the results obtained from analyzes with interface elements.

A FINITE ELEMENT ANALYSIS OF NON-PRESCRIBED CRACK PROPAGATION IN CONCRETE

Abstract In this paper a finite element model for the study of crack propagation in concrete is presented. The approach is based on Hillerborgs concepts of discrete crack formation and fracture energy dissipation. Interface elements with initial zero thickness are inserted along the interelement boundaries and no special remeshing procedures are required. Cracks open along those interface elements at which an approximate version of the usual criterion for mode I fracture is satisfied. The constitutive behaviour and the energy dissipation along the opening interface elements are corrected to take into account the difference between the mesh-dependent directions of the opening interfaces and the predicted direction of crack formation. Numerical results for a shear beam test and a pullout test are presented and comparisons between computed and experimental results are made.

Generalisation of non-iterative methods for the modelling of structures under non-proportional loading

Abstract Localisation of initially distributed cracking is a numerical challenging task, which is difficult to accomplish with conventional iterative methods, e.g. the Newton–Raphson method (Crisfield in Comput Aided Anal Des Concr Struct (1):331–358, 1984). A total approach, such as the sequentially linear approach (SLA), has been used to overcome convergence problems. However, the use of a total approach in combination with non-proportional loading raises important difficulties due to: (1) an incomplete description of the material loading history; and (2) incremental nonlinear behaviour due to the rotation of the principal stress directions. In this manuscript, two existing methods adopting combined total and incremental approaches are extended to non-proportional loading conditions. In these methods, preferential use of the incremental approach is made and the total approach is adopted only when critical bifurcations points are found. Comparison with a purely total non-iterative method (SLA) is performed, and the numerical results are validated using experimental tests available in literature.

Numerical modelling of concrete beams reinforced with pre-stressed CFRP

In this paper, a numerical simulation is presented on the behaviour of concrete beams, reinforced with pre-stressed CFRP. The numerical results are compared to experimental results. Nonlinear material behaviour is considered, namely: the inelastic compressive concrete behaviour, the elasto-plastic behaviour of steel reinforced bars, the bond-slip relationship between the concrete and the internal steel reinforced bars, the mode-II fracture interface between the concrete and the pre-stressed CFRP and concrete cracking. Cracking in concrete is modelled according to a discrete crack approach: micro-cracking is assumed to localize at fictitious cracks with initial zero width. Two different approximations are adopted: (i) the fictitious cracks are embedded within the finite elements, giving rise to a discrete strong discontinuity formulation and (ii) main cracks, similar to the experimentally observed, are introduced, using interface elements, along the element boundaries, since the beginning of the analysis. A non-iterative sequentially-linear approach is adopted in order to avoid convergence problems. The aim of the present analysis is to try to better understand the failure mechanisms found in the experimental tests. Despite the complexity of the multiple nonlinear aspects of the behaviour of the structure, it is concluded that the numerical results are similar and are close to those observed experimentally.

Modeling the behavior of reinforced concrete beams strengthened with FRP

The strengthening of reinforced concrete structures with fiber reinforced polymers (FRP) is particularly attractive due to their mechanical properties. The understanding of the premature failure modes is of great importance. Since rupture is frequently found to occur at the interface FRP- -concrete, there is a clear need to study the nature of the bonding so as to develop techniques to permit its design modeling. The stress distribution in shear test models does not precisely match the one obtained in flexural reinforcement; in the latter, according to various authors [1], [2], in addition to the stresses tangential to the interface, normal stresses are also important. In this paper, a numerical model is presented to describe the behavior of reinforced concrete beams strengthened with FRP. This model is based on previous studies focused both on: i) the distribution of shear stresses at the interface FRP-concrete and on ii) the stress concentration at the plate ends in flexural models. Furthermore, the importance of the flexural cracks in the premature rupture of the element is also analyzed. The behavior of reinforced concrete beams strengthened with both FRP laminates and sheets is considered. The FRP-epoxy-concrete arrangement and the flexural cracks are modeled with interface elements with initial zero thickness, using a discrete approach and a localized damage model. A softening behavior is adopted to simulate the stress transfer along the FRP-concrete interface. The importance of considering the mixed mode of fracture is discussed. Mention is also made to some of the main mathematical models found in the literature.

New Developments in the Study of Strong Embedded Discontinuities in Finite Elements

In this paper, a new formulation of strong embedded discontinuities in f inite elements is introduced. The displacement jumps are obtained at additional nodes, located along the surface of discontinuity. These additional nodes and the corresponding degrees of freedom ar global, instead of local as assumed in previous embedded formulations. As a consequence: i ) th jump displacement field is non-homogeneous within each parent element and ii) the jumps are continuous across element boundaries. Simple examples are presented and compare d with the results obtained from experiments and with other formulations, namely the discrete approach with int erface elements (discrete-interface approach), the partition of unity method and previous embedded formula ti ns. Introduction Recently, alternative approaches to the embedded discontinuity formulati on h ve been proposed in the literature, namely the extended finite element method and the partition of unity method [1]. In the latter methods the displacement jumps, which occur at the disconti nuity surfaces, are bound to be continuous along element boundaries. This is achieved through the use of addi ti n l global degrees of freedom, located at the usual element nodes. In this way, the concept of embedded discontinuity is somehow lost, although it is still possible to trace the exact location of the discontinuities in the mesh. In this paper, the embedded discontinuity approach [2,3] and the partition of unity method [1] are studied within an unified framework, which covers both the variational form ulation and the numerical implementation. Furthermore, a new embedded discontinuity formul ation is proposed, in which non-homogeneous displacement jump fields are allowed in each eleme nt crossed by a discontinuity. In this new approach, additional degrees of freedom are al so introduced in the parent elements as done in [1]. However, these degrees of freedom are not l ocated at the usual element nodes; instead, the additional degrees of freedom are located at new nod s inserted along the discontinuity surface. These new nodes are global, instead of local as ssumed in previous embedded formulations [2,3]. As a consequence, continuous jumps across the elem ent boundaries can be obtained, similar to the work presented in [1]. Kinematics of a strong discontinuity Consider a domain Ω, with boundary ∂Ω, crossed by a surface of discontinuity Γd. The total displacement field is the sum of a regular part û on Ω and a discontinuous part corresponding to the displacement jump [ |u(x)|], localized at the discontinuity surface Γd: u(x) = û(x) + Γd [|u(x)|], (1) where Γd is defined as: Γd = HΓd − (1−r), 0 ≤ r ≤1 (2) Key Engineering Materials Online: 2003-10-15 ISSN: 1662-9795, Vols. 251-252, pp 109-114 doi:10.4028/www.scientific.net/KEM.251-252.109

Modelling of Reinforced Concrete Beams Strengthened With Pre-Stressed CFRP

In this work a numerical simulation on the behaviour of reinforced concrete beams, strengthened with pre-stressed CFRP at the lateral faces of the beams is presented. The numerical results are compared to experimental results obtained from a testing campaign made in 2004 by Franca [1]. In this simulation, several hypotheses are adopted related to the material and numerical models, namely: i) cracked concrete is modelled using a discrete cracking approach and strong discontinuities embedded in the finite elements; ii) an elastoplastic behaviour is adopted for concrete under compression; iii) reinforced bars and CFRP are also modelled using an elastoplastic stress-strain relationship; iv) the bond-slip relationship adopted between the concrete and reinforcement in tension is based on the MODEL CODE 1990 and v) a mode-II bond-slip relationship is adopted between the concrete and CFRP, ac-cording to the work presented by Costa[2].

On the use of non-iterative methods in cohesive fracture

Localisation of initially diffused cracking in softening materials is a numerically challenging task, which is difficult to accomplish with conventional iterative methods. The sequentially linear approach has been used to overcome convergence problems. However, the use of such a total approach raises some difficulties, namely can not be treated in a straightforward way and significant load decays may occur. In this manuscript, new non-iterative numerical methods are proposed to overcome the difficulties mentioned above: (i) an improved total non-proportional analysis and (ii) a purely incremental approach. Similar to the Sequentially Linear Approach, the evolution of the material response is based on the control of the damage state. A mixed-mode discrete damage model is used to exemplify the implementation of these methods. Finally, examples are presented to illustrate the improvements obtained with these new techniques.

Modelling Mixed-Mode Fracture using an Energy-based Damage Model

In this work, a new damage model for mixed-mode fracture in the scope of the discrete crack approach is introduced. An energy-based internal damage variable is adopted. In the model, deformation-driven loading surfaces are defined and the dual loading surfaces in the traction space are derived. Under proportional loading, it is found that the constitutive relationship is symmetric. Further enrichment of the energy-based variable on the traction field is also introduced, allowing for a better approximation of: i) the limit surface defined in traction space and ii) non-proportional loading. However, in this case symmetry of the constitutive tensor is lost.