A localized mapped damage model for orthotropic materials
Introduction
The mechanical behaviour of anisotropic materials involves properties that vary from point to point, due to composite or heterogeneous nature, type and arrangement of constituents, presence of different phases or material defects. A macroscopic continuum model aimed at the phenomenological description of anisotropic materials should account for (i) the elastic anisotropy, (ii) the strength anisotropy (or yield anisotropy, in case of ductile materials) and (iii) the brittleness (or softening) anisotropy [1].
Several materials can be considered, with an acceptable degree of approximation, to be orthotropic, even though some of them are not so in the whole range of behaviour. Modelling the elastic orthotropy does not present big difficulties, since it is possible to use the general elasticity theory [2]. On the other hand, the need to model the strength and nonlinear orthotropic behaviour requires the formulation of adequate constitutive laws, which can be based on such theories as plasticity or damage. In particular, although several failure functions have been proposed, the choice of a suitable orthotropic criterion still remains a complex task.
One of the more popular attempts to formulate orthotropic yield functions for metals in the field of plasticity theory is due to Hill [3], [4], who succeeded in extending the von Mises [5] isotropic model to the orthotropic case. The main limitation of this theory is the impossibility of modelling materials that present a behaviour which not only depends on the second invariant of the stress tensor, i.e. the case of geomaterials or composite materials. On the other hand, Hoffman [6] and Tsai–Wu [7] orthotropic yield criteria are useful tools for the failure prediction of composite materials.
For the description of incompressible plastic anisotropy, not only yield functions [8] and phenomenological plastic potentials [9] have been proposed over the years. Other formulation strategies have been developed, related to general transformations based on theory of tensor representation [10], [11]. A particular case of this general theory, which is based on linearly transformed stress components, has received more attention. This special case is of practical importance because convex formulations can be easily developed and, thus, stability in numerical simulations is ensured. Linear transformations on the stress tensor were first introduced by Sobotka [12] and Boehler and Sawczuck [13]. For plane stress and orthotropic material symmetry, Barlat and Lian [14] combined the principal values of these transformed stress tensors with an isotropic yield function. Barlat et al. [15] applied this method to a full stress state and Karafillis and Boyce [16] generalized it as the so-called isotropic plasticity equivalent theory with a more general yield function and a linear transformation that can accommodate other material symmetries. Betten [17], [18] introduced the concept of mapped stress tensor to express the behaviour of an anisotropic material by means of an equivalent isotropic solid (mapped isotropic problem). The same approach was later refined by Oller et al. [19], [20], [21], [22], [23] with the definition of transformation tensors to relate the stress and strain tensors of the orthotropic space to those of a mapped space, in which the isotropic criterion is defined. The stress and strain transformation tensors are symmetric and rank-four and establish a one-to-one mapping of the stress/strain components defined in one space into the other and vice versa (Fig. 1). The constitutive law and the damage criterion are explicitly expressed only in the isotropic mapped space. In this way, it is possible to use standard isotropic models in calculations, with all the related computational benefits, while the information concerning the real orthotropic properties of the material is included in the transformation tensor. The parameters that define the transformation tensor can be calibrated from adequate experimental tests. The implementation of this theory into the framework of the standard FE codes is straightforward.
The aforementioned approach based on mapped tensors was principally addressed to Plasticity problems. Recently it has been extended to Continuum Damage Mechanics (CDM) constitutive laws by Pelà et al. [24], [25] and applied to the study of masonry structures.
This paper explores the application of the model also to generic orthotropic materials. The underlying theory applied to CDM is recovered and its theoretical consistency and flexibility to different applications are stressed. The proposed mapped damage model is then used to simulate the failure loci of masonry, fibre-reinforced composites and wood. The main novelty of this research is the combination of the mapped damage model with the local crack-tracking technique proposed by Cervera et al. [26]. The purpose of this improvement of the original approach is the FE analysis of tensile cracking phenomena in orthotropic materials. The combination of the mapped tensor theory with a crack-tracking algorithm poses some issues that are addressed in this paper.
The introduction of local or global crack-tracking techniques into the framework of standard finite elements and constitutive models [25], [26], [27], [28] has revealed to be a satisfactory solution to some of the major drawbacks of the classical Smeared Crack Approach (SCA) [29]. In addition to modelling the tensile damage as a smeared quantity spreading over large regions of the FE mesh, the SCA presents other well-known disadvantages. Firstly, the smeared damage propagation depends on mesh-size and mesh-bias, with a consequent lack of objectivity in the numerical results when different spatial discretizations are considered. Secondly, crack locking can be observed especially in bending problems, when the advancing flexural crack experiences a sudden “about-turn”. The effectiveness of crack-tracking techniques to avoid mesh dependency and locking problems has been demonstrated in Refs. [25], [26], [27], [28].
The crack-tracking procedure labels the finite elements which can damage and prevents the others from failing. A correction of spurious changes of crack propagation direction is carried out. These features of the method allow the analyst to avoid the aforementioned problems usually found in classical SCA, without increasing excessively the implementation effort or the computational cost. Crack-tracking algorithms are also employed in E-FEM and X-FEM to establish which elements lie in the discontinuity path and need to be enriched [30]. Despite the wide diffusion of the aforementioned procedures, it is worth noting that the introduction of mixed approaches in the field of Computational Failure Mechanics does not require any crack-tracking method [31], [32], [33].
Benchmark numerical examples are presented to check the capability of the numerical model to reproduce the correct crack paths in a material with different inclinations of the axes of orthotropy. The FE simulation of mixed mode fracture experimental tests on brick masonry members is discussed. The model is able to predict the failure load and the cracking path in orthotropic materials subject to complex stress states.
The material is modelled by considering a macro-scale approach and it is represented as a homogeneous continuum. No distinction is made among components if a composite material, e.g. FRP or masonry, is analysed. An alternative treatment is the use of any theory of homogenization [34], [35].
Tensor notation is used in this paper. The material coordinate system, which coincides with the principal axes of orthotropy of the solid, is denoted by axes 1 and 2 in the two-dimensional case, see Fig. 2. Tensors and vectors referred to that local coordinate system are marked by apex (′). The angle θ indicates the inclination between the material and the global coordinate systems (xy) and it is measured counter clockwise from the x-axis to the 1-axis. Finally, apex (*) is assigned to variables related to the mapped isotropic space.
Section snippets
Mapped damage model
The orthotropic mapping of CDM constitutive laws has been presented in Refs. [1], [24], [25]. In this section, the basics of the method are recovered and its thermodynamic consistency is demonstrated. The flexibility of the procedure for the application to generic orthotropic materials is stressed.
Local crack-tracking technique for damage localization in orthotropic materials
The local crack-tracking technique proposed in [26] was successfully applied to 2D three-noded standard elements with the aim of simulating the propagation of localized cracks in isotropic quasi-brittle materials. The algorithm was validated by comparison with benchmark tests, experimental results and finally used for the pushover analysis of the representative bay structure Mallorca Cathedral [49], showing its usefulness even for large scale structures.
The crack-tracking technique proposed in
Validation examples
This section presents the validation of the proposed model by means of comparisons with experimental data of orthotropic materials. Firstly, the orthotropic model is used to reproduce the directional strength of wood, the failure envelopes of composite laminates and masonry. Such applications show how to set the parameters of the model and demonstrate the wide applicability of the method to different orthotropic materials. Secondly, the damage model combined with the local crack-tracking
Conclusions
A novel methodology has been presented to simulate numerically the tensile crack propagation in orthotropic materials. An implicit orthotropic damage criterion is formulated by defining an isotropic criterion in a mapped space. Linear transformations for stress and strain tensors from the orthotropic space to the isotropic mapped one are established. The different behaviours along the material axes can be reproduced by means of a very simple formulation, taking advantage of the well-known
Acknowledgments
This research has received the financial support from the Ministerio de Educación y Ciencia of the Spanish Government and the ERDF (European Regional Development Fund) through the research project MICROPAR (Identification of mechanical and strength parameters of structural masonry by experimental methods and numerical micro-modelling, ref num. BIA2012-32234).
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