An orthotropic damage model with internal sliding and friction for masonry-like material

Highlights

• Thermodynamics of irreversible processes to define the non-linear process.

• Fixed directional approach to describe orthotropic damage.

• Damage–sliding coupling to account for hysteretic loops along shear directions.

• Representative local and structural case studies are exposed and discussed.

Abstract

Quasi-brittle materials like masonry exhibit mechanical properties and develop nonlinearities that are mainly driven by their joints. In the present work, a Continuum Damage Mechanics point of view is considered to describe the macroscopic behavior of masonry. Based on damage models developed for ceramic matrix composite materials, a fixed directional damage approach is proposed. From this formulation, unilateral effect, as well as internal sliding and friction coupled with damage, are introduced. Numerical examples of the response of the model for different loading cases involving cyclic and non-proportional loadings are carried out and compared to experimental results.

Introduction

Running bond masonry is a widely used structural material in bridge and tunnel heritage, and a large amount of these are still in service. They are facing the regular increase in railway traffic, and natural hazards are the main unpredictable causes of their failure. In order to better prevent the impact of natural hazards, overall seismic ones, on masonry railway infrastructure, robust material models are needed.

A running bond masonry is constituted of quasi-rectangular-shaped blocks linked by mortar joints. For a high aspect ratio of the blocks, it leads to an orthotropic elastic behavior of the masonry at the macroscale [1]. This quasi-brittle material develops cracks mainly in mortar joints [2], [3]. For cyclic uniaxial loading, one can observe unilateral effects due to the closure of the developed cracks. Under cyclic shear loading, hysteretic dissipation develops [4], [5], [6], [7], [8]. This last phenomenon can be linked to internal sliding in the cracks generated at the mesoscale as for concrete [9].

As developed for instance in [10], [11], [12], different scales and different modeling strategies can be considered to describe the mechanical behavior of masonry. Among the first modeling strategy to evaluate the failure of masonry structure, one can find the work of Heyman [13]. He applied the limit analysis method to masonry, considering simple hypotheses ($i.e.,$ no tensile strength and infinite compressive strength for masonry, and no possible sliding between blocks) in order to evaluate the strength of masonry arch. In this framework, several developments have been made later on to improve the representativeness of the masonry behavior like the derivation of the ultimate strength of masonry from a homogenization approach [14] or the definition of numerical tools to evaluate the limit states of a block assembly [15], [16]. In order to describe the failure of masonry structures and to describe explicitly the cracking in masonry, the discrete element method developed originally by Cundall [17] has shown its efficiency (e.g. [18]). In this general framework, some specific developments have been made, like, for instance, the description of interactions between bodies through the non-smooth contact dynamics method [19], [20].

For evaluating the response of masonry structures considering moderate loadings without provoking complete failure, homogeneous description of the masonry at the macroscale within a continuous description has shown its efficiency (see for instance [21], [22]). Different modeling strategies can be found to describe masonry as a homogeneous continuous medium. Due to its low resistance under tensile stress, a category of models proposes the perfectly no-tension material hypothesis [23]. It allows to determine the maximum capacity of a structure; nevertheless, it needs dedicated numerical strategies [24] and does not allow to investigate the softening response of a structure under seismic loadings. The nonlinear behavior of masonry can also be described by classical continuum theory as smeared cracks (e.g. [25]), plasticity (e.g. [26]), damage (e.g. [27], [28]) or the coupling of damage and plasticity (e.g. [29] or [30] with application to masonry structures in [21]). As shown in the last example, numerous models in this continuum mechanics framework are inspired by models initially developed for concrete. Unless there is a robust and efficient description of nonlinear phenomena in quasi-brittle materials, these models generally miss the description of the anisotropic nature of the nonlinearities in masonry media. To assess this anisotropic nonlinear behavior of cracking phenomenon, an orthotropic damage continuum framework has been proposed by some authors [31], [32]. In [31], the authors build a damage tensor for in-plane problems based on a combination of scalar damage variables associated with the normal and parallel directions of the bed joints. The influence of these scalar damage variables on shear response is derived from equilibrium at the macroscale using the effective stress tensor and friction angles. In [32], using mapped tensors, the authors build an orthotropic damage model from an isotropic one. Both models are able to describe the progressive orthotropic degradation of masonry; nevertheless, no mechanism is introduced to describe the hysteretic loops observed for cyclic shear loadings. Finally, the anisotropic behavior of the masonry can also be obtained through lower scale informed multiscale approaches. An efficient approach in this framework that includes various nonlinear local phenomena like friction without inducing large computational time is proposed by [33] using the Transformation Field Analysis (TFA) [34].

The development of crack families associated with specific material direction can also be observed in ceramic matrix composites. To describe this nonlinear behavior and the effect of crack families on the response at the macroscale, the concept of fabric tensor is considered (e.g. [35]). The consistent decomposition of the degradation in a thermodynamical framework allows to develop a rigorous and numerically robust model. Furthermore, the coupling between damage and other mechanisms like plasticity for sliding in cracks (e.g. [36]) can be easily performed in this formalism.

The present paper proposes to develop a novel approach with respect to existing continuous models for masonry by considering a description of the orthotropic damage through a decomposition per direction and associated crack family. Furthermore, coupling between damage and friction is introduced to describe the hysteric loop observed during cyclic shear loading. These aspects are of main importance when dealing with the dynamic response of structures subjected to medium earthquakes.

This paper is divided into three parts. The first one concerns the theoretical formulation (Section 2). Different assumptions about the formulation and the constitutive equations are developed. First of all, the mechanical framework and the coupling between elasticity, damage and plasticity are explained. Then, the evolution of internal variables during loading/unloading conditions is presented. The formulation of an orthotropic damage model and the introduction of plasticity to describe friction under cyclic shear loading represent a new contribution to masonry modeling. The second part deals with the numerical implementation and algorithm (Section 3). The last part highlights numerical validation and applications.