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A Nonlinear Interface Element for 3D Mesoscale Analysis
of Brick-Masonry Structures
L. Macorini
1
, B.A. Izzuddin
2
Abstract
This paper presents a novel interface element for the geometric and material nonlinear
analysis of unreinforced brick-masonry structures. In the proposed modelling approach, the
blocks are modelled using 3D continuum solid elements, while the mortar and brick-mortar
interfaces are modelled by means of the 2D nonlinear interface element. This enables the
representation of any 3D arrangement for brick-masonry, accounting for the in-plane stacking
mode and the through-thickness geometry, and importantly it allows the investigation of both
the in-plane and the out-of-plane response of unreinforced masonry panels. A co-rotational
approach is employed for the interface element, which shifts the treatment of geometric
nonlinearity to the level of discrete entities, and enables the consideration of material
nonlinearity within a simplified local framework employing first-order kinematics. In this
respect, the internal interface forces are modelled by means of elasto-plastic material laws
based on work-softening plasticity and employing multi-surface plasticity concepts.
Following the presentation of the interface element formulation details, several experimental-
numerical comparisons are provided for the in-plane and out-of-plane static behaviour of
brick-masonry panels. The favourable results achieved demonstrate the accuracy and the
significant potential of using the developed interface element for the nonlinear analysis of
brick-masonry structures under extreme loading conditions.
Keywords: non linear interface element, cohesive model, multi-surface plasticity, geometric
nonlinearity, brick-masonry, in-plane and out-of-plane behaviour.
1
Marie Curie Research Fellow, Department of Civil and Environmental Engineering, Imperial College London.
2
Professor of Computational Structural Mechanics, Department of Civil and Environmental Engineering,
Imperial College London.
2
Introduction
Nonlinear interface elements represent an effective tool to model interaction among different
constitutive components of solids and to capture failure mechanisms in a large variety of
structural systems. Interface elements were initially employed for simulating discontinuities
in rock mechanics [1] and cracks in brittle materials like concrete [2], while more recently
they have been used to model delamination and fracture in multi-layered composites [3,4].
In order to accurately reproduce most of the physical phenomena associated with interface
failure, the interface elements have to be coupled with accurate material models that relate
tractions with separation displacements. Some main features of such constitutive relations
were introduced in the first cohesive zone models tailored to analyse crack propagation in
either ductile or brittle materials (a detailed review can be found in [5]). At present, the most
advanced cohesive models are based on either softening plasticity [6] or damage mechanics
[7]. They account for the interaction between opening and sliding fracture modes and allow
the description of delamination, decohesion and loss of friction at the interfaces between
different bodies and at the fracture process zones in solid elements.
The use of interface elements is particularly effective when the locus of potential damage and
fracture is known a priori, which is typically related to the inherent texture of the analysed
structure. This is the case for unreinforced brick-masonry (URM) where bricks are arranged
in an orderly manner so as to form structural elements. Experimental outcomes [8] and
inspection of failure modes of real URM structures show that cracks usually run along brick-
mortar interfaces and can then continue through bricks following continuous paths. These
intrinsic features of the brick-masonry structural behaviour suggested the use of nonlinear
interface elements for detailed finite element analyses of URM panels under monotonic and
cyclic loads [9,10]. In such structural models zero-thickness interfaces are employed to
represent the nonlinear behaviour of mortar and brick-mortar interface as well as potential
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cracks in bricks. Damage and fracture are assumed to occur at the interfaces only, whereas
the connected continuous elements are characterized by a linear elastic behaviour.
Most of the mesoscale models for URM developed so far, account for the in-plane stacking
mode of bricks and mortar only, and are aimed at investigating the in-plane nonlinear
response of masonry walls. Such models cannot be effectively employed to assess the
structural performance under complex loading conditions as in the case of earthquakes, when
panels in URM buildings are loaded simultaneously by both in-plane and out-of-plane
actions.
In order to define more general analysis tools for mesoscale analysis of URM elements
characterized by complex brick arrangements (quarry masonry, multi-leaf walls, etc.), both
the in-plane stacking mode and the through-thickness geometry should be represented. Just a
few models, presented very recently, consider a detailed description for the 3D texture of
URM. These advanced models are based either on the use of the finite element (FE)
continuum approach, where the progressive failure is examined at the level of constituents
employing solid elements [11] or using 3D kinematic FE limit analysis [12]. The use of the
former strategy is typically applicable to only small representative volume elements (RVEs)
of masonry, because of the extremely high computational cost, while the latter provides a
good approximation for the URM maximum capacity but does not represent all the main
structural response features (initial stiffness, progressive damage, post-peak behaviour etc.).
In addition, the aforementioned models account for only material nonlinearity, and do not
consider geometric nonlinearity, such as due to large displacements which can be relevant
especially when analysing the out of plane behaviour of URM panels. This was confirmed in
recent tests [13] that showed how the out-of-plane failure can be governed by geometric
instabilities that arise when the URM walls rock out-of-plane under dynamic loads.
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A novel 2D nonlinear interface element is presented in this work, which is used in an
accurate mesoscale description for the geometric and material nonlinear analysis of URM
structures. After presenting the main features of the 3D mesoscale approach for modelling
URM structures, the formulation of the 2D nonlinear interface element is detailed. A
corotational approach is used to account for geometric nonlinearity, and a multi-surface
softening plasticity model is employed to model all the relevant failure modes: opening in
tension, sliding in shear/tension and shear/compression and crushing in compression.
In order to demonstrate the applicability and accuracy of the proposed modelling approach,
several studies are undertaken on masonry panels in the final part of the paper, where
favourable comparisons are achieved between experimental outcomes and numerical
predictions.
1. 3D mesoscale model for brick-masonry
In this work, the finite element method is used for mesoscale analysis of URM structures.
Adopting a similar approach to that developed by Lourenço & Rots [9] for the in-plane static
analysis of single-leaf masonry panels, the blocks are modelled using continuous elements
while the mortar and the brick-mortar interfaces are modelled by means of nonlinear interface
elements. Furthermore, zero-thickness interface elements are also arranged in the vertical
mid-plane of all blocks along the direction of the shorter horizontal dimension so as to
account for possible unit failure in tension and shear (Fig. 1). While Lourenço & Rots [9]
used linear 2D planar elements for bricks with nonlinear 1D interface elements, the proposed
approach utilises 20-noded 3D elastic continuum solid elements and 16-noded 2D nonlinear
interface elements, both accounting for large displacements (Fig. 2). This allows the
representation of any 3D arrangement for brick-masonry and to model both initial and
damage induced anisotropy.
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The material nonlinearity that marks the behaviour of bricks, mortar and brick-mortar
interfaces is represented through the discrete approach founded on the principles of nonlinear
fracture mechanics [5]. The use of nonlinear interface elements to model potential crack or
slip planes, allows increased accuracy of the numerical solution simply through mesh
refinement [6]. In the present context, the softening post-peak behaviour does not lead to
mesh-dependency, since it is directly related to the fracture energy which is an intrinsic
material property. This avoids the need for addressing the localisation of the solution that
usually arises when the continuum approach featuring smeared-cracking models with
softening laws is used [11,14].
However special attention should be paid to determining the global solution because of the
brittle nature of the interface model. When cracks develop and spread along the structure, the
elastic energy stored in the bulk material connected to a damaged interface has to be
redistributed into other elements, leading in certain cases to very sharp snap-backs and
solution jumps in the static global response [15]. Specific numerical techniques based on the
arc-length method [16] can be used to successfully capture the actual global behaviour. In
any case, the use of a fine mesh often allows the determination of a smoother solution, and
dynamic analysis techniques can help overcome much of the numerical problems because the
suddenly released elastic energy is gradually transformed into kinetic and viscous energy.
According to the finite element method, the boundary value problem for any URM mesoscale
model corresponds to a set of local and global nonlinear equations. The local evolution
equations are functions of internal variables and define the central problem of computational
plasticity at quadrature point level [17], while the global algebraic equations express the
equilibrium conditions. The numerical solution is obtained using an incremental-iterative
strategy and the backward Euler scheme at local level. At each time or pseudo-time
increment, in the case of either dynamic or static analysis respectively, the computation is
![Figure 15. Wall loaded out-of-plane : (a) deformed shape (displacement scale=100) and (b) plastic work contours at the end of the analysis, (c) cracks paths surveyed after the test [30].](/figures/figure-15-wall-loaded-out-of-plane-a-deformed-shape-1tovlw9f.webp)
![Figure 8. URM wall loaded in plane: (a) boundary conditions [29], (b) employed meshes.](/figures/figure-8-urm-wall-loaded-in-plane-a-boundary-conditions-29-b-1uua1imr.webp)