Micromechanical models to guide the development of synthetic ‘brick and mortar’ composites

This paper describes a micromechanical analysis of the uniaxial response of composites comprising elastic platelets (bricks) bonded together with thin elastic perfectly plastic layers (mortar). The model yields closed-form results for the spatial variation of displacements in the bricks as a function of constituent properties, which can be used to calculate the effective properties of the composite, including elastic modulus, strength and work-to-failure. Regime maps are presented which indicate critical stresses for failure of the bricks and mortar as a function of constituent properties and brick architecture. The solution illustrates trade-offs between elastic modulus, strength and dissipated work that are a result of transitions between various failure mechanisms associated with brick rupture and rupture of the interfaces. Detailed scaling relationships are presented with the goal of providing material developers with a straightforward means to identify synthesis targets that balance competing mechanical behaviors and optimize material response. Ashby maps are presented to compare potential brick and mortar composites with existing materials, and identify future directions for material development.     

Closed-form expressions for the macroscopic flexural rigidity coefficients of periodic brickwork

Approximate expressions for the macroscopic out-of-plane elastic coefficients of brick masonry with a regular pattern are derived in closed form using a homogenization approach for periodic media. Following an approach similar to the Method of Cells for fiber reinforced composites, a (piecewise-)differentiable expression depending on very a limited number of degrees of freedom and fulfilling suitable periodicity conditions is proposed for the microscopic transverse displacement field over any Representative Volume Element (RVE). Some of the equilibrium conditions at the interfaces between bricks and mortar joints are also fulfilled. By averaging the moment and curvature fields over the RVE, the macroscopic bending stiffness coefficients can be explicitly obtained. Using the FE solution of a masonry panel subjected to elementary load conditions as a benchmark, the proposed approach is found to accurately match the numerically obtained stiffness coefficients, for masonry elements of different geometry and different mechanical properties. In several instances, the proposed expressions agree with the numerical predictions better than other analytical expressions available in the literature.     

爆炸冲击震动对砖墙破坏作用的数值模拟

砖墙在爆炸冲击震动作用下的动力反应非常复杂,本构关系很难精确建立。本文阐述了砖墙几种常用的有限元模拟方法,分析各种方法的优缺点,确定采用一种砖块和砂浆分开的精细化建模的三维砖墙有限元模型;通过LS-DYNA软件,得到砖墙在水平爆炸冲击震动荷载下的破坏过程,计算结果与实验现象很好。研究表明:该种分析模型综合考虑了砖块和砂浆之间复杂的相互作用,并且对砂浆层进行了单独建模,保证了砖墙在数值模拟上的真实性和正确性,因此可以准确地模拟出实验中砖块之间砂浆层的损伤积累破坏。     

A multi-scale enriched model for the analysis of masonry panels

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.     

Optimization design of strong and tough nacreous nanocomposites through tuning characteristic lengths

How nacreous nanocomposites with optimal combinations of stiffness, strength and toughness depend on constituent property and microstructure parameters is studied using a nonlinear shear-lag model. We show that the interfacial elasto-plasticity and the overlapping length between bricks dependent on the brick size and brick staggering mode significantly affect the nonuniformity of the shear stress, the stress-transfer efficiency and thus the failure path. There are two characteristic lengths at which the strength and toughness are optimized respectively. Simultaneous optimization of the strength and toughness is achieved by matching these lengths as close as possible in the nacreous nanocomposite with regularly staggered brick-and-mortar (BM) structure where simultaneous uniform failures of the brick and interface occur. In the randomly staggered BM structure, as the overlapping length is distributed, the nacreous nanocomposite turns the simultaneous uniform failure into progressive interface or brick failure with moderate decrease of the strength and toughness. Specifically there is a parametric range at which the strength and toughness are insensitive to the brick staggering randomness. The obtained results propose a parametric selection guideline based on the length matching for rational design of nacreous nanocomposites. Such guideline explains why nacre is strong and tough while most artificial nacreous nanocomposites aere not.     

Limit states for brick masonry based on homogenization approach

In this paper, a procedure is developed for assessing the strength of brick masonry based on homogenization theory. The approach invokes a lower bound analysis whereby plastically admissible stress fields are constructed in the constituents involved, subject to periodic boundary conditions and static equilibrium requirements. The critical load is obtained by solving a constrained optimization problem. The analysis employs a set of specific loading histories such as axial tension, pure shear and biaxial tension–compression at different orientations of bed joints. The performance of this approach is verified against numerical solutions based on finite element analysis. In the second part of this paper, a methodology is outlined for identification of a macroscopic failure criterion that incorporates a critical plane approach. A quantitative verification of this criterion is carried out for different loading conditions and the results are compared with the experimental data available in the literature.     

Compatible model for herringbone bond masonry: Linear elastic homogenization,failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.     

Experimental Investigation and Finite Element Analysis of Fatigue Crack Growth in Pipes Containing a Circumferential Semi-elliptical Crack Subjected to Bending

In this paper, a circumferential external surface flaw in a metallic round pipe under cyclic bending loading is considered. Because of very rapid changes in the geometrical parameters around the crack front region, the mesh generation of this region must be done with great care. This may lead to an increase in the run time which makes it difficult to reach valid results and conclusions. Because of the advantages of the sub-modeling technique in problems which need very high mesh density, this method is used. Stress intensity factors in mode I condition are determined using three-dimensional finite element modeling with 20 node iso-parametric brick elements in the ANSYS 9.0 standard code and the singular form of these finite elements at the crack front. In order to estimate the analysis error, the structural parameter error in energy norm criterion was used. Because of the advantages of non-dimensional analysis, this method is employed, and the stress intensity factors are normalized. For the analysis of the fatigue crack growth, the Paris law is used. The propagation path of the surface flaw is obtained from the diagram of aspect ratio versus relative crack depth. The fatigue crack growth analysis (the relative crack depth against loading cycles diagram) of different initial crack aspect ratio under cyclic loading is also considered. Fatigue shape development of initially semi-elliptical external surface defects is illustrated. The effect of the Paris exponent (material constant) on fatigue crack propagation is shown as well. Moreover, the fatigue crack growth of several specimens is assessed experimentally using a manually-constructed experimental set up. Finally, the experimental results obtained by cyclic bending loading tests are compared with the numerical results. The experimental results show good conformity with the finite element results.     

Foam mechanics: nonlinear response of an elastic 3D-periodic microstructure

The compressive response of a 3D open-cell foam with periodic tetrakaidecahedral cells is studied through combined theoretical and numerical efforts. Under compressive loading the response is characterized by an extended load plateau following the relatively sharp rise to a maximum load. Several processes of loading have been simulated numerically using appropriately nonlinear kinematics. The onset of failure under macroscopic loading conditions is shown to be the reason of the load plateau. A failure surface is defined in macroscopic stress space by the onset of the first buckling-type instability encountered along proportional load paths. The analysis is carried out through two methods. The first one consists in increasing specimen size with periodic boundary conditions leading to the termed microfailure surface. The second one consists in considering both periodic and nonperiodic displacements variations on a minimum unit cell. The resulting failure surfaces are shown to coincide. Moreover, the postbuckling analysis has been carried out for two particular loadings: the uniaxial compression and the uniaxial deformation.     

Lattice Spring Models

Solid mechanics can be addressed by a Lattice Spring Model whose major ingredients are briefly described. It is applied to solve the dynamic equations of motions and the static equations derived by homogenization. Results relative to the macroscopic properties of solids are successfully compared to the ones obtained by analytical methods and by other techniques of numerical calculations. Wave velocities derived by direct simulations are in good agreement with the ones derived by homogenization.     

Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example

In the context of a special problem, this paper investigates the possibility of modeling dissipative mechanical response in solids on the basis of the equilibrium theory of finite elasticity for materials that may lose ellipticity at large strains. Quasi-static motions for such materials are in general dissipative if the associated equilibrium fields involve discontinuous displacement gradients. For the problem treated, consideration of such deformations is shown to lead naturally to an internal variable formalism similar to those used to describe macroscopic plastic behavior arising from microstructural effects. For quasi-static motions which are maximally dissipative in a specified sense, this formalism leads to a mechanical response which resembles that associated with the pseudo-elastic effect in shape-memory alloys.     

Experimental investigations of the joint-mortar behaviour

An important failure mode of the masonry walls is the shearing process in joint mortar. In order to understand better this phenomenon, an experimental study is carried out on half brick couplet specimen. Load/Unload shear tests are performed to assess the type of the shear behaviour of the joint mortar. The cohesion and the internal friction angle are then derived from linear regression while assuming Mohr–Coulomb criterion. In particular, the influence of holes on the joint behaviour is studied by comparing results obtained with both solid and hollow bricks. In both cases, the experimental results show that there is not any stiffness degradation even in the softening regime. Actually, the shear modulus remains constant. Hence, the joint behaviour is considered to be elastoplastic, independently on the brick type. However, it seems that the presence of holes increases the stiffness but does not affect the internal friction angle of the joint mortar.     

Refined models of elastic beams undergoing large in-plane motions: Theory and experiment

Accurate mechanical models of elastic beams undergoing large in-plane motions are discussed theoretically and experimentally. Employing the geometrically exact theory of rods with appropriate kinematic assumptions and asymptotic arguments, two approximate models are obtained—a relaxed model and its constrained version—that describe extensional and bending motions and neglect shear deformations. These models are shown to be suitable to predict, via an asymptotic approach, closed-form nonlinear motions of beams with general boundary conditions and, in particular, with boundary conditions that longitudinally constrain the motions. On the other hand, for axially unrestrained or weakly restrained beams, an inextensible and unshearable model is presented that describes bending motions only. The perturbations about the reference configuration up to third order are consistently derived for all beam models. Closed-form solutions of the responses to primary-resonance excitations are obtained via an asymptotic treatment of the governing equations of motion for two different beam configurations; namely, hinged–hinged (axially restrained) and simply supported (axially unrestrained) beams. In particular, considering the present theory and the existing theories, variations of the frequency–response curves with the beam slenderness or the relative boundary mass are investigated for the lowest modes. The fidelity of the proposed nonlinear models is ascertained comparing the theoretically obtained frequency–response curves of the first mode with those experimentally obtained.     

An investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure

An analysis of the effects of microscopic instabilities on the homogenized response of heterogeneous solids with periodic microstructure and incrementally linear constitutive law is here carried out. In order to investigate the possibility to obtain a conservative prediction of microscopic primary instability in terms of homogenized properties, novel macroscopic constitutive stability measures are introduced, corresponding to the positive definiteness of the homogenized moduli tensors relative to a class of conjugate stress–strain pairs.Numerical simulations, addressed to hyperelastic microstructural models representing cellular solids and reinforced composites, are worked out through the implementation of an innovative one-way coupled finite element formulation able to determine sequentially the principal equilibrium solution, the incremental equilibrium solutions providing homogenized moduli and the stability eigenvalue problem solution, for a given monotonic macrostrain path. Both uniaxial and equibiaxial loading conditions are considered.The exact microscopic stability region in the macrostrain space, obtained by taking into account microstructural details, is compared with the macroscopic stability regions determined by means of the introduced macroscopic constitutive measures. These results highlight how the conservativeness of the adopted macroscopic constitutive stability measure with respect to microscopic primary instability, strictly depends on the type of loading condition (tensile or compressive) and the kind of microstructure.     

Kinematic FE limit analysis homogenization model for masonry walls reinforced with continuous FRP grids

A homogenization model for periodic masonry structures reinforced with continuous FRP grids is presented. Starting from the observation that a continuous grid preserves the periodicity of the internal masonry layer, rigid-plastic homogenization is applied directly on a multi-layer heterogeneous representative element of volume (REV) constituted by bricks, finite thickness mortar joints and external FRP grids. In particular, reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints and the FRP grid is applied on the external surfaces of the REV. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces.A FE strategy is adopted at a cell level, modeling joints and bricks with six-noded wedge shaped elements and the FRP grid through rigid infinitely resistant truss elements connected node by node with bricks and mortar. A possible jump of velocities is assumed at the interfaces between contiguous wedge and truss elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for FRP bars some formulas available in the literature are adopted to reproduce the delamination of the truss from the support.Two meaningful structural examples are considered to show the capabilities of the procedure proposed, namely a reinforced masonry deep beam (0°/90° continuous reinforcement) and a masonry beam in simple flexion for which experimental data are available. Good agreement is found between present model and alternative numerical approaches.     

Development of plasticity and damage in vibrating structural elements performing guided rigid-body motions

Summary  A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems. Received 1 September 1999; accepted for publication 9 March 2000