Jan Eliáš

Lattice modeling of aggregate interlocking in concrete

In this paper, we study a mixed-mode fracture process using a conventional two dimensional lattice model with incorporated meso-level internal material structure. Simple elasto-brittle elements of the network are divided into three phases according to a projected grain layout. The stiffness of any element that fulfils a failure criterion is removed. As a new feature of the otherwise standard lattice approach, we added the recovery of normal stiffness when a severed element enters the compressive regime. This enhancement enables capture of the shear resistance of an existing crack caused by crack roughness, i.e. what is termed aggregate interlocking. We demonstrate this enhancement via the simulation of mixed-mode experiments on concrete performed at a laboratory at the Technical University of Denmark. Double notched concrete specimens were initially pre-cracked in tension. Then, various combinations of tensile and shear load (normal and tangential to the crack plane) were applied. Simulated crack patterns and load–displacement curves are compared to the experimental observations.

Computation of Probability Distribution of Strength of Quasibrittle Structures Failing at Macrocrack Initiation

AbstractEngineering structures must be designed for an extremely low failure probability, Pf<10-6. To determine the corresponding structural strength, a mechanics-based probability distribution model is required. Recent studies have shown that quasibrittle structures that fail at the macrocrack initiation from a single representative volume element (RVE) can be statistically modeled as a finite chain of RVEs. It has further been demonstrated that, based on atomistic fracture mechanics and a statistical multiscale transition model, the strength distribution of each RVE can be approximately described by a Gaussian distribution, onto which a Weibull tail is grafted at a point of the probability about 10-4 to 10-3. The model implies that the strength distribution of quasibrittle structures depends on the structure size, varying gradually from the Gaussian distribution modified by a far-left Weibull tail applicable for small-size structures, to the Weibull distribution applicable for large-size structures. Com...

Modification of the Audze-Eglājs criterion to achieve a uniform distribution of sampling points

Audze-Eglźjs criterion for DoE provides nonuniformly distributed experimental points.Simple improvement of AE criterion that ensures uniform distribution is presented.The improvement lies in considering periodicity of the design space.Extensive calculations verifying biased and improved uniform sampling are included. Display Omitted The Audze-Eglźjs (AE) criterion was developed to achieve aźuniform distribution of experimental points in aźhypercube. However, the paper shows that the AE criterion provides strongly nonuniform designs due to the effect of the boundaries of the hypercube. We propose aźsimple remedy that lies in the assumption of periodic boundary conditions. The biased behavior of the original AE criterion and excellent performance of the modified criterion are demonstrated using simple numerical examples focused on (i) the uniformity of sampling density over the design space and, (ii) statistical sampling efficiency measured through the ability to correctly estimate the statistical parameters of functions of random variables. An engineering example of reliability calculation is presented, too.

Generalization of load–unload and force-release sequentially linear methods

Two different non-iterative solving approaches for mechanical systems composed of elements with saw-tooth constitutive laws have been described in the literature. The first algorithm called the load–unload (L–U) method removes the whole external load after every rupture event and linearly increases it again from zero until the next rupture event occurs. The second algorithm, called the force-release (F-R) method, keeps the load unchanged after a rupture and redistributes the stresses from damaged elements until an equilibrium state has been reached. Then, the load may increase again. It is known that these two algorithms yield different results with respect to crack paths and force–deflection curves, even for proportional loading paths. It will be shown that these two algorithms can be taken as extreme cases of a general approach that is developed in this paper. The general method is based on the simultaneous redistribution of stresses from damaged elements together with the scaling of external load. Both the L–U and F-R algorithms can be reproduced by the general method and thus compared within one framework. The general method also allows the tracing of snap-backs and enables the simulation to be controlled indirectly, e.g. via crack mouth opening displacement. It is applicable to both proportional and non-proportional load paths.

Boundary Layer Effect on Behavior of Discrete Models

The paper studies systems of rigid bodies with randomly generated geometry interconnected by normal and tangential bonds. The stiffness of these bonds determines the macroscopic elastic modulus while the macroscopic Poisson’s ratio of the system is determined solely by the normal/tangential stiffness ratio. Discrete models with no directional bias have the same probability of element orientation for any direction and therefore the same mechanical properties in a statistical sense at any point and direction. However, the layers of elements in the vicinity of the boundary exhibit biased orientation, preferring elements parallel with the boundary. As a consequence, when strain occurs in this direction, the boundary layer becomes stiffer than the interior for the normal/tangential stiffness ratio larger than one, and vice versa. Nonlinear constitutive laws are typically such that the straining of an element in shear results in higher strength and ductility than straining in tension. Since the boundary layer tends, due to the bias in the elemental orientation, to involve more tension than shear at the contacts, it also becomes weaker and less ductile. The paper documents these observations and compares them to the results of theoretical analysis.

A Probabilistic Crack Band Model for Quasibrittle Fracture

This paper presents a new crack band model (CBM) for probabilistic analysis of quasibrittle fracture. The model is anchored by a probabilistic treatment of damage initiation, localization, and propagation. This model regularizes the energy dissipation of a single material element for the transition between damage initiation and localization. Meanwhile, the model also takes into account the probabilistic onset of damage localization inside the finite element (FE) for the case where the element size is larger than the crack band width. The random location of the localization band is related to the random material strength, whose statistics is described by a finite weakest link model. The present model is applied to simulate the probability distributions of the nominal strength of different quasibrittle structures. It is shown that for quasibrittle structures direct application of the conventional CBM for stochastic FE simulations would lead to mesh-sensitive results. To mitigate such mesh dependence, it is essential to incorporate the strain localization mechanism into the formulation of the sampling distribution functions of material constitutive parameters.

Improved formulation of Audze-Eglājs criterion for space-filling designs

The Audze–Eglājs (AE) criterion was developed to achieve uniform distribution of experimental points in a hypercube. However, the paper shows that the AE criterion provides strongly nonuniform designs due to the effect of boundaries of the hypercube. We propose a simple remedy that lies in the assumption of periodic boundary conditions. The biased behavior of the original AE criterion and excellent performance of the modified criterion is demonstrated using simple numerical examples focused on (i) uniformity of the samples density over the design space and, (ii) statistical sampling efficiency measured through the ability to correctly estimate statistics of functions of random variables.

Interplay of probabilistic and deterministic internal length in simulations of concrete fracture

The contribution presents simulations of fracture in concrete beams loaded in three-point bending via discrete model. The size and distribution of discrete units reflect the concrete heterogeneity and together with properties of their contacts provide an internal length scale. Due to a random placement of the discrete units, the model mimics natural randomness of material structure arising from its random heterogeneity. A second random component of the model considered is additional fluctuation of material parameters at contacts between the units. This randomness reflects variations in material properties due to mixing, drying, etc. and it is considered in a form of random field with its own internal length scale provided in a form of autocorrelation length. By changing the ratios between the autocorrelation length and the internal length arising from heterogeneities, one can observe strong effects of this ratio on structural strength when cracks propagate from smooth surface. For deeply notched beams, the spatial strength fluctuations only affects the variance of the peak load and no significant effect observed on its mean value. The described dependency of structural strength on the ratio between the two characteristic lengths is demonstrated and elucidated.

Fracture Simulations of Concrete Using Discrete Meso-level Model with Random Fluctuations of Material Parameters

The paper presents numerical simulations of concrete fracture performed on beams of variable size and notch depth using the stochastic meso-level discrete model. The model includes a substantial part of randomness in concrete heterogeneity by accounting for the largest grains when assembling the lattice geometry. The remaining randomness, caused by finer particles and the non-uniformity of the mixing process, is introduced by random fluctuations of material parameters represented by a random field. The results of the stochastic meso-level discrete model are compared with published fracture experiments performed on concrete beams loaded in three-point bending. The effects of randomness in connection with different beam size and notch depth are discussed, as well as observed differences in dissipated energy.

Simulations of Bending Experiments of Concrete Beams by Stochastic Discrete Model

The paper describes results of numerical simulations of experiments on concrete beams loaded in three-point bending. Stochastic lattice-particle model has been applied in which the material was represented by discrete particles of random size and location. Additional spatial variability of material properties was introduced by stationary autocorrelated random field. Three different types of geometrically similar beams were modeled: half-notched, fifth-notched and unnotched, each in four different sizes. The deterministic and stochastic model parameters were identified via automatic procedure based on comparison to a subset of experimental data, so that the adequacy of the model response could be validated by comparison with the remaining experimental data.