Thermodynamics and Analysis of Predicted Responses of a Phase Field Model for Ductile Fracture

The fundamental idea in phase field theories is to assume the presence of an additional state variable, the so-called phase field, and its gradient in the general functional used for the description of the behaviour of materials. In linear elastic fracture mechanics the phase field is employed to capture the surface energy of the crack, while in damage mechanics it represents the variable of isotropic damage. The present paper is concerned, in the context of plasticity and ductile fracture, with a commonly used phase field model in fracture mechanics. On the one hand, an appropriate framework for thermodynamical consistency is outlined. On the other hand, an analysis of the model responses for cyclic loading conditions and pure kinematic or pure isotropic hardening are shown.

Probabilistic Mesoscale Analysis of Concrete Beams Subjected to Flexure

In this paper, the probabilistic behavior of plain concrete beams subjected to flexure is studied using a continuous mesoscale model. The model is two-dimensional where aggregate and mortar are treated as separate constituents having their own characteristic properties. The aggregate is represented as ellipses and generated under prescribed grading curves. Ellipses are randomly placed so it requires probabilistic analysis for model using the Monte Carlo simulation with 20 realizations to represent geometry uncertainty. The nonlinear behavior is simulated with an isotropic damage model for the mortar, while the aggregate is assumed to be elastic. The isotropic damage model softening behavior is defined in terms of fracture mechanics parameters. This damage model is compared with the fixed crack model in macroscale study before using it in the mesoscale model. Then, it is used in the mesoscale model to simulate flexure test and compared to experimental data and shows a good agreement. The probabilistic behavior of the model response is presented through the standard deviation, moment parameters and cumulative probability density functions in different loading stages. It shows variation of the probabilistic characteristics between pre-peak and post-peak behaviour of load-CMOD curves.

Fracture simulations using edge-based smoothed finite element method for isotropic damage model via Delaunay/Voronoi dual tessellations

In this study, an effective numerical framework for fracture simulations is proposed using the edge-based smoothed finite element method (ES-FEM) and isotropic damage model. The duality between the Delaunay triangulation and Voronoi tessellation is utilized for the mesh construction and the compatible use of the finite element solution with the Voronoi-cell lattice geometry. The mesh irregularity is introduced to avoid calculating the biased crack path by adding random variation in the nodal coordinates, and the ES-FEM elements are defined along the Delaunay edges. With the Voronoi tessellation, each nodal mass is calculated and the fractured surfaces are visualized along the Voronoi edges. The rotational degrees of freedom are implemented for each node by introducing the elemental formulation of the Voronoi-cell lattice model, and the accurate visualizations of the rotational motions in the Voronoi diagram are achieved. An isotropic damage model is newly incorporated into the ES-FEM formulation, and the equivalent elemental length is introduced with an additional geometric factor to simulate the consistent softening behaviors with reducing the mesh sensitivity. The full matrix form of the smoothed strain-displacement matrix is constructed for optimal use in the element-wise computations during explicit time integration, and parallel computing is implemented for the enhancement of the computational efficiency. The simulated results are compared with the theoretical solutions or experimental results, which demonstrates the effectiveness of the proposed methodology in the simulations of the quasi-brittle fractures.

Analysis of randomly damaged triangular beam lattice: elastic field and effective properties

The decay of elastic moduli of the triangular beam lattice with randomly missed links is studied. Random low-intensity damage is considered with no more than 10% of the links removed. Unidirectional and isotropic damage models are considered. Approximate analytical formulas for elastic moduli of damaged lattice are suggested and numerically verified. Models of isotropic and scalar damage are discussed; the variation of the Poisson coefficients as a result of damage is studied. Beams of different thicknesses are examined; the dependence of moduli on thickness is investigated. Random damage is simulated using a method based on the discrete Fourier transform; the mean value and standard deviation are calculated.

A geometrically inspired model for brittle damage in compressible elastomers

Regularized continuum damage models such as those based on an order parameter (phase field) have been extensively used to characterize brittle damage of compressible elastomers. However, the prescription of the surface integral and the degradation function for stiffness lacks a physical basis. In this article we propose a continuum damage model that draws upon the postulate that a damaged material could be mathematically described as a Riemannian manifold. Working within this framework with a well defined Riemannian metric designed to capture features of isotropic damage, we prescribe a scheme to prevent damage evolution under pure compression. The result is a substantively reduced stiffness degradation due to damage before the peak response and a faster convergence rate with the length scale parameter in comparison with a second order phase field formulation that involves a quadratic degradation function. We also validate this model using results of tensile experiments on double notched plates.

Selected Concrete Models Studied Using Willam’s Test

Willam’s test is a quick numerical benchmark in tension–shear regime, which can be used to verify inelastic (quasi-brittle) material models at the point level. Its sequence consists of two separate steps: uniaxial tension accompanied with contraction—until the tensile strength is attained; and next for softening (cracking) of the material—tension in two directions together with shear. A rotation of axes of principal strains and principal stresses is provoked in the second stage. That kind of process occurs during the analysis of real concrete structures, so a correct response of the material model at the point level is needed. Some familiar concrete models are selected to perform Willam’s test in the paper: concrete damaged plasticity and concrete smeared cracking—distributed in the commercial ABAQUS software, scalar damage with coupling to plasticity and isotropic damage—both implemented in the FEAP package. After a brief review of the theory, computations for each model are discussed. Passing or failing Willam’s test by the above models is concluded based on their results, indicating restrictions of their use for finite element computations of concrete structures with predominant mixed-mode fracture.