Milan Jirásek

Comparative study on finite elements with embedded discontinuities

The recently emerged idea of incorporating strain or displacement discontinuities into standard finite element interpolations has triggered the development of powerful techniques that allow efficient modeling of regions with highly localized strains, e.g. of fracture process zones in concrete or shear bands in metals or soils. Following the pioneering work of Ortiz, Leroy, and Needleman, a number of studies on elements with embedded discontinuities were published during the past decade. It was demonstrated that local enrichments of the displacement and/or strain interpolation can improve the resolution of strain localization by finite element models. The multitude of approaches proposed in the literature calls for a comparative study that would present the diverse techniques in a unified framework, point out their common features and differences, and find their limits of applicability. There are many aspects in which individual formulations differ, such as the type of discontinuity (weak/strong), variational principle used for the derivation of basic equations, constitutive law, etc. The present paper suggests a possible approach to their classification, with special attention to the type of kinematic enhancement and of internal equilibrium condition. The differences between individual formulations are elucidated by analyzing the behavior of the simplest finite element – the constant-strain triangle (CST). The sources of stress locking (spurious stress transfer) reported by some authors are analyzed. It is shown that there exist three major classes of models with embedded discontinuities but only one of them gives the optimal element behavior.

Nonlocal models for damage and fracture: Comparison of approaches

Abstract The paper analyzes nonlocal constitutive models used in simulations of damage and fracture processes of quasibrittle materials. A number of nonlocal formulations found in the literature are classified according to the type of variable subjected to nonlocal averaging. Analytical and numerical solutions of a simple one-dimensional localization problem are presented. It is shown that some of the formulations inevitably lead to residual stresses even at very late stages of the deformation process and, consequently, they are not capable of modeling complete separation in a widely open macroscopic crack. The mechanisms leading to this specific type of stress locking are explained based on a theoretical analysis of the nonlocal constitutive equations. It is also pointed out that the nonlocal approach distorts the shape of the stress-strain diagram, which has to be taken into account when designing an appropriate local softening law.

Comparison of integral-type nonlocal plasticity models for strain-softening materials

The paper analyzes and compares a number of softening plasticity models regularized by nonlocal averaging. To highlight the fundamental properties and gain insight into the regularizing effect of various formulations, the localization problem is examined in the one-dimensional setting. It is shown that some of the theoretically appealing formulations are not genuine localization limiters, and that a localized plastic zone of nonzero measure is obtained only with softening laws that take into account the effect of both the local and the nonlocal cumulative plastic strain. The evolution of the plastic zone is studied, and formulations suitable for the description of the entire deformation process up to the complete loss of cohesion are identified. The effect of boundaries on the shape of the plastic strain profile and on the dissipated energy is analyzed. Attention is also paid to the thermodynamic aspects of nonlocal plasticity, especially to the consistent extension of the concept of generalized standard materials to nonlocal continua.

Embedded crack model. Part II: combination with smeared cracks

The paper investigates the behaviour of finite elements with embedded displacement discontinuities that represent cracks. Examples of fracture simulations show that an incorrect separation of nodes due to a locally mispredicted crack direction leads to a severe stress locking, which produces spurious secondary cracking. As a possible remedy the paper advocates a new concept of a model with transition from a smeared to an embedded (discrete) crack. An additional improvement is achieved by reformulating the smeared part as non-local. Various criteria for placing the discontinuity are compared, and the optimal technique is identified. Remarkable insensitivity of the resulting model to mesh-induced directional bias is demonstrated. It is shown that the transition to an explicit description of a widely opening crack as a displacement discontinuity improves the behaviour of the combined model and remedies certain pathologies exhibited by regularized continuum models. Copyright

Embedded crack model. I : Basic formulation

The recently emerged idea of enriching standard finite element interpolations by strain or displacement discontinuities has triggered the development of powerful techniques that allow efficient modelling of regions with highly localized strains, e.g. of fracture zones in concrete, or shear bands in metals or soils. The present paper describes a triangular element with an embedded displacement discontinuity that represents a crack. The constitutive model is formulated within the framework of damage theory, with crack closure effects and friction on the crack faces taken into account. Numerical aspects of the implementation are discussed. In a companion paper, the embedded crack approach is combined with the more traditional smeared crack approach. Copyright

A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses

Microplane models are based on the assumption that the constitutive laws of the material may be established between normal and shear components of stress and strain on planes of generic orientation (so-called microplanes), rather than between tensor components or their invariants. In the kinematically constrained version of the model, it is assumed that the microplane strains are projections of the strain tensor, and the stress tensor is obtained by integrating stresses on microplanes of all orientations at a point. Traditionally, microplane variables were defined intuitively, and the integral relation for stresses was derived by application of the principle of virtual work. In this paper, a new thermodynamic framework is proposed. A free-energy potential is defined at the microplane level, such that its integral over all orientations gives the standard macroscopic free energy. From this simple assumption, it is possible to introduce consistent microplane stresses and their corresponding integral relation to the macroscopic stress tensor. Based on this, it is shown that, in spite of the excellent data fits achieved, many existing formulations of microplane model were not guaranteed to be fully thermodynamically compliant. A consequence is the lack of work conjugacy between some of the microplane stress and strain variables used, and the danger of spurious energy dissipation/generation under certain load cycles. The possibilities open by the new theoretical framework are developed further in Part II companion paper.

Consistent tangent stiffness for nonlocal damage models

Abstract This paper deals with the computational analysis of strain localization problems using nonlocal continuum damage models of the integral type. The general framework for a consistent derivation of the “nonlocal” tangent stiffness is presented. The properties of the tangent stiffness matrix are discussed and the corresponding assembly procedure is described. The quadratic rate of convergence of the Newton–Raphson iteration procedure is demonstrated and the efficiency of the proposed technique is compared to the standard approach based on the secant or elastic stiffness matrices. In this context, performance of direct and iterative solvers for the linearized equilibrium equations is also examined.

Process zone resolution by extended finite elements

A new numerical technique for the computational resolution of highly localized strains in narrow damage process zones of quasibrittle materials is proposed. Objective description of localization due to softening is provided by a nonlocal damage model. The extended finite element method is exploited for an adaptive enrichment of the standard displacement approximation by regularized Heaviside functions that are close to the exact localization mode. An accurate closed-form expression for the width of the enriched zone is derived by localization analysis under uniaxial stress. Numerical examples show that satisfactory results can be obtained even on coarse basic meshes with only a few added degrees of freedom.

Macroscopic fracture characteristics of random particle systems

This paper deals with determination of macroscopic fracture characteristics of random particle systems, which represents a fundamental but little explored problem of micromechanics of quasibrittle materials. The particle locations are randomly generated and the mechanical properties are characterized by a triangular softening force-displacement diagram for the interparticle links. An efficient algorithm, which is used to repetitively solve large systems, is developed. This algorithm is based on the replacement of stiffness changes by inelastic forces applied as external loads. It makes it possible to calculate the exact displacement increments in each step without iterations and using only the elastic stiffness matrix. The size effect method is used to determine the dependence of the mean macroscopic fracture energy and the mean effective process zone size of two-dimensional particle systems on the basic microscopic characteristics such as the microscopic fracture energy, the dominant inhomogeneity spacing (particle size) and the coefficients of variation of the microstrength and the microductility. Some general trends are revealed and discussed.

R-curve modeling of rate and size effects in quasibrittle fracture

The equivalent linear elastic fracture model based on an R-curve (a curve characterizing the variation of the critical energy release rate with the crack propagation length) is generalized to describe both the rate effect and size effect observed in concrete, rock or other quasibrittle materials. It is assumed that the crack propagation velocity depends on the ratio of the stress intensity factor to its critical value based on the R-curve and that this dependence has the form of a power function with an exponent much larger than 1. The shape of the R-curve is determined as the envelope of the fracture equilibrium curves corresponding to the maximum load values for geometrically similar specimens of different sizes. The creep in the bulk of a concrete specimen must be taken into account, which is done by replacing the elastic constants in the linear elastic fracture mechanics (LEFM) formulas with a linear viscoelastic operator in time (for rocks, which do not creep, this is omitted). The experimental observation that the brittleness of concrete increases as the loading rate decreases (i.e. the response shifts in the size effect plot closer to LEFM) can be approximately described by assuming that stress relaxation causes the effective process zone lenght in the R-curve expression to decrease with a decreasing loading rate. Another power function is used to describe this. Good fits of test data for which the times to peak range from 1 sec to 250000 sec are demonstrated. Furthermore, the theory also describes the recently conducted relaxation tests, as well as the recently observed response to a sudden change of loading rate (both increase and decrease), and particularly the fact that a sufficient rate increase in the post-peak range can produce a load-displacement response of positive slope leading to a second peak.