Numerical studies in dynamic fracture mechanics

This paper provides a summary of recent studies concerning numerical modeling of dynamic crack-propagation, Both stationary mesh as well as moving mesh finite-element procedures are examined. Simple procedures, using a moving mesh of conventional isoparametric elements in conjunction with certain path-independent integrals for the evaluation of stress-intensity factors for a dynamically propagating crack are presented.

---

Résumé Le mémoire fournit des synthèses des études récentes relatives à la modélisation numérique de la propagation de fissures dynamiques. On examine, à la fois, le maillage stationaire et le maillage mobile utilisés dans les procédures d'éléments finis. On présente des procédures simples utilisant un maillage mobile d'éléments conventionnels isoparamétriques utilisé avec certaines intégrales indépendantes du parcours, en vue d'évaluer les facteurs d'intensité de contrainte dans le cas d'une fissure en cours de propagation dynamique.

     

A criterion for dynamic fracture strength of materials

A phenomenological criterion for damage to materials due to short duration stresses is proposed. Based on the proposed criterion, an equation is obtained to find the time to failure corresponding to an applied stress history. It is shown that the equation is in agreement with several experimental and theoretical studies. The criterion is generalized to the case of repeated loading and an equation is derived to calculate cumulative damage under repeated loading. The predictions of the equation are in qualitative agreement with some test data.     

Finite element crack growth algorithm for dynamic fracture

A new finite element crack growth algorithm has been developed to simulate dynamic fracture. In this algorithm, pseudo elements with very high initial density are placed below the crack plane and the density is reduced to zero in a gradual manner as the crack passes the element. A number of linear elastic and elasto-viscoplastic problems have been carried out to test the new algorithm. The results are compared with some of the existing crack growth models.     

Framework for a generalized four-stage fracture model of cement-based materials

In cement-based materials the full range from brittle to ductile fracture can be achieved by changing the material structure, the loading conditions, the specimen size and/or the boundary conditions. Considering just the material, at one side of the spectrum hardened cement paste behaves brittle, whereas at the other side, new fibre reinforced cements may behave ductile. Structural conditions affect the brittleness/ductility as well, and by simply changing the loading (uniaxial tension, uniaxial and confined compression, etc.), the specimen/structure size or by changing the boundary conditions the full range from brittle to ductile response can be observed. Basically there is no difference in behaviour between the various loading cases and the same four-stage fracture process can always be identified. The four ‘universal’ stages are the linear elastic regime, the microcrack regime (before the maximum load is reached), the macrocrack regime (viz. the first, usually steep part of the softening curve), and the bridging stage. Microcracks are defined as cracks that can be arrested by elements in the material structure, whereas macrocracks can only be delayed/ arrested by means or structural measures at a larger scale than the material structure. In this paper it is tried to develop a unified view on fracture of materials belonging to this broad class, which may be seen as conceptual framework for an all encompassing fracture model for cementitious materials.     

Cohesive modeling of dynamic fracture in functionally graded materials

The dynamic fracture of functionally graded materials (FGMs) is modeled using an explicit cohesive volumetric finite element scheme that incorporates spatially varying constitutive and failure properties. The cohesive element response is described by a rate-independent bilinear cohesive failure model between the cohesive traction acting along the cohesive zone and the associated crack opening displacement. A detailed convergence analysis is conducted to quantify the effect of the material gradient on the ability of the numerical scheme to capture elastodynamic wave propagation. To validate the numerical scheme, we simulate dynamic fracture experiments performed on model FGM compact tension specimens made of a polyester resin with varying amounts of plasticizer. The cohesive finite element scheme is then used in a parametric study of mode I dynamic failure of a Ti/TiB FGM, with special emphasis on the effect of the material gradient on the initiation, propagation and arrest of the crack.     

Effect of random defects on dynamic fracture in quasi-brittle materials

We propose an asynchronous spacetime discontinuous Galerkin (aSDG) method combined with a novel rate-dependent interfacial damage model as a means to simulate crack nucleation and propagation in quasi-brittle materials. Damage acts in the new model to smoothly transition the aSDG jump conditions on fracture surfaces between Riemann solutions for bonded and debonded conditions. We use the aSDG method’s powerful adaptive meshing capabilities to ensure solution accuracy without resorting to crack-tip enrichment functions and extend those capabilities to support fracture nucleation, extension and intersection. Precise alignment of inter-element boundaries with flaw orientations and crack-propagation directions ensures mesh-independent crack-path predictions. We demonstrate these capabilities in a study of crack-path convergence as adaptive error tolerances tend to zero. The fracture response of quasi-brittle materials is highly sensitive to the presence and properties of microstructural defects. We propose two approaches to model these inhomogeneities. In the first, we represent defects explicitly as crack-like features in the analysis domain’s geometry with random distributions of size, location, and orientation. In the second, we model microscopic flaws implicitly, with probabilistic distributions of strength and orientation, to drive nucleation of macroscopic fractures. Crack-path oscillation, microcracking, and crack branching make numerical simulation of dynamic fracture particularly challenging. We present numerical examples that explore the influence of model parameters and inhomogeneities on fracture patterns and the aSDG model’s ability to capture complex fracture patterns and interactions.     

Recent application of caustics on experimental dynamic fracture studies

Combining the caustic method with high‐speed photography is an efficient optical measurement technique to study the dynamic fracture behaviours of homogenous and isotropic material. In the last decade, the main emphasis is extended to study dynamic fracture of anisotropic material and dynamic propagation of multi‐cracks and interface cracks in practical engineering materials. In this paper, the recent advances and applications about the dynamic caustic method in China are reviewed, such as impact response and dynamic fracture of composite materials (fibre composites, functionally gradient materials and nanometre composites), and dynamic interaction and propagation of multi‐cracks and interface cracks. Particularly, some new numerical methods were developed to solve the complicated caustic equations by introducing both the maximum characteristic size and the relevant angles in caustic patterns. Also, some important experimental results in fracture mechanics are described, and the potential research prospects about dynamic caustics are included as well.     

Special features of dynamic fracture of brittle materials in the mode of ultimate fracture velocities

Based on the model of spherical cavity expansion in brittle materials, which involves the concept of ultimate fracture velocities, we consider a stress-strain state in the elastic precursor zone and in the region of material broken by radial cracks. We have noted the mechanical effects of dynamic overload and fracture retardation, which arise within this model. Institute of Problems of Materials Science, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 20–26, March–April, 2000.     

Quasi-static and dynamic fracture behaviour of rock materials: phenomena and mechanisms

An experimental investigation is conducted to study the quasi-static and dynamic fracture behaviour of sedimentary, igneous and metamorphic rocks. The notched semi-circular bending method has been employed to determine fracture parameters over a wide range of loading rates using both a servo-hydraulic machine and a split Hopkinson pressure bar. The time to fracture, crack speed and velocity of the flying fragment are measured by strain gauges, crack propagation gauge and high-speed photography on the macroscopic level. Dynamic crack initiation toughness is determined from the dynamic stress intensity factor at the time to fracture, and dynamic crack growth toughness is derived by the dynamic fracture energy at a specific crack speed. Systematic fractographic studies on fracture surface are carried out to examine the micromechanisms of fracture. This study reveals clearly that: (1) the crack initiation and growth toughness increase with increasing loading rate and crack speed; (2) the kinetic energy of the flying fragments increases with increasing striking speed; (3) the dynamic fracture energy increases rapidly with the increase of crack speed, and a semi-empirical rate-dependent model is proposed; and (4) the characteristics of fracture surface imply that the failure mechanisms depend on loading rate and rock microstructure.     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

A micromechanical constitutive model for dynamic damage and fracture of ductile materials

This paper proposes a detailed theoretical analysis of the development of dynamic damage in plate impact experiments for the case of high-purity tantalum. Our micro-mechanical model of damage is based on physical mechanisms (void nucleation and growth). The model is aimed to be general enough to be applied to a variety of ductile materials subjected to high tensile pressure loading. In this respect, the work of Czarnota et al. (J Mech Phys Solids 56:1624–1650, 2008) has been extended by introducing the concept of nucleation law and by entering a nonlinear formulation of the elastic response based on the Mie-Grüneisen equation of state. This later aspect allows us to consider high impact velocities. All model parameters are directly assessed by experimental measurements to the exception of the nucleation law which is characterized by the way of an inverse identification method using three free-surface velocity profiles (at low, intermediate and high impact velocities). It is shown that the nucleation law can be consistently determined in the range of operating pressures. The nucleation law being identified, the development of internal damage happens to be a natural outcome of the modelling. The model is applied to predict damage development and free-surface velocity profiles for various test conditions. The variety and the quality of results support the physical basis (in particular micro-inertia effects) upon which the proposed model of dynamic damage is based.     

Computational dynamic fracture mechanics

This paper provides a review on the state of the art in computational dynamic fracture mechanics. The following important essential ingredients in computational dynamic fracture mechanics are included: (i) fundamental aspects of dynamic fracture mechanics, (ii) types of fracture simulation, (iii) computational models of dynamic crack propagation, and (iv) use of dynamic J-integral in computational models. In the item (i), a special attention is focused on the asymptotic eigen fields for various states of dynamic crack tips, which provide the foundation of dynamic fracture mechanics as Williams' asymptotic eigen solutions provided the foundation of static linear fracture mechanics. In the item (ii), a new concept of mixed-phase simulation is presented for general nonself-similar crack propagation, in addition to the generation-phase and application-phase simulations. A comprehensive summary of computational models for dynamic crack propagation is given in the item (iii). Finally, in the item (iv) several attractive features of the dynamic J-integral are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

A generalized theory of fracture mechanics

A fracture-mechanics theory is developed which gives fracture criteria for solids in general, without limitations as to their linearity, elastic behaviour or infinitesimal strain. Besides the “standard” results of the theory which reduce to familiar forms like the Griffith equation for linear, elastic solids, several new results emerge from the theory. These include a relationship between the surface work and the true surface energy of the solid, an explanation of certain departures from standard fracture mechanics obtained with inelastic materials, and a prediction and explanation of the phenomenon of notch brittleness. Further applications of the theory, such as adhesive failure and fatigue, will be explored in a subsequent paper.     

Numerical solutions of three dimensional dynamic crack problems and simulation of dynamic fracture phenomena by a “non-standard” finite difference method

A brief presentation of the novel features of a non-standard finite difference method (HEMP and HEMP 3D computer codes) is given. To demonstrate the capability of the HEMP 3D code with the extrapolation sub-routine for solving dynamic problems in linear fracture mechanics, through-thc-thickness cracks and semielliptic surface cracks of a section of cylindrical pipe under Heaviside-function time dependence pressure applied to the interior surface are considered and their dynamic stress intensity factors are calculated. The dynamic fracture phenomenon caused by the penetration of a rigid cylindrical punch into the center of a cylindrical wafer of linear-elastic brittle material is simulated numerically by using the HEMP code with the incorporation of the method of equivalent free-surface boundary conditions and the method of artificial velocity. A simple physical model that crack will initiate and propagate in any region where a critical stress has been reached and the crack will propagate in a direction perpendicular to the maximum principal stress is chosen here.     

Evaluation by indentation of fracture toughness of ceramic materials

A transition fracture mode from Palmqvist to median has been observed in a number of ceramic materials. A new expression to determine the fracture toughness (K _IC) by indentation is presented. The K _IC values calculated by this formula are independent of the crack profile (median or Palmqvist) and of the applied load. This formula has been obtained by modifying the universal curve of Evans and Charles to incorporate Palmqvist and median cracks over a wide range of loads in the case of brittle materials with different mechanical properties (elastic properties: E, v, K _IC).