Phase field modeling of fracture in anisotropic brittle solids

A phase field model of fracture that accounts for anisotropic material behavior at small and large deformations is outlined within this work. Most existing fracture phase field models assume crack evolution within isotropic solids, which is not a meaningful assumption for many natural as well as engineered materials that exhibit orientation-dependent behavior. The incorporation of anisotropy into fracture phase field models is for example necessary to properly describe the typical sawtooth crack patterns in strongly anisotropic materials. In the present contribution, anisotropy is incorporated in fracture phase field models in several ways: (i) Within a pure geometrical approach, the crack surface density function is adopted by a rigorous application of the theory of tensor invariants leading to the definition of structural tensors of second and fourth order. In this work we employ structural tensors to describe transverse isotropy, orthotropy and cubic anisotropy. Latter makes the incorporation of second gradients of the crack phase field necessary, which is treated within the finite element context by a nonconforming Morley triangle. Practically, such a geometric approach manifests itself in the definition of anisotropic effective fracture length scales. (ii) By use of structural tensors, energetic and stress-like failure criteria are modified to account for inherent anisotropies. These failure criteria influence the crack driving force, which enters the crack phase field evolution equation and allows to set up a modular structure. We demonstrate the performance of the proposed anisotropic fracture phase field model by means of representative numerical examples at small and large deformations.     

弹塑性体脆断相场模型的一致性无单元伽辽金法

采用无单元伽辽金法（EFG）对弹塑性体脆性断裂的相场模型进行了数值实现。利用无单元法便于构建高阶近似函数的优势,位移和相场均采用二阶移动最小二乘（MLS）近似。刚度阵的数值积分采用更为高效的二阶一致三点积分格式QC3（Quadratically Consistent 3-point integration scheme）。本构算法采用Newton-Raphson迭代和弹塑性一致性切线模量。数值结果表明了本文方法模拟弹塑性体脆性断裂的有效性。     

Simulation of spallation in brittle solids

A new model of dynamic fracture for brittle materials based on Perzyna's[1] idea and Seaman's experimental results[2] is developed. Numerical simulations of metal uranium[3] spalling process in its impact tests are carried out with the model. Fair agreement between the computation and experimental data has been obtained.     

Porosity dependence of strength in brittle solids

The effect of pore size and pore volume fraction on strength in brittle solids is evaluated. The analysis considers that the strength degradaton of a solid containing a large number of spherical pores is due to a strong effect of porosity on Young's modulus. Each pore is assumed to possess radial or annular flaws emanating from the pore surface whose lengths are considered to be independent of pore size. The effect of stress concentration induced by the presence of the pore is included in the equation for strength through the Young's modulus dependence of porosity originally developed using the concept of crack opening displacement. It is shown that the strength of a solid containing spherical pores is controlled by the pore size, pore volume fracton and the radial (or annular) crack size to pore size ratio. Predicted variation of strength with pore volume fraction is tested against experimental data for glass and polycrystalline alumina.     

Micro-cracks informed damage models for brittle solids

A class of micro-cracks informed damage models for describing the softening behavior of brittle solids is proposed, in which damage evolution is treated as a consequence of micro-crack propagation. The homogenized stress–strain relation in the cracked microscopic cell defines the degradation tensor, which can be obtained by the equivalence between the averaged strain energy of the microscopic cell and the strain energy density of the homogenized material. This energy equivalence relationship serves as an energy bridging vehicle between the damaged continuum and the cracked microstructure. Several damage evolution equations are obtained by this energy bridging method. The size effect of the micro-cracks informed damage law is characterized through the microscopic cell analysis, and the proper scaling of the characterized damage evolution functions to eliminate mesh dependency in the continuum solution is introduced.     

Bridging micro to macroscale fracture properties in highly heterogeneous brittle solids: weak pinning versus fingering

The effect of strong toughness heterogeneities on the macroscopic failure properties of brittle solids is investigated in the context of planar crack propagation. The basic mechanism at play is that the crack is locally slowed down or even trapped when encountering tougher material. The induced front deformation results in a selection of local toughness values that reflect at larger scale on the material resistance. To unravel this complexity and bridge micro to macroscale in failure of strongly heterogeneous media, we propose a homogenization procedure based on the introduction of two complementary macroscopic properties: An apparent toughness defined from the loading required to make the crack propagate and an effective fracture energy defined from the rate of energy released by unit area of crack advance. The relationship between these homogenized properties and the features of the local toughness map is computed using an iterative perturbation method. This approach is applied to a circular crack pinned by a periodic array of obstacles invariant in the radial direction, which gives rise to two distinct propagation regimes: A weak pinning regime where the crack maintains a stationary shape after reaching an equilibrium position and a fingering regime characterized by the continuous growth of localized regions of the fronts while the other parts remain trapped. Our approach successfully bridges micro to macroscopic failure properties in both cases and illustrates how small scale heterogeneities can drastically affect the overall failure response of brittle solids. On a broader perspective, we believe that our approach can be used as a powerful tool for the rational design of heterogeneous brittle solids and interfaces with tailored failure properties.     

Uniqueness in brittle fracture mechanics

It is shown that the boundary initial value problem of brittle fracture mechanics has, at most, one solution in the class of all vector fields having finite energy at each instant. No definite assumption on the elasticity tensor is made.     

A new damage model for microcrack-weakened brittle solids

In the present paper, a micromechanically based damage model for microcrack-weakened solids is developed. The concept of the domain of microcrack growth (DMG) is defined and used to describe the damage state and the anisotropic properties of brittle materials. After choosing an appropriate fracture criterion of microcrack, we obtain the analytical expression of DMG under a monotonically increasing proportional plane stress. Under a complex loading path, the evolution equation of DMG and the overall effective compliance tensor of damaged materials are given. The project supported by National Natural Science Foundation of China     

On possible causes of brittle fracture

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 137–141, May–June, 1988.     

A nonlocal theory for brittle fracture

In this paper, a nonlocal theory of fracture for brittle materials has been systematically developed, which is composed of the nonlocal elastic stress fields of Griffith cracks of mode-I, II and III, the asymptotic forms of the stress fields at the neighborhood of the crack tips, and the maximum tensile stress criterion for brittle fracture. As an application of the theory, the fracture criteria of cracks of mode-I, II, III and mixed mode I–II, I–III are given in detail and compared with some experimental data and the theoretical results of minimum strain energy density factor.     

Kinematic features of the brittle fracture phenomenon

Summary Normal-fracture surfaces of various high-polymers were investigated with the aid of light microscopy. The common fracture pattern is discussed in relation with the kinematic fracture picture introduced bySmekal (1–3). Attention is paid to radial traces and hyperbolic traces, as well as to other morphological features. With the aid of the morphological picture and results of mechanical surface roughness measurements, the three-dimensional density (volume population) of fracture sources could, at polymethylmethacrylate, be roughly estimated. The geometry of fracture sources of polymethylmethacrylate was elucidated by means of electron microscopy. Supplementary experiments performated at normal-fracture surfaces of polymethylmethacrylate showed the effect of a 90°-rotation of the largest principal stress upon the critical condition of inhomogeneities.

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Zusammenfassung Bruchflächen verschiedener Hochpolymeren wurden lichtmikroskopisch untersucht. Die Bruchfiguren werden an Hand des kinematischen Bildes des Bruchvorgangs, wie esSmekal entwickelte, betrachtet. Besondere Aufmerksamkeit wurde den radialen und hyperbolischen Spuren sowie den anderen morphologischen Zügen gewidmet. Mit Hilfe des morphologischen Bildes und der Ergebnisse von mechanischen Messungen der Oberflächenrauhigkeit konnte grob für Polymethylmethacrylat die 3-dimensionale Dichte (Volumenhäufigkeit) der Quellen für den Bruch abgeschätzt werden. Die geometrische Form der Bruchquellen für Polymethylmethacrylat wurde elektronenmikroskopisch geklärt. Ergänzende Experimente an Normal-Bruchflächen von Polymethylmethacrylat zeigen den Effekt einer 90°-Drehung der größten Hauptspannung auf die kritische Inhomogenitätsbedingung.

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Presented at the conference Flow, Fatigue and Failure held in Leeds on January 8th and 9th, 1959, by the British Society of Rheology.     

Domain switching effect on fracture of piezoelectric solids

Rational design of smart sensors and actuators that consist of piezoelectric solids requires a thorough understanding of the constitutive behavior of this material under mechanical and electrical loading. Domain switching is the cause of significant nonlinearity in the constitutive behavior of piezoelectric solids, which may be enhanced in the presence of cracks. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The corresponding finite element equations are derived and applied in the solution of the piezoelectric center crack problems. The effects of domain switching are evaluated on the near tip stress intensity factors.     

The mixed mode brittle fracture criteria in siliding mode fracture

It is well-known that the present mixed mode brittle fracture criteria are all theopening mode fracture criterion.We consider that mixed mode brittle fracture of slidingmode fracture exists too.Hence we propose three criteria of mixed mode brittle fracture ofsliding mode fracture;:the radial shearing stress criterion,the maximum shearing stresscriterion and the distortional strain-energy-density criterion.Thus,we can overall explainthe phenomena of brittle fracture in the structural elements with cracks.     

A theory for brittle fracture in compression

A symmetric cylindrical body composed of brittle material, if compressed along its generators may collapse progressively, firstly through catastrophic loss of material adjacent to its lateral surface(s). With increase of the compressive load the reduced cylinder, unable to resist, sunders. The first stage of this type of failure can be explained by the assumed existence of latent interior cracks which open abruptly on a curved, symmetric interior surface when the total energy of the body reaches a critical value determined by the geometry of the initial fracture surface.     

Phase-field models for brittle and cohesive fracture

In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown.     

Uniqueness and minimum theorems for a multifield model of brittle solids

Some minimum theorems potentially useful to construct numerical schemes related to quasi-static evolution of damage in brittle elastic solids are proposed. The approach is that of multifield theories, with a second-order damage tensor describing the microcrack density. The use of damage entropy flux and damage pseudo-potential are both investigated.     

Scattered fracture of porous materials with brittle skeleton

A model of damage accumulation in a porous medium with a brittle skeleton saturated with a compressible fluid is formulated in the isothermal approximation. The model takes account of the skeleton elastic energy transformation into the surface energy of microcracks. In the case of arbitrary deformations of an anisotropic material, constitutive equations are obtained in a general form that is necessary and sufficient for the objectivity and thermodynamic consistency principles to be satisfied. We also formulate the kinetics equation ensuring that the scattered fracture dissipation is nonnegative for any loading history. For small deviations from the initial state, we propose an elastic potential which permits describing the principal characteristics of the behavior of a saturated porous medium with a brittle skeleton. We study the acoustic properties of the material under study and find their relationship with the strength criterion depending on the accumulated damage and the material current deformation. We consider the problem of scattered fracture of a saturated porous material in a neighborhood of a spherical cavity. We show that the cavity failure occurs if the Hadamard condition is violated.