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.     

Stable growth of fracture in brittle aggregate materials

Fracture of concrete is analyzed by combining the resistance curve (R-curve) approach with linearly elastic solutions for the energy release rate resulting from the quasi-static crack model of Wnuk, analogous to the D-BCS model of a stationary crack used in describing quasi-brittle fracture in metals. The R-curve, representing the crack length dependence of the energy consumed per unit fracture extension, is calculated using the concept of the energy separation rate associated with a finite crack growth steps. To simplify calculations, the tensile stress transmitted across the nonlinear zone ahead of the fracture front is assumed to be uniformly distributed over the entire nonlinear zone, even though in reality it must be a gradually declining stress resulting in strain-softening; and an infinite elastic medium loaded at infinity is assumed. These assumptions permit an easy solution with the help of Green's function for an infinite elastic medium. Application to bodies of finite size then requires assuming the nonlinear zone (fracture process zone) to be negligible with regard to specimen dimensions, crack length and ligament length. Even though this assumption is not always realistic, the end results, which are of practical importance, appear reasonable. The analysis leads to a nonlinear first-order ordinary differential equation for the R-curve, which is integrated numerically. The R-curves calculated in this manner can be closely fitted to data from previous fracture tests. Only two parameters, characterizing the initial and the final lengths of the nonlinear zone, need to be adjusted to test data.     

基于COMSOL的脆性材料相场断裂模型

相场法通过一系列微分方程描述材料断裂过程,避免了繁琐的裂纹面追踪,在模拟裂纹的萌生、扩展和分叉等方面具有优势。介绍了基于相场法的脆性材料断裂模型,给出了脆性材料断裂问题相场法控制方程的推导过程,提出了基于分步迭代法在COMSOL中实现脆性材料相场断裂模型的方法。再现了脆性材料单元模型和单边缺口平板受拉及受剪作用下的开裂过程,模拟的裂纹扩展路径与已有文献的结果相近,验证了程序的合理性。针对脆性材料相场断裂模型包含的诸多参数,采用Morris法对影响荷载-位移关系的脆性材料断裂模型参数进行了全局敏感性分析,结果表明,杨氏模量(E)、临界能量释放率(G_c)和位移增量(Δu_x)是影响模型荷载-位移关系输出结果的主要参数。基于COMSOL实现的相场断裂模型能够有效模拟脆性材料的裂纹萌生和扩展断裂过程,模型参数E,G_c和Δu_x对材料断裂性能的提升或模型参数反演效率的提高具有重要影响。     

Structural-kinetic stochastic model of fracture and durability of brittle materials

structural-kinetic stochastic approach to the study of the brittle fracture of materials was developed. This approach was used to determine the conditions for passive (latent) and active development of brittle fracture. An asymptotic expression was obtained for the average durability of brittle materials at low stresses at which passive development of fracture occurs.     

Energy-wave approach in the mechanics of brittle dynamic fracture

Moscow. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 118–123, March–April, 1994.     

脆性材料破坏模拟的网格生成算法

脆性材料的破坏过程具有随机性,当前的网格生成算法没有充分考虑脆性材料破坏时裂纹扩展和碎块生成的随机性。在Persson网格生成算法与Delaunay随机网格剖分理论基础上,提出了一种可根据模拟需要动态控制网格品质的网格生成算法。通过对随机分布点的Delauna三角化,生成初始网格,然后将网格体系比拟为桁架结构,网格节点即为桁架节点。桁架节点在虚拟力作用下可动态调整位置,并最终达到整个体系受力平衡。对Persson 算法中的尺寸分布函数和收敛条件进行了修正,从而提高了收敛速度,并适用于任意形状对象的网格剖分。 基于VC++平台开发了算法程序。通过实例对算法进行了验证,表明算法能够满足脆性材料破碎模拟的需要。     

3-D numerical simulation of fracture processes in heterogeneous brittle materials

By using the lattice model combined with finite element methods and statistical techniques, a numerical approach is developed to establish mechanical models of three-dimensional heterogeneous brittle materials. A special numerical code is introduced, in which a lattice model and statistical approaches are used to simulate the initial heterogeneity of material properties. The size of displacement-load step is adaptively determined so that only few elements would fail in each load step. When the tensile principal strain in an element exceeds the ultimate strain of this element, the element is considered broken and its Young's modulus is set to be very low. Some important behaviors of heterogeneous brittle materials are indicated using this code. Load-displacement curves and figures of three-dimensional fracture patterns are also numerically obtained, which are similar to those observed in laboratory tests.     

Principles of the mechanics of brittle fracture of materials with initial stresses

Conclusions for Fracture Criteria 1. With an increase in compressive stresses along cracks in materials with initial stresses, there is a reduction in the breaking loads corresponding to brittle fracture for normal-rupture and transverse-shear cracks. 2. When the initial stresses reach values corresponding to surface instability of the half-space (within the framework of plane strain), the breaking loads vanish for normal-rupture and transverse-shear cracks.The mechanical phenomenon corresponding to the second conclusion has the following explanation. Due to the use of the linearized theory at the indicated values of the initial stresses, a body with a crack is in a state of neutral equilibrium. In connection with this, it is not necessary to apply an additional finite load in order to remove the body from this state.Several other results on the mechanics of brittle fracture of materials with initial stresses have been published in studies mentioned in the survey [1].Corresponds to the text of a paper presented at the 6th International Conference on Fracture (New Delhi, India, 1984).Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 23, No. 10, pp. 34–39, October, 1987.     

Effect of rock shapes on brittle fracture using Smoothed Particle Hydrodynamics

Breakage of rocks or particulates plays a major role in various industries, such as mineral and ore processing. Many of the processes used for fracturing materials in these industries have the requirement to produce specified size and/or shape of the products. Numerical modelling can assist in understanding and predicting complex fracture processes, and can be used in designing the equipment and setting the process parameters to ensure desired product quality. In this paper, a mesh-free numerical method, called Smoothed Particle Hydrodynamics (SPH), is extended to predict impact fracture of rocks. SPH is a particle based Lagrangian method which is particularly suited to the analysis of fracture due to its capacity to model large deformation and track the free surfaces generated. A continuum damage model is used to predict the fracture of rocks. Evolution of damage is predicted using the strain history of each particle. Damage inhibits the transmission of tensile stress between particles, and once it reaches unity, the particle is unable to transmit tensile stress, resulting in a macro-crack. Connected macro-cracks lead to complete fragmentation.Firstly, an Unconfined Compressive Strength (UCS) test under uniaxial compression of a rock sample is modelled using SPH and compared against experiments to validate the capability of SPH for prediction of fracture in rocks. The SPH prediction matched the well-known experimentally observed diagonal fracture pattern. SPH is subsequently used to simulate brittle fracture of rocks during impact. Rock specimens of different shapes are examined to determine the effects of shape on both the fracture pattern and the energy dissipation during impact fracture. Rock shape is found to have considerable influence on the fracture process, fragment sizes, energy dissipation, and post-fracture motion of the fragments.     

Computation of V-notch shape factors in four-point bend specimen for fracture tests on brittle materials

Four-point bend (FPB) specimen is an important test sample in mixed mode fracture study of notched components made from brittle materials like rocks, brittle polymers, ceramics, etc. On the other hand, the notch stress intensity factors (NSIFs) are vital parameters in brittle fracture assessment of V-notched structures. Therefore, computation of NSIFs in FPB specimens is of practical interest to engineers and researchers. Since the available methods for calculating the NSIFs are often cumbersome and need complicated calculations, it is preferred to show them as a set of dimensionless parameters for the FPB specimen. In this research, the finite element method coupled with a recently developed algorithm called FEOD is employed to calculate the NSIFs of a FPB specimen for several V-shape notches and for different combinations of mode I and mode II. The obtained NSIFs are then converted to dimensionless parameters called notch shape factors and are illustrated in a number of figures. It is shown that depending on the notch depth and the location of loading points, full mode mixity from pure mode I to pure mode II can be provided in the FPB specimen. The numerical results obtained in this research are verified by using very limited results reported earlier in literature.     

Genesis of the multiscale approach for materials with microstructure

This paper presents an overview of the origin of multiscale approaches in mechanics. While the pioneer molecular models of linear elastic bodies by Navier, Cauchy and Poisson were contradicted by experiments, the phenomenological energetic approach by Green still seems suitable for simple materials only. Voigt’s molecular model, here reinterpreted in the light of contemporary mechanics, reconciled the two approaches providing a conceptual guideline for developing constitutive models based on a direct link between continuum and discrete solid mechanics. Such a theoretical background proves to be especially suitable for new complex materials. An example referred to masonry-like materials is given.     

Probabilistic approach and size effect in brittle materials under multiaxial stress states

Summary The problem of evaluating the strength of brittle materials demands a statitistical approach owing to the considerable dispersion shown by these materials, particularly if subject to stress states in which the tensile component is predominant. For this purpose, the Weibull theory can be conveniently applied, which is based on the concept of the weakest link in a chain allowing the determination of both the failure probability of a model under certain loading conditions and the corresponding mean strength.In the present work the Weibull theory is applied, by means of of the Stanley's approach, to the case of multiaxial stress states in order to deal with some problems not thoroughly investigated so far, among which the effect of the shear stresses on the flexural strength and the indirect evaluation of the tensile strength by means of the Brazilian test.In conclusion, the theory is applied to assess the difference between the tensile strength values obtained by means of different tests. The results achieved are then compared with experimental data.

---

Sommario Il problema della resistenza dei materiali fragili richiede un approccio statistico a causa della forte dispersione propria di questi materiali, in particolare se soggetti a uno stato tensionale prevalentemente di trazione. A questo scopo può essere applicata la teoria di Weibull, basata sul concetto dell'anello più debole di una catena, che consente di determinare la probabilità di crisi di un modello per una determinata condizione di carico e la corrispondente resistenza media.In questo lavoro la teoria di Weibull viene applicata al caso degli stati di tensione pluriassiali con l'approccio di Stanley, allo scopo di indagare su alcuni problemi non ancora affrontati per questa via, tra cui l'influenza delle tensioni tangenziali sulla resistenza a flessione e la determinazione indiretta della resistenza a trazione mediante prova Brasiliana.A conclusione, la teoria viene impiegata per valutare la differenza tra i valori di resistenza a trazione ottenibili con diversi tipi di prova. I risultati vengono posti a confronto con dati sperimentali.

---

Signification of fundamental symbols used P _f probability of failure - applied stress - _u location parameter - ₀ scale parameter - m shape parameter (Weibull modulus) - (z) Gamma function - V total volume of the model - unit volume - _f mean tensile strength in flexure - _b mean indirect tensile strength (Brazilian test) - _t mean direct tensile strength - _tv mean direct tensile strength of unit volume - _c mean cylindrical compressive strength. Some results of this work were presented at the following meetings:- 10th National Meeting of A.I.A.S., Associazione Italiana per l'Analisi delle Sollecitazioni, Cosenza (Italy), September 22–25, 1982;- 4th I.C.A.S.P., International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Firenze (Italy), June 13–17, 1983.This work was supported by the C.N.R.     

Photoelasto-plastic studies on brittle fracture of high-polymer solids

The mechanism of brittle fracture of high-polymer solids is experimentally investigated under one-or two-dimensional stress states by a new photoelastoplastic method suggested by the author. The application of the photoelasto-plastic method on the brittle-fracture problem is based on the principle that breaking stress can be computed in brittle fracture by the measurement of the fringe orderN _B of isochromatic lines at fracture point. Bending under three-point and four-point loads, and the plane problems, some having stress concentration and others being under contacting load, are examined by using rigid polyester cast resin containing styrol as a model specimen; and, in conclusion, the brittle fracture of high-polymer solids under one- or two-dimensional stress states is decided by the constant tensile stress, whose magnitude depends only upon the material used as a model specimen, and is larger than its ultimate tensile strength. Many kinds of factors in fracture are defined, and stress-concentration factors in fracture are compared with stress-concentration factors in elasticity. A new photoelasto-plastic simple method for the determination of stress-concentration factors in elasticity is suggested by utilization of the experimental results on this brittle fracture of high-polymer solids and is examined on the perforated plane problem having finite width under tension in comparison with theoretical analysis and the experimental results by other measuring methods.     

On possible causes of brittle fracture

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

脆性材料动态强度应变率效应

通过求解波动方程,结合有限结构-时间破坏准则,得到了动态载荷下脆性材料单轴拉伸强度应变 率效应的解析表达式。分析结果表明:材料动态强度的应变率效应具有明显的结构响应特征,即材料动态强 度除与应变率和静态参数相关外,还显著地依赖于外载荷结构及其与材料间的相互作用,因而动态强度不是 表征材料动态破坏的内禀材料性质;此外,由于在不同的外载荷条件下材料将表现出不同的动态强度,这是导 致实验结果离散性大的内在因素。