Phase-field modeling of hydraulic fracture

In this work a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the phase-field parameter are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.     

Sharp-Interface Nematic-Isotropic Phase Transformations With Flow

We develop a sharp-interface theory for phase transformations between the isotropic and uniaxial nematic phases of a flowing liquid crystal. Aside from conventional evolution equations for the bulk phases and corresponding interface conditions, the theory includes a supplemental interface condition expressing the balance of configurational momentum. As an idealized illustrative application of the theory, we consider the problem of an evolving spherical droplet of the isotropic phase surrounded by the nematic phase in a radially-oriented state. For this problem, the bulk and interfacial equations collapse to a single nonlinear second-order ordinary differential equation for the radius of the droplet—an equation which, in essence, expresses the balance of configurational momentum on the interface. This droplet evolution equation, which closely resembles a previously derived and extensively studied equation for the expansion of contraction of a spherical gas bubble in an incompressible viscous liquid, includes terms accounting for the curvature elasticity and viscosity of the nematic phase, interfacial energy, interfacial viscosity, and the ordering kinetics of the phase transformation. We determine the equilibria of this equation and study their stability. Additionally, we find that motion of the interface generates a backflow, without director reorientation, in the nematic phase. Our analysis indicates that a backflow measurement has the potential to provide an independent means to determine the density difference between the isotropic and uniaxial nematic phases.     

基于统一相场理论的早龄期混凝土化-热-力多场耦合裂缝模拟与抗裂性能预测

受水化反应和热量传输等过程影响, 混凝土在养护阶段会发生受约束收缩变形, 并由此在结构内引发较大的拉应力, 而此时混凝土力学性能往往还处于较低水平, 容易导致建造期混凝土结构即出现裂缝等病害. 这种早龄期混凝土裂缝对核安全壳、桥梁隧道、地下结构、水工或海工结构等重大土木工程和基础设施的全生命周期完整性、耐久性和安全性造成严重影响. 为了准确预测早龄期混凝土抗裂性能并量化裂缝演化对混凝土结构行为的不利影响, 亟需开展化-热-力多场耦合环境下的混凝土裂缝建模与抗裂性能分析研究. 针对这一需求, 本工作在前期提出的固体结构损伤破坏统一相场理论基础上, 考虑开裂过程与水化反应、热量传输等之间的相互影响, 建立裂缝相场演化特征(包括基于强度的裂缝起裂准则、基于能量的裂缝扩展准则和基于变分原理的扩展方向判据等)与混凝土水化度和温度之间的定量联系, 提出混凝土化-热-力多场耦合相场内聚裂缝模型, 发展相应的多场有限元数值实现算法并应用于若干验证算例. 数值模拟结果表明, 上述多场耦合相场内聚裂缝模型合理地考虑了水化反应、热量传输、力学行为以及裂缝演化之间的耦合效应, 揭示了早龄期混凝土热膨胀变形和自收缩变形的相互竞争机理, 且分析结果不受裂缝尺度和网格大小等数值参数的影响, 实现了早龄期裂缝演化全过程准确模拟和抗裂性能定量预测, 有望在混凝土结构早龄期裂缝预测和控制方面发挥重要作用.      

Investigation on thermomechanical fracture in the framework of configurational forces

A configurational force approach is developed for providing a fresh look onto classical aspects of thermomechanical fracture. The theoretical framework is based on the finite deformation and makes no restrictions on the material response. The integral form of configurational force balance at the crack tip is constructed, and the concentrated configurational body force is decomposed into the inertial and internal parts. The energy release rate is evaluated through the generalized second law of thermodynamics applicable to configurational force system. The theoretical investigation shows that the negative of the projection of the internal configurational force concentrated at the crack tip along the direction of crack propagation plays the role of the energy release rate and acts directly in response to crack propagation. This finding enables us to deal with the thermomechanical fracture problems in material space.     

Simulation of crack propagation in fiber-reinforced bulk metallic glasses

Based on a phase-field model for deformation in bulk metallic glasses (BMGs), shear band formation and crack propagation in the fiber-reinforced BMG are investigated. Ideal unbroken fibers embedded in the BMG matrix are found to significantly influence the shear banding and crack propagation in the matrix. The crack propagation affected by fibers’ length and orientation is quantitatively characterized and is described by micromechanics models for composite materials. Furthermore, fractures in some practical fiber-reinforced BMG composites such as tungsten-reinforced Zr-based BMG are simulated. The relation between the enhanced fracture toughness and the mechanical properties of fiber reinforcements is determined. Different fracture modes of BMG-matrix composites are identified from the systematic simulation studies, which are found to be consistent with experiments. The simulation results suggest that the phase-field modeling approach could be a useful tool to assist the fabrication and design of BMG composites with high fracture toughness and ductility.     

Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions

We present a family of phase-field models for fracture in piezoelectric and ferroelectric materials. These models couple a variational formulation of brittle fracture with, respectively, (1) the linear theory of piezoelectricity, and (2) a Ginzburg–Landau model of the ferroelectric microstructure to address the full complexity of the fracture phenomenon in these materials. In these models, both the cracks and the ferroelectric domain walls are represented in a diffuse way by phase-fields. The main challenge addressed here is encoding various electromechanical crack models (introduced as crack-face boundary conditions in sharp models) into the phase-field framework. The proposed models are verified through comparisons with the corresponding sharp-crack models. We also perform two dimensional finite element simulations to demonstrate the effect of the different crack-face conditions, the electromechanical loading and the media filling the crack gap on the crack propagation and the microstructure evolution. Salient features of the results are compared with experiments.     

脆性固体中内聚断裂点阵列的扩张行为及间隔影响

建立一个一维模型, 分析脆性材料中多个等间距虚拟断裂点在均匀应变率拉伸作用下的扩张断裂过程. 采用线弹性波动方程组描述材料内部动力学关系, 采用线性内聚力断裂模型(linear cohesive fracture model)描述虚拟断裂点的扩张行为, 根据初始均匀拉伸条件和虚拟裂纹等间距假设给出定解条件, 形成一个初边值问题. 采用Laplace变换方法求解控制方程组, 得到虚拟断裂点扩张过程中内聚应力随时间变化曲线, 以及发生完全断裂的临界时间和单位裂纹体(碎片)的临界膨胀位移. 在此基础上分析应变率和裂纹间距对碎裂发生时间及单元裂纹体临界膨胀位移的影响. 在假设脆性材料在自然碎裂过程中单元裂纹体临界膨胀位移最小的基础上,进一步研究应变率对碎片尺度的影响.      

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Experimental Study on the Pore Structure Fractals and Seepage Characteristics of a Coal Sample Around a Borehole in Coal Seam Water Infusion

In this study, a two-dimensional fully coupled computational model is developed for simulation of proppant settlement in hydro-fractures with the use of the extended finite element framework. The porous domain is governed by the well-known \((\mathbf{u}-p)\) formulation, which consists of the momentum balance equation of the bulk, in conjunction with the momentum balance and continuity equations of the pore fluid. The hydro-fracture inflow is modeled as a 1D flow on the basis of the Darcy law, in which fracture permeability is incorporated by means of the cubic law. Contact constraints are elaborated to eliminate the overlap of fracture edges and the leak-off flow. Proppant settlement is conducted on the basis of Stokes’ law for particle terminal velocity, in which the effects of fracture walls, concentration, viscosity and bridging are incorporated into the model. A fixed-point algorithm is introduced to achieve the optimum combination for the proppant injection. Using the extended finite element method, the strong discontinuity in the displacement field due to crack body, as well as the weak discontinuity in the pressure field due to leakage, is included in the model with the use of the Heaviside and modified level set enrichment functions, respectively. The robustness and versatility of the proposed numerical algorithm in determining the optimum proppant injection is examined through several numerical simulations.     

Subcritical crack growth in weak interface under mixed mode in two dimension

Constitutive equations, which govern subcritical crack growth within a weak brittle interface, are derived assuming mixed-mode loading, i.e., both tensile and shear stresses acting at the crack tip. The subcritical crack growth is assumed to be caused by the classical activation mechanism of fluctuation fracture kinetics. To derive the constitutive equations, two approaches are developed. The first approach is process-zone-detail-independent (PZI), which ignores any details of the process zone, i.e., a near-crack-tip zone of significant damage resulting in fracture, and takes into account only the process zone length. The second approach is process-zone-detail-dependent (PZD), which takes into account some details of the process-zone structure. After some general considerations including 3D case, the detailed consideration is given for 2D case, particularly, for plane strain. Illustrative calculated examples of the obtained theoretical results are presented.     

Wave propagation in fractured porous media

A theory of wave propagation in fractured porous media is presented based on the double-porosity concept. The macroscopic constitutive relations and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. In microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law in fractured porous media. The macroscopic constitutive relations of elastic porous media saturated by one or two fluids and saturated fractured porous media can be obtained from the constitutive relations developed in the paper. In the simplest case, the final set of governing equations reduce to Biot's equations containing the same parameters as of Biot and Willis.Now at Izmir Institute of Technology, Anafartalar Cad. 904, Basmane 35230, Izmir, Turkey.     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

Initiation and stable growth of penny-shaped crack in aging viscoelastic transversally isotropic material

Considered is the long-term cracking of an aging transversally isotropic material containing a Mode I penny-shaped crack under remotely applied tensile stress. The aging material properties are described by the Boltzmann–Volterra’s linear theory for integral operators with non-difference kernels. It applied to wood, concrete, some polymers and rocks. Only the symmetric case is considered where the crack lies in the plane of isotropy. The modified Leonov–Panasyuk–Dugdale’s crack model is used with a constant process zone assuming that the critical opening displacement is the fracture criterion. Volterra’s principle is applied to derive the equations of subcritical crack growth. Numerical calculations are made for subcritical crack growth for the specific example of transversally isotropic material simulating the behavior of reinforced concrete.     

CRACK INTERACTING WITH AN INDIVIDUAL INCLUSION BY THE FRACTURE CRITERION OF CONFIGURATIONAL FORCE

基于构型力概念提出一种可判断裂纹起裂以及裂纹扩展方向的新断裂准则.该准则假设当构型合力值达到一个临界值时裂纹开始扩展,而裂纹扩展的方向则为构型合力的矢量方向.基于此断裂准则,本文开发构型力的有限元计算方法,实现对裂纹扩展的数值模拟,并着重对工程中常见的含孔洞/夹杂结构的裂纹扩展问题展开研究.研究结果表明,基于构型力的裂纹扩展准则可以很好地预测裂纹与孔/夹杂的干涉作用,其数值模拟结果与实验结果相符,从而验证了该裂纹扩展模拟方法的有效性.通过对裂纹和夹杂（圆孔、软夹杂、硬夹杂）干涉问题的数值模拟表明,裂纹前端夹杂对裂纹的扩展具有重要影响.裂纹的扩展方向与裂纹和夹杂的相对位置、以及夹杂类型密切相关.软夹杂和圆孔会吸引裂纹向其扩展,而硬夹杂会排斥裂纹扩展,裂纹在扩展过程中会绕开硬夹杂.当裂纹与夹杂夹角较小时,夹杂对裂纹扩展的影响作用明显,当夹角较大时,夹杂对裂纹扩展的影响较小;特别当裂纹与夹杂夹角为45°时,软夹杂和圆孔可能会抑制裂纹的扩展,使裂纹扩展发生止裂.研究结果有助于认清含孔洞/夹杂结构中的裂纹扩展或止裂问题,对于工程中的断裂问题具有重要指导意义.     

基于统一相场理论的锂电池电极颗粒断裂模拟研究

锂电池充放电过程中,锂离子的脱出和嵌入会引发电极颗粒的不均匀体积变化和机械应力.上述锂离子扩散过程诱发的应力与电极颗粒的尺寸大小、截面形状和冲放电速率有关,可能会导致电极颗粒出现裂缝起裂、扩展甚至断裂等力学失效,对锂离子电池的容量和循环寿命等性能产生不利影响.为准确模拟并预测电极颗粒的力学失效过程,在笔者前期提出的统一相场理论框架内进一步考虑化学扩散、力学变形和裂缝演化等耦合过程,建立化学–力学耦合相场内聚裂缝模型,发展相应的多场有限元数值实现算法,并应用于二维柱状和三维球体锂电池电极颗粒的力学失效分析.由于同时涵括了基于强度的起裂准则、基于能量的扩展准则以及基于变分原理的裂缝路径判据,这一模型不仅适用于带初始缺陷电极颗粒的开裂行为模拟,而且适用于无初始缺陷电极颗粒的损伤破坏全过程分析.数值计算结果表明,相场内聚裂缝模型能够模拟锂离子扩散引发的电极颗粒裂缝起裂、扩展、汇聚等复杂演化过程,可为锂离子电池电极颗粒的力学失效预测和优化设计提供有益的参考.     

Subcritical growth of high-cycle fatigue cracks in finite thin isotropic plates

The problem of constructing a two-stage model of the motion of a fatigue crack in finite thin isotropic plates under symmetric tension-compression is formulated, and a method for its solution is considered. The two-stage nature is regarded as the presence of the incubation and propagation stages. The model is constructed by jointly considering the resolving equations of the theory of elasticity and the evolutionary equations of the mechanics of continuous damage. The damage function's attaining the critical value is considered as a criterion of initiation of local fracture and movement of a fatigue crack. Plates containing central and lateral cracks are considered. The effect of the level of stresses, the finiteness of the plate, the initial length of the crack, and the behavior of the length of the plastic zone on fracture kinetics is evaluated. The present work was carried out with financial support from the European Community in accordance with the INTAS-UA 95-0202 International Project. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 106–116, July, 2000.     

Energy release rate of small cracks in hyperelastic materials

The energy release rate of a small crack in an infinite hyperelastic medium, and subjected to large strain multiaxial loading conditions, is derived by considering the balance of configurational stresses acting on two planes: one cutting the center of the crack face, and the other at an infinite distance in front of the crack tip. The analysis establishes that the energy release rate of a small crack is always proportional to the size of the crack, irrespective of the loading conditions and the crack orientation. The balance of configurational stresses is illustrated for several benchmark cases including simple extension, pure shear and equibiaxial extension, and for perpendicular and inclined cracks.     

Thermo‐hydrodynamic‐mechanical multiphase flow model for CO2 sequestration in fracturing porous media

In this paper, we introduce a fully coupled thermo‐hydrodynamic‐mechanical computational model for multiphase flow in a deformable porous solid, exhibiting crack propagation due to fluid dynamics, with focus on CO₂ geosequestration. The geometry is described by a matrix domain, a fracture domain, and a matrix‐fracture domain. The fluid flow in the matrix domain is governed by Darcy's law and that in the crack is governed by the Navier–Stokes equations. At the matrix‐fracture domain, the fluid flow is governed by a leakage term derived from Darcy's law. Upon crack propagation, the conservation of mass and energy of the crack fluid is constrained by the isentropic process. We utilize the representative elementary volume‐averaging theory to formulate the mathematical model of the porous matrix, and the drift flux model to formulate the fluid dynamics in the fracture. The numerical solution is conducted using a mixed finite element discretization scheme. The standard Galerkin finite element method is utilized to discretize the diffusive dominant field equations, and the extended finite element method is utilized to discretize the crack propagation, and the fluid leakage at the boundaries between layers of different physical properties. A numerical example is given to demonstrate the computational capability of the model. It shows that the model, despite the relatively large number of degrees of freedom of different physical nature per node, is computationally efficient, and geometry and effectively mesh independent. Copyright © 2017 John Wiley & Sons, Ltd.     

一维六方准晶压电材料反平面Ⅲ型裂纹电塑性分析

论文基于一维六方准晶压电材料反平面问题的基本方程,对Ⅲ型裂纹的电塑性区进行分析.采用条带模型并假设在电塑性区的切应力保持为常数,得到了电塑性区大小的表达式.这种假设方式消除了电场和应力在裂纹尖端的奇异性,这与实际情况相符合,也为一维六方准晶压电材料的断裂分析提供了理论基础.