Elastic-brittle Materials Research Articles

Nucleation of Fracture: The First-Octant Evidence Against Classical Variational Phase-Field Models

Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses.

Fourth-order phase field modelling of brittle fracture with strong form meshless method

This study aims to find a solution for crack propagation in 2D brittle elastic material using the local radial basis function collocation method. The staggered solution of the fourth-order phase field and mechanical model is structured with polyharmonic spline shape functions augmented with polynomials. Two benchmark tests are carried out to assess the performance of the method. First, a non-cracked square plate problem is solved under tensile loading to validate the implementation by comparing the numerical and analytical solutions. The analysis shows that the iterative process converges even with a large loading step, whereas the non-iterative process requires smaller steps for convergence to the analytical solution. In the second case, a single-edge cracked square plate subjected to tensile loading is solved, and the results show a good agreement with the reference solution. The effects of the incremental loading, length scale parameter, and mesh convergence for regular and scattered nodes are demonstrated. This study presents a pioneering attempt to solve the phase field crack propagation using a strong-form meshless method. The results underline the essential role of the represented method for an accurate and efficient solution to crack propagation. It also provides valuable insights for future research towards more sophisticated material models.

A Griffith description of fracture for non-monotonic loading with application to fatigue

With the fundamental objective of establishing the universality of the Griffith energy competition to describe the growth of large cracks in solids not just under monotonic but under general loading conditions, this paper puts forth a generalization of the classical Griffith energy competition in nominally elastic brittle materials to arbitrary non-monotonic quasistatic loading conditions, which include monotonic and cyclic loadings as special cases. Centered around experimental observations, the idea consists in: (i) viewing the critical energy release rate Gcnot as a material constant but rather as a material function of both space X and time t, (ii) one that decreases in value as the loading progresses, this solely within a small region Ωℓ(t) around crack fronts, with the characteristic size ℓ of such a region being material specific, and (iii) with the decrease in value of Gc being dependent on the history of the elastic fields in Ωℓ(t). By construction, the proposed Griffith formulation is able to describe any Paris-law behavior of the growth of large cracks in nominally elastic brittle materials for the limiting case when the loading is cyclic. For the opposite limiting case when the loading is monotonic, the formulation reduces to the classical Griffith formulation. Additional properties of the proposed formulation are illustrated via a parametric analysis and direct comparisons with representative fatigue fracture experiments on a ceramic, mortar, and PMMA.

Fourth-order phase-field simulation of cracks using strong form meshless method

The prediction of crack propagation in fracture mechanics is still a fundamental challenge. In recent decades, phase-field formulation emerged as a powerful tool for the simulation of crack evolution, which offers a continuous representation of cracks from intact to broken material. For the first time, the present study combines a fourth-order phase field and a strong-form meshless method to investigate crack propagation in brittle elastic material. The local radial basis function collocation method, structured with the augmented polyharmonic splines, is used to solve the coupled mechanical and phase-field models. Due to the smooth nature of the fourth-order phase-field model, the sharp transition at the crack surface can be avoided. It is found that the presented method successfully predicts the crack propagation for a material subjected to tensile loading. The obtained results agree well with the benchmark case. Further study will include elastoplastic material behaviour and applications to crack formation in different steel processing steps.

Аспекты численного моделирования процессов разрушения упруго-хрупких тел

Understanding the nucleation and evolution of microdefects in solid bodies is important to ensure the reliability and safety of critical structures and to identify their strength and deformation resources. In numerical modeling, failure zones can be represented as areas with significantly underestimated rigid characteristics by analogy with the method of variable elastic parameters used in solving the boundary-value problems of the theory of plasticity. However, the formal application of numerical algorithms of plasticity does not always lead to an adequate description of failure processes especially in elastic-brittle bodies. This paper considers some aspects of the numerical simulation of failure processes, such as the calculation of a stress-strain state after reducing the rigidity of finite elements under constant boundary conditions by organizing an appropriate iteration procedure, and the selection of the maximum number of finite elements fractured per iteration, the value of a loading step, and the discretization degree of the computational domain. The influence of the above aspects on the results of failure simulation is illustrated by comparing the numerical solutions to the problem of deformation of the strip made of elastic-brittle material with the edge stress concentrator, which were obtained by different algorithms. The loading diagrams were plotted, and the implementation of the post-critical stage at the macro level was demonstrated. Failure kinetics was analyzed for different variants of implementation of the iterative procedure and at a variable number of elements fractured per iteration. It has been found that, in order to get an accurate description of deformation and failure processes, the automatic selection of a loading step seems to be more reasonable. Analysis has indicated that the discretization degree of the computational domain has a strong impact on the modeling results. This suggests that the finite element size should correspond to a certain strength constant of a material having the dimensions of length.

The strength of the Brazilian fracture test

Since its introduction in the 1940s until present times, the so-called Brazilian test has been embraced by practitioners worldwide as a method of choice to indirectly measure the tensile strength of concrete, rocks, and other materials with a large compressive strength relative to their tensile strength. This is because of the ease that the test affords in both the preparation of the specimen (a circular disk) and the application of the loads (two platens compressing the specimen between them). Yet, this practical advantage has to be tempered by the fact that the observations from a Brazilian test — being an indirect experiment in the sense that it involves not uniform uniaxial tension but non-uniform triaxial stress states throughout the specimen — have to be appropriately interpreted to be useful. The main objective of this paper is to carry out a complete quantitative analysis of where and when fracture nucleates and propagates in a Brazilian test and thereby establish how to appropriately interpret its results. We do so by deploying the phase-field fracture theory of Kumar et al. (2020), which has been recently established as a complete theory of fracture capable of accurately describing the nucleation and propagation of cracks in linear elastic brittle materials under arbitrary quasistatic loading conditions. The last section of this paper puts forth a new protocol to deduce the tensile strength of a material from a Brazilian test that improves on the current ISRM and ASTM standards.

A finite crack growth energy release rate for elastic-plastic fracture

The concept of energy release rate proposed by Griffith originally used for elastic brittle material was extended for ductile material by Irwin and Orowan using the sum of the surface energy and the work of plastic dissipation as fracture resistance. This fracture resistance, however, depends on the load type and specimen geometry, which hinders its wide application. In contrast, in this paper, a finite crack growth energy release rate is proposed by assuming a finite crack growth, which is conveniently applicable to finite element analysis. The effect of the plastic dissipation on the crack growth is considered in the driving force. Elastic-plastic finite element analysis is used to apply the present energy release rate. Compared to the experiment results, it is shown that for a fixed finite crack growth, a constant critical energy release rate can well predict the stable crack growths and residual strengths of sheets with single and various collinear cracks subjected to different types of loads. The present finite crack growth energy release rate is conceptually simple, has a solid physical foundation, and would be appealing for ductile crack growth analysis.

Phase field modeling on fracture behaviors of elastomers considering deformation-dependent and damage-dependent material viscosity

Elastomers have attracted great research interests owing to their high compliance and large deformation capacity. However, with inevitable flaws, elastomers are susceptible to rupture, leading to reduction in stretchability and loss of functionality. Though phase field model (PFM) provides a feasible tool in fracture simulation on brittle elastic materials, numerical study on the fracture of elastomers is still at a tentative stage with challenges stemming from the extremely large deformation and the nonlinear material properties. In addition, recent experimental observations on the insensitivity of the onset of rupture to loading rates suggest another knowledge gap in revealing the role of the rate-dependent viscoelasticity in fracture. To capture such phenomenon and fill the research gap, this work will propose a finite element (FE) framework by incorporating the polymer dynamics into the PFM. For the first time, the driving force to the fracture of viscoelastic elastomers is identified and the micro-mechanism of the material viscosity is further elucidated with the consideration of polymer chain breakage. The disentanglement of polymer chains in the fracture process is considered in the model to release the constraints on polymer chain reptation from nearby chains, capturing the inherent damage-dependent as well as the deformation-dependent material viscosity. The simulation results demonstrate good agreement with existing experimental data in both loading rate sensitivity tests and geometry sensitivity tests. With the variation of loading rates, obvious differences in stress responses are observed, while the critical fracture stretches almost keep constant. This work provides a deeper understanding on the fracture mechanisms of viscoelastic materials and is expected to provide guidance for the better design and full potential applications of elastomer-based devices.

Simulation of fully coupled hydro-mechanical behavior based on an analogy between hydraulic fracturing and heat conduction

The hydraulic fracturing process involves complex coupling problems of the deformation and fracturing of solid and fluid flow in fractures. In this paper, a simple numerical tool for simulating the coupled hydro-mechanical behavior is developed by using an analogy between hydraulic fracturing and heat conduction. Using this analogy, the numerical implementation can be simply realized in the framework of a commercial finite element package ABAQUS with two user material subroutines to fully characterize the complex coupling problems of hydraulic fracturing. It is thus considerably simplified without any new development of user-defined element and fluid–solid solver but directly with the commercial finite element package. The proposed numerical tool along with the virtual crack method for elastic–brittle material is firstly verified by the classical 2D plain strain KGD analytical solution and then applied to several representative hydraulic fracturing problems and laboratory tests to illustrate its advantageous features. The present method is aiming to develop a useful numerical tool to study the coupled hydro-mechanical behavior of hydraulic fracturing. However, it is not limited to any specific materials and can be extended to other constitutive models that featured with the fully coupling problems of fluid and solid due to its simplicity.

Flexural behavior of RC continuous beams having hybrid reinforcement of steel and GFRP bars

Fiber reinforced polymers (FRP) have been considered to be the alternative material to steel in reinforced concrete structures (RC) with the advantages of corrosion resistance, non-conductivity, and high strength-to-weight ratio. FRP-RC beam is featured with higher deformability and larger crack width. However, there is no recognizable ductility in FRP-RC members. Due to the lack of the ductility of FRP-RC beam, it was suggested using hybrid reinforcement of steel and FRP to achieve a preferable strength and ductility. The flexural behavior of the concrete beams reinforced with hybrid reinforcement was investigated through series of six continuous beams. The positive moment GFRP reinforcement ratios of 0.75% and 1.25% were used and the steel reinforcement ratios were ranged to determine the optimum percentage to work efficiently with GFRP. The experimental results showed that increasing GFRP positive moment reinforcement ratio to be more than 1.4 [Formula: see text] increased the flexural capacity of the section but with less ductile behavior. However, increasing the steel positive moment reinforcement ratio leaded to the increase of both the flexural capacity and the ductile behavior in a condition that the steel positive moment reinforcement ratio was [Formula: see text]. It was shown that the flexural capacity predication of the hybrid section using the semi brittle elastic material method to present the de-bonding of moment that occurred at the failure stage had a good accuracy with the experimental results.

Fracture modeling by the eigenfracture approach for the implicit material point method framework

AbstractThe material point method (MPM) is efficiently applied for the simulation of structures undergoing large deformations where fracture and crack initiation are expected. The eigenfracture approach is introduced in the paper at hand for the implicit MPM to model crack development and propagation in static and dynamic fracture of brittle elastic materials. Eigenfracture is an energetic fracture formulation applied in the postprocessing step of the implicit MPM, making its implementation relatively straightforward. Furthermore, the driving energy used to check crack propagation is evaluated using the representative crack elements (RCE), by which the crack is modeled as a discrete phenomenon. The RCE approach shows more realistic results compared to other split models. Additionally, the fracture description of reinforced materials within the MPM is also presented in this article by coupling truss finite elements to the MPM, considering the bond stress‐slip constitutive model. Two‐ and three‐dimensional problems in static and dynamic applications are presented to assess the efficacy of the approach.

Improved peridynamic model and its application to crack propagation in rocks.

The conventional bond-based peridynamics (BB-PD) model is suitable for simulating the failure mode of homogeneous elastic-brittle materials. However, the strain hardening and subsequent strain softening characteristics of rocks under loading cannot be reflected. In addition, the fracture mechanisms of rock materials under tension and compression are completely different. To solve these problems, this paper proposes an improved BB-PD model using different fracture criteria in the tensile and compression stages of the bond based on previous improved models, and a critical failure condition obeying the Weibull distribution is introduced to reflect the heterogeneity of the rock. The crack propagation processes of an intact rock specimen, rock specimen with a single pre-existing flaw and rock specimens with two and three pre-existing flaws under compressive loading are simulated using the model, and its feasibility is verified by comparing with the results of previous laboratory tests. Next, the effects of the inclination angle and length on the wing crack propagation length are studied. Finally, the changes in the crack aggregation modes under different rock bridge inclination angles are simulated. Eight types of crack aggregation modes are found, and the conditions under which they may occur are analysed. The improved model proposed can effectively simulate the crack propagation and coalescing processes and has a wide application prospect for rock fracture simulations.

The revisited phase-field approach to brittle fracture: application to indentation and notch problems

In a recent contribution, Kumar et al. (J Mech Phys Solids 142:104027, 2020) have introduced a comprehensive macroscopic phase-field theory for the nucleation and propagation of fracture in linear elastic brittle materials under arbitrary quasistatic loading conditions. The theory can be viewed as a natural generalization of the phase-field approximation of the variational theory of brittle fracture of Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998) to account for the material strength at large. This is accomplished by the addition of an external driving force—which physically represents the macroscopic manifestation of the presence of inherent microscopic defects in the material—in the equation governing the evolution of the phase field. The main purpose of this paper is to continue providing validation results for the theory by confronting its predictions with direct measurements from three representative types of experimentally common yet technically challenging problems: (i) the indentation of glass plates with flat-ended cylindrical indenters and the three-point bending of (ii) U-notched and (iii) V-notched PMMA beams.

Experiment and Numerical Simulation of Damage Progression in Transparent Sandwich Structure under Impact Load

Crack initiation and propagation is a long-standing difficulty in solid mechanics, especially for elastic brittle materials. A new type of transparent sandwich structure, with a magnesium–aluminum spinel ceramic glass as the outer structure, was proposed in this paper. Its dynamic response was studied by high-speed impact experiments and numerical simulations of peridynamics under impact loads, simultaneously. In the experiments, a light gas cannon was used to load the projectile to 180 m/s, and the front impacted the transparent sandwich structure. In the numerical simulations, the discontinuous Galerkin peridynamics method was adopted to investigate the dynamic response of the transparent sandwich structure. We found that both the impact experiments and the numerical simulations could reproduce the crack propagation process of the transparent sandwich structure. The radial cracks and circumferential cracks of the ceramic glass layer and the inorganic glass layer were easy to capture. Compared with the experiments, the numerical simulations could easily observe the damage failure of every layer and the splashing of specific fragments of the transparent sandwich structure. The ceramic glass layer and the inorganic glass layer absorbed the most energy in the impact process, which is an important manifestation of the impact resistance of the transparent sandwich structure.

Energy Release Rate Formulations for Non-conventional Fracture Test Geometries

Non-conventional fracture test geometries such as single cantilever bending, clamped beam bending and clamped wire bending have been developed over the years to suit the needs of micro- and nano-scale testing. The same geometries can also be used at the macro-scale if testing standards are well established. Development of length scale compatible geometries helps the materials engineer to seamlessly extract material properties across multiple length scales. This article proposes the crack driving force (or energy release rate G) criteria for the above three geometries. These geometries have found widespread use at the micro-scale and can be adapted to the macro-scale. Use of a global energy-based criterion instead of the local stress-based criteria (stress intensity factor K) has its own advantages, especially in multi-phase materials and interface-dominated structures which display an R-curve. Aspects of crack stability even under load control arising because of the geometric factors, especially the beam or wire aspect ratio, are discussed in this context. Validation of the compliance approach is carried out by comparing it to the J-integral extracted directly for a linear elastic material using extended finite element modeling and also using the analytical formulation for a double-cantilever beam specimen. Experimental evidence of the validity of the solutions in determining the fracture energy of a linear elastic brittle material, PMMA, as a homogeneous model system is shown.

Multiple cracking model in a 3D GraFEA framework

In this work, a thermodynamically consistent three-dimensional (3D) small strain-based theory to describe the deformation and fracture in quasi-brittle and brittle elastic solids is presented. The description of fracture at a material point resembles the microplane fracture approach developed by Bažant et al. (J Eng Mech 126(9):944–953, 2000, J Eng Mech 122(3): 245–254, 1996), but the present theory has the following novel features: (a) a probabilistic description of fracture propagation is used, developing evolution equations for the probability of a microcrack occurring at a given location and (b) a kinematical approach to modeling crack opening and closing. The new 3-D constitutive theory, in which elements were recently proposed by Srinivasa et al. (Mech Adv Mater Struct 80(27–30):2099–2108, 2020), has been computationally implemented within a Graph-based Finite-Element Analysis (GraFEA) framework developed by Reddy and colleagues (Khodabakhshi et al. in Meccanica 51:3129–3147, 2016, Acta Mech 230:3593–3612, 2019), and it has also been implemented into the dynamics-based Abaqus/Explicit (Reference manuals. Simulia-Dassault Systemes, 2020) finite element program through a vectorized user–material subroutine interface. Our computational approach for fracture modeling is intra-element-based, which is central to the GraFEA approach rather than inter-element fracture, as is done in cohesive zone-based numerical methods, together with selective non-locality where the non-locality is only for probability evolution motivated by population dynamic models that allows us to perform efficient implementation of the code without special elements or other numerical artifacts. Several homogeneous deformation cases for fracture in cementitious and brittle elastic materials were modeled, and the response obtained from the constitutive theory and its finite element implementation are qualitatively similar to that obtained in the literature. In particular, we show that our computational procedure is able to model crack closure in solids in a robust, relatively simple and elegant manner instead of relying on a previously developed method of decomposing the stored energy into “positive” and “negative” portions (Amor et al. in J Mech Phys Solids 57(8):1209–1229, 2009, Miehe et al. in Int J Numer Meth Eng 83:1273–1311, 2010).

Modeling of damage and cracking in heterogeneous rock-like materials by phase-field method

In this work, a phase-field method is applied to modeling damage and cracking in rock-like materials by considering spatial variability of materials heterogeneities. The formulation of phase-field method is first summarized for elastic brittle materials. The effective elastic properties of rocks are determined as functions of mineral compositions by using a linear homogenization method. The randomly distribution of mineral inclusions is assumed to follow the Weibull distribution. A series of five numerical specimens are chosen to investigate the onset and propagation of localized cracks. Comparisons between numerical results and experimental data are also presented for overall stress-strain responses.

Inverse approach for flax yarns mechanical properties identification from statistical mechanical characterization of the fabric

Plant fiber properties, such as morphological and mechanical, are characterized by a large dispersion. Thereby, a statistical analysis is needed to obtain consistent results. An experimental study, conducted on 50 flax yarns, shows that the flax yarn properties (Young's modulus, tensile strength and diameter) follow Gaussian distributions. This approach is obviously reliable, however time-consuming, to get relevant information. An alternative could be the identification of the yarn mechanical properties using an inverse approach, based on tensile tests conducted on flax fabric reinforcement. The aim of this study is to develop a numerical method that allows to identify the statistical distributions of flax yarn properties based on tensile tests conducted on fabric specimens. The proposed strategy relies on two assumptions. On the one hand, fabric is constituted of several yarns acting like springs in parallel. On the other hand, yarns are considered as brittle-elastic materials. Hence, a yarn breaks when the load reaches its failure strength, leading to a loading redistribution to the intact yarns. A comparison of the numerical and experimental results of flax fabric tensile behavior, and of the flax property statistical distributions allows to confirm the performance of this strategy. The results show that the flax fabric tensile behavior is correctly described by the proposed modeling strategy. The average and the standard deviation of the Young's modulus, the tensile strength and the diameter of flax yarns identified from fabric tensile tests via inverse approach are close to those obtained experimentally on individual yarns. Indeed, the measured average diameter of unitary yarns are 244.4±19.2μm and the fitted one is 2474.8±17.4μm, the fitting failure strength 292.3±33.4MPa are close to the experimental 271.2±47.5MPa. The identified Young's modulus is 9.4±0.9GPa is lower than the experimental 10.8±1.3GPa.

A simple explicit model constructed from a homogenization solution for the large-strain mechanical response of elastomeric syntactic foams

This paper introduces a homogenization-based constitutive model for the large-strain mechanical response of elastomeric syntactic foams subject to arbitrary quasistatic loading and unloading conditions. Based on direct observations from experiments, this class of emerging foams are considered to be made of a nonlinear elastic matrix filled with a random isotropic distribution of hollow thin-walled spherical shells – commonly termed microspheres or microballoons – each having the same mean diameter that are made of an elastic brittle material that is much stiffer than the elastomeric matrix, typically, either glass or a hard polymer. Accordingly, such underlying microballoons behave effectively as rigid particles initially. Along a given loading path, however, they may fracture (in the case of glass microballoons) or buckle (in the case of polymer microballoons) at which point they abruptly transition to behave effectively as vacuous pores. On that account, the proposed constitutive model corresponds to a homogenization solution for the nonlinear elastic response of particle-filled porous elastomers – precisely, elastomers embedding both rigid spherical particles and vacuous spherical pores of equal monodisperse size – wherein the volume fraction of pores corresponds to the volume fraction of fractured or buckled microballoons and hence is not a fixed parameter but rather an internal variable of state that evolves as a function of the loading history. After the general presentation of the model, where its theoretical and practical features are discussed, its descriptive and predictive capabilities are showcased via comparisons with experimental data for silicone syntactic foams filled with glass microballoons.

A three dimension lattice-spring model with rotational degree of freedom and its application in fracture simulation of elastic brittle materials

Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model with rotational degree of freedom (3D-LSMR). This 3D-LSMR contains nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of system. Compared with the general lattice spring model, 3D-LSMR adds the rotational and tangential degrees of freedom, whose spring parameters are obtained by finite element parameter mapping. The traditional lattice spring model only considers the normal interaction between two points (the Poisson’s ratio is fixed), which limits the application of the lattice spring model in a wider range of materials. Some scholars proposed to add shear spring into the traditional model to solve the problem of fixed Poisson’s ratio, but the rotation invariance could not be maintained. The main reason is that the shear spring can’t distinguish the difference of tangential velocity (or displacement) between the two particles due to the common rotation or shear. Therefore, the rotation of the whole rigid body may cause incorrect generation of additional strain energy inside the shear spring. 3D-LSMR model is able to maintain rotation invariance and reproduces different Poisson’s ratios because of the spring parameters with rotational degrees of freedom and the rotation effect of lattice points. In this paper, the finite element stiffness matrix with rotational freedom is derived by using 3D-LSMR as the basic mechanical model, and the parameter mapping theory of spring stiffness coefficient is introduced. By means of numerical simulation, the large deflection of slender cantilever beam, elastic symmetrical collision and the simulation of dynamic fracture for concrete L-specimen are checked and calculated respectively, and then the correctness of the model is verified.