An isotropic damage model for concrete

The studies on modeling of concrete fracture by employing finite element methods have been carried out for the last thirty years. Among these constitutive models, isotropic damage concepts have been found more advantageous for its numerical practicability on modeling of concrete fracture compared to smeared crack applications. In this paper, anew damage function is proposed by simplifying the calculated curves for the damage function in a previous study. A new relation for the fracture energy of concrete is obtained by using this new damage function and the suitability of the recommended equation with the previously proposed relations is verified. Besides that, a realistic damage bounding surface for concrete elements subjected to biaxial stress states is proposed. Finally, the successful results of the applications of the damage model to a plain concrete specimen and a shear panel are illustrated.     

Mesh objective continuum damage models for ductile fracture

During machining processes, the work piece material is subjected to high deformation rates, increased temperature, large plastic deformations, damage evolution and fracture. In this context the Johnson‐Cook failure model is often used even though it exhibits pathological mesh size dependence. To remove the mesh size sensitivity, a set of mesh objective damage models is proposed based on a local continuum damage formulation combined with the concept of a scalar damage phase field. The first model represents a mesh objective augmentation of the well‐established element removal model, whereas the second one degrades the continuum stress in a smooth fashion. Plane strain plate and hat specimens are used in the finite element simulations, with the restriction to the temperature and rate independent cases. To investigate the influence of mesh distortion, a structured and an unstructured meshes were used for the respective specimen. For structured meshes, the results clearly show that the pathological mesh size sensitivity is removed for both models. When considering unstructured meshes, the mesh size sensitivity is more complex as revealed by the considered hat‐specimen shear test. Nevertheless, the present work indicates that the proposed models can predict realistic ductile failure behaviors in a mesh objective fashion. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonlinear three–dimensional finite–element modelling of reinforced–concrete beams: Computational challenges and experimental validation

Three–dimensional nonlinear finite–element models have been developed to investigate the loading–unloading–reloading behaviour of two reinforced–concrete beams under four–point bending using explicit dynamics in ABAQUS. The damaged–plasticity model proposed by Lubliner and collaborators was employed for the plain concrete, and elastic–perfectly plastic models were employed for the steel reinforcement. A perfect bond was assumed between the steel rebars and concrete, whereby the bond–slip behaviour, as well as damage along crack patterns, were modelled through concrete damage. The influence of the shape of the tension–softening law on the numerical load–deflection response was studied by considering bi–linear, exponential and linear post–failure stress–displacement and stress–strain relationships. The effect of modelling steel rebars with truss or beam elements was also investigated. Structured meshes of linear hexahedral elements either with incompatible modes or with reduced integration, and unstructured meshes of either linear or ‘modified’ quadratic tetrahedral elements were considered. In terms of load–deflection curves, both the structured and the unstructured meshes gave results in very good agreement with test results. In terms of crack patterns, results predicted by the structured meshes exhibited some mesh bias, which was less pronounced with the unstructured meshes. In the post–yield phase, if a geometrically nonlinear model is used, discrepancies were found when truss elements were used for steel rebars, whereas good agreement was found if the bending stiffness of the rebars is included using beam elements. This is a non–obvious result that may be important to consider when studying the progressive collapse of RC structures.     

混凝土试样在静态载荷作用下断裂过程的数值模拟研究

提出了一个模拟混凝土断裂过程的细观力学模型,并应用该模型从混凝土的细观非均匀性结构出发,对混凝土试样在单轴和双轴静态载荷作用下的断裂过程进行了数值模拟,给出双轴载荷作用下混凝土的强度包络面。数值模型结果较好地模拟了混凝土试样从裂纹萌生、扩展到宏观裂纹形成的整个断裂过程,与实验结果表现出较好的一致性。     

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

This paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.     

Biaxial tension fatigue response of concrete

Concrete structures such as rigid airport pavements are subjected to repeated high-amplitude loads resulting from passing aircraft. The resulting stress-state in concrete is a biaxial combination of compression and tension. It is of interest to understand the response of plain concrete to such loading conditions, which will enable development of realistic material models for implementation in mechanistic pavement design procedures.The objective of this work is to characterize the quasi-static and low-cycle fatigue response of concrete subjected to biaxial stresses in the biaxial tension region, where the principal tensile stress is larger than or equal in magnitude when compared with the principal compressive stress. An experimental investigation of material behavior in the biaxial tension region is conducted. The experimental setup consists of the following test configurations: (a) notched concrete beams tested in three-point bend configuration, and (b) hollow concrete cylinders subjected to torsion.Failure of concrete in the biaxial tension region is shown to be a local phenomenon under quasi-static and fatigue loading, wherein the specimen fails owing to a single crack. The crack propagation is studied using the equivalent elastic crack concept. It is observed that the crack growth rate in constant amplitude fatigue loading exhibits a two-phase process: a deceleration phase followed by an acceleration stage. The crack growth in the acceleration stage is shown to follow Paris law. The model parameters obtained from uniaxial fatigue tests are shown to be sufficient for predicting the considered biaxial fatigue response.     

ECC双轴压力学性能及破坏准则试验研究

利用真三轴液压伺服试验机,对3种强度的高延性纤维增强水泥基复合材料(Engineered CementitiousComposites,ECC)进行了双轴压试验研究,得到了ECC在双轴压应力状态下的破坏形态、极限强度和应力-应变曲线,并与普通混凝土的试验结果进行对比,分析了不同应力比下ECC的破坏机理和主压强度的变化规律。试验结果表明,双轴压应力状态下,ECC强度包络线与普通混凝土有一定差别,强度较低时两者结果相近,但强度较高时相同应力比下ECC主压强度增幅明显小于混凝土。最后,基于Kupfer等建立的普通混凝土的强度准则和试验结果,提出了ECC的双轴压强度破坏准则。     

Coarsening unstructured meshes by edge contraction

A new unstructured mesh coarsening algorithm has been developed for use in conjunction with multilevel methods. The algorithm preserves geometrical and topological features of the domain, and retains a maximal independent set of interior vertices to produce good coarse mesh quality. In anisotropic meshes, vertex selection is designed to retain the structure of the anisotropic mesh while reducing cell aspect ratio. Vertices are removed incrementally by contracting edges to zero length. Each vertex is removed by contracting the edge that maximizes the minimum sine of the dihedral angles of cells affected by the edge contraction. Rarely, a vertex slated for removal from the mesh cannot be removed; the success rate for vertex removal is typically 99.9% or more. For two‐dimensional meshes, both isotropic and anisotropic, the new approach is an unqualified success, removing all rejected vertices and producing output meshes of high quality; mesh quality degrades only when most vertices lie on the boundary. Three‐dimensional isotropic meshes are also coarsened successfully, provided that there is no difficulty distinguishing corners in the geometry from coarsely‐resolved curved surfaces; sophisticated discrete computational geometry techniques appear necessary to make that distinction. Three‐dimensional anisotropic cases are still problematic because of tight constraints on legal mesh connectivity. More work is required to either improve edge contraction choices or to develop an alternative strategy for mesh coarsening for three‐dimensional anisotropic meshes. Copyright © 2003 John Wiley & Sons, Ltd.     

3D meso-mechanical analysis of concrete specimens under biaxial loading

In recent years, the mechanics of materials group at ETSECCPB‐UPC has developed an approach for meso‐mechanical analysis of concrete using zero‐thickness interface elements in 2D and more recently in 3D. In this methodology, the meso‐structure is generated with in‐house developed computer programs based on Voronoï/Delaunay theory. In the analysis, continuum elements are assumed linear elastic. Non‐linearity and fracture phenomena are made possible by the systematic use of zero‐thickness interface elements inserted on a priori determined potential fracture planes. In this paper, the results obtained for a 3D specimen under biaxial loading are presented. The results turn out to be very satisfactory and, in particular, it is observed that even the specimens which contain a reduced number of aggregates (14 in the present calculations) lead to a realistic failure envelope under biaxial loading, and they also capture the tendencies of cracking and fracture orientations observed in experiments for different rates of biaxial loading. The special limit case of biaxial loading under restrained out‐of‐plane deformations is also analysed, leading to practically elastic behaviour as shown by available experimental evidence.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

含孔CFRP正交层合板的双轴拉伸力学行为

采用加载臂开槽的中心开孔等厚度十字形试样,实验研究了正交对称铺层碳纤维增强聚合物基复合材料（CFRP）层合板在双轴拉伸载荷作用下的力学行为,分析了3种双轴加载比对其拉伸强度和破坏行为的影响。研究表明:纤维被切断的铺层部分在拉伸作用下容易与其相邻铺层脱粘,导致层合板承载力下降;等双轴加载时,在孔边的被切断纤维与连续纤维间基体在横向拉伸和纵向剪切组合作用下首先开裂;非等双轴加载时,在垂直于快速拉伸方向的铺层中沿孔边应力集中处先出现基体裂纹;随着加载比的增大,快速拉伸方向的细观结构损伤随载荷的增大发展更快,刚度下降更快,破坏时主裂纹的扩展方向更趋于垂直于快速拉伸方向;强度包络线的分析表明快速拉伸方向的拉伸强度随加载比的增大呈缓慢增大的趋势。      

Automatic generation of anisotropic quadrilateral meshes on three‐dimensional surfaces using metric specifications

A new algorithm for constructing full quadrilateral anisotropic meshes on 3D surfaces is proposed in this paper. The proposed method is based on the advancing front and the systemic merging techniques. Full quadrilateral meshes are constructed by systemically converting triangular elements in the background meshes into quadrilateral elements.By using the metric specifications to describe the element characteristics, the proposed algorithm is applicable to convert both isotropic and anisotropic triangular meshes into full quadrilateral meshes. Special techniques for generating anisotropic quadrilaterals such as new selection criteria of base segment for merging, new approaches for the modifications of the background mesh and construction of quadrilateral elements, are investigated and proposed in this study. Since the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space, the mesh generator is robust and no expensive geometrical computation that is commonly associated with direct quadrilateral mesh generation schemes is needed. Copyright © 2002 John Wiley & Sons, Ltd.     

On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture

In this paper, strong discontinuities embedded in finite elements are used to model discrete cracking in quasi-brittle materials. Special attention is paid to (i) the constitutive models used to describe the localized behaviour of the discontinuities, (ii) the enforcement of the continuity of the crack path and (iii) mixed-mode crack propagation. Different constitutive relations are adopted to describe the localized behaviour of the discontinuities, namely two damage laws and one plasticity law. A numerical algorithm is introduced to enforce the continuity of the crack path. In the examples studied, an objective dissipation of energy with respect to the mesh is found. Examples of mode-I and mixed-mode crack propagation are presented, namely a double notch tensile test and a single-edge notched beam subjected to shear. In the former case different crack patterns are obtained depending on the notch offset; in the latter case special emphasis is given to the effect of shear on the global structural response. In particular, both the peak load and the softening response of the structure are related to the amount of shear tractions allowed to develop between crack faces. The results obtained are compared to experimental results. As a general conclusion, it is found that crack path continuity allows for the development of crack patterns similar to those found in experiments, even when reasonably coarse meshes are used.     

Shear behavior model for steel fiber-reinforced concrete members without transverse reinforcements

Due to the complex shear mechanism of steel fiber-reinforced concrete (SFRC) members, there is lack of comprehensive shear behavior models for SFRC members. The shear behavior model, based on a smeared crack model, requires the tensile stress–strain constitutive equation of SFRC membrane subjected to biaxial stresses. After SFRC panel tests under biaxial stresses were recently conducted, it has been possible to create a more complete smeared crack model for estimating the shear behavior of SFRC members. It is, however, very difficult to conduct such experiments for different types of steel fibers, various amount of steel fibers, different ranges of concrete strengths, etc. Thus, in this study, steel fibers are modeled as average direct tensile contribution elements in a modified smeared crack truss model, considering directionality and distribution of fibers. In this way, only simple bond tests are required to reflect the effects of different characteristics of SFRC. In addition, the shear contribution of steel fibers can be obtained considering the bond failure of steel fibers. The proposed model was compared to the test results of 8 SFRC panels and 80 SFRC beams, and the shear behavior of the SFRC members was well estimated.     

钢筋混凝土双向受弯梁次生扭矩的分析研究

探讨了次生扭矩对钢筋混凝土双向受弯梁抗剪承载力的影响。提出逐级加载、待定系数的方法计算钢筋混凝土双向受弯梁的次生扭矩,编制了钢筋混凝土双向受弯梁次生扭矩的全过程分析计算机程序。计算值与试验结果的对比分析表明,该方法合理可行。通过对次生扭矩影响抗剪承载力的主要因素的分析,获得一些有参考价值的结论,并提出在工程实践中,计算承受集中荷载的有腹筋双向受弯梁的抗剪强度时,次生扭矩影响应予以考虑。     

Probability density distribution of the orientation of strength-controlling flaws from multiaxial loading using the unit-sphere stochastic strength model for anisotropy

Models that predict the failure probability of monolithic glass and ceramic components under multiaxial loading have been developed by authors such as Batdorf, Evans, and Matsuo. These “unit-sphere” failure models assume that the strength-controlling flaws are randomly oriented, noninteracting planar microcracks of specified geometry but of variable size. The purpose of this paper is to describe a formulation of the probability density distribution of the orientation of critical strength-controlling flaws that results from an applied load. This distribution is a function of the multiaxial stress state, the shear sensitivity of the flaws, the Weibull modulus, and the strength anisotropy. Examples are provided showing the predicted response on the unit sphere for various stress states for isotropic and transversely isotropic (anisotropic) materials—including the most probable orientation of critical flaws for offset uniaxial loads with strength anisotropy. The author anticipates that this information could be used to determine anisotropic stiffness degradation or anisotropic damage evolution for individual brittle (or quasi-brittle) composite material constituents within finite element or micromechanics-based software.     

Failure prediction for a glass/epoxy cruciform specimen under static biaxial loading

Material models were developed to predict the mechanical behavior of glass/epoxy multidirectional laminates under complex stress states. An incremental plane stress analysis was performed, taking into account the anisotropic material non-linearity, separate damage onset conditions and distinct post-failure stiffness degradation rules. Theoretical formulations were implemented in a shell element of the 1st order shear deformation theory. Numerical results were validated via comparison with test data from cruciform specimens subjected to static biaxial tensile loading. Local strain gauge and full-field strain measurements, obtained using the Digital Image Correlation (DIC) technique, corroborated numerical predictions. Improved strength and failure mode results were derived when, in addition to stiffness reduction, compressive strength degradation in the fiber direction was also considered.     

The characteristics of an inclined crack in anisotropic paperboard under biaxial loading

In this paper, a fracture mechanics method was applied for the evaluation of crack behaviour in anisotropic paperboard subjected to biaxial uniform loading. The experiment was performed to determine the crack propagation angle and the fracture strength of paperboard under biaxial loading with the cruciform specimen optimized by FEM simulation. The effects of biaxial loads on the critical stress ratio and crack propagation angle for various inclination angles were investigated. The experimental results were compared with theoretical results, which were calculated by using the Normal Stress Ratio Criteria. The experimental results for crack propagation angle and critical stress show good agreement with theoretical results.     

Characterization of mixed mode crack opening in concrete

In real concrete structures cracks often open in mixed mode after their initiation. To capture the direct material behavior of a mixed mode crack opening a stiff biaxial testing machine, capable of imposing both normal and shear loads on a given crack area, has been applied. The opening and sliding components of the mixed mode displacement are measured using a custom made orthogonal gauge, and the measurements are used directly as the closed loop control signals. A double notch, concrete specimen is used for the crack investigation. The tests are divided into two steps, a pure Mode I opening step, where a macro crack is initiated in the specimen followed by the mixed mode opening step. The high stiffness of the set-up together with the closed control loop ensures a stable crack initiation followed by a controllable mixed mode opening. The deep notches result in a plane crack, only influenced by material aspects such as the aggregate size and concrete strength. Despite the occurrence of a few, local, secondary cracks during the mixed mode crack opening, the results can be treated as the mixed mode material point behavior of a crack in concrete. Results are reported for a range of mixed mode angles and for varying initial Mode I openings of the crack.     

Microplane damage model for fatigue of quasibrittle materials: Sub-critical crack growth,lifetime and residual strength

In contrast to metals and fine grained ceramics, fatigue in concrete and other quasibrittle materials occurs in a large fracture process zone that is not negligible compared to the structure size. This causes the fatigue to be combined with triaxial softening damage whose localization is governed by a finite material characteristic length. A realistic model applicable to both has apparently not yet been developed and is the goal of this paper. Microplane model M7, shown previously to capture well the nonlinear triaxial behavior of concrete under a great variety of loadings paths, is extended by incorporating a new law for hysteresis and fatigue degradation, which is formulated as a function of the length of the path of the inelastic volumetric strain in the strain space. The crack band model, whose band width represents a material characteristic length preventing spurious localization, is used to simulate propagation of the fatigue fracture process zone. Thus the fatigue crack with its wide and long process zone is simulated as a damage band of a finite width. For constant amplitude cycles, the model is shown to reproduce well, up to several thousands of cycles, the Paris law behavior with a high exponent previously identified for concrete and ceramics, but with a crack growth rate depending on the structure size. Good agreement with the crack growth histories and lifetimes previously measured on three-point bend beams of normal and high strength concretes is demonstrated. The calculated compliance evolution of the specimens also matches the previous experiments. The model can be applied to load cycles of varying amplitude, to residual strength under sudden overload and damage under nonproportional strain tensor variation. Application to size effect in fatigue is relegated to a follow-up paper, while a cycle-jump algorithm for extrapolation high-cycle fatigue with millions of cycles remains to be researched.