Evolution of fatigue damage using the fatigue damage map method

Using only readily available material properties and the concept of the fatigue damage map the work attends to develop a methodology able to predict crack tip propagation rates characterising each of the fatigue stages, namely crack arrest, microstructurally and physically short crack (Stage I), long crack growth (Stage II) and Stage III growth. Crack tip propagation rates are predicted through the combination of an Elasto-Plastic crack tip opening displacement model and the crack blunting theory. The methodology provides a theoretical framework for a 3D version of the fatigue damage map for damage tolerance design. In addition, the work attends to explain a number of fundamental problems starting from crack tip propagation discrepancies, found in the short cracking stage, as well as to provide an insight towards the effect of grain size on the fatigue limit/threshold stress intensity factor.     

Principles of the micromechanics of material damage. 1. Long-term damage

The principles of the theory of long-term damage based on the mechanics of stochastically inhomogeneous media are set out. The process of damage is modeled as randomly dispersed micropores resulting from the destruction of microvolumes. A failure criterion for a single microvolume is associated with its long-term strength dependent on the relationship of the time to brittle failure and the difference between the equivalent stress and the Huber-von Mises failure stress, which is assumed to be a random function of coordinates. The stochastic elasticity equations for porous media are used to determine the effective moduli and the stress-strain state of microdamaged materials. The porosity balance equation is derived in finite-time and differential-time forms for given macrostresses or macrostrains and arbitrary time using the properties of the distribution function and the ergodicity of the random field of short-term strength as well as the dependence of the time to brittle failure on the stress state and the short-term strength. The macrostress-macrostrain relationship and the porosity balance equation describe the coupled processes of deformation and long-term damage __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 108–121, February 2007. For the centenary of the birth of G. N. Savin.     

Investigation of the damage variable basic issues in continuum damage and healing mechanics

Consistent mathematics and mechanics are used here to properly interpret the damage variable within the confines of the concept of reduced area due to damage. In this work basic issues are investigated for the damage variable in conjunction with continuum damage and healing mechanics. First, the issue of the additive decomposition of the damage variable into damage due to voids and damage due to cracks in continuum damage mechanics is discussed. The accurate decomposition is shown to be non-additive and involves a term due to the interaction of cracks and voids. It is shown also that the additive decomposition can only be used for the special case of small damage. Furthermore, a new decomposition is derived for the evolution of the damage variable. The second issue to be discussed is the new concept of independent and dependent damage processes. For this purpose, exact expressions for the two types of damage processes are presented. The third issue addressed is the concept of healing processes occurring in series and in parallel. In this regard, systematically and consistently, the equations of healing processes occurring either consecutively or simultaneously are discussed. This is followed by introducing the new concept of small healing in damaged materials. Simplified equations that apply when healing effects are small are shown. Finally, some interesting and special damage processes using a systematic and original formulation are presented.     

Initiation of damage with implicit gradient-enhanced damage models

Through a mixed local and nonlocal formulation, a continuous transition from a local to a nonlocal behavior is obtained. Initially driven by the local action, the damage driving quantity at a material point in non-damaged materials is later influenced by the nonlocal effect only if the occurrence of damage is effectively found in the surroundings. Qualitative resemblance is hence expected in the prediction of damage initiation by the non-regularized and regularized damage models. Numerical simulations with simple tests and comparisons with existing nonlocal damage models show the qualitative improvement of the mixed formulation in the prediction of damage initiation for 1D and 2D damaged structures.     

From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models

We propose a construction method of non-homogeneous solutions for the traction problem of an elastic damaging bar. This bar has a softening behavior that obeys a gradient damaged model. The method is applicable for a wide range of brittle materials. For sufficiently long bars, we show that localization arises on sets whose length is proportional to the material internal length and with a profile that is also a material characteristic. From its onset until the rupture, the damage profile is obtained either in a closed form or after a simple numerical integration depending on the model. Thus, the proposed method provides definitions for the critical stress and fracture energy that can be compared with experimental results. We finally discuss some features of the global behavior of the bar such as the possibility of a snapback at the onset of damage. We point out the sensitivity of the responses to the parameters of the damage law. All these theoretical considerations are illustrated by numerical examples.     

ABAQUS混凝土损伤塑性模型中损伤因子的率相关性及实现方法

ABAQUS程序中最常用的混凝土损伤塑性（concrete damage plasticity, CDP）模型无法实现损伤因子与应变率相关。为了准确描述混凝土材料在高应变率下的损伤特性,基于CDP模型定义了新的应变率场变量,编制了VUSDFLD用户子程序,开发了能够考虑损伤因子率相关性的改进的CDP（modified CDP,MCDP）模型。MCDP模型采用能量法求解混凝土拉压损伤因子,主求解程序能够随着应变率场变量的变化而自动更新不同应变率对应的损伤参数,计算得到的混凝土单轴静态加载结果与CDP模型吻合较好。MCDP模型对高应变率下动态压缩性能的模拟结果表明:混凝土材料在不同应变率下的拉压损伤对其动态力学性能有显著影响,编制的VUSDFLD子程序和MCDP模型能够有效地解决损伤应变率相关的模拟难题,可以准确地模拟爆炸荷载作用下钢筋混凝土梁的动态响应,为预测爆炸冲击等强动载作用下混凝土结构的响应和破坏提供了更可靠的技术途径。     

Modelling unilateral damage effect in strongly anisotropic materials by the introduction of the loading mode in damage mechanics

Damage weakens the mechanical characteristics of materials. But this weakening can disappear if the cracks close again: this is called the unilateral effect of damage. We propose a model of this phenomenon using damage mechanics in the case of a diffuse network of identical microcracks. The microcracks state is defined with two internal state variables. These two variables are control parameters of the geometry of the microcracks. They also define the loading mode in damage mechanics as in fracture mechanics. In order to limit the anisotropy induced by the microcracks, hence by the loading, we suppose that the geometry of damage spreads into preferential directions. Therefore, this model is essentially applied to materials with a strong anisotropy where the defects follow the constituents of the material. An application is given for composite laminates.     

Characterization of story-level seismic damage using an energy-based damage model

An energy-based damage model currently used for seismic analysis of structures is modified to ensure a positive value in the damage index at all levels of inelastic response. At the ultimate limit state, the modified model gives the same plastic strain energy capacity as the previous damage model. Testing of small-scale cantilever specimens showed different strength deterioration parameters for coldrolled and hot-rolled steel and composite systems of double cantilevers. The strength deterioration parameter for the composite system is smaller than that of the individual cantilevers. Various weighted-average rules for combining local member damage indices into story-level damage indices are compared with measured story-level damage indices. Based on testing of small-scale steel cantilevers, the current combination rules predict the story-level damage reasonably well near the ultimate limit state but tend to underestimate the story-level damage in the less severely damaged states. A combination rule based on best fitting of the experimental data obtained in this study is presented.     

基于损伤诱导矢量及元素位置矢量诊断结构损伤

通过引入损伤诱导矢量和元素位置矢量,作者构造一种采用模态测试数据诊断结构损伤的新方法。利用损伤诱导矢量能迅速排除无损伤元素,缩小识别范围。这里认为结构损伤是由一个或多个元素的损伤引起的,采用元素损伤矢量ｈ１＝（ａｉ,ｂｉ,ｃｉ）描述元素ｉ及损伤程度及类型。该方法构造简单,能大幅度降低计算量。文中以浅拱为例,给出各种损伤情况与下的诊断结果,以说明其有效性。     

NDE of metal damage: ultrasonics with a damage mechanics model

This research describes a nondestructive method for the quantitative estimation of property variations due to damage in metal materials. The method employs a damage mechanics model, which accounts for stiffness degradation and damage evolution of a metal medium with a measurement of ultrasonic velocity. In order to describe the progressive deterioration of materials prior to the initiation of macrocracks, we have developed a new damage mechanics model. Thereafter, a finite element model valid for numerically describing such damage process has been developed by ABAQUS/Standard code, and correlations between damage state, elastic stiffness and plastic strain could be found by the results of the finite element simulation. The property variations due to damage evolution are calculated based on the Mori–Tanaka theory, and then the ultrasonic velocity can be predicted by Christoffel’s equation. When the measured velocity is coupled with the theoretically predicted velocity, the unknown damage variable is solved, from which other residual properties are determined by the predictions of damage model. The proposed technique is performed on type 304 stainless steel bars. The numerical results obtained by the simulation were compared with experimental ones in order to verify the validity of the proposed finite element model and good agreement was found. It is shown that the damaged properties of metals can be estimated accurately by the proposed method.     

Stress-based nonlocal damage model

Progressive microcracking in brittle or quasi-brittle materials, as described by damage models, presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results. Such regularization methods, which describe interactions between points, provide some general properties (including objectivity and the non-alteration of a uniform field) as well as drawbacks (damage initiation, free boundary).A modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution. The orientation and intensity of nonlocal interactions are modified in accordance with the stress state. The fundamental framework of the original nonlocal method has been retained, making it possible to maintain the method’s advantages. The modification is introduced through the weight function, which in this modified version depends not only on the distance between two points (as for the original model) but also on the stress state at the remote point.The efficiency of this novel approach is illustrated using several examples. The proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques. Damage initiation and propagation in mode I as well as shear band formation are analyzed herein.     

First-order gradient damage theory

Taking the strain tensor, the scalar damage variable, and the damage gradient as the state variables of the Helmholtz free energy, the general expressions of the firstorder gradient damage constitutive equations are derived directly from the basic law of irreversible thermodynamics with the constitutive functional expansion method at the natural state. When the damage variable is equal to zero, the expressions can be simplified to the linear elastic constitutive equations. When the damage gradient vanishes, the expressions can be simplified to the classical damage constitutive equations based on the strain equivalence hypothesis. A one-dimensional problem is presented to indicate that the damage field changes from the non-periodic solutions to the spatial periodic-like solutions with stress increment. The peak value region develops a localization band. The onset mechanism of strain localization is proposed. Damage localization emerges after damage occurs for a short time. The width of the localization band is proportional to the internal characteristic length.     

A simple model for damage in the elastic range

A brittle material presents significant loss of strength after being subjected to a process of repeated loading and unloading. This effect is usually described in a generic way as the body being damaged. A very simple method for treating this damage is to assume that, aside from the main cracks which characterize the limit strength of the material, other cracks also exist which modify its fundamental stress state and which therefore influence the stress intensity factor of the main cracks.     

Investigation of shear damage considering the evolution of anisotropy

The damage that occurs in shear deformations in view of anisotropy evolution is investigated. It is widely believed in the mechanics research community that damage (or porosity) does not evolve (increase) in shear deformations since the hydrostatic stress in shear is zero. This paper proves that the above statement can be false in large deformations of simple shear. The simulation using the proposed anisotropic ductile fracture model (macro-scale) in this study indicates that hydrostatic stress becomes nonzero and (thus) porosity evolves (increases or decreases) in the simple shear deformation of anisotropic (orthotropic) materials. The simple shear simulation using a crystal plasticity based damage model (meso-scale) shows the same physics as manifested in the above macro-scale model that porosity evolves due to the grain-to-grain interaction, i.e., due to the evolution of anisotropy. Through a series of simple shear simulations, this study investigates the effect of the evolution of anisotropy, i.e., the rotation of the orthotropic axes onto the damage (porosity) evolution. The effect of the evolutions of void orientation and void shape onto the damage (porosity) evolution is investigated as well. It is found out that the interaction among porosity, the matrix anisotropy and void orientation/shape plays a crucial role in the ductile damage of porous materials.     

On the description of anisotropic damage in composite laminates

A general anisotropic damage theory of cracked laminates is formulated here. The deformation of composite laminates is composed of matrix elastic strains, pseudo-elastic damage strains due to cracking and permanent damage strains due to interlaminar slip. The surface of damage initiation is constructed according to the concept of linear elastic fracture mechanics for the virgin material. After the initial damage, a pesudo-elastic damage can be used to describe the damage behaviour if interlaminar slip is negligible. Damage evolution, load induced anisotropy and interlaminar intralaminar interaction for composite laminates are examined; the latter can perturb the normality structure of damage strain rate. Explicit expressions are given for pseudo-elastic (or secant) moduli of the damaging composite laminates, under a non-interacting assumption imposed on the cracks between different families.     

Modelling of the elasto-viscoplastic damage behaviour of glassy polymers

Constitutive equations are proposed in order to describe the elasto-viscoplastic damage behaviour of polymers. The behaviour is well accounted for by a modified Bodner–Partom model comprising hydrostatic and void evolution terms. The true stress–strain and volumetric strain behaviour of typical rubber-toughened glassy polymers (RTPMMA and HIPS) were experimentally determined at constant local true strain rate by using a video-controlled technique. Successful agreement is obtained between experimental results and the proposed model.