Defects in gradient micropolar elasticity—I: screw dislocation

A gradient micropolar elasticity is proposed based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium. This theory is an extension of the theory of micropolar elasticity with couple stresses together with gradient elasticity in a way that in addition to hyper stresses, hyper couple stresses also appear. In particular, the strain energy, besides its dependence upon the distortion and bend-twist terms of a micropolar medium (Cosserat continuum), depends also on distortion and bend-twist gradients. Using a simplified but rigorous version of this gradient theory, we can connect it to Eringen's nonlocal micropolar elasticity. In addition, it is used to study a screw dislocation in gradient micropolar elasticity. One important result is that we obtained nonsingular expressions for the force and couple stresses. The components of the force stress have maximum values near the dislocation line and those of the couple stress have maximum values at the dislocation line.     

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

Mode I and mode II crack tip asymptotic fields with strain gradient effects

The strain gradient effect becomes significant when the size of fracture process zone around a crack tip is comparable to the intrinsic material lengthl, typically of the order of microns. Using the new strain gradient deformation theory given by Chen and Wang, the asymptotic fields near a crack tip in an elastic-plastic material with strain gradient effects are investigated. It is established that the dominant strain field is irrotational. For mode I plane stress crack tip asymptotic field, the stress asymptotic field and the couple stress asymptotic field can not exist simultaneously. In the stress dominated asymptotic field, the angular distributions of stresses are consistent with the classical plane stress HRR field; In the couple stress dominated asymptotic field, the angular distributions of couple stresses are consistent with that obtained by Huang et al. For mode II plane stress and plane strain crack tip asymptotic fields, only the stress-dominated asymptotic fields exist. The couple stress asymptotic field is less singular than the stress asymptotic fields. The stress asymptotic fields are the same as mode II plane stress and plane strain HRR fields, respectively. The increase in stresses is not observed in strain gradient plasticity for mode I and mode II, because the present theory is based only on the rotational gradient of deformation and the crack tip asymptotic fields are irrotational and dominated by the stretching gradient. The project supported by the National Natural Science Foundation of China (19704100), National Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20), CAS K.C. Wong Post-doctoral Research Award Fund and Post-doctoral Science Fund of China     

Generalization of Saint-Venant's principle to micropolar continua

The mathematical formulation and proof of Saint-Venant's principle as given by Toupin for non-polar solids is generalized to the case of micropolar elasticity. On one end of a micropolar cylinder of arbitrary length and cross-section we apply a system of self-equilibrated stresses and couple stresses. We first prove that the norms of the stress and couple stress tensors are bounded by the energy density. By means of Rayleigh's principle for the lowest natural eigenfrequency for a slice of the cylinder we then prove that the energy, stored in the cylinder beyond a certain distance from the loaded end, has an exponential decrease with this distance, thus establishing Saint-Venant's principle for the system.     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

Mode I crack tip field with strain gradient effects

A plane strain mode I crack tip field with strain gradient effects is investigated. A new strain gradient theory is used. An elastic-power law hardening strain gradient material is considered and two hardening laws, i. e. a separation law and an integration law are used respectively. As for the material with the separation law hardening, the angular distributions of stresses are consistent with the HRR field, which differs from the stress results^[19]; the angular distributions of couple stresses are the same as the couple stress results^[19]. For the material with the integration law hardening, the stress field and the couple stress field can not exist simultaneously, which is the same as the conclusion^[19], but for the stress dominated field, the angular distributions of stresses are consistent with the HRR field; for the couple stress dominated field, the angular distributions of couple stresses are consistent with those in Ref. [19]. However, the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only, while the crack tip field of mode I is dominated by the tension gradient, which will be shown in another paper. Supported by the National Science Foundation of China (No. 19704100), Science Foundation of Chinese Academy of Sciences (Project KJ951-1-20), CAS K. C. Wong Post-doctoral Research Award Fund and the Post Doctoral Science Fund of China.     

Finite element solutions for plane strain mode I crack with strain gradient effects

In this paper, a new phenomenological theory with strain gradient effects is proposed to account for the size dependence of plastic deformation at micro- and submicro-length scales. The theory fits within the framework of general couple stress theory and three rotational degrees of freedom ω_i are introduced in addition to the conventional three translational degrees of freedom u_i. ω_i is called micro-rotation and is the sum of material rotation plus the particles' relative rotation. While the new theory is used to analyze the crack tip field or the indentation problems, the stretch gradient is considered through a new hardening law. The key features of the theory are that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the term of stretch gradient is represented as an internal variable to increase the tangent modulus. In fact the present new strain gradient theory is the combination of the strain gradient theory proposed by Chen and Wang (Int. J. Plast., in press) and the hardening law given by Chen and Wang (Acta Mater. 48 (2000a) 3997). In this paper we focus on the finite element method to investigate material fracture for an elastic-power law hardening solid. With remotely imposed classical K fields, the full field solutions are obtained numerically. It is found that the size of the strain gradient dominance zone is characterized by the intrinsic material length l₁. Outside the strain gradient dominance zone, the computed stress field tends to be a classical plasticity field and then K field. The singularity of stresses ahead of the crack tip is higher than that of the classical field and tends to the square root singularity, which has important consequences for crack growth in materials by decohesion at the atomic scale.     

Photoelastic analysis of the stress field surrounding a fatigue crack

A photelastic analysis was carried out on plane polyester specimens containing a fatigue crack, in order to study the effect of plastic yielding around the crack tip on the elastic stress distribution in the vicinity of the crack. In general, results were in good agreement with values calculated for the case of a sharp-tipped crack. However, very near the crack tip, principal stresses obtained experimentally were slightly lower than calculated stresses, probably due to the bluntness of the fatigue crack. Also lines of constant stress tended to move behind the crack tip, in contrast with the calculated stresses, which occurred further forward over the field of investigation.     

分子动力学模拟含不同初始缺陷单晶镍的断裂行为与应力演化

断裂是一个跨尺度复杂的物理过程,对宏观尺度的断裂行为已有深入的研究和发展,然而对微观尺度的断裂行为及断裂过程中应力场的变化缺乏深入的理解。本文通过分子动力学模拟,研究了具有不同初始缺陷(尖锐裂纹、钝裂纹和孔洞)的单晶镍的断裂行为和应力分布特征。结果表明,不同的初始缺陷导致了不同的断裂机制、断裂强度和抗断裂性能。含初始孔洞的单晶镍样品有最高的断裂强度和最强的抗断裂性能,这与孔洞扩展过程中堆积层错的形成密切相关。其次是含初始钝裂纹的样品,在裂纹扩展过程中出现由[100]超位错发射引起的裂尖钝化;含尖锐裂纹的样品表现为脆性断裂,裂尖原子没有出现微结构的变化,其强度和抗断裂性能最低。此外,不同的初始缺陷也会导致断裂过程中应力分布的变化,对含有尖锐裂纹的脆性断裂试样,高应力(拉伸应力、平均应力和米塞斯应力)总是出现在扩展裂纹的裂尖。而对于含有钝裂纹或孔洞的韧性断裂试样,高应力不仅分布在裂尖,也分布在位错发射和堆积层错形成的区域,在裂纹/孔洞扩展之前,应力随着加载时间的增加而迅速增加,而一旦裂纹或孔洞开始扩展,应力增加非常缓慢或几乎不增加,但拉伸应力值始终大于平均应力和米塞斯应力值。这表明,在I型...     

Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity

This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behavior of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localized surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results.     

Numerical study of contact forces for crack closure analysis

Plasticity induced crack closure (PICC) has been widely studied using numerical models. Different numerical parameters can be considered to quantify the opening level, namely one based on the analysis of contact stresses at minimum load. A modified version of this parameter is proposed here, based on nodal contact forces instead of contact stresses. The predictions were found to be similar to those obtained from the contact status of 2nd node behind crack tip. The PICC_contact parameter was also found to be very consistent and adequate for parametric studies of the influence of different physical parameters. The contributions to the opening stress intensity factor of different points along crack flank were found to strongly decrease with distance to crack tip. The cumulative K_open between the crack tip and a distance of 0.1 mm was found to vary from 30% to 100%, increasing with stress ratio, R. Finally, a K solution was developed for punctual forces applied on crack flank and compared with a literature solution for infinite plates. A good agreement was found for plane strain state but significant differences of about 10% were found for plane stress state.     

Quasicontinuum simulation of single crystal nano-plate with a mixed-mode crack

The quasicontinuum (QC) method is employed to simulate a nickel single crystal nano-plate with a mixed-mode crack. Atomic stresses near the crack tip are fitted according to the elastoplastic fracture mechanics equations. It is found that the atomic stress fields neighboring the crack tip are also singular and controlled by the atomic stress intensity factors. And then the critical energy release rates for brittle and ductile fracture are computed and compared in order to predict crack propagation or dislocation emission. Four possible slip directions at the crack tip are pointed out. Finally, the slip direction around the crack tip is determined by the shear stress and it is well consistent with the atomic pictures from the QC simulation.     

Reverse shear on inclined planes at the tip of a sharp crack

Plane strain plastic yielding at a crack tip has been represented by edge dislocations with Burgers vectors parallel to symmetrical planes inclined at 70° and 45° to the plane of the crack. The plastic displacement and the stresses near the crack tip were calculated by a numerical method and the effect of a reduction in applied stress was determined. Removal of the whole or a part of the initial load produces reverse shear in regions of the slip band nearest the crack tip. The amount of reverse shear depends only on the reduction in the load and not on its initial value. The reverse shear is associated with the presence of negative dislocations and the stresses near the crack tip may become compressive even though the applied (remote) stress is still tensile. The degree and extent of compression depends on the reduction in applied stress and on its original value. It is argued that the residual compressive stresses produced under fluctuating loads may produce crack closure and crack arrest. The effect of residual plasticity in a slip band left behind a growing crack has been estimated. It is shown that after an overload the excess residual plasticity opposing crack opening rises to a maximum value when the crack tip has advanced some distance from the point where the overload was applied.     

Singular behavior at the tip of a crack in an elastic-plastic material with plastic orthotropy

The two-dimensional stress field at the tip of a crack in a plastically orthotropic material is analyzed by the total deformation theory of plasticity in conjunction with the J-integral. A model of a plastically orthotropic material is constructed by the use of the theory proposed by R. Hill (1950) and the uniaxial stress-strain relation suggested by W. Ramberg and W.R. Osgood (1943). It is found that the stresses in the vicinity of the crack tip have a singularity of the same order as that in the case of isotropic materials, but their amplitudes are greatly influenced by the plastic orthotropy. Numerical work is carried out for two typical metals, and the effect of the plastic orthotropy is examined for the stress field, the crack opening displacement, the strain energy density, and the shape of the elastic-plastic boundary.     

Characterization of crack tip stresses in plane-strain fracture specimens having weld center crack

Fracture toughness of metals depends strongly on the state of stress near the crack tip. The existing standards (like R-6, SINTAP) are being modified to account for the influence of stress triaxiality in the flaw assessment procedures. These modifications are based on the ability of so-called ‘constraint parameters’ to describe near tip stresses. Crack tip stresses in homogeneous fracture specimens are successfully described in terms of two parameters like J–Q or J–T. For fracture specimens having a weld center crack, strength mismatch ratio between base and weld material and weld width are the additional variables, along with the magnitude of applied loading, type of loading, and geometry of specimen that affect the crack tip stresses. In this work, a novel three-parameter scheme was proposed to estimate the crack tip opening stress accounting for the above-mentioned variables. The first and second parameters represent the crack tip opening stress in a homogeneous fracture specimen under small-scale yielding and are well known. The third parameter accounts for the effect of constraint developed due to weld strength mismatch. It comprises of weld strength mismatch ratio (M, i.e. ratio of yield strength of weld material to that of base material), and a plastic interaction factor (I_p) that scales the size of the plastic zone with the width of the weld material. The plastic interaction factor represents the degree of influence of weld strength mismatch on crack tip constraint for a given mismatch ratio. The proposed scheme was validated with detailed FE analysis using the Modified Boundary Layer formulation.     

弹塑性疲劳微弯裂纹尖端塑性区问题

主要研究疲劳载荷作用下弹塑性裂纹弯曲延伸扩展问题.通过分析论证,比较精确地研究了疲劳载荷作用下弯曲延伸裂纹尖端塑性区域边界上交变应力的分布状况.综合考虑了疲劳作用应力,塑性区域交变应力,利用二阶摄动方法,研究计算了弯曲延伸裂纹尖端塑性区域的范围,并预测了疲劳载荷作用下弹塑性裂纹扩展路径.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

On the effects of characteristic lengths in bending and torsion on Mode III crack in couple stress elasticity

The problem of a stationary semi-infinite crack in an elastic solid with microstructures subject to remote classical K_III field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales. The stress and displacement fields turn out to be strongly influenced by the ratio between the characteristic lengths. Moreover, the symmetric stress field turns out to be finite at the crack tip, whereas the skew-symmetric stress field displays a strong singularity. Ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. The asymptotic fields dominate within this zone, which however has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. In this limiting case the full-field solution recovers the classical K_III field with square-root stress singularity. Outside the zone where the total shear stress is negative, the full-field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude can be adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion. Moreover, the occurrence of a sharp crack profile denotes that the crack becomes stiffer with respect to the classical elastic response, thus revealing that the presence of microstructures may shield the crack tip from fracture.     

Photoelastic study of the stresses near a cleavage crack

Photoelastic analysis has been used to determine the stress distribution near the tip of a cleavage crack. The crack bisects a slender rectangular bar along its length, and forces are applied at one end of it tending to open it out. The region of high stresses is localized in a region of the order of the half-width of the bar in size. Very near the crack tip, the stresses decrease in proportion to the inverse square root of the distance from the crack tip. The maximum gradient of the principal tensile stress lies at an angle of 70 to 80 deg from the plane of the crack. Contour maps of the principal stresses, the 45-deg shear stress and the maximum tensile stress are presented.