Internal fe approximation of spaces of divergence-free functions in three-dimensional domains

The space of divergence-free vector functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements having the same property. An easy way of generating basis functions in these subspaces is shown.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

On a duality of stress and deformation fields in finite elasticity

For plane deformations of an isotropic compressible hyperelastic material, the first Piola-Kirchoff stress potentials are shown to be a possible plane deformation for another material of the same type. An explicit formula connecting the strain-energy functions of the two materials is derived.

---

Zusammenfassung Für Ebenedeformationen eines isotropisch zusammendrückbaren hyperelastischen Materials erweisen sich die ersten Piola-Kirchhoff-Stress-Potientale als eine mögliche Ebenedeformation eines anderen Materials desselben Typs. Eine explizite Formel, die die Deformationsenergie-Functionen der beiden Materialien verbindet, wird abgeleitet.

     

The near tip stress and strain fields in cracked single crystals

The numerical analyses of stationary mathematically sharp Mode I crack in FCC and BCC crystals with elastic-ideally plastic (EIP) and fast hardening saturation (FHS) law are carried out in the present paper. From the calculated results, it is shown that: for the cases of small strain, EIP crystal cracks, the features of concentrated deformation patterns and the stress state in near-crack tip deformation fields are identical to the earlier analytical solutions, but along the angular sector boundaries, there exist narrow complex stress zones. The overall characteristics of deformation patterns for the cases of EIP and FHS are similar. The behaviours of crack tip opening can be characterized by crack-tip-opening-displacement (CTOD). For the case of FHS, finite deformation BCC crystal crack, our calculations are qualitatively in agreement with recent experimental observations. The project supported by National Natural Science Foundation of China     

A study on stress intensity factors and singular stress fields of three-dimensional interface crack

By using the finite-part integral concepts and limit technique, the hypersingular integrodifferential equations of three-dimensional (3D) planar interface crack were obtained; then the dominant-part analysis of 2D hypersingular integral was further used to investigate the stress fields near the crack front theoretically, and the accurate formulae were obtained for the singular stress fields and the complex stress intensity factors. After that, a numerical method is proposed to solve the hypersingular integrodifferential equations of 3D planar interface crack, and the problem of elliptical planar crack is then considered to show the application of the method. The numerical results obtained are satisfactory. Project supported by the Foundation of Solid Mechanics Open Research Laboratory of State Education Commission at Tongji University and the National Natural Science Foundation.     

Boundary contour analysis for surface stress recovery in 2-D elasticity and Stokes flow

Summary A variant of the boundary element method, called the boundary contour method, offers a further reduction in dimensionality. Consequently, boundary contour analysis of 2-D problems does not require any numerical integration at all. In a boundary contour analysis, boundary stresses can be accurately computed using the approach proposed in Ref. [1]. However, due to singularity, this approach can be used only to calculate boundary stresses at points that do not lie at an end of a boundary element. Herein, it is shown that a technique based on the displacement/velocity shape functions can overcome this drawback. Further, the approach is much simpler to apply, requires less computational effort, and provides competitive accuracy. Numerical solutions and convergence study for some well-known problems in linear elasticity and Stokes flow are presented to show the effectiveness of the proposed approach. This research was supported in part by the 2004 Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities and by the University of South Alabama Research Council.     

Atomistic and continuum studies of stress and strain fields near a rapidly propagating crack in a harmonic lattice

Large-scale atomistic simulations of a mode I crack propagating in a harmonic lattice are presented. The objective of this work is to study the stress and strain fields near a rapidly propagating mode I crack. The asymptotic continuum mechanics solutions of the elastic fields are compared quantitatively with molecular-dynamics simulation results for different crack velocities. It is observed that both atomistic stress and atomistic strain can be successfully related to the corresponding continuum quantities. The study reveals that the atomistic simulation results agree well with the continuum theory predictions, which suggests that the continuum theory can be applied for nano-scale dynamic problems.     

The stress-strain fields at crack tip in axially cracked cylindrical shells and the calculation of stress intensity factors

A perturbation solution for stress-strain fields (including modes I, II, III) at crack tip in axially cracked cylindrical shells is given. The analysis, using 10th-order differential equations which take the transverse shear deformations into account, involves perturbation in a curvature parameter λ², (λ²=[12(1-v ²)]^1/2 a ²/Rh). Stress intensity factors for finite size cylindrical shells under bending and internal pressure loading are evaluated. A good accuracy can be obtained without using fine meshes in a region near the crack tip. Besides, the influence of the finite size and the shearing stiffness on bulging factors, which are commonly used in engineering, are analyzed.     

Three dimensional K-T z stress fields around the embedded center elliptical crack front in elastic plates

Through detailed three-dimensional (3D) finite element (FE) calculations, the out-of-plane constraints T_z along embedded center-elliptical cracks in mode I elastic plates are studied. The distributions of T_z are obtained near the crack front with aspect ratios (a/c) of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z decreases from an approximate value of Poisson ratio ν at the crack tip to zero with increasing normalized radial distances (r/a) in the normal plane of the crack front line, and increases gradually when the elliptical parameter angle ϕ changes from 0° to 90°at the same r/a. With a/c rising to 1.0, T_z is getting nearly independent of ϕ and is only related to r/a. Based on the present FE calculations for T_z, empirical formulas for T_z are obtained to describe the 3D distribution of T_z for embedded center-elliptical cracks using the least squares method in the range of 0.2≤a/c≤1.0. These T_z results together with the corresponding stress intensity factor K are well suitable for the analysis of the 3D embedded center-elliptical crack front field, and a two-parameter K-T_z principle is proposed. The project supported by the National Natural Science Foundation of China (50275073) The English text was polished by Keren Wang.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

EXACT SOLUTIONS OF NEAR CRACK LINE FIELDS FOR MODE I CRACK UNDER PLANE STRESS CONDITION IN AN ELASTIC－PERFECTLY PLASTIC SOLID

ＥＸＡＣＴＳＯＬＵＴＩＯＮＳＯＦＮＥＡＲＣＲＡＣＫＬＩＮＥＦＩＥＬＤＳＦＯＲＭＯＤＥＩＣＲＡＣＫＵＮＤＥＲＰＬＡＮＥＳＴＲＥＳＳＣＯＮＤＩＴＩＯＮＩＮＡＮＥＬＡＳＴＩＣ－ＰＥＲＦＥＣＴＬＹＰＬＡＳＴＩＣＳＯＬＩＤＥＸＡＣＴＳＯＬＵＴＩＯＮＳＯＦＮＥＡ...     

Shear and normal stress relaxation in polymer blends: stress predictions from interface velocity and Laplace pressure terms

We derived for the first time the relationships among shear stress and normal stress differences for ellipsoidal interfaces under large step shear strains considering interface velocity term and Laplace pressure term in the expression of the stress tensor for mixtures of two Newtonian fluids. In the derivation, orientation angle of the interface is assumed to be given by the affine deformation assumption and is independent of time based on experimental results for blends with 0.048 ≤ K ≤ 0.54 where K is the ratio of droplet viscosity to matrix viscosity. For ellipsoidal droplets, the shear stress is only proportional to the first normal stress difference. On the other hand, for spheroidal droplets, proportionality among the shear stress, the first and the second normal stress differences was derived, and the ratio of the second normal stress difference to the first normal stress difference was given as a function of step strain. The shear stress and the first normal stress difference obtained experimentally satisfy the derived relationship, indicating applicability of the stress expression for polymer blends.     

???????????????????????????????????

对于连续截面直梁平面横力弯曲情形, 分析了剪切变形引起横截面翘曲的影响. 基于位移模式假定, 用材料力学方法得到相应的正应力与切应力表达式, 并以均布力作用的矩形截面简支梁为例说明截面翘曲对于应力的影响. 该分析方法及其结果适用于材料力学教学.     

三轴应力场中不同形状孔洞的长大及其新模型

对不同形状孔洞在从光滑试样到裂纹试样这样广泛三轴应力场中的长大规律,本文通过控制体胞宏观应力三维度的方法进行了精确的有限元分析,计算结果表明：（1）孔洞的体积改变和形状变化是孔洞演化的两种基本机制,在不同的三轴应力场中,这两种机制的作用不同;（2）现有模型对孔洞长大规律的描述是不准确的,由它们得到的临界孔洞扩张比参数HGC与临界孔洞体积分数fc不具备一一对应关系,因此不以很好地反映也洞的实际扩张。在此基础上,提出了一个描述孔洞长大的新模型,与四种常用的现有模型相比,该模型不仅能更好地描述不同三轴应力场中孔洞的长大,而且能反映不同应力三维度水平下材料破坏模式的变化。     

压应力场中爆生裂纹分布与扩展特征实验分析

采用动静组合加载实验装置和数字激光焦散线实验系统,进行了0、3、6、9 MPa等4种压应力场中PMMA试件的爆破致裂实验,分析了沿静态主应力方向扩展的裂纹运动学和力学行为。实验结果表明:首先,静态竖向载荷在预制炮孔周围产生应力集中,在炮孔壁上下端部处出现最大拉应力;随后,在动态爆炸载荷的叠加作用下,裂纹优先在炮孔壁上最大拉应力位置处起裂,并沿最大主应力方向扩展;裂纹扩展过程中,静态竖向载荷越大,裂纹扩展速度越大,且裂纹尖端应力强度因子值越大。     

Interface crack-tip generalized stress field and stress intensity factors in transversely isotropic piezoelectric bimaterials

Transversely isotropic piezoelectric (TIP) bimaterials with an impermeable interface crack have been classified [Int. J. Frac. 119 (2003) L41] into two classes corresponding to the vanishing of the two singularity parameters or κ. It is shown in the present paper that the related eigenvalue problems for either =0 or κ=0 are not degenerate. The crack-tip generalized stress fields are obtained subsequently. A new definition of crack-tip intensity factors is presented for interface cracks in practical TIP bimaterial of practical interest. Such defined intensity factors are real numbers, which dominate the maximum crack-tip stress singularity and do not generate any phase angle change under any dimension system transformation for physical quantities.     

On the stress distribution in a prestretched simply supported strip containing two neighboring circular holes under forced vibration

The three-dimensional linearized theory of elastic waves in initially stressed bodies under plane strain is used to study the influence of the initial stretching of a simply supported plate strip with two neighboring circular holes on the stress concentration around the holes caused by additional uniformly distributed dynamic (time-harmonic) normal forces acting on the upper face. The corresponding problem is formulated and solved by the finite-element method. Numerical results on the stress concentration around the holes and the influence of the initial stretching on this concentration are presented Published in Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 135–140, October 2007.     

Assurance of reliable time limits in fatigue depending on choice of failure simulation: Energy density versus stress intensity

The principle of least variance is applied to evaluate the reliability of the design conditions of the Runyang cable-stayed bridge. Monitored fatigue load in service data are analyzed in conjunction with the specimen fatigue crack growth data for bridge steel. Aside from size differences, the interactive effects of material behavior with load amplitude and frequency would vary with the depicted physical model for the reliability of life prediction. Based on the same crack growth history in time or cycle, the two choice selected for comparison are stress intensity factor (SIF) range, and the strain energy density (SED) range. Reliability is found to depend on the trade off between load amplitude and frequency. Considered are high-amplitude; low-frequency and low-amplitude; high-frequency. In each case, the chances are the reliable time span of fatigue crack growth will not coincide with the useful portion of bridge life, simply because the load frequency must be anticipated as an educated estimate. It is subject to change. Conversion of the crack length fatigue cycle history to the corresponding time history requires the specification of load frequency that can set the time span of the useful life. This is demonstrated for the Runyang bridge, where approximately 30 MPa and 8 MPa would correspond to the high and low fatigue load, respectively.Significant variances were found for the SIF and SED models. The difference can be attributed to the inclusion of the mean stress in the SED that is more forgiving since it accounts for both the stress and strain effects, in contrast to the SIF model that leaves out the strain and the mean stress. Since the principle of least variance refers to the average of the R-integrals, the results based on the linear sum (LS) and root mean square (RMS) will differ quantitatively, but not qualitatively. The obvious mismatch of the fatigue load used to determine the material property and that for the bridge design can be adjusted and absorbed into the appropriate choice for the load frequency, a compensating factor not realized up to now. To this end, the weighted functions in the R-integrals further emphasize long run effects of the least variance reliability analysis. Attention is called to Changeability in addition to determinability and probability for predicting the time to failure. That is to better anticipate the change in the fatigue load frequency, to which the assistance of health monitoring should provide.     

Elastic stress distributions for hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings

Closed-form solutions are developed for the stress fields induced by circumferential hyperbolic and parabolic notches in axisymmetric shafts under torsion and uniform antiplane shear loading. The boundary value problem is formulated by using complex potential functions and two different coordinate systems, providing two classes of solutions. It is also demonstrated that some solutions of linear elastic fracture and notch mechanics reported in the literature can be derived as special cases of the general solutions proposed herein.Finally the analytical frame is used to link the Mode III notch stress intensity factor to the maximum shear stress at the notch tip, as well as to give closed-form expressions for the strain energy averaged over a finite size volume surrounding the notch root.