The theory of non-steady state fracture kinetics analysis—1: General theory of crack propagation

The kinetics theory of thermally activated time dependent crack propagation is extended to describe the crack size distribution in non-steady state. The distribution is represented by a series of n differential equations, each expressing the rate of crack tip concentration change over the system of n consecutive energy barriers. The general solution for the set of homogeneous linear first order differential equations developed in this report is of the form ρ_i = Σ_jc_i,je^{−λj_t + C_i}, where ρ_i is the crack tip concentration in the ith valley. The theory takes into consideration the discrete character of solids in contrast with the usual continuum models of fracture studies. The analysis is readily applicable to Regions I and II of stress corrosion cracking.     

Some problems in the J-integral

For a blunt crack the j-integral is path dependent on contours which are very close to the crack tip even for elastic material. Using the incremental J-integral theory we introduce a new parameter J_t, characterizing the behavior of a crack tip and prove that the J-integral is almost path independent on contours whose radii are greater than several COD if σ_ij,1Δε_ij — ε_ijΔσ_ij = 0 in plastic regions for elasto-plastic material.     

J-integral under transient temperature state

For a cracked plate under a transient temperature distribution, theJ-integral is expressed in the form of a line integral by using the convolution integral. The J_I-integral is calculated for a center-cracked steel plate under thermal and mechanical loading conditions and the calculated values are in good agreement with previous results. The effect of the inertia term on the J_I-integral is negligible for a steel but not for a glass. For the glass plate, the rates of J_I-integral value to time increase if the values of material properties such as specific heat, thermal conductivity, thermal diffusivity and Young's modulus as well as crack length and temperature difference on the cracked edge increase.     

通电瞬时板内半无限长裂纹尖端域的应力场

以导电弹性体的麦克斯威尔方程为出发点,借助于边界条件和初始条件,推得了在向含半无限长直线裂纹的无限大导电薄板内通入电流的瞬时,裂纹尖端附近电流密度的表达式。在此基础上,得到了裂纹尖端区域处温度和应力的具体表达式。算例表明,在电流所产生的焦耳热源的作用下,裂尖区域处的温度将瞬时升高,并伴有压应力的产生,从而可达到阻止裂纹扩展的目的。     

Stress analysis in a cracked infinite electrostrictive plate with local saturation electric fields

Based on the electric circular saturation zone near an ideal crack tip, the approximate complete solution for electric and stress field in a cracked electrostrictive plate under general loading at infinity is carried out. The SIFs are then obtained. We find that the stress distributions in front of the crack tip can be divided into four different regions. The fracture behavior is closely related to these distributions.     

Strain energy density fracture criterion in elastodynamic mixed mode crack propagation

Sih's fracture criterion based on strain energy density, S, for mixed mode crack extension under static loading is extended to dynamic mixed mode, K_I and K_II, crack propagation. Influence of the second order term, σ_ox, which represents the non-singular constant stress acting parallel to the direction of crack propagation, on the S distribution surrounding the crack tip, is demonstrated. Numerical studies show that positive σ_ox enhances the fracture angle and negative σ_oxreduces the fracture angle irrespective of the sign of K_II/K_I, when S is measured at a critical distance r_c from the crack tip. This fracture criterion is verified by the crack curving results of dynamic photoelastic fracture specimens. Omission of σ_ox term leads to predicted fracture angles which are at variance with experimental data.     

A comparison between the elastic strain energy and the plastic work at the crack tip

For ductile fractures it becomes progressively imperative to include the plastic components of the quantities defining each particular fracture criterion, in order to derive a relationship, which not only fits the experimental data but also satisfies some appropriate physical principle. In this paper the plastic components of stresses entering in the experessions for the specific plastic work were derived from the plastic singular solution described by the HRR stress field. In order to ascertain the accuracy of such approximate solutions for criteria based on energy balance, a comparison between the components of elastic strain energy density, and its total value, with the specific plastic work (W_p) was undertaken, in order to derive certain conclusions about the importance of the contribution of energy component and the specific plastic work. The analysis was based on the typical case of a transverse internal crack in plate under plane-stress conditions, which is submitted to a mode I deformation. While the dilatational (T_v) and the distortional (T_D) components of elastic SED were calculated for impending plasticity the specific plastic work (W_p) was evaluated by assuming ssy conditions and the theory of HRR field using the plastic singular solution for mode I. It was shown that the contributions of T_v, T_D, T_e (T_e = t_v + T_D) and W_P for different strain hardening exponents were varying with ductile materials, presenting a strong influence of W_p. The conclusion was that for highly strain-hardened materials the contribution of the elastic SED and particularly of T_v is important to the mode of fracture of the plate.     

A higher-order approximation for the T-criterion of fracture in biaxial fields

The T-criterion of fracture is based on the principle that crack propagates when the maximum value of the distribution of the dilatational component of strain energy density T_v, evaluated along contour lines of constant distortional energy density T_D around the crack tip, attains a limiting value T_vo The angle of this maximum defines also the direction of initiation of crack propagation. Then, the study of the distribution of T_v around the crack tip presents a special interest for understanding mechanisms of fracture.

In this investigation an exhaustive theoretical analysis of the distribution of t_v-component around the tip of crack under in-plane modes of loading was undertaken. The T_v-distribution was evaluated along the elastic-plastic boundary, developed around the crack tip for impending plasticity, according to the Mises yield condition (T_D = T_D0 = const.). The mode of loading of the cracked plate was assumed biaxial with different biaxiality ratios k and a two-term approximation for the respective complex stress function was considered, according to the studies of Liebowitz et al.[1], instead of only the singular term considered up-to-now.

It was found that the T_v-distribution along the Mises initial elastic-plastic boundary presents always a maximum in front of the crack tip, whose position and magnitude depend on the biaxiality factor k and the angle of loading β. The position and the magnitude of this maximum for the two-term approximation of φ(z) showed differences in some regions with the respective values for the singular solution.     

Parameter of local cyclic strain distribution

The local cyclic strain distribution near the crack tip was investigated by the Computer Aided Fine Grid Method. This method makes it possible to measure continuously every in-plane component of local cyclic strain distribution. It was found that the magnitude of the local cyclic strain distribution near the crack tip was varied by the applied cyclic load level and the material, but the shape of the local cyclic strain distribution near the crack tip was experimentally scarcely altered.

The local cyclic strain field near the crack tip could be written as the following equation. Δε_eq(r, θ) = ΔA · f(θ) · r⁻¹ A single parameter ΔA, which characterizes the local cyclic strain field near the crack tip, was newly proposed by authors.     

载流金属材料裂纹尖端电磁应力的数值模拟

用数值模拟方法研究了金属裂纹尖端电磁应力的分布情况,给出了电流分布、磁场分布和电磁应力的分布.模拟结果表明,金属材料裂纹尖端受的电磁应力是最大的,并且这个力的大小随着电流密度的增加而增大,裂纹尖端的电磁力指向金属的内部.通过具体算例表明,在金属能承受的电流密度下,金属材料裂纹尖端的电磁应力约能达到1MPa的数量级.因此,在研究电磁场处理金属裂纹时,不能忽略电磁应力.     

导电薄板内裂纹尖端区域的电磁应力

为了研究电磁应力对导电薄板内裂纹尖端的作用,从基本电磁理论出发,通过对导体表面所受电场力的分析,推得了导电薄板内裂纹边缘处电场力的表达式.在此基础上,通过导电薄板内裂纹尖端区域磁场的确定,得到裂纹尖端区域的电磁应力表达式.裂纹尖端电磁应力的计算表明,金属薄板中裂纹尖端的电磁应力是由裂纹尖端指向金属内部的压应力,并且当电流密度为103～104A/mm2的数量级时,裂纹尖端的压应力数值可达数兆帕到数百兆帕.因此,在研究裂纹止裂问题上,其影响不容忽视.     

Growth mechanism of stress corrosion cracking in high strength steel

Stress corrosion crack growth rates are measured at sveral stress intensity levels for low-tempered 4340 steel in 0.1N H₂SO₄ solution. The characteristics of the growth rates are divided into three regions of stress intensity factors: Region I near K_1SCC; Region III near unstable fracture toughness, K_1SC; and Region II, which lies between the two. K_1SCC is the value of K at which no crack growth can be detected after 240 hr.

In order to explain these experimental results, the crack initiation analysis reported in a previous paper is extended to the growth rates. A detached crack initiates and grows at the tip of an already existing crack. When the detached crack reaches the tip of the main crack, the process repeats as a new existing crack.

A relationship between crack growth rate, v, and stress intensity factor, K, is obtained as a function of b/a and a = b + d, where b is the distance from the tip of the main crack to the detached crack, and d is the ydrogen atom saturated domain.

The experimental data are in good agreement with the theoretical values in Region II when a = 0.02 mm, b/a = 0.8, c₁/c₀ = 2.8 for 200°C tempered specimens and a = 0.015 mm, b/a = 0.7, c₁/c₀ = 3.0, ρ_b = 0.055 mm for 400°C tempered specimens, where ρ_b is a fictitious notch radius. The plateau part in Region II for 400°C tempered specimens is also successfully explained by the present theory. For Region III, the value of b/a will be almost equal to 1 because v → ∞ for b/a → 1. On the other hand, for Region I, b/a will be zero, since the value of v becomes negligibly small and no crack growth is observable.     

Finite element modeling of melting crack tip under thermo-electric Joule heating

The finite element modeling of the melting crack tip under thermo-electric Joule heating is presented in this paper. It is necessary to consider the AC input, temperature-dependent material constants and crack contact conditions. The shape of the melting crack tip zone may be either circular or elliptical. As the crack length relative to the plate width becomes too small, the crack tip may not melt. Finally, this paper demonstrates that the electric current density factor cannot be used to estimate the crack tip temperature and melting condition.     

Effect of stress triaxiality levels in crack tip regions on the characteristics of void growth and fracture criteria

The behaviour of void growth in the crack tip regions of four specimen geometries with different stress triaxiality levels have been investigated by the FEM method and experimental observations in plane strain and plane stress cases respectively. It was found that the shape change of growing voids, the configurations of a blunting crack tip and the sizes of decreasing ligament between the void and the crack tip are strongly dependent upon the stress triaxiality levels. Under the condition of plane stress, the stress triaxiality on the ligaments of cracks are nearly the same for different specimen geometries, also the void growth, crack tip blunting asnd decreasing of ligament size are identical for various specimens with increasing load levels, which lead to the conclusion that the J_i-value is independent of specimen geometries. However, in the plain strain case, different void growth, crack tip blunting and decreasing in ligament size for various stress triaxiality levels directly caused the J_i-value to be dependent on the specimen geometries. It was found that when the void is linked to the blunting crack tip by the extrapolation to the zero ligament from FEM calculations, the J_i values, measured experimentally, are underestimated slightly.     

A new method for evaluation of stress intensities for interface cracks

A new method is presented for calculating the values of K_I and K_II in the elasticity solution at the tip of an interface crack. The method is based on an evaluation of the J-integral by the virtual crack extension method. Expressions for calculating K_I and K_II by using the displacements and the stiffness derivative of the finite element solution and asymptotic crack tip displacements are derived. The method is shown to produce very accurate solutions even with coarse element mesh.     

An algorithm for the eigenvalues of a corner with mixed boundary conditions

Displacements and stresses near a corner with one side fixed and the other side free can be written as series expansions of eigenfunctions of the form r^λjf_j(γ). The same is true of a point on a smooth boundary with a sudden change of boundary conditions from free to fixed. The eigenvalues λ_j are in general complex numbers that are required in order of ascending real part. An algorithm is presented that computes any number of λ_j in order of ascending real part.     

Phase field simulation of domain structures in cracked ferroelectrics

The fracture of ferroelectrics is a complex process which is influenced by various factors, among which are the domain switching near the crack tip, the crack face boundary conditions and the applied electric field. Domain switching near crack tips induces major local nonlinearity, while the crack face boundary conditions vary considerably due to different working conditions. In this work, a phase field model and a generalization of the configurational force theory into this model are used to investigate the microstructure around the crack tip and to quantitatively study the influence of the applied electric field and the crack face boundary conditions (permeable, impermeable, semi-permeable and energetically consistent). Evaluation of the fracture properties is done by the nodal configurational force at the crack tip based on the generalized configurational force theory. Results show that the induced domain structure relies significantly on the loading and on the surface boundary conditions. Among the four different conditions considered, the energetically consistent conditions lead to the smallest crack driving force, and the permeable conditions lead to the largest crack driving force. Calculations also show that positive electric fields tend to inhibit fracture, whereas negative electric fields tend to promote fracture.     

In situ observation of electric-field-induced domain switching and crack propagation in poled PMNT62/38 single crystals

It is important to in situ observe the crack propagation and the corresponding domain switching in ferroelectrics subjected to electric loading since domain switching has been widely assumed to play a critical role in the electric-field-induced crack growth. In this investigation, we in situ observed the crack propagation and the domain switching in PMNT62/38 single crystals poled along the [001] orientation. An experimental setup was designed and constructed to investigate the crack propagation and the domain switching in thin plate specimens with pre-crack subjected to electric field by using polarized light microscope (PLM). The pre-crack began to propagate forward accompanied by the appearance of domain switching zones near the crack tip and the disappearance of switched zones behind the crack tip at the unipolar electric field of E = 0.8E_C. The results indicate that the structure mismatch of the adjacent switched zones with different polarizations stimulated by the intensive electric field near the crack tip results in the electric-field-induced crack growth.     

Analysis of the microstructure near the crack tip of ASME Gr.92 steel after creep crack growth

The microstructures near to and remote from the tip of a crack in ASME Gr.92 steel were investigated after creep crack growth at 873 and 898 K, focusing on the martensitic lath, the dislocation structure, and precipitates. After creep, the mean lath width near the crack tip was obviously larger than that of the virgin material, whereas the lath width remote from the crack tip was only slightly larger than that of the virgin material. The mean dislocation density near the crack tip markedly decreased after creep, whereas only a small change was observed in the dislocation density remote from the crack tip. The mean size of M₂₃C₆ particles near the crack tip after creep was larger than that of the virgin material, whereas their mean size remote from the crack tip was almost the same as that of the virgin material.     

Energy density approach for calculation of three-dimensional inelastic strain and stress at the crack tip in compact tension specimens of polycarbonate

An energy-based method is utilized for calculating elastic-plastic strains and stresses near fatigue crack tip in specimens of Merlon polycarbonate. The stress redistribution caused by the plastic yielding around the crack tip is taken into account so that theoretical crack tip strain is improved. The estimated values of crack tip strain based on an energy density approach are compared with experimental results obtained from an embedded grid moire technique and embedded strain gages. Large-scale yielding seems to dominate near the crack tip. In fact, the measured strain is in agreement with the elastic solution, which means, in reality, only small-scale yielding takes place near the crack tip. The strain in the mid-plane (plane strain) is found to be higher than in the surface plane (plane stress). The experimental and theoretical results are in good agreement.