The crack extension resistance and fracture properties are studied in detail for quasi-brittle materials like concrete with a softening traction-separation law by investigating the complete fracture process. The computed samples are the three-point bending notched beams of concrete with different sizes tested by other researchers. The softening traction-separation law which was proposed by Reinhardt et al. based on direct tension tests for normal concrete materials was chosen in the computations. Different distribution shapes of the cohesive force on the fictitious crack zone were considered for the corresponding loading states. The computations were mainly based on the analytic solutions for this problem using Gauss-Chebyshev quadrature to achieve the integration which is singular at the integral boundary. The crack extension resistance curves in terms of stress intensity (KRcurves) were determined by combining the crack initiation toughness Kini Ic that is the inherent toughness of the material needed to resist the crack initiation in the case that is in the lack of an extension of the main crack with the contribution due to the cohesive force along the fictitious crack zone during the complete processes of fracture. The situation of crack propagation can be judged by comparing KR-curves of crack extension resistance with the stress intensity factor curves which were calculated using the lengths of the extending crack and the corresponding loads at each loading states, e.g., when the crack extension resistance curve (KR-curve) is higher than the stress intensity factor curve, the crack propagation is stable; otherwise, it is unstable. In the computation, the obtained relationship between the crack tip opening displacement CTOD and the amount of crack extension for the complete fracture process is in agreement with the testing results of other researchers.

In this paper, an attempt is made to determine the double-K fracture parameters Kini Ic and Kun Ic using three-point bending notched beams. First, based on the knowledge from extensive investigations which showed that the nonlinearity of P-CMOD curve is mainly associated with crack propagation, a linear asymptotic superposition assumption is proposed. Then, the critical effective crack length ac is analytically evaluated by inserting the secant compliance cs into the formula of LEFM. Furthermore, an analytical result of a fictitious crack with cohesive force in an infinite strip model was obtained. The double-K fracture parameters Kini Ic and Kun Ic as well the critical crack tip opening displacement CTODc were analytically determined. The experimental evidence showed that the double-K fracture parameters Kini Ic and Kun Ic are size-independent and can be considered as the fracture parameters to describe cracking initiation and unstable fracture in concrete structures. The testing method required to determine Kini Ic and Kun Ic is quite simple, without unloading and reloading procedures. So, for performing this test, a closed-loop testing system is not necessary.

This paper shows how the double-K fracture parameters Kini Ic and Kun Ic can be determined for concrete using CT-specimens and wedge splitting specimens. The experimental results collected from the fracture tests on the very large size CT-specimens and small size wedge splitting specimens carried out by many researchers are utilized to investigate the characters of the obtained double-K fracture parameters Kini Ic and Kun Ic. It was found that the double-K fracture parameters Kini Ic and Kun Ic determined from fracture tests on the large size CT-specimens are size-independent. And the values of Kini Ic and Kun Ic determined from small size wedge splitting specimens with same dimensions are independent of the relative preformed notch length a0=D. However, when the dimensions of small size wedge splitting specimens change from 150

采用标准三点弯曲梁砼试件，利用试验中测得的最大荷载Pmax及对应的裂缝口张开位移CMODc计算了砼裂缝亚临界扩展量Dac。提出了由虚拟裂缝闭合力s（x）产生的应力强度因子cKI的实用解析方法，进而计算了砼双K断裂参数ini KIC及un KIC。采用这一方法，只需在试验中测得最大荷载Pmax及对应的裂缝口张开位移CMODc、砼弹性模量及抗拉强度即可确定双K断裂参数并且无需对cKI进行数值积分。大量的计算分析表明，这一实用解析方法不但可使计算过程进一步简化，而且具有很高的精度。

In this paper a mode It fracture testing method has been developed for wood from analytical, experimental and numedcaI investigations. Analytical resufls obtained by other researchers showed that the specimen geometry and loading type used for the proposed mode Ⅱ testing method results in only mode Ⅱ stress intensity and no mode Ⅰ stress intensity at the crack tip Experiments have been carded out to determine mode II fracture toughness A'iic and fracture energy GIIF from the test data collected from both spruce (picea afiies) and popIar (popuIus nigra) specimens. It was found that there existed a very good relation between fracture toughness Kiic and fracture energy GIlF when the influence of orthotropic stiffness EI in mode Ⅱ was taken into account. It verified that for this mode Ⅱ testing method the formula of LEFM can be employed for calculating mode Ⅱ fracture toughness even for highly orthotropic materiaIs like wood In the numerical studies for the tested spruce specimen, the crack propagation process, stress and strain fields in front of crack tips and the stress distributions along the ligament have been investigated in detail. It can be seen that the simulated crack propagating process along the ligament is a typical s hear cracking pattem and the development of cracks along the ligament is due to shear stress concentrations at the crack tips of the specimen. It has been shown that this mode Ⅱ fracture testing method is suitable for measuring mode Ⅱ fracture toughness/(ElC for highly orthotropic materials like wood.