Škec Leo, Alfano Giulio, Jelenić Gordan
1Department of Mechanical and Aerospace Engineering, Brunel University London, Kingston Lane, Uxbridge, UB8 3PH UK.
2Faculty of Civil Engineering, University of Rijeka, Radmile Matejčić 3, Rijeka, 51000 Croatia.
Int J Fract. 2019;215(1):1-37. doi: 10.1007/s10704-018-0324-5. Epub 2018 Nov 14.
In this work we develop a complete analytical solution for a double cantilever beam (DCB) where the arms are modelled as Timoshenko beams, and a bi-linear cohesive-zone model (CZM) is embedded at the interface. The solution is given for two types of DCB; one with prescribed rotations (with steady-state crack propagation) and one with prescribed displacement (where the crack propagation is not steady state). Because the CZM is bi-linear, the analytical solutions are given separately in three phases, namely (i) linear-elastic behaviour before crack propagation, (ii) damage growth before crack propagation and (iii) crack propagation. These solutions are then used to derive the solutions for the case when the interface is linear-elastic with brittle failure (i.e. no damage growth before crack propagation) and the case with infinitely stiff interface with brittle failure (corresponding to linear-elastic fracture mechanics (LEFM) solutions). If the DCB arms are shear-deformable, our solution correctly captures the fact that they will rotate at the crack tip and in front of it even if the interface is infinitely stiff. Expressions defining the distribution of contact tractions at the interface, as well as shear forces, bending moments and cross-sectional rotations of the arms, at and in front of the crack tip, are derived for a linear-elastic interface with brittle failure and in the LEFM limit. For a DCB with prescribed displacement in the LEFM limit we also derive a closed-form expression for the critical energy release rate, . This formula, compared to the so-called 'standard beam theory' formula based on the assumptions that the DCB arms are clamped at the crack tip (and also used in standards for determining fracture toughness in mode-I delamination), has an additional term which takes into account the rotation at the crack tip. Additionally, we provide all the mentioned analytical solutions for the case when the shear stiffness of the arms is infinitely high, which corresponds to Euler-Bernoulli beam theory. In the numerical examples we compare results for Euler-Beronulli and Timoshenko beam theory and analyse the influence of the CZM parameters.
在这项工作中，我们针对双臂被建模为铁木辛柯梁且在界面处嵌入双线性内聚区模型（CZM）的双悬臂梁（DCB），开发了一个完整的解析解。给出了两种类型DCB的解：一种是规定了转角的（具有稳态裂纹扩展），另一种是规定了位移的（裂纹扩展是非稳态的）。由于CZM是双线性的，解析解在三个阶段分别给出，即：（i）裂纹扩展前的线弹性行为，（ii）裂纹扩展前的损伤增长，以及（iii）裂纹扩展。然后利用这些解推导出界面为线弹性且具有脆性破坏（即裂纹扩展前无损伤增长）的情况以及界面为无限刚性且具有脆性破坏（对应线弹性断裂力学（LEFM）解）的情况的解。如果DCB的臂是剪切可变形的，我们的解正确地捕捉到这样一个事实，即即使界面是无限刚性的，它们在裂纹尖端及其前方也会发生转动。对于具有脆性破坏的线弹性界面以及在LEFM极限情况下，推导了定义界面处接触牵引力分布以及裂纹尖端处和前方臂的剪力、弯矩和横截面转角的表达式。对于在LEFM极限下具有规定位移的DCB，我们还推导了临界能量释放率的封闭形式表达式。与基于DCB臂在裂纹尖端处被夹紧这一假设的所谓“标准梁理论”公式（该公式也用于确定I型分层断裂韧性的标准中）相比，这个公式有一个额外的项，该项考虑了裂纹尖端处的转动。此外，我们给出了臂的剪切刚度为无限大（对应欧拉 - 伯努利梁理论）情况下的所有上述解析解。在数值示例中，我们比较了欧拉 - 伯努利梁理论和铁木辛柯梁理论的结果，并分析了CZM参数的影响。
