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, $$G_c$$Gc. 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.

中文翻译：

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具有双线性准脆性和脆性界面的双悬臂梁试样的完整解析解决方案

在这项工作中，我们为双悬臂梁 (DCB) 开发了一个完整的解析解，其中臂被建模为 Timoshenko 梁，并在界面处嵌入了双线性内聚区模型 (CZM)。给出了两种类型的 DCB 的解决方案；一种具有规定的旋转（具有稳态裂纹扩展），一种具有规定的位移（裂纹扩展不是稳态）。因为 CZM 是双线性的，所以解析解分三个阶段分别给出，即（i）裂纹扩展前的线弹性行为，（ii）裂纹扩展前的损伤增长和（iii）裂纹扩展。然后，这些解用于导出界面为线弹性脆性破坏情况下的解（即 裂纹扩展前无损伤增长）和具有脆性破坏的无限刚性界面的情况（对应于线弹性断裂力学 (LEFM) 解决方案）。如果 DCB 臂是可剪切变形的，我们的解决方案正确地捕捉到了这样一个事实：即使界面是无限刚性的，它们也会在裂纹尖端和裂纹尖端旋转。定义界面处接触牵引力分布的表达式，以及裂纹尖端处和裂纹尖端前臂的剪切力、弯矩和横截面旋转，是为具有脆性破坏的线弹性界面导出的，在LEFM 极限。对于在 LEFM 限制中具有规定位移的 DCB，我们还推导出临界能量释放率 $$G_c$$Gc 的封闭形式表达式。这个公式，与基于 DCB 臂夹在裂纹尖端的假设（也用于确定模式 I 分层中的断裂韧性的标准）的所谓“标准梁理论”公式相比，有一个附加项，它考虑考虑裂纹尖端的旋转。此外，我们为臂的剪切刚度无限大的情况提供了所有提到的解析解，这对应于 Euler-Bernoulli 梁理论。在数值例子中，我们比较了 Euler-Beronulli 和 Timoshenko 梁理论的结果，并分析了 CZM 参数的影响。此外，我们为臂的剪切刚度无限大的情况提供了所有提到的解析解，这对应于 Euler-Bernoulli 梁理论。在数值例子中，我们比较了 Euler-Beronulli 和 Timoshenko 梁理论的结果，并分析了 CZM 参数的影响。此外，我们为臂的剪切刚度无限大的情况提供了所有提到的解析解，这对应于 Euler-Bernoulli 梁理论。在数值例子中，我们比较了 Euler-Beronulli 和 Timoshenko 梁理论的结果，并分析了 CZM 参数的影响。