基于等效黏性弹簧的黏弹性Timoshenko裂纹梁弯曲解析解

考虑裂纹的缝隙和黏性效应，将梁中横向裂纹等效为黏弹性扭转弹簧，利用广义Delta函数，给出了Laplace变换域内裂纹梁的等效抗弯刚度，得到了具有任意开闭裂纹数目且满足标准线性固体黏弹性本构的Timoshenko梁在时间域内的弯曲变形显式解析通解．在此基础上，通过两个数值算例，分析了时间、梁跨高比和裂纹深度等参数对黏弹性Timoshenko开裂纹梁弯曲变形的影响．结果表明：裂纹黏性对Timoshenko裂纹梁的弯曲具有显著的影响．相比于裂纹的弹性扭转弹簧模型，考虑裂纹黏性效应的黏弹性Timoshenko裂纹梁在裂纹处挠度尖点和转角跳跃现象十分明显．另外，由于横向剪切引起的附加变形，Timoshenko裂纹梁的稳态挠度与Euler-Bernoulli梁挠度的差值为常数，其大小与裂纹模型、梁跨高比或裂纹深度无关，这些结果对梁裂纹无损检测具有指导意义．

Analytical Bending Solution of Viscoelastic Timoshenko Cracked Beam Based on Equivalent Viscoelastic Spring

Considering the effects of gap and viscoelasticity of crack, and regarding the transverse crack in the beam as a viscoelastic rotational spring, the equivalent flexural rigidity of the cracked beam in Laplace transformed domain is presented by a generalized Dirac delta function, and a general explicit analytical solution in time domain for the bending deformation of the Timoshenko beam having an arbitrary number of cracks and satisfying the standard linear viscoelastic constitutive relation is derived. On this basis, the influences of time, span-height ratio of beam and crack depth, etc. on the bending deformation of viscoelastic Timoshenko beam with open-cracks are examined by two numerical examples. It is shown that the crack viscoelasticity has remarkable influence on the bending of the Timoshenko cracked beam. Compared with the elastic rotational spring model of crack, the phenomena of deflection cusp and jump of rotational angle at the crack location of the viscoelastic Timoshenko cracked beam is very evident. Furthermore, due to the additional deformation caused by transversal shear, the differences between the stationary deflections of the Timoshenko cracked beam and those of the Euler-Bernoulli cracked beam are constants, which are independent of the crack model, the span-height ratio of the beam or the crack depth. These results are helpful for guiding non-destructive crack identification on beams.