中文摘要：

基于图像子区的数字图像相关方法需采用合适的形函数来近似目标图像子区的真实变形。由于实际测量时目标子区的局部变形往往是未知的,实际采用的不同阶次（零阶、一阶和二阶）的形函数不可避免地产生误匹配（欠匹配和过匹配）问题,从而引入位移测量的系统或随机误差。尽管由欠匹配形函数引起的系统误差已被充分认识,由过匹配形函数引起的位移误差仍缺少理论解释。本文首先推导出采用一阶和二阶形函数的数字图像相关方法的随机误差理论公式,随后采用一系列数值实验验证了理论公式的准确性。结果显示：过匹配形函数不会引入额外的系统误差,但会增加随机误差,且二阶形函数的随机误差是一阶形函数的二倍。考虑到由欠匹配一阶形函数引入的系统误差往往远大于过匹配二阶形函数的随机误差,因此在未能确知变形的情况下,推荐使用二阶形函数。

英文摘要：

Digital image correlation method（DIC）,which is based on image sub region,should use the appropriate shape function to approximate the real deformation of target image sub region.Since in most practical measurement,the local deformation of target sub region is generally unknown,and the practically adopted shape functions with defferent orders of Taylor＇s expansion（e.g.,zero-order,first-order and second-order）inevitably produce mismatch（undermatched or overmatched）problems,resulting in additional systematic error or random error in displacement measurement.Although the systematic error due to undermatched shape functions has been thoroughly studied,but the displacement measurement error associated with overmatched shape functions is still lack of theoretical analysis.In this work,theoretical formula for random error associated with first-order and secondorder shape functions adopted in DIC method was derived first.Then,a series of numerical experiments were adopted to verify the correctness of theoretical formula.Experimental results reveal that overmatched shape function will not induce additional systematic error,but will increase random error,and that the random error from secon-order shape function is two times of that from first-order shape function.In addition,taking into account that the systematic error due to undermatched firstorder shape function is often far larger than the random error due to overmatched second-order shape function,so under unknown deformation condition,second-order shape function is recommanded in practical application.