We consider a variant of the classical Biot problem concerning the wrinkling of a compressed hyperelastic half-space. The traction-free surface is no longer flat but has a localized ridge or trench that is invariant in the $x_1$-direction along which the wrinkling pattern is assumed to be periodic. With the $x_2$-axis aligned with the depth direction, the localized imperfection is assumed to be slowly varying and localized in the $x_3$-direction, and an asymptotic analysis is conducted to assess the effect of the imperfection on the critical stretch for wrinkling. The imperfection introduces a length scale so that the critical stretch is now weakly dependent on the wave number. It is shown that the imperfection increases the critical stretch (and hence reduces the critical strain) whether the imperfection is a ridge or trench, and the amount of increase is proportional to the square of the maximum gradient of the surface profile.

我们考虑了有关压缩超弹性半空间起皱的经典比奥问题的一种变体。无牵引力的表面不再是平坦的，而是具有局部的脊或沟槽，在$X_{\mathrm{1个}}$方向被认为是周期性的。随着$X_2$轴与深度方向对齐，假定局部缺陷缓慢变化并局限在 $X_3$方向，并进行渐近分析以评估瑕疵对临界皱纹的影响。缺陷引入了长度标度，因此临界拉伸现在几乎不依赖于波数。结果表明，无论缺陷是脊还是沟槽，缺陷都会增加临界拉伸（从而减小临界应变），并且增加量与表面轮廓最大梯度的平方成正比。