Indentation scaling relationships provide normalized guidance for measuring and predicting mechanical properties in indentation experiments. At the nano-scale, the material size-effect is significant, while conventional scaling relationships fail to depict this phenomenon in nanoindentation precisely. In the present research, cross-scale indentation scaling relationships are investigated using a strain gradient theory. The nanoindentation response is found to be sensitive to different material parameters, including the material intrinsic length, yield stress, and work-hardening exponent across size-scales. If the strain gradient effect is ignored, the nanoindentation scaling relationships approach the macroscopic conventional ones. The cross-scale indentation scaling relationships obtained in the form of dimensionless functions in this work provide quantitative references to instrumented indentation tests on multiple size-scales, coinciding well with experimental results. The understanding of nanoindentation hardness is enhanced by the present work.

压痕比例关系为压痕实验中测量和预测机械性能提供了标准化的指导。在纳米尺度上，材料的尺寸效应是显着的，而常规的尺度关系却无法在纳米压痕中精确地描述这种现象。在本研究中，使用应变梯度理论研究了跨尺度压痕尺度关系。发现纳米压痕响应对不同的材料参数敏感，包括不同材料尺度的材料固有长度，屈服应力和加工硬化指数。如果忽略应变梯度效应，则纳米压痕缩放关系接近宏观的常规关系。在这项工作中以无量纲函数形式获得的跨尺度压痕比例关系为定量测量压痕测试提供了定量参考，这些压痕测试在多个尺寸尺度上均与实验结果吻合。纳米压痕硬度的理解通过当前的工作得以增强。