The small-scale yielding fracture of plastic-hardening metals is a well-understood theory, essentially conceived by Hutchinson, Rice and Rosengren (hence the name HRR theory). However, even though specimens of rather different sizes have been tested to verify the small-scale yielding theory, an analytical scaling law for the size effect transition from elastic–plastic behavior through small- and large-scale yielding to fracture process zone has apparently not been formulated. Such a scaling law would be useful for the design as well as measurement of mode-I ductile fracture properties of metals, and is the aim of this study. Unlike the fracture of quasibrittle materials such as concrete or composites, the modeling of plastic-hardening materials is complicated by a millimeter scale singular yielding zone that forms between the micrometer-scale fracture process zone (FPZ) and the elastic (unloading) material on the outside. Essential for the large-scale transitional size effect is the effective yielding zone size, which is here calculated from the equivalence of the virtual works within the plastic-hardening zone and elastic singular stress fields within the transition zone, and is shown to depend on the crack-parallel $T$-stress. The size effect analysis requires taking into account not only the dissipation in the FPZ delivered by the $J$-integral flux of energy through the yielding zone, but also the energies released from the structure and from the unloaded band of plasticized material trailing the advancing yielding zone. Equating the rates of energy releases and energy dissipation leads to an approximate energetic size effect (scaling) law that matches the calculated small- and large-size asymptotic behaviors, when the crack ligament contains the yielding zone.. The law is similar to that for quasibrittle fracture but its coefficients depend on the fracture energy and the yielding zone size in a different way. This law, reducible to linear regression, can be exploited for size effect testing of fracture energy (or critical $J$-integral) and effective size $r_p$ of the yielding zone. An effect of high crack-parallel stress $T$ on $r_p$ is likely but is relegated to future study, as it would not affect the scaling law derived. For testing of the transition from the small-size range (large-scale yielding) to the large-size range (small-scale yielding), a modified size effect method, requiring nonlinear optimization, is developed. The size effect law is verified by scaled tests of notched specimens of aluminum.

塑性硬化金属的小规模屈服断裂是一个很好理解的理论，基本上由Hutchinson，Rice和Rosengren构想（因此得名HRR理论）。但是，尽管已经测试了大小不同的试样以验证小规模屈服理论，但尺寸效应从弹塑性行为通过小规模和大规模屈服过渡到断裂过程区的解析尺度定律显然没有被制定。这样的缩放定律对于金属的I型延性断裂特性的设计和测量将是有用的，并且是本研究的目的。与混凝土或复合材料等准脆性材料的断裂不同，毫米级的奇异屈服区形成在微米级的断裂加工区（FPZ）和外部的弹性（卸载）材料之间，从而使塑性硬化材料的建模变得复杂。有效的屈服区尺寸对于大规模的过渡尺寸效应至关重要，该屈服区尺寸是根据塑性硬化区中的虚拟工件与过渡区中的弹性奇异应力场的等效性计算得出的，并且取决于屈服强度。平行裂纹$Ť$-强调。尺寸效应分析不仅要考虑到$Ĵ$-通过屈服区的能量的积分通量，还包括从结构和从前进的屈服区后面的增塑材料的卸载带释放的能量。当裂纹韧带包含屈服区时，使能量释放和能量耗散率相等，会导致近似的能量尺寸效应（定标）定律与所计算的小尺寸和大尺寸渐近行为相匹配。该定律类似于准脆性断裂，但其系数以不同的方式取决于断裂能和屈服带大小。该定律可简化为线性回归，可用于断裂能（或临界）的尺寸效应测试$Ĵ$-积分）和有效尺寸 ${\mathrm{[R}}_p$屈服区。高裂纹平行应力的影响$Ť$ 上 ${\mathrm{[R}}_p$可能但由于不影响所得出的缩放定律而被归入未来研究。为了测试从小尺寸范围（大规模屈服）到大尺寸范围（小规模屈服）的过渡，开发了需要非线性优化的改进尺寸效应方法。尺寸效应定律通过对铝制缺口样品的定标测试进行验证。