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2026 Vol.36, Issue 2 Preview Page

Original Article

Statistical Variability Analysis of Subcritical Crack Growth Index in Constant Stress-Rate Testing and Minimum Specimen Number Estimation Using Propagation-of-Error Method

일정 응력속도 시험에서 임계하 균열성장 지수의 통계적 변동성 분석 및 오차 전파법을 이용한 최소 시편수 산정

Tae Young Ko1*
고 태영1*
1Associate Professor, Kangwon National University Department of Energy and Resources Engineering
1강원대학교 에너지자원공학과 부교수
*Corresponding Author
30 April 2026. pp. 149-170
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Abstract

In constant stress-rate testing, the statistical variability of the subcritical crack growth index (n) is governed by four parameters: the expected index n, the number of specimens per stress rate (N), the stress-rate layout coefficient (CR), and the Weibull modulus (m). While previous studies have relied on Monte Carlo simulations to characterize this variability, a practical analytical solution for pre-test design has been lacking. This study employs the propagation-of-error method to derive an analytical expression for the coefficient of variation, CV(n), which is subsequently inverted to propose a minimum-specimen-number estimation formula. The proposed analytical model was validated against bootstrap resampling (1,000 iterations) using experimental data from Coconino sandstone grooved-disc specimens (N = 40 per stress rate, m = 9.07, n = 37.0). While the estimation formula slightly underestimates the bootstrap results at N = 10, it yields conservative and safe-side estimates for N ≥ 13. According to the derived estimation formula, achieving a target CV(n) ≤ 30% under the standard ASTM C1368 three-decade stress-rate range requires at least 13 specimens per stress rate for Coconino sandstone, compared to the 10-specimen minimum established for ceramics with higher Weibull moduli. Bootstrap analysis further indicates that the distribution of n is strongly right-skewed at small sample sizes but approaches normality as N increases. Furthermore, a subset analysis of the four experimental stress rates (0.01, 0.1, 1, and 10 MPa/s) demonstrates that when the full four-rate protocol is impractical, utilizing only the extremum rates (0.01 and 10 MPa/s) retains 90% of the layout coefficient. This alternative yields a minimal n-estimate bias of +1.5% and attains a comparable level of precision by using 15 specimens per stress rate to meet the CV ≤ 30% criterion. Conversely, relying solely on adjacent stress rates introduces severe estimation biases (up to 108% in the present dataset) and must be avoided. While theoretically applicable to any brittle material conforming to a Weibull distribution, the empirical validation of the proposed estimation formula is currently limited to Coconino sandstone.

Keywords
Subcritical crack growth
Constant stress-rate test
Propagation-of-error
Weibull distribution
Minimum specimen number

일정 응력속도 시험(constant stress-rate test)에서 임계하 균열성장 지수 n의 통계적 변동성은 n, 응력속도당 시편 수 N, 응력속도 배치 계수 CR, 재료의 와이블 계수 m의 네 가지 인자에 의해 결정된다. 기존 연구는 이러한 상관관계를 규명하기 위해 몬테카를로(Monte Carlo) 시뮬레이션에 주로 의존하였으며, 실험 설계 단계에서 선제적으로 활용할 수 있는 해석적 수식을 제시하지 못했다. 이에 본 연구에서는 오차 전파법(propagation-of-error method)을 적용하여 변동계수 CV(n)를 n, m, N, CR을 입력으로 하는 해석적 관계식을 유도하고, 이를 역산하여 목표 정밀도를 달성하는 최소 시편수 산정식을 도출하였다. 유도된 해석적 관계식의 타당성을 검증하기 위해, 코코니노 사암(Coconino sandstone)의 홈이 파인 디스크(GD) 시편을 이용한 일정 응력속도 시험 결과(응력속도당 N=40, m=9.07, n=37.0)를 바탕으로 1,000회의 부트스트랩(bootstrap) 재추출 결과와 내부적 교차 비교를 수행하였다. 도출된 산정식의 예측값은 N = 10에서는 부트스트랩 결과를 약간 과소추정하였으나, N ≥ 13 구간부터는 보수적인 추정을 보였다. ASTM C1368의 3자릿수 응력속도 범위 하에서 CV(n) ≤ 30%를 달성하기 위한 최소 시편수는 응력속도당 13개로 산출되었으며, 이는 강도 산포가 적은 세라믹 재료를 기준으로 권고된 10개보다 더 엄격한 기준이다. 또한, 부트스트랩 분석 결과 시편 수가 적을수록 n의 확률밀도함수가 강한 양의 비대칭성(right-skewed)을 나타내나, 시편 수가 증가할수록 정규분포에 근접함이 확인되었다. 4개 실험 응력속도(0.01, 0.1, 1, 10 MPa/s)에 대한 서브셋 분석 결과, ASTM이 권고하는 4개 응력속도 시험의 적용이 현실적으로 제한될 경우, 양 끝단의 두 응력속도(0.01 및 10 MPa/s)만을 조합하더라도 CR의 90%를 유지하며 추정 편향을 최소화(+1.5%)하여 기존 방식에 준하는 정밀도를 확보할 수 있음이 입증되었다. 반면, 인접한 두 응력속도만 조합할 경우 막대한 추정 오차가 발생하므로(본 자료 기준 최대 108%), 실험 설계 시 이를 반드시 피해야 한다. 본 연구에서 제시된 산정식은 와이블 분포를 따르는 취성 재료 전반에 이론적으로 적용할 수 있으나, 본 연구에서의 실증적 검증은 코코니노 사암에 국한되었다.

키워드
임계하 균열성장
일정 응력속도 시험
오차 전파법
와이블 분포
최소 시편수
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Information
  • Publisher :Korean Society for Rock Mechanics and Rock Engineering
  • Publisher(Ko) :한국암반공학회
  • Journal Title :Tunnel and Underground Space
  • Journal Title(Ko) :터널과 지하공간
  • Volume : 36
  • No :2
  • Pages :149-170
  • Received Date : 2026-04-03
  • Revised Date : 2026-04-20
  • Accepted Date : 2026-04-21
  • DOI :https://doi.org/10.7474/TUS.2026.36.2.149
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