斷裂是橫跨土木、機(jī)械和航空工程等廣泛領(lǐng)域的結(jié)構(gòu)系統(tǒng)中最常見(jiàn)的失效模式之一。防止斷裂引起的破壞是結(jié)構(gòu)設(shè)計(jì)的主要關(guān)注點(diǎn)，并且歷史上一直在推動(dòng)理論和實(shí)驗(yàn)方法的發(fā)展，以預(yù)測(cè)結(jié)構(gòu)破壞的成核和傳播。

由于在多個(gè)空間尺度上同時(shí)存在多個(gè)物理過(guò)程，因此斷裂力學(xué)的一般主題本身非常復(fù)雜，而動(dòng)態(tài)裂紋的特定主題由于裂紋表面粗糙，不穩(wěn)定性和分支而可能更具挑戰(zhàn)性。在過(guò)去的幾十年中，由于數(shù)值方法的迅速發(fā)展，動(dòng)態(tài)裂縫的分析取得了重大進(jìn)展。從動(dòng)態(tài)建模角度看，可用于破壞分析的方法可以分為兩類，即離散和連續(xù)。離散方法計(jì)算步驟非常繁雜，連續(xù)體方法的高計(jì)算成本都是制約動(dòng)態(tài)破壞研究的重要因素。

來(lái)自美國(guó)普渡大學(xué)機(jī)械工程學(xué)院的Ray W. Herrick實(shí)驗(yàn)室的Sansit Patnaik教授團(tuán)隊(duì)，通過(guò)可變和分?jǐn)?shù)階算子對(duì)彈性動(dòng)力學(xué)問(wèn)題進(jìn)行了重新公式化，推導(dǎo)出一種獨(dú)特且功能強(qiáng)大的方法來(lái)模擬在動(dòng)態(tài)載荷下固體中裂縫的成核和擴(kuò)展。

其推導(dǎo)的最終動(dòng)態(tài)裂縫公式是完全演化的，因此可以分析復(fù)雜的裂縫模式，而無(wú)需對(duì)損傷位置和生長(zhǎng)路徑進(jìn)行任何先驗(yàn)假設(shè)，也無(wú)需使用任何算法來(lái)數(shù)值跟蹤演化的裂縫表面。該方法避免了額外的偏微分方程來(lái)預(yù)測(cè)損傷場(chǎng)的演化，從而顯著降低了復(fù)雜性和計(jì)算成本。

通過(guò)三個(gè)基準(zhǔn)實(shí)驗(yàn)，即1）Kalthoff-Winkler實(shí)驗(yàn)中涉及雙缺口試樣的沖擊剪切載荷，（2）動(dòng)態(tài)裂紋分支實(shí)驗(yàn)，（3）約翰-沙赫（John-Shah）實(shí)驗(yàn)的對(duì)比，驗(yàn)證了該理論計(jì)算框架的準(zhǔn)確性與有效性。本研究提出的基于變階力學(xué)的的理論框架，動(dòng)態(tài)斷裂的研究提供了強(qiáng)大的分析計(jì)算工具。

該文近期發(fā)表于npj Computational Materials 7： 27 （2021），英文標(biāo)題與摘要如下。

Variable-order fracture mechanics and its application to dynamic fracture

Sansit Patnaik & Fabio Semperlotti

This study presents the formulation， the numerical solution， and the validation of a theoretical framework based on the concept of variable-order mechanics and capable of modeling dynamic fracture in brittle and quasi-brittle solids.

More specifically， the reformulation of the elastodynamic problem via variable and fractional-order operators enables a unique and extremely powerful approach to model nucleation and propagation of cracks in solids under dynamic loading.

The resulting dynamic fracture formulation is fully evolutionary， hence enabling the analysis of complex crack patterns without requiring any a priori assumption on the damage location and the growth path， and without using any algorithm to numerically track the evolving crack surface.

The evolutionary nature of the variable-order formalism also prevents the need for additional partial differential equations to predict the evolution of the damage field， hence suggesting a conspicuous reduction in complexity and computational cost.

Remarkably， the variable-order formulation is naturally capable of capturing extremely detailed features characteristic of dynamic crack propagation such as crack surface roughening as well as single and multiple branching.

The accuracy and robustness of the proposed variable-order formulation are validated by comparing the results of direct numerical simulations with experimental data of typical benchmark problems available in the literature.

編輯：jq