Complexity and evolution in fracture mechanics

Professor Yokobori has long been involved in a wide range of disciplines in fracture. He has studied extensively high temperature creep induced rupture, fatigue rupture, interaction of creep and fatigue, fatigue crack growth under extreme temperatures, micro-mechanics of dislocations and dislocation clusters using both experiments and theories. Based on these studies, he put forward the philosophy of holism in fracture theory and founded the international academic journal “Strength, Fracture and Complexity”. He also gave honorary lectures on complex behavior of fracture in the International Conference on Fracture and disseminated these wise philosophical ideas to the fracture community.

He took the lead in founding the International Congress on Fracture (ICF). Together with other founding members, he drafted the Statutes of ICF, which sets out the goals, membership, organizational structure and operational mechanism of this academic body ensuring its long flourishing. For more than 50 years after the establishment of the ICF, fourteen conferences have been held to help exchange ideas, advance academic research, promote young researchers, opening up a journey towards a century-old society.

The philosophy and methodology put forward by Professor Yokobori also guide numerous researchers to study and advance fracture at the methodological level. As the objects of fracture study changed from brittle, quasi-brittle materials, to smart materials, low dimensional natural and natural-biological materials, the original analytical methods based on Newtonian mechanics gradually evolved to the holistic view of complex systems (or a combination of both). From a phenomenological point of view, fracture means that a material being divided into two or more. While in most applications, we want materials and structures to be integral. Studying the law of fracture and ensuring the completeness of materials and structures are the tasks of fracture research. The self-healing and self-growth phenomena in nature and biology have broadened the vision of the researchers in fracture. The goal of fracture research is to make human life safer and the researches should seek human well being.

Looking back over the past half century, the disciplines are more closely linked and the complexity of the system under study rises. The methodology of the holism theory for complex systems is becoming more and more concerned by researchers. A particularly important research aspect is how to describe the evolution of realistic phenomena. When mechanics or fracture mechanics are intersecting with other disciplines, a critical question is raised when quantitatively describing the many variables in these interdisciplinary subjects: One must provide the equation that describes the evolution of the system variables (especially for coupled systems). The new equation (or system of equations) that links the variables of different systems (or multiple coupled systems) is the equation of evolution. The equation of evolution has to answer two fundamental questions: the first is to specify which variables determine the variation of the state variables; the second is to provide the functional forms that the state variables follow.

A few examples of the equation of evolution have been discussed and illustrated in my honorary lecture on the Fourteenth International Conference on Fracture in 2017 [4], such as:

(a) Use the set theory of mathematics to predict the constitutive behavior of the quasi-brittle material via considering the domain of crack growth at the mesoscale [8];

(b) Through nanoscale friction tests, reveal chemical bond formation as a viable mechanism for the evolution effect in rock friction, which are critical for understanding the sliding behavior of faults during earthquakes [2];

(c) Quantitatively describe the transformation-induced toughening and evolution by introducing the internal variable representing the volume fraction of martensitic transformation at the mesoscale and adopting the irreversible thermodynamics [5];

(d) Capture the evolution behavior of domain switch volume fraction for ferroelectric materials with unconventional butterfly curve under cyclic electrical loading by extending the one-dimensional monotonic external loading scheme [9];

(e) Through cooperation with researchers in medical sciences, obtain a non-monotonic, highly non-linear bone evolution laws for describing bone injure and healing evolution when bones are subjected to external force and electromagnetic loads via observation and data fitting [3];

(f) Describe the enzymatic and collagen degradation behavior using the theory of irreversible thermodynamics and the maximum energy dissipation rate, which can help improve the drug deliver in tumor cells and reduce the residual stress of the extra-cellular matrix (ECM) [7].

The subjects in the above limited case studies have been extended from relatively simple quasi-brittle materials to complex biological systems. Example-(e) and example-(f) are typical examples of complex systems. This shows that the scope of fracture mechanics has been extended to from brittle and elastic-plastic materials to a wide range of geological materials and biomaterials, from man-made engineering materials to natural materials, faults, iceberg, and even open, non-equilibrium systems such as human body. The rich subjects invite more in-depth studies in fracture mechanics.

It is a more fundamental and more difficult and time-consuming task to study the damage evolution [1,6] from the perspective of statistical mechanics and statistical physics. I think this type of approach should be encouraged.

To study complex systems and to study the coupling of the systems, it is necessary to deduce the quantitative equations for evolution so that enough governing equations can be obtained for describing the realistic problems. This is the essential purpose why we have to discuss “complexity and evolution”.

Takeo Yokobori and his research group have long been working at Tohoku University in Sendai, Japan. I visited Sendai several times, either attending the ICF Executive Committee (2011) meeting and the annual meeting of the Japan Accreditation Board of Engineering Education (JABEE), or participating in the VAMAS High Temperature Strength of Materials Project organized by Professor Yokobori. During my visits to Sendai, I met Professor Yokobori several times and had the privilege to listen to his lectures, talk with him about how to better boost the ICF and better shape a scientific-democratic-harmonic academic atmosphere in the community. As the founding president and a senior member, he always gave very important advice.

Sendai is a city symbolizing the friendship between China and Japan. Mr. Xun Lu, an eminent Chinese writer, had a famous article, “Mr. Fujino”, describing how Mr. Lu met his teacher when he studied medicine in Japan and how the teacher Mr. Fujino guided him to grow up. This article has been included in textbooks in Chinese secondary and high schools for many years. I have read this article myself and was deeply touched by the benevolent teacher. During my visit to Sendai, I devoted myself to the place where Mr. Xun Lu was studying, which naturally reminded me about Mr. Takeo Yokobori. Professor Yokobori has made enormous contributions in scientific discoveries and advancement of research methodology as well as academic organizations to the international fracture community. His contributions deserve to be forever remembered by future generations and the development of fracture will be engraved with the footprints of Mr. Takeo Yokobori.