This work presents a new method to estimate the stress intensity factor (K) in notches from direct measurements of Crack Opening Profile (COP), defined as the opening of the notch faces along the entire notch front. This method was introduced by Allahdiniyan and Taylor (2025) for application to sharp cracks. In the present work, we extended the method to notches considering U-shaped and elliptical notches in three configurations: a single notch at the centre of a finite-width plate, a single edge notch on the plate and double-edge notches. Finite Element Analysis demonstrated that crack-based equations work reasonably well in modelling the COP and estimating K from COP for notches which were small (in relation to plate width) and sharp (small ratio of root radius to length). For blunter notches the COPs were significantly different, leading to new, correct formulae to describe them. This approach models COP with errors <5% for sharper notches and < 10% for blunter notches. COP-derived K closely matches equivalent K from the J-integral method in most cases (max error 24% in challenging situations). Practical applicability was demonstrated on notched PMMA specimens using 2D Digital Image Correlation, proving the equations effectively predict COP and local K values. This study demonstrates the practical value of the COP method for estimating K through simple displacement measurements without requiring stress analysis or complex simulations, with broad applicability across engineering scenarios.

The Theory of Critical Distances comprises several methodologies that allow fracture, fatigue, and stress corrosion cracking phenomena to be analyzed. Such methodologies are usually referred to as the Point Method (PM), the Line Method (LM), the Area Method (AM), and the Volume Method (VM). All of them provide analyses where the corresponding material resistance (e.g., fracture toughness, fatigue threshold, and stress corrosion cracking threshold) is used together with an additional material parameter with length units (the critical distance, L). Moreover, the accuracy of these four approaches is very similar, but the PM and the LM have a much simpler application. When dealing with fracture processes, the TCD allows fracture conditions for structural materials in the presence of notch-type defects to be established, and simple formulas for estimating the apparent fracture toughness (i.e., the fracture resistance in the presence of notches) to be obtained. This paper provides a number basic reasonings related to both the PM and/or the LM formulations that allow different straightforward conclusions to be derived, with significant theoretical and practical implications. Real cases with experimental results are also included, exemplifying what is discussed in the theoretical analysis.

The limpet Patella vulgata increases in size throughout its life, leading to a distribution of sizes for a given population and location. This paper presents a theoretical model in which the increase in shell size after maturity is modelled as a response to risks which are stochastic in nature and size-related. Among these is the risk of impact damage caused by rocks and other projectiles during storms. Data were collected to determine the size distribution and growth rates of limpet shells as well as the sizes of potential projectiles at a site on the Irish coast. Impact strength (energy to cause failure) was measured as a function of shell size. A theoretical model was developed which was able to accurately predict the size distribution (relative number of shells of a given size) for mature limpets (shell length of 25 mm and above). The model was able to demonstrate the value of continued growth: it predicted that if growth were to stop at maturity (25 mm), then the density of limpets at this site would decrease by 37%. Comparison with published data from other sites showed a consistent relationship between size distribution and the potential for storm-related impacts. Possible effects of climate change (increasing the number and energy of storms) could also be estimated. The model can potentially be expanded to cover other size-related risks (predation, food supply, etc.). This work is a contribution to our understanding of the strategies that organisms have evolved to resist the effects of mechanical forces.

In 1978 Rod Smith published a paper, co-authored by Keith Miller, on the subject of notch fatigue limits. Of the many valuable contributions which Rod made to our field, this paper stands out for me. I believe it to be an excellent example of good science, bringing clarity to a field which was (and still is) characterized by complexity. In what follows I introduce the paper and in particular a type of plot used in it, which I propose to name the Smith-Miller diagram. This diagram is of great practical use in engineering design for the assessment of stress concentration features and also has applications in forensic failure analysis and in undergraduate teaching, areas in which Rod Smith also excelled.

Hair can become damaged and break as a result of mechanical actions such as brushing. Individual hairs (known as hair fibres) are usually tested to failure in tension but this does not reflect the type of loading to which they are normally subjected. Previously, we proposed a new test - the Moving Loop fatigue test - which simulates the extreme bending of tangled hair during repeated brushing. Previous results showed that this test is capable of generating longitudinal splits, simulating the phenomenon of split ends. In the present paper we report further results from this test method, expanding the number of hair types investigated and demonstrating the dependence on applied force and effects of environmental treatments (combinations of hydration and heating). In addition to recording the number of cycles to failure we also used interrupted testing to investigate the mechanisms of damage initiation and propagation. We found that cracks can initiate in one of three interfaces - cuticle/cuticle, cuticle/cortex and cortex/cortex. The first two result in splits which start at or near the hair surface and propagate across the hair fibre to cause fracture, whilst the cortex/cortex-initiated splits propagate along the hair to macroscopic lengths. We attribute these differences in behaviour to differing anisotropy of hair strength, due to varying bond strengths in the cell-membrane complexes in these three interfaces. Computer simulation using finite element analysis provided insights into the distribution of tensile and shear stress and the effects of anisotropy on failure modes.

The material properties of individual micro-struts are critical to the overall success of lattice structures. These properties can be significantly compromised by defects inherited from powder bed fusion processes. Among these defects, porous inclusions are well understood to have a detrimental effect on mechanical properties; posing a high risk to the implant under loading. While the majority of these defects can be avoided through optimisation of printing parameters, this has generally only been done for traditional bulk components with no in-designed porosity. Furthermore, a number of studies have observed changes in the frequency of such porous inclusions as feature size is reduced, indicating a size effect. This also suggests that the optimal parameters for bulk material are not necessarily translatable to the individual micro-struts which build the lattice. In this study, the relationship between parameter optimisation and feature size was investigated. Here, a higher energy density input was required for processing micro-strut lattices with an optimised relative density, than it was for bulk components. This could be attributed to faster rates of heat loss in micro-strut samples on account of their increased surface-to-volume ratio. The consequential improvement in mechanical properties was also assessed. An increase in both strength and stiffness could be largely attributed to an increase in the percentage volume of load bearing material, while improvements in failure strain were largely driven by minimisation of stress concentrations around the irregular pore morphologies. Fatigue properties did not improve beyond the effects of yielding. Rather, crack initiation was dominated by surface defects; which on account of their surface free energy, experience a much higher stress intensity factor.

Potential effects of climate change include greater extremes of temperature and increased severity of storms. Many plants have evolved to resist the challenges of winter (freezing, dehydration, and wind) in a process known as cold hardening. Sensing reducing temperatures, they make structural changes at the cellular level to increase their mechanical resistance and prevent damage. Previous work on this topic, though extensive, has been conducted under laboratory conditions rather than in the field, and while many workers have observed changes to cell wall thickness and composition, which imply increased mechanical strength, few have actually measured strength or any other parameter describing structural integrity. This paper describes experiments on a model system designed to measure the structural integrity of leaf laminae from plants growing naturally in the field over extended periods, allowing seasonal variations to be captured. Standard engineering properties-tensile strength and fracture toughness-were measured for leaves of Griselinia littoralis on 19 separate occasions over a 12-month period. Toughness (rather than strength) was found to be the controlling mechanical property. Toughness values were found to change significantly during the year, by more than a factor of 2. Toughness correlated strongly with average daily soil temperature, but with a lag of about 1-2 weeks, suggesting that this is the time needed for structural adjustments to take place. Highest toughness values occurred in winter, confirming cold hardening. Increasing temperature in the spring was associated with decreasing toughness, but in the summer, when highest temperatures occurred, toughness increased again. This apparent "hot hardening" may be a response to dehydration. Results imply that a given leaf is able to both increase and decrease its toughness in response to temperature changes, demonstrating excellent plasticity of response. This case study of a single species establishes a method of reliably measuring changes in a plant's structural integrity due to cold hardening and other seasonal variations, which may be used to investigate the effects of climate change and other variables.

Splitting of hair, creating ‘split ends’, is a very common problem which has been extensively documented. However, the mechanics underlying the splitting phenomenon are poorly understood. This is partly owing to the lack of a test in which splitting can be generated and quantified under laboratory conditions. We developed three new tests, known as ‘loop tensile’, ‘moving loop’ and ‘moving loop fatigue’, aiming to simulate the mechanical environment of tangles of hair strands during combing. We tested straight strands of human hair, comparing low-quality hair (from a subject who experienced split ends) with hair from a control (non-splitting) subject. Significant differences were found, especially in the moving loop fatigue test where the low-quality hair failed in fewer cycles. Splitting occurred in both types of hair, but with the crucial difference that in the low-quality hair, splits originated inside the hair strand and propagated longitudinally over considerable distances, while in the control hair, splits originated at the strand surface and remained short. Bleaching of the control hair changed its behaviour, making it similar to that of the low-quality hair. Some simple calculations emphasized the role of longitudinal shear stress and shear stress intensity in generating microcracks which could then propagate within the moving loop, paving the way for a future theoretical model of the splitting mechanism.

Mechanical forces applied over a period of time tend to cause fatigue failure in natural organisms and in engineering structures. Here, the theoretical approach known as Continuum Damage Mechanics is applied to study fatigue damage development in trees. It is found that growth in the form of an annual ring of new material is a very effective strategy to limit fatigue damage, due to the fact that, over time, each ring moves inside the trunk, reducing stress. If (as is generally assumed) the tree grows so as to keep the bending stress on its trunk constant, then fatigue failure will be effectively impossible until the tree is very old. One interpretation of this finding is that high cycle fatigue simply never occurs in trees: they don't accumulate fatigue damage but rather fail by instantaneous overload or low cycle fatigue during a single storm. Another interpretation is that the bending stress is maybe not kept constant but changes as the tree grows, which would be a more efficient strategy making the best use of material. These findings are considered using data from the literature and their implications for the creation of biomimetic products are discussed. Possible experiments to test these theoretical predictions are suggested.

Plant stems have evolved to withstand mechanical forces and survive damaging events. This study investigates the ability of plant stems to resist the impacts of external damage on their mechanical integrity. The resistance is defined as defect tolerance in this study (since damage or defect tolerance is a property of a structure relating to its ability to sustain defects safely until repair can take place). We developed a new approach for quantifying defect tolerance as a structural attribute, based on relative changes to various mechanical properties including slope of elastic regions, energy to yield and maximum force. Stems from fuchsia (Fuchsia magellanica), elder (Sambucus nigra) and ash (Fraxinus excelsior) were tested in three-point bending. Results were compared to idealised engineering materials obtained from finite element analysis. Three-point bend tests showed the defect tolerance of plant stems is exceptional, exceeding the peak behaviour derived from simulations of idealised materials. This was attributed to: material anisotropy, differences between tensile and compressive behaviour and inbuilt residual stress. Imaging techniques (microscopy and microCT) contributed to understanding how both structural and material properties determine performance. This advances our understanding of plants superior defect tolerance and may assist in the development of future engineering materials.

Powder Bed Fusion (PBF) additive manufacturing techniques have enabled the fabrication of geometrically complex porous metallic lattice structures. Despite the many advantages of this approach, surface and near surface defects inherited from the fabrication process act to limit their structural and mechanical integrity. Consequently, surface cleaning techniques such as chemical etching have been increasingly adopted post process. Although the benefits of this technique have been captured across a range of lattice structures, the existing body of work is typically a characterisation of the engineering mechanical properties of lattices, rather than the bulk mechanical properties of the material. From a design perspective, characterisation of such properties is fundamental to prediction of mechanical performance and failure mechanisms.In this study, standardised micro-strut dog-bone geometries were adopted as a means of isolating geometrical variations between lattice designs; allowing us to study the bulk mechanical properties of the material. Incre-mental etching was conducted and changes to both the surface and morphological properties of the micro-structs was characterised. Improvements in these factors was associated with a corresponding increase in mechanical properties. Both strength and stiffness increased following removal of process inherited surface defects, which can be linked to a removal of non-load bearing material. Failure strain and fatigue resistance also improved following surface etching, although similar fracture surfaces were observed in both groups. Together these findings demonstrate the benefits of chemical etching for improving the mechanical properties of additively manufactured Ti-6Al-4V lattice structures.

This work aims to apply the Theory of Critical Distances (TCD) to the fatigue assessment of additively manufactured (AM) Ti-6Al-4V material produced via the selective laser melting (SLM) process. Modified alternatives to traditional TCD methods are considered. In this sense, it is sought to develop a fatigue prediction model that is better suited to assessing the impact of multiple stress-rising features which are located in close proximity to each other. Hereby, consideration has been given to modelling process-inherent surface roughness in combination with an internally positioned artificial defect, shaped as a feature that is reminiscent of a pore. Simultaneously, the research also seeks to circumnavigate a potential issue with respect to the current TCD methodology. This concerns the matter of applying TCD practices to components whereby the area of interest for conducting stress-distance analytics is on a size scale that is smaller than that of the critical distance length parameter itself. Several different strategies were attempted as a way to try and achieve meaningful modifications to the TCD process. Results show that it is possible to overcome such challenges that can often present themselves during the fatigue appraisal of AM metal parts. In this sense, the optimal novel strategy that was experimented with returned average error margins of 13.7% or better. It is anticipated that such models may assist in further optimising the accuracy of service life evaluation for metallic AM components that are intended for industry.

The Theory of Critical Distances (TCD) has undoubtedly represented a breakthrough in the brittle failure assessment of engineering materials containing defects, crack, or notches. The basic idea on which the simplest formulation of the TCD is based is to evaluate an effective stress at a characteristic distance from the tip of the defect/crack/notch and compare it with an inherent fracture strength. Is the critical distance related to the material (micro) structure? Whereas a correlation was already proved for homogeneous materials, the current attention to nonhomogeneous ones has brought the question back to the fore. The goal of the present work is therefore twofold: (i) to extend the use of the TCD, through the simple yet effective Point Method (PM), for the static failure assessment of inhomogeneous materials, such as cellular, biological, and additively manufactured (AM) materials; and (ii) to look for a correlation between critical distance and internal (micro) structure.

Previous research has presented the concept of self-engineering (SE) systems that aim to identify and preserve system functions autonomously. Examples of self-engineering responses include self-healing, self-repair, self-adapting and self-reconfiguration. Biology already utilises many of these responses to repair and survive, greater understanding of complexity in these biological systems could improve future bioinspired designs. This paper provides a novel systematic evaluation of the complexity of SE biological systems. Eight biological self-engineering systems identified are evaluated using Axiomatic design and complexity. The key functional requirements and design parameters for each biological system are identified. Design matrices were used to highlight different types of complexity. A further evaluation of eight SE biological systems is performed using the SE complexity theory; nine experts and 23 students used the complexity theory to complete a ranking exercise. The results of the ranking were analysed and compared, with a final normalised mean plotted for each factor and biological system. From the analysis of both studies, proposed design rules are presented to help designers handle complexity while creating new self-engineering systems inspired by biology.

EDITORIAL article Front. Plant Sci., 24 March 2023Sec. Plant Development and EvoDevo Volume 14 - 2023 | https://doi.org/10.3389/fpls.2023.1181342

Selective Laser Melting is an additive manufacturing practice that permits the production of metal alloy-based parts. While facilitating the design of complex geometry, SLM leads to the fabrication of a unique material structure that showcases distinct behavioural characteristics relative to their traditional methods of material manufacture. Defects that are innate to SLM inspire the presence of a compositional outlook that is inhomogeneous in nature and only serves to hinder part efficiency. Thus, the Theory of Critical Distances offers a refreshed proposal to evaluating notched Ti-6Al-4V material produced by additive manufacturing processes. Key principles of the theory’s working mechanisms are outlined. Subsequently, symmetrical notches of contrasting size are assessed. Findings reveal that the Theory of Critical Distances is adequately compatible with accurate fatigue prediction of SLM Ti-6Al-4V in its as-built state. Additionally, fracture surface analysis reveals that crack initiation is predominantly a surface-based phenomenon. Hereby, increased focus must be given to the quality of processed material that is located at the externalities of additively manufactured components, in order to enhance their service life capabilities. This will induce an increasingly uniform material structure that will allow for more predictable behavioural characteristics.

Robust and simple failure prediction is a relevant and active field of research. The present work aims at unifying the Theory of Critical Distances (TCD) and Averaged Strain Energy Density (ASED) criterion. The proposed Energy TCD is formulated along with analytical expressions for the characteristic length scales and validated using experimental data. On this data set, the accuracy of the Energy TCD is similar to that of the classical methods. Most importantly, the presented methods allow to transfer the point-based calibration technique from notched specimens, which is common for the TCD, to the Energy TCD ASED criterion.

The Theory of Critical Distances (TCD) is a well-established method for predicting the effect of notches and other stress concentration features on the fracture and fatigue strength of materials and structures. It has been applied to fibre composite laminate materials but only in a limited manner: previous work was largely limited to two particular cases: circular holes and sharp cracks, both being loaded in axial tension. Long-fibre composites are increasingly replacing metals in many applications, but they are highly sensitive to stress concentration and damage. The present work aimed to develop a more complete picture of notch effects and to investigate the accuracy of the TCD as a predictive tool for this class of materials. Using a quasi-isotropic carbon-fibre/epoxy laminate, we carried out experiments on samples containing two types of stress concentration (notches and corners), varying notch length from 1 to 10 mm and root radius from 0.5 to 20 mm giving stress concentration (Kt) factors from 1.16 to 3.92. Samples were tested in axial tension, in axial torsion and also in mixed tension/torsion loadings. Finite element analysis (FEA) was used to determine stresses in the vicinity of the notches. Predictions using the TCD with a constant value of the critical distance L, were found to be very accurate for the specimens loaded in tension. Specimens loaded in torsion tended to fail at applied torques which were higher than predicted by the TCD, by factors of the order of two. This difference may be explained by the three-dimensional nature of these stress concentrations. The torsion failures could be predicted accurately by using different constants from those used for tension, but this solution is not satisfactory in the general case and should be investigated further.

This work shows an analysis of several models of multiaxial fatigue for notches: Navarro-Rios’ model, which analyses the interaction between the crack and its associated plastic zone with the material microstructural barriers, and three models that combine a critical volume method for notches with a critical plane model for multiaxial fatigue in unnotched solids. Specifically, the application of these models for the prediction of the fatigue limit for a plate with a circular hole subjected to axial, shear and in-phase biaxial cyclic loading is studied. The effects of two parameters are analysed: the radius of the hole and the relationship between the torsional and axial fatigue limits. For all the analysed models, cases are observed in which an increase in the hole radius produces an increase in the predicted fatigue limit, that is, the evolution of the fatigue limit with an increasing hole radius is not always monotonically decreasing, as would be expected. These effects, which we have called “humps” because of their appearance on the prediction graphs, mainly occur in shear loading. No humps were observed in the studied experimental results, but the number of available experimental results is too small to assure this tendency. The results shown in the work indicate that a greater knowledge of the physics of multiaxial fatigue in notches is necessary to achieve models that are capable of providing increasingly accurate predictions.

Additive manufacturing techniques such as selective laser melting enable the production of customised components with high geometrical freedom. However, SLM results in a material condition with different properties to their conventionally manufactured counterparts. The presence of process-inherent defects can significantly impact the degradation of part performance. Hereby, a novel approach to assessing notched SLM Ti-6Al-4V material via a critical distance theory is presented. Geometrical notches of varying size are evaluated. Results show that the Theory of Critical Distances is appropriately applicable to fatigue prediction of SLM Ti-6Al-4V in its as-built state.