• Notch Signalling in Breast Development and Cancer

      Edwards, Abigail; Brennan, Keith; email: keith.brennan@manchester.ac.uk (Frontiers Media S.A., 2021-07-06)
      The Notch signalling pathway is a highly conserved developmental signalling pathway, with vital roles in determining cell fate during embryonic development and tissue homeostasis. Aberrant Notch signalling has been implicated in many disease pathologies, including cancer. In this review, we will outline the mechanism and regulation of the Notch signalling pathway. We will also outline the role Notch signalling plays in normal mammary gland development and how Notch signalling is implicated in breast cancer tumorigenesis and progression. We will cover how Notch signalling controls several different hallmarks of cancer within epithelial cells with sections focussed on its roles in proliferation, apoptosis, invasion, and metastasis. We will provide evidence for Notch signalling in the breast cancer stem cell phenotype, which also has implications for therapy resistance and disease relapse in breast cancer patients. Finally, we will summarise the developments in therapeutic targeting of Notch signalling, and the pros and cons of this approach for the treatment of breast cancer.
    • Notes on the genus Chinattus Logunov, 1999 from India, Pakistan and Nepal (Arachnida: Araneae: Salticidae).

      Logunov, Dmitri V; email: dmitri.v.logunov@manchester.ac.uk (2021-07-29)
      A new speciesChinattus mikhailovi sp. nov. (♂♀, from Pakistan, Peshawar)is diagnosed, described and illustrated. New records of Chinattus validus (Xie, Peng et Kim, 1993) from Nepal, India (Himachal Pradesh) and Vietnam, and C. chichila Logunov, 2003 from Nepal are presented. The collecting localities of all three species are mapped. A brief discussion of the state of knowledge of the genus Chinattus Logunov, 1999 is provided as well.
    • Nothing about us without us: involving patient collaborators for machine learning applications in rheumatology.

      Shoop-Worrall, Stephanie J W; orcid: 0000-0002-9441-5535; email: stephanie.shoop-worrall@manchester.ac.uk; Cresswell, Katherine; Bolger, Imogen; Dillon, Beth; Hyrich, Kimme L; orcid: 0000-0001-8242-9262; Geifman, Nophar; Members of the CLUSTER consortium (2021-07-05)
      Novel machine learning methods open the door to advances in rheumatology through application to complex, high-dimensional data, otherwise difficult to analyse. Results from such efforts could provide better classification of disease, decision support for therapy selection, and automated interpretation of clinical images. Nevertheless, such data-driven approaches could potentially model noise, or miss true clinical phenomena. One proposed solution to ensure clinically meaningful machine learning models is to involve primary stakeholders in their development and interpretation. Including patient and health care professionals' input and priorities, in combination with statistical fit measures, allows for any resulting models to be well fit, meaningful, and fit for practice in the wider rheumatological community. Here we describe outputs from workshops that involved healthcare professionals, and young people from the Your Rheum Young Person's Advisory Group, in the development of complex machine learning models. These were developed to better describe trajectory of early juvenile idiopathic arthritis disease, as part of the CLUSTER consortium. We further provide key instructions for reproducibility of this process.Involving people living with, and managing, a disease investigated using machine learning techniques, is feasible, impactful and empowering for all those involved. [Abstract copyright: © Author(s) (or their employer(s)) 2021. Re-use permitted under CC BY. Published by BMJ.]
    • Novel methodology to assess the effect of contouring variation on treatment outcome

      Jenkins, Alexander; Mullen, Thomas Soares; Johnson‐Hart, Corinne; Green, Andrew; McWilliam, Alan; Aznar, Marianne; van Herk, Marcel; Vasquez Osorio, Eliana; email: eliana.vasquezosorio@manchester.ac.uk (2021-04-24)
      Purpose: Contouring variation is one of the largest systematic uncertainties in radiotherapy, yet its effect on clinical outcome has never been analyzed quantitatively. We propose a novel, robust methodology to locally quantify target contour variation in a large patient cohort and find where this variation correlates with treatment outcome. We demonstrate its use on biochemical recurrence for prostate cancer patients. Method: We propose to compare each patient’s target contours to a consistent and unbiased reference. This reference was created by auto‐contouring each patient’s target using an externally trained deep learning algorithm. Local contour deviation measured from the reference to the manual contour was projected to a common frame of reference, creating contour deviation maps for each patient. By stacking the contour deviation maps, time to event was modeled pixel‐wise using a multivariate Cox proportional hazards model (CPHM). Hazard ratio (HR) maps for each covariate were created, and regions of significance found using cluster‐based permutation testing on the z‐statistics. This methodology was applied to clinical target volume (CTV) contours, containing only the prostate gland, from 232 intermediate‐ and high‐risk prostate cancer patients. The reference contours were created using ADMIRE® v3.4 (Elekta AB, Sweden). Local contour deviations were computed in a spherical coordinate frame, where differences between reference and clinical contours were projected in a 2D map corresponding to sampling across the coronal and transverse angles every 3°. Time to biochemical recurrence was modeled using the pixel‐wise CPHM analysis accounting for contour deviation, patient age, Gleason score, and treated CTV volume. Results: We successfully applied the proposed methodology to a large patient cohort containing data from 232 patients. In this patient cohort, our analysis highlighted regions where the contour variation was related to biochemical recurrence, producing expected and unexpected results: (a) the interface between prostate–bladder and prostate–seminal vesicle interfaces where increase in the manual contour relative to the reference was related to a reduction of risk of biochemical recurrence by 4–8% per mm and (b) the prostate's right, anterior and posterior regions where an increase in the manual contour relative to the reference contours was related to an increase in risk of biochemical recurrence by 8–24% per mm. Conclusion: We proposed and successfully applied a novel methodology to explore the correlation between contour variation and treatment outcome. We analyzed the effect of contour deviation of the prostate CTV on biochemical recurrence for a cohort of more than 200 prostate cancer patients while taking basic clinical variables into account. Applying this methodology to a larger dataset including additional clinically important covariates and externally validating it can more robustly identify regions where contour variation directly relates to treatment outcome. For example, in the prostate case we use to demonstrate our novel methodology, external validation will help confirm or reject the counter‐intuitive results (larger contours resulting in higher risk). Ultimately, the results of this methodology could inform contouring protocols based on actual patient outcomes.
    • Novel Reviews

      Toivanen, Anna-Leena; email: anna-leena.toivanen@uef.fi; Taylor, Joanna E.; email: joanna.taylor@manchester.ac.uk (Berghahn Books, 2021-03-01)
      Michèle Rakotoson, Elle, au printemps (Saint-Maur: Sépia, 1996), 122 pp.Kathleen Jamie, Surfacing (UK: Sort of Books, 2019), 240 pp., £7.99. Kathleen Jamie (ed.), Antlers of Water: Writing on the Nature and Environment of Scotland (Edinburgh: Canongate Books, 2020), 232pp., £20.
    • Novel Rotational Combination Regimen of Skin Topicals Improves Facial Photoaging: Efficacy Demonstrated in Double-Blinded Clinical Trials and Laboratory Validation

      DiNatale, Lisa; Idkowiak-Baldys, Jolanta; Zhuang, Young; Gonzalez, Anthony; Stephens, Thomas J.; Jiang, Lily I.; Li, Weiping; Basson, Rubinder; Bayat, Ardeshir; email: ardeshir.bayat@manchester.ac.uk; email: ardeshir.bayat@uct.ac.za (Frontiers Media S.A., 2021-09-17)
      Topical antiaging products are often a first-line intervention to counter visible signs of facial photoaging, aiming for sustained cosmetic improvement. However, prolonged application of a single active topical compound was observed clinically to lead to a plateau effect in improving facial photoaging. In view of this, we set out to reduce this effect systematically using a multi-tiered approach with laboratory evidence and clinical trials. The objective of the study was to evaluate the effects of active topical ingredients applied either alone, in combination, or in a rotational manner on modulation of facial photoaging. The study methodology included in vitro, organotypic, and ex vivo skin explants; in vivo biopsy study; as well as clinical trials. We demonstrate for the first time that a pair of known antiaging ingredients applied rotationally, on human dermal fibroblasts, maximized pro-collagen I production. Indeed, rotational treatment with retinol and phytol/glycolic acid (PGA) resulted in better efficacy than application of each active ingredient alone as shown by explants and in vivo biopsy study, with penetration of active ingredients confirmed by Raman spectroscopy. Furthermore, two split-face, randomized, double-blinded clinical trials were conducted, one for 12 months to compare treated vs. untreated and the other for 6 months followed by a 2-month regression to compare treated vs. commercially marketed products. In both studies, rotational regimen showed superior results to its matching comparison as assessed by clinical grading and image analysis of crow's feet wrinkles. In conclusion, rotational regimen using retinol and PGA is effective in treating facial photoaging signs with long-lasting benefits.
    • Now, Imagine an Actually Existing Unicorn: On Russellian Worries for Modal Meinongianism

      de Jong, Andreas; email: andreas.dejong@manchester.ac.uk (Springer Netherlands, 2020-04-23)
      Abstract: Modal Meinongianism provides the semantics of sentences involving intentional verbs Priest (Towards Nonbeing, Oxford University Press, Oxford, 2016). To that end, Modal Meinongianism employs a pointed non-normal quantified modal logic model. Like earlier Meinongian views Modal Meinongianism has a characterisation principle (QCP), that claims that any condition whatsoever is satisfied by some object in some world. Recently, Everett (The nonexistent, Oxford University Press, Oxford, 2013, 169, p. 36) has proposed an argument against QCP that, if successful, gives rise to problems identical to those Russell (Mind 14:530–538, 1905, p. 533) raised for Naïve Meinongianism, namely that it allows for true contradictions, and allows us to define anything into existence. Everett claims that the ordinary meanings of “actual” license an inference pattern, such that if an object satisfies Actual A at some world, then that object satisfies A in the actual world. Given that actual world is the designated point of evaluation for truth simpliciter, QCP would fall prey to Russell’s criticisms. As opposed to Everett, I argue that, even if we grant Everett the assumption that “actual” is a modal indexical that rigidly refers to the actual world, it does not conform to the inference pattern above. This is because when an object satisfies Actual A at some world, this alters the assertoric force of “actual”, because “actual” is interpreted in the scope of some modal or intentional operator. I also explain that Everett’s proposed example carries existential commitment because the problematic noun-phrase occurs outside the scope of a modal or intentional operator.
    • Nuclear spin relaxation as a probe of zeolite acidity: a combined NMR and TPD investigation of pyridine in HZSM-5.

      Robinson, Neil; orcid: 0000-0002-0893-2190; Bräuer, Pierre; York, Andrew P E; D'Agostino, Carmine; orcid: 0000-0003-3391-8320; email: carmine.dagostino@manchester.ac.uk (2021-06-30)
      The relative surface affinities of pyridine within microporous HZSM-5 zeolites are explored using two-dimensional 1H nuclear magnetic resonance (NMR) relaxation time measurements. The dimensionless ratio of longitudinal-to-transverse nuclear spin relaxation times T1/T2 is shown to exhibit strong sensitivity to the silica/alumina ratio (SAR) of these zeolites, which is indicative of material acidity. This trend is interpreted in terms of increased pyridine surface affinity with decreasing SAR. Temperature programmed desorption (TPD) analysis corroborates this observation, revealing a distinct increase in the heat of desorption associated with adsorbed pyridine as a function of decreasing SAR. A direct correlation between NMR and TPD data suggests NMR relaxation time analysis can be a valuable tool for the non-invasive characterisation of adsorption phenomena in microporous solids.
    • Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

      Green, Charles Wing Ho; email: 1604518@chester.ac.uk; Liu, Yanzhi; email: 39036@llu.edu.cn; Yan, Yubin; orcid: 0000-0002-5686-5017; email: y.yan@chester.ac.uk (MDPI, 2021-10-27)
      We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr, j=0, 1, 2, …, N with a≥1 and r≥1, where loga=logt0logt1⋯logtN=logT is a partition of [logt0, logT]. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes logtj=loga+logtNaj(j+1)N(N+1), j=0, 1, 2, …, N. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., DCHa, tαy(t)∉C1[a, T] with α∈(0, 2), the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r≥1. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.
    • Nutrition knowledge and dietary intake of hurlers

      Murphy, John; orcid: 0000-0002-8337-722X; O’Reilly, James (SAGE Publications, 2020-11-26)
      The current study investigated the association between sports nutrition knowledge and dietary quality in a sample of adult Irish male hurling players. Nutrition knowledge was measured by the validated Sports Nutrition Knowledge Questionnaire (SNKQ). Diet quality was measured by the Australian Recommended Food Score (ARFS) calculated from food frequency questionnaire data. Analysis of variance and linear modelling were used to assess associations between variables. A total of 265 (129 elite, 136 sub-elite) players were recruited. No significant difference in nutrition knowledge (SNKQ) was found between groups. Results showed a significant difference (p = 0.02; d = 0.39 ± 0.25; small) in food score (ARFS) between groups. A weak, positive association (r = 0.3, p = 0.007) was found between nutrition knowledge and food score. Elite level players, aged 28–32, with college degrees, that have previously received nutritional guidance displayed the highest levels of both nutrition knowledge and food score. Higher levels of nutrition knowledge and food score were expected in elite players, however were only found in food score. Nutrition knowledge does contribute to dietary quality although future interventions should focus on specific gaps in knowledge such as how to meet total energy/carbohydrate requirements.
    • Nutritional parameters and outcomes in patients admitted to intensive care with COVID-19: a retrospective single-centre service evaluation

      Eden, Timothy; McAuliffe, Shane; orcid: 0000-0002-7166-4299; Crocombe, Dominic; Neville, Jonathan; Ray, Sumantra (BMJ Publishing Group, 2021-08-05)
      Background: COVID-19 is an inflammatory syndrome caused by novel coronavirus SARS-CoV-2. Symptoms range from mild infection to severe acute respiratory distress syndrome (ARDS) requiring ventilation and intensive care. At the time of data collection, UK cases were around 300 000 with a fatality rate of 13% necessitating over 10 000 critical care admissions; now there have been over 4 million cases. Nutrition is important to immune function and influences metabolic risk factors such as obesity and glycaemic control, as well as recovery from acute illnesses. Poor nutritional status is associated with worse outcomes in ARDS and viral infections, yet limited research has assessed pre-morbid nutritional status and outcomes in patients critically unwell with COVID-19. Objectives: Investigate the effect of body mass index (BMI), glycaemic control and vitamin D status on outcomes in adult patients with COVID-19 admitted to an intensive care unit (ICU). Methods: Retrospective review of all patients admitted to a central London ICU between March and May 2020 with confirmed COVID-19. Electronic patient records data were analysed for patient demographics; comorbidities; admission BMI; and serum vitamin D, zinc, selenium and haemoglobin A1c (HbA1c) concentrations. Serum vitamin D and HbA1c were measured on admission, or within 1 month of admission to ICU. Primary outcome of interest was mortality. Secondary outcomes included time intubated, ICU stay duration and ICU-related morbidity. Results: Seventy-two patients; 54 (75%) men, mean age 57.1 (±9.8) years, were included. Overall, mortality was 24 (33%). No significant association with mortality was observed across BMI categories. In the survival arm admission, HbA1c (mmol/mol) was lower, 50.2 vs 60.8, but this was not statistically significant. Vitamin D status did not significantly associate with mortality (p=0.131). However, 32% of patients with low vitamin D (<25 IU/L) died, compared with 13% of patients with vitamin D levels >26 IU/L. Serum zinc and selenium, and vitamin B12 and folate levels were measured in 46% and 26% of patients, respectively. Discussion/conclusion: Increased adiposity and deranged glucose homeostasis may potentially increase risk of COVID-19 infection and severity, possibly relating to impaired lung and metabolic function, increased proinflammatory and prothrombotic mechanisms. Vitamin D deficiency may also associate with poorer outcomes and mortality, supporting a possible role of vitamin D in immune function specific to pulmonary inflammation and COVID-19 pathophysiology. There are plausible associations between raised BMI, glycaemic control, vitamin D status and poor prognosis, as seen in wider studies; however, in this service evaluation audit during the first wave of the pandemic in the UK, with a limited data set available for this analysis, the associations did not reach statistical significance. Further research is needed into specific nutritional markers influencing critical care admissions with COVID-19.
    • Nutritional parameters and outcomes in patients admitted to intensive care with COVID-19: a retrospective single-centre service evaluation

      Eden, Timothy; McAuliffe, Shane; orcid: 0000-0002-7166-4299; Crocombe, Dominic; Neville, Jonathan; Ray, Sumantra (BMJ Publishing Group, 2021-08-05)
      Background: COVID-19 is an inflammatory syndrome caused by novel coronavirus SARS-CoV-2. Symptoms range from mild infection to severe acute respiratory distress syndrome (ARDS) requiring ventilation and intensive care. At the time of data collection, UK cases were around 300 000 with a fatality rate of 13% necessitating over 10 000 critical care admissions; now there have been over 4 million cases. Nutrition is important to immune function and influences metabolic risk factors such as obesity and glycaemic control, as well as recovery from acute illnesses. Poor nutritional status is associated with worse outcomes in ARDS and viral infections, yet limited research has assessed pre-morbid nutritional status and outcomes in patients critically unwell with COVID-19. Objectives: Investigate the effect of body mass index (BMI), glycaemic control and vitamin D status on outcomes in adult patients with COVID-19 admitted to an intensive care unit (ICU). Methods: Retrospective review of all patients admitted to a central London ICU between March and May 2020 with confirmed COVID-19. Electronic patient records data were analysed for patient demographics; comorbidities; admission BMI; and serum vitamin D, zinc, selenium and haemoglobin A1c (HbA1c) concentrations. Serum vitamin D and HbA1c were measured on admission, or within 1 month of admission to ICU. Primary outcome of interest was mortality. Secondary outcomes included time intubated, ICU stay duration and ICU-related morbidity. Results: Seventy-two patients; 54 (75%) men, mean age 57.1 (±9.8) years, were included. Overall, mortality was 24 (33%). No significant association with mortality was observed across BMI categories. In the survival arm admission, HbA1c (mmol/mol) was lower, 50.2 vs 60.8, but this was not statistically significant. Vitamin D status did not significantly associate with mortality (p=0.131). However, 32% of patients with low vitamin D (<25 IU/L) died, compared with 13% of patients with vitamin D levels >26 IU/L. Serum zinc and selenium, and vitamin B12 and folate levels were measured in 46% and 26% of patients, respectively. Discussion/conclusion: Increased adiposity and deranged glucose homeostasis may potentially increase risk of COVID-19 infection and severity, possibly relating to impaired lung and metabolic function, increased proinflammatory and prothrombotic mechanisms. Vitamin D deficiency may also associate with poorer outcomes and mortality, supporting a possible role of vitamin D in immune function specific to pulmonary inflammation and COVID-19 pathophysiology. There are plausible associations between raised BMI, glycaemic control, vitamin D status and poor prognosis, as seen in wider studies; however, in this service evaluation audit during the first wave of the pandemic in the UK, with a limited data set available for this analysis, the associations did not reach statistical significance. Further research is needed into specific nutritional markers influencing critical care admissions with COVID-19.
    • Occlusion: is there a third way? A discussion paper.

      Davies, Stephen; email: stephen.j.davies@manchester.ac.uk; Dawson, Philip; Weerapperuma, Indika; Waring, David; Beddis, Hannah; Leven, Johanna; Aminian, Amin; Rayarel, Ashish; Cowan, Callum; Crawford, Charles; et al. (2021-08-13)
      This paper does not set out to describe the reasons why a new concept of dental care should be deemed as acceptable and recognised as mainstream. Rather, the starting point of this paper is the belief that some dentists who are increasing the overall vertical dimension for worn dentition patients are not using the protocols of the traditional 're-organised approach'. If adhesive direct restorations are used, there seems to be anecdotal indication that despite not restoring in the terminal hinge axis, it can have a successful outcome.So, while not criticising this approach simply because it does not follow orthodoxy, the paper has two objectives. It hopes to stimulate some debate and research on this subject. Furthermore, by suggesting some parameters for what might be considered a new approach, it aims to improve patient outcome. [Abstract copyright: © 2021. The Author(s), under exclusive licence to the British Dental Association.]
    • Off the couch: a psychiatrist’s candid stories of hope and healing

      Loewenthal, Kate Miriam; orcid: 0000-0001-7667-7809 (Informa UK Limited, 2021-07-09)
    • Off the couch: a psychiatrist’s candid stories of hope and healing

      Loewenthal, Kate Miriam; orcid: 0000-0001-7667-7809 (Informa UK Limited, 2021-07-09)
    • Older adults’ construal of sedentary behaviour: Implications for reducing sedentary behaviour in older adult populations

      McGowan, Laura J; orcid: 0000-0002-4054-9300; email: laura.mcgowan@manchester.ac.uk; Powell, Rachael; French, David P; orcid: 0000-0002-7663-7804 (SAGE Publications, 2020-03-01)
      Older adults are the most sedentary age group, with sedentary behaviour having negative health-related consequences. There is currently limited understanding of how older adults view sedentary behaviour. This study investigated older adults’ understanding of the concept of sedentary behaviour. Semi-structured interviews were conducted with 22 community-dwelling older adults in urban North-West England, selected to be diverse in socio-economic background and activity levels. Interviews were recorded and transcribed verbatim. An inductive thematic analysis was conducted. Participants often construed sedentary behaviour as synonymous with a lack of physical activity, and many perceived reducing sedentary behaviour and increasing moderate-to-vigorous physical activity to be the same thing. Participants perceived the term ‘sedentary’ to have negative connotations and were often judgemental of people who engaged in high levels of sedentary behaviour. Most participants considered reducing sedentary behaviour to be of value, though more active individuals were unconvinced that reducing sedentary behaviour has value beyond the benefits of being physically active. Interventions may wish to provide education to address the misconception that increasing moderate-to-vigorous physical activity is necessary in order to reduce sedentary behaviour. Educating older adults on the independent health consequences of sedentary behaviour may also prove beneficial.
    • On the Derivation of Multisymplectic Variational Integrators for Hyperbolic PDEs Using Exponential Functions

      Kosmas, Odysseas; orcid: 0000-0002-7047-9438; email: odysseas.kosmas@manchester.ac.uk; Boom, Pieter; orcid: 0000-0001-7437-277X; email: pieter.boom@manchester.ac.uk; Jivkov, Andrey P.; orcid: 0000-0002-3454-7341; email: andrey.jivkov@manchester.ac.uk (MDPI, 2021-08-25)
      We investigated the derivation of numerical methods for solving partial differential equations, focusing on those that preserve physical properties of Hamiltonian systems. The formulation of these properties via symplectic forms gives rise to multisymplectic variational schemes. By using analogy with the smooth case, we defined a discrete Lagrangian density through the use of exponential functions, and derived its Hamiltonian by Legendre transform. This led to a discrete Hamiltonian system, the symplectic forms of which obey the conservation laws. The integration schemes derived in this work were tested on hyperbolic-type PDEs, such as the linear wave equations and the non-linear seismic wave equations, and were assessed for their accuracy and the effectiveness by comparing them with those of standard multisymplectic ones. Our error analysis and the convergence plots show significant improvements over the standard schemes.
    • On the Geometric Description of Nonlinear Elasticity via an Energy Approach Using Barycentric Coordinates

      Kosmas, Odysseas; orcid: 0000-0002-7047-9438; email: odysseas.kosmas@manchester.ac.uk; Boom, Pieter; email: pieter.boom@manchester.ac.uk; Jivkov, Andrey P.; orcid: 0000-0002-3454-7341; email: andrey.jivkov@manchester.ac.uk (MDPI, 2021-07-19)
      The deformation of a solid due to changing boundary conditions is described by a deformation gradient in Euclidean space. If the deformation process is reversible (conservative), the work done by the changing boundary conditions is stored as potential (elastic) energy, a function of the deformation gradient invariants. Based on this, in the present work we built a “discrete energy model” that uses maps between nodal positions of a discrete mesh linked with the invariants of the deformation gradient via standard barycentric coordinates. A special derivation is provided for domains tessellated by tetrahedrons, where the energy functionals are constrained by prescribed boundary conditions via Lagrange multipliers. The analysis of these domains is performed via energy minimisation, where the constraints are eliminated via pre-multiplication of the discrete equations by a discrete null-space matrix of the constraint gradients. Numerical examples are provided to verify the accuracy of the proposed technique. The standard barycentric coordinate system in this work is restricted to three-dimensional (3-D) convex polytopes. We show that for an explicit energy expression, applicable also to non-convex polytopes, the general barycentric coordinates constitute fundamental tools. We define, in addition, the discrete energy via a gradient for general polytopes, which is a natural extension of the definition for discrete domains tessellated by tetrahedra. We, finally, prove that the resulting expressions can consistently describe the deformation of solids.
    • On Thresholds for Dynamic Strength in Solids

      Bourne, N. K.; orcid: 0000-0002-8883-1196; email: neil.bourne@manchester.ac.uk (Springer International Publishing, 2021-04-20)
      Abstract: The limits of elastic behaviour change with the nature of the impulse applied to a target and the size of volume interrogated by a measurement, since it is the pre-existing defects sampled within its rise that determine the response observed. This review considers a range of solids of different material classes and tracks the development of the strength of the material during shock loading, from yield at the Hugoniot elastic limit, across the weak shock regime, to its transition to strong shock behaviour. It is shown that at this stress, the weak shock limit (WSL), the shear component of the applied stress exceeds the theoretical strength of the material. Beyond this threshold, there are a number of new responses that confirm a transition from an inhomogeneous to a homogeneous state. Further, whilst strength rises across the weak shock regime, it saturates at the WSL. For instance, failure in shocked glasses transitions from localised fracture initiated at target boundaries to a global failure at this threshold at the theoretical strength. Sapphire′s strength asymptotes to the theoretical strength of the strongest direction in its lattice. Finally, the fourth-power dependence of strain rate upon stress appears to be a consequence of the homogeneous flow in the strong shock regime. This review suggests that µ/2π is a good approximation for the unrelaxed theoretical strength of solids at increasing stresses beyond the WSL. The methodology unfolded here represents a new means to experimentally determine the ultimate shear strength of solids.
    • On Thresholds for Dynamic Strength in Solids

      Bourne, N. K.; orcid: 0000-0002-8883-1196; email: neil.bourne@manchester.ac.uk (Springer International Publishing, 2021-04-20)
      Abstract: The limits of elastic behaviour change with the nature of the impulse applied to a target and the size of volume interrogated by a measurement, since it is the pre-existing defects sampled within its rise that determine the response observed. This review considers a range of solids of different material classes and tracks the development of the strength of the material during shock loading, from yield at the Hugoniot elastic limit, across the weak shock regime, to its transition to strong shock behaviour. It is shown that at this stress, the weak shock limit (WSL), the shear component of the applied stress exceeds the theoretical strength of the material. Beyond this threshold, there are a number of new responses that confirm a transition from an inhomogeneous to a homogeneous state. Further, whilst strength rises across the weak shock regime, it saturates at the WSL. For instance, failure in shocked glasses transitions from localised fracture initiated at target boundaries to a global failure at this threshold at the theoretical strength. Sapphire′s strength asymptotes to the theoretical strength of the strongest direction in its lattice. Finally, the fourth-power dependence of strain rate upon stress appears to be a consequence of the homogeneous flow in the strong shock regime. This review suggests that µ/2π is a good approximation for the unrelaxed theoretical strength of solids at increasing stresses beyond the WSL. The methodology unfolded here represents a new means to experimentally determine the ultimate shear strength of solids.