Marlène Gamelon
email: marlene.gamelon@ntnu.no

Hunting, predation and senescence in boars

Keywords: age-specific mortality, cause-specific mortality, competing risks, harvest, individual monitoring, life history strategy, sex-specific mortality, survival, Sus scrofa, ungulate

Identifying the factors causing mortality over life is of great importance in gerontology, ecology and evolution. I first present some general methods to properly estimate age-related causes of mortality. Second, I report some of the studies that have quantified the main causes of mortality affecting wild boar over life, an emblematic game species that has dramatically increased in abundance and distribution across Europe. Hunting is the major cause of mortality, followed by natural mortality (e.g. starvation, senescence) and predation, all these factors of mortality being age-and/or sex-specific. Third, I discuss why it is important to examine how some sources of mortality can interact. Finally, I show how hunting can select for accelerated life history that allows wild boar to compensate high mortality probabilities with high reproductive output.

Glossary

Adults: Individuals older than 2 years of age.

Hunting mortality probability: the annual probability for an individual to die killed by hunting.

Juveniles: Individuals between 0 and 1 year of age, also called piglets.

Mortality due to predation: the annual probability for an individual to die killed by a nonhuman predator. Main predators for the wild boar are red foxes and wolves.

Natural mortality probability: the annual probability for an individual to die but not from human and non-human predation.

Natural survival probability: the probability for an individual to survive from one year to the next one, in absence of human and non-human predation.

Overall survival probability: the probability for an individual to survive from one year to the next one and thus not to die from hunting, predation or natural causes.

Subadults: Individuals between 1 and 2 years of age, also called yearlings.

Introduction

Identifying the factors causing mortality over life is of great importance in gerontology. This is indeed the first required step to understand the underlying mechanisms shaping mortality, such as genetic mechanisms and environmental factors. The work from Horiuchi and Wilmoth [START_REF] Horiuchi | Age patterns of the life table aging rate for major causes of death in Japan, 1951-1990[END_REF] is a nice illustrative study showing the major causes of death over life for females in Japan between 1951 and 1990. The same approach used on males and females in France between 1979 and 1994 indicates that malignant neoplasms, hypertensive disease or liver cirrhosis are some of the main causes of mortality between ages 30 and 54. At older ages, death rates increase drastically caused by infection diseases and heart failure (see Figure 2 in Horiuchi et al. [START_REF] Horiuchi | Differential Patterns of age-related mortality increase in middle age and old age[END_REF] showing the age pattern of the cause-of-death structure).

A better understanding of the sources of mortality affecting individuals from birth to old ages is also crucial in ecology and evolution. For instance, Forrester and Wittmer [START_REF] Forrester | A review of the population dynamics of mule deer and black-tailed deer Odocoileus hemionus in North America[END_REF] reviewed 48 studies to identify the major causes of mortality affecting mule deer and blacktailed deer Odocoileus hemionus at different ages. They found that predation is the main cause of mortality for all ages. Also, malnutrition as well as diseases can severely affect survival and more generally the growth of the populations. In addition of affecting population dynamics, the multiples sources of mortality act as selective pressures shaping life history traits. Studies led on harvested animal populations have provided valuable insight on the key role of age-specific mortality on life history evolution. For instance, there is growing evidence that high hunting mortality on the adult class can induce evolutionary changes such as earlier age at maturity and reduced body size [START_REF] Proaktor | Evolutionary responses to harvesting in ungulates[END_REF]. On the contrary, a removal of nonmature juvenile individuals may mimic natural mortality and predation pattern and thus limit undesirable evolutionary responses due to harvesting [START_REF] Milner | Hunting Bambi -evaluating the basis for selective harvesting of juveniles[END_REF]. Ultimately, a better knowledge of the causes of mortality affecting wild vertebrates over life might help elaborating appropriate management actions to control populations in a sustainable way.

As most ungulate populations in temperate areas, wild boar (Sus scrofa) abundance and distribution have increased over the last decades across Europe [START_REF] Apollonio | European ungulates and their management 21st century[END_REF][START_REF] Massei | Wild boar populations up, numbers of hunters down? A review of trends and implications for Europe[END_REF]. Changes in agricultural practices, reduced hunting pressure, global change and land abandonment have favoured the expansion of this emblematic game species [START_REF] Massei | The environmental impact of wild boar[END_REF]. When they overturn soil to feed, wild boars lead to important damages to crops and influence plant, animal, fungi and aquatic communities (see [START_REF] Barrios-Garcia | Impact of wild boar (Sus scrofa) in its introduced and native range: a review[END_REF] for a review). They are also reservoirs of several diseases with some of them being possibly transmitted to humans, such as brucellosis or leptospirosis [START_REF] Meng | Wild boars as sources for infectious diseases in livestock and humans[END_REF].

Controlling wild boar populations has thus become an important goal for wildlife managers and hunting has long been proposed as a tool to achieve it.

Interestingly, this species exhibits an unusual life history strategy among ungulates rendering its regulation challenging. Firstly, females are particularly fecund, being able to produce up to 14 piglets in a single litter [START_REF] Servanty | Litter size and fetal sex ratio adjustment in a highly polytocous species: the wild boar[END_REF], at young ages (from their first year of life, 12) after having reached only 33-41% of their asymptotic adult body mass [START_REF] Servanty | Pulsed resources and climate-induced variation in the reproductive traits of wild boar under high hunting pressure[END_REF]. Secondly, the generation time, i.e. the mean age of mothers at childbirth, is close to 2 years in some heavily hunted populations whereas it is around 6 years for similar-sized ungulates [START_REF] Servanty | Influence of harvesting pressure on demographic tactics: implications for wildlife management[END_REF]. This short generation time characterizing wild boar life history is typically observed in passerine birds or rodents. This indicates an especially fast turnover, with compensation for reduced survival in heavily hunted environment by reallocation of resources to reproduction [START_REF] Gamelon | High hunting pressure selects for earlier birth date: wild boar as a case study[END_REF][START_REF] Servanty | Influence of harvesting pressure on demographic tactics: implications for wildlife management[END_REF].

The purpose of this chapter is to identify the sources of mortality affecting wild boar over life, discuss how they can interact and shape life history traits. First, I present some general methods to properly estimate age-related causes of mortality. Second, I report some of the studies that have quantified the main causes of mortality affecting wild boars from young to older ages. Based on this literature survey, I discuss the type of mortality an individual is likely to experience according to its age and sex. Third, I argue why it is important to examine how some sources of mortality can interact. Fourth, I discuss how wild boar population growth rates can increase despite a somewhat low survival.

Estimating cause-specific mortality

The multiple sources of mortality that affect human and non-human individuals compete and their probability of wining depends on the strength of the different causes of mortality [START_REF] Ergon | The utility of mortality hazard rates in population analyses[END_REF]. Statisticians have thus long been interested in properly decomposing mortality into its different sources, the so-called "competing risks". Sometimes, the fate of an individual (i.e. alive or dead) and the cause of death are precisely known. This is the case in medicine, human demography, in plants and captive animal populations. Also, in free-ranging animals, the recent advances in GPS and radiotelemetry technology may allow to identify the cause of death [START_REF] Tomkiewicz | Global positioning system and associated technologies in animal behaviour and ecological research[END_REF]. Estimating age-and cause-specific mortality probability is thus straightforward with classical competing risk statistical models (see [START_REF] Heisey | A review of methods to estimate cause-specific mortality in presence of competing risks[END_REF] for a review of methods to estimate cause-specific mortality in presence of competing risks; see also the books from [START_REF] Crowder | Classical competing risks[END_REF][START_REF] Collett | Modelling survival data in medical research[END_REF]).

However, in non-captive animal populations, knowing the fate of an individual is challenging. For instance, an individual alive is not necessarily detected by the observer. On top of that, when the individual dies, identifying the cause of death is difficult because death is never observed in a wild population. Thanks to the development of multi-state capturerecapture (CR) models, estimating cause-specific mortality probability when detection is imperfect and the cause of death is unobservable is now possible [START_REF] Pradel | Multievent: an extension of multistate capture-recapture models to uncertain states[END_REF]. Briefly, these methods are applicable to the study of marked animals that are recaptured several times during their life and recovered from at least one known cause of mortality (hunting for instance, see [START_REF] Gamelon | High hunting pressure selects for earlier birth date: wild boar as a case study[END_REF][START_REF] Servanty | Assessing whether mortality is additive using marked animals: a Bayesian state-space modeling approach[END_REF] for case studies on wild boar). For example, Koons et al. [START_REF] Koons | Methods for studying cause-specific senescence in the wild[END_REF] estimated age-and cause-specific mortality probability, namely the mortality probability due to hunting vs. the mortality probability due to human-unrelated causes, on lesser snow geese Chen caerulescens caerulescens and roe deer Capreolus capreolus by fitting multi-state CR models. Noticeably, mortality probability, defined as the probability for an individual to die during a given time interval, is commonly used in the fields of statistical modelling of CR data. On the contrary, mortality hazard rate, corresponding to the latent intensity of deadly events that an individual is exposed to, is classically used in medicine and human demography (see [START_REF] Ergon | The utility of mortality hazard rates in population analyses[END_REF] for a discussion on the use of mortality hazard rates instead of mortality probabilities).

Hunting

From a literature survey, I report some of the studies that have quantified the main causes of mortality affecting wild boars from young to older ages using individual monitoring (table ).

<Table near here>

Hunting is one of the major causes of mortality for the wild boar in Europe [START_REF] Massei | Wild boar populations up, numbers of hunters down? A review of trends and implications for Europe[END_REF][START_REF] Keuling | Mortality rates of wild boar Sus scrofa L. in central Europe[END_REF].

The annual probability for an individual to be killed by hunting ranges from 11 to almost 50% in the reviewed studies (table). It is noteworthy that this probability is often agedependent. For instance, thanks to a long-term individual monitoring and the use of CR models, it has been shown that this source of mortality increases with increasing ages in males at Châteauvillain in France [START_REF] Toïgo | Disentangling natural from hunting mortality in an intensively hunted wild boar population[END_REF]. In the Nature Reserve of Somiedo in Spain, adults are also those that are preferentially removed by hunting [START_REF] Nores | Wild boar Sus scrofa mortality by hunting and wolf Canis lupus predation: an example in northern Spain[END_REF]. This age-specific pattern of human-induced mortality may be explained by the reluctance to shoot juveniles, a general feature observed among hunters [START_REF] Saether | Effective size of harvested ungulate populations[END_REF]. Moreover, in wild boar, the probability to be killed by hunting depends on the sex of the individual, subadult and adult males being more likely to die from hunting than females. This sex-specific pattern of human-induced mortality may result from contrasting abilities to escape from hunters. Indeed, wild boar live in matrilineal social groups with one large female leading a group ( 27) composed of juveniles with limited movement abilities [START_REF] Merli | Wild boar under fire: the effect of spatial behaviour, habitat use and social class on hunting mortality[END_REF]. Contrary to solitary subadult and adult males, family groups tend to favour coppice habitats rather than bushlands during the hunting season [START_REF] Saïd | Sex effect on habitat selection in response to hunting disturbance: the study of wild boar[END_REF]. This can explain why subadult and adult males are more likely to be hunted than females. Sex-specific patterns of humaninduced mortality may also simply result from hunting rules that orientate hunting pressure on individuals with specific phenotypic characteristics. At Châteauvillain in France for instance, hunters have to pay a financial penalty if they shoot females larger than 50 kg [START_REF] Gamelon | Making use of harvest information to examine alternative management scenarios: a body weight-structured model for wild boar[END_REF].

As a consequence, they prefer shooting a solitary male instead of shooting in a group, thus increasing mortality on subadult and adult males and relaxing the hunting pressure on adult females.

Predation

In some of the studied areas, human is not the only predator. In one Italian study site where both red foxes (Vulpes vulpes) and wolves (Canis lupus) are present, the monitoring of 164 wild boars tagged with radio-collars or transmitters equipped with a mortality sensor has shown that 2 of them died from predation (table, 28). In Poland, the analysis of wolf fecal samples has confirmed that wild boar is part of the wolf diet [START_REF] Wł | Prey selection and predation by wolves in BiaŁowieŻa Primeval Forest, Poland[END_REF]. In a recent review, Mori et al. [START_REF] Mori | What does the wild boar mean to the wolf?[END_REF] highlights that wild boar even constitutes the largest frequency of ungulate prey in wolf diet across Italy, before roe deer, red deer (Cervus elaphus) and livestock [START_REF] Mori | What does the wild boar mean to the wolf?[END_REF].

However, for the wild boar, mortality due to predation does not exceed 0.06 and thus remains small compared with human-induced mortality (table). Remarkably, this cause of mortality mainly affects wild boar at young ages. In Italy, they represent 77% of the wolf diet and this proportion even reaches 94% in Poland. Once again, such an age-specific pattern of predatorinduced mortality may be explained by the low abilities of juveniles to escap when facing a predator.

Other sources of mortality

Wild boar can die from natural causes (i.e. not from human or non-human predation).

Natural mortality remains low, especially at adult stages (see table), translating to high natural survival. This is expected among ungulates, where the average natural adult survival probability was estimated to be 0.88 in males (see [START_REF] Williams | Pleiotropy, natural selection, and the evolution of senescence[END_REF] for an analysis among 18 species) and may exceed 0.95 in females [START_REF] Gaillard | Temporal variation in fitness components and population dynamics of large herbivores[END_REF]. The reported studies (table) provide estimates of natural survival at adulthood but age-specific survival patterns are ignored. However, a decline of natural survival with increasing ages (hereafter actuarial senescence) may occur as a result of the decline of the forces of natural selection with age [START_REF] Medawar | An unsolved problem of biology[END_REF], the selection of genes with a beneficial effect early in life that are deleterious later on [START_REF] Williams | Pleiotropy, natural selection, and the evolution of senescence[END_REF] as well as high fertility translating to high rate of senescence [START_REF] Hamilton | The moulding of senescence by natural selection[END_REF].

Senescence is pervasive in the wild (see [START_REF] Nussey | Senescence in natural populations of animals: widespread evidence and its implications for bio-gerontology[END_REF] for a review) and wild boar is not an exception [START_REF] Gamelon | Do age-specific survival patterns of wild boar fit current evolutionary theories of senescence?[END_REF]. Indeed, during an 18-year period at Castelporziano in Italy, 1783 juveniles and subadults were marked with ear-tags, released after handling and recaptured later on. The three oldest monitored individuals were 13 years of age. Thanks to this long-term individual monitoring, we estimated natural survival probabilities for each age and sex [START_REF] Gamelon | Do age-specific survival patterns of wild boar fit current evolutionary theories of senescence?[END_REF]. We found a decrease of survival with age from age 3 onwards, with males exhibiting lower natural survival than females. Sex-specific reproductive tactics that increase mortality risks for males [START_REF] Bonduriansky | Sexual selection, sexual conflict and the evolution of ageing and life span[END_REF] can explain this between-sex difference. Also, harsh environmental conditions such as droughts may affect male survival stronger than female survival [START_REF] Toïgo | Causes of sex-biased adult survival in ungulates: sexual size dimorphism, mating tactic or environment harshness?[END_REF]. However, it is noteworthy that this between-sex difference in adult survival remains quite weak (about 10%, as in humans). Compared to other ungulates, wild boar females have earlier actuarial senescence for their body size. The high and early fertility of wild boar females may advance the age from which the decline of the forces of natural selection occurs [START_REF] Gamelon | Do age-specific survival patterns of wild boar fit current evolutionary theories of senescence?[END_REF].

In the first year of life, natural mortality can be higher than at adulthood (table). At Castelporziano for instance, juvenile survival probability was only 0.68 [0.64; 0.72] (39). The major cause of natural mortality that has been identified is caused by severe winters (see [START_REF] Melis | Biogeographical variation in the population density of wild boar (Sus scrofa) in western Eurasia[END_REF][START_REF] Vetter | What is a mild winter? regional differences in withinspecies responses to climate change[END_REF] for large-scale studies). Snow cover and low temperatures make food resources in the soil hardly accessible and increase juvenile mortality through disease and/or starvation. For some authors, milder winters and thus enhanced juvenile survival might explain the recent expansion of the species observed in the last decades [START_REF] Vetter | What is a mild winter? regional differences in withinspecies responses to climate change[END_REF]. At Castelporziano, characterized with a typical Mediterranean climate, severe droughts might explain such a low piglet survival [START_REF] Focardi | Survival of wild boars in a variable environment: unexpected life-history variation in an unusual ungulate[END_REF]. In the second year of life, we found that natural survival reaches up to 0.85 [0.77; 0.91] for females and drops to 0.57 [0.50; 0.64] for males. Such a high natural mortality for subadult males has already been reported in another site, at Châteauvillain in France [START_REF] Gamelon | High hunting pressure selects for earlier birth date: wild boar as a case study[END_REF]. At this period of their life, males often disperse from their natal area to become solitary [START_REF] Truvé | Timing and distance of natal dispersal for wild boar Sus scrofa in Sweden[END_REF] and face with particularly increased mortality risks (e.g. starvation, collision with vehicles).

Interaction between causes of mortality

It is clear from the three previous sections that different sources of mortality affect wild boars from birth to older ages (see Figure for a schematic summary). These causes of mortality are age-specific. For instance, while predators preferentially remove piglets, hunters remove relatively more subadults and adults. They are also sex-specific, with higher mortality risks due to starvation and collision with vehicles for males than for females. Finally, they depend on the location of the studied population. For instance, the probability of dying from starvation because of winter harshness or from predation by wolves obviously depends on the study area. <Figure near here> Importantly, these sources of mortality can be additive. Basically, it means that they are independent on each other and the overall mortality probability corresponds to the sum of all causes of mortality. In other words, an individual that dies from hunting would have survived in absence of hunting. But sources of mortality can be dependent on each other and interact. They might be compensatory, i.e. negatively correlated such as the overall mortality probability is lower than the sum of all causes of mortality. For instance, in a harvested duck population, Hepp et al. [START_REF] Hepp | Physiological condition of autumn-banded mallards and its relationship to hunting vulnerability[END_REF] showed that the individuals with the lowest body condition and thus with the lowest natural survival were more likely to be killed by hunting. In that case, an increase of hunting mortality leads to a reduction in natural mortality because the "strongest" individuals remain in the population. In that respect, predation by non-human predators is directly comparable, because it mainly affects vulnerable individuals in a population such as juveniles and senescents that often exhibit the highest natural mortality probability. Also, an increase of hunting mortality or mortality due to predation might reduce population density and thus disease transmission or competition among individuals. This ultimately leads to a decrease of natural mortality [START_REF] Boyce | Seasonal compensation of predation and harvesting[END_REF]. Sources of mortality may be depensatory, i.e. positively correlated such as the overall mortality probability is higher than the sum of all causes of mortality. In that case, hunting can increase the mortality due to other causes. For instance, trophy hunting that aims at removing individuals with the largest horns/antlers potentially removes the individuals that perform the best in the population [START_REF] Mysterud | Can compensatory culling offset undesirable evolutionary consequences of trophy hunting?[END_REF], leading to an increase of natural survival. Hunting might also influence the way individuals use their habitat. In willow ptarmigan Lagopus lagopus for instance, individuals spend more time in dense forests where foraging opportunities and availability of food resources are limited, thus leading to a decrease of body condition [START_REF] Brøseth | Disturbance effects of hunting activity in a willow ptarmigan Lagopus lagopus population[END_REF] and potentially high natural mortality. This change of habitat also increases the risk of predation [START_REF] Brøseth | Disturbance effects of hunting activity in a willow ptarmigan Lagopus lagopus population[END_REF]. As a consequence, understanding how hunting interacts with other causes of mortality has become a central goal in animal ecology ( [START_REF] Péron | Compensation and additivity of anthropogenic mortality: life-history effects and review of methods[END_REF], see e.g. ( 50) for a case study on willow ptarmigan).

In wild boar, whether mortalities are additive, compensatory or depensatory is a question that has been addressed in the population of Châteauvillain, in France. In that area, hunting constitutes the main source of mortality, followed by natural mortality (see table ).

Servanty et al. [START_REF] Servanty | Assessing whether mortality is additive using marked animals: a Bayesian state-space modeling approach[END_REF] found that these two sources of mortality are depensatory, natural mortality increasing with hunting mortality. Several factors can explain this result. First, selective hunting with the removal of the "best" individuals in the population may lead to increased natural mortality probability. However, hunters are posted around a given area and wait for animals startled by beaters and dogs. It is thus unlikely that they actually assess the phenotypic quality of the individuals when they are flushed out of the vegetation. Second, increased emigration in response to hunting can wrongly be interpreted as higher natural mortality. However, wild boar is sedentary [START_REF] Truvé | Timing and distance of natal dispersal for wild boar Sus scrofa in Sweden[END_REF] and it is unlikely that increasing hunting pressure is associated with increasing emigration rates. The only plausible explanation for an increased natural mortality with high hunting mortality is crippling loss. Individuals wounded or killed by hunting but never retrieved and recovered by hunters are considered as dead from natural causes. This can explain higher mortality probabilities when the probability to be killed by hunting is high [START_REF] Servanty | Assessing whether mortality is additive using marked animals: a Bayesian state-space modeling approach[END_REF]. Whether depensatory mortalities are a common feature among wild boar populations or are specific to the studied population remains to be investigated.

Hunting mortality and life history strategy

Wild boar survival is strongly affected by hunting at all ages. In addition, predation, starvation, disease, collision with vehicles and finally senescence at older ages are additional sources of mortality for the species. All these causes of mortality may result in a particularly low overall survival probability. At Châteauvillain in France, overall survival probability is close to 0.45 for females and drops to 0.23 for adult males [START_REF] Toïgo | Disentangling natural from hunting mortality in an intensively hunted wild boar population[END_REF] leading to a somewhat short lifespan for such a long-lived species. In that context, one can wonder how wild boar numbers can still increase throughout Europe despite such a low overall survival. This is because fecundity has become the focus of all selective pressures and particularly of hunting pressure. At Châteauvillain in France, birth dates have advanced by up to 12 days in a 22-year period, selected by a high hunting pressure. This allows juveniles born early in the season, to grow for longer and thus reach the threshold body mass to reproduce early in life, in their first year of life [START_REF] Gamelon | High hunting pressure selects for earlier birth date: wild boar as a case study[END_REF]. These findings of earlier age at first breeding are in line with a demographic analysis (see 14) that has compared two wild boar populations in contrasting environments: the French population suffering from a high hunting pressure and the lightly hunted Italian population at Castelporziano in Italy (see table ). This analysis shows that the lightly hunted population has a typical demography of long-lived species with a high contribution of adult survival to the population growth rate and a generation time of 3.6 years. However, the heavily hunted population has a typical demography of short-lived species such as passerines, with the highest contribution for juvenile survival and exhibits an accelerated life history with a shorter generation time close to 2 years [START_REF] Servanty | Influence of harvesting pressure on demographic tactics: implications for wildlife management[END_REF]. Therefore, in environments characterized with a hunting mortality probability, wild boar females compensate by reallocating resources to reproduction early in life. This explains why an increase of hunting mortality probability does not translate to reduced population growth rates in wild boar, contrary to other ungulate species such as roe deer. In that respect, wild boar exhibits an unusual life history strategy among ungulates. Table -Summary of some studies reporting causes of mortality that affect wild boar over life. Location of the studied population, annual mortality probabilities due to hunting, predation or other causes (e.g. disease, starvation) and the reference of the study are provided. Potential age-and sex-related variations in mortality probabilities are also reported with the associated sentences extracted from the papers.

Others:

"Natural mortality of adults was similar for males and females". [START_REF] Toïgo | Disentangling natural from hunting mortality in an intensively hunted wild boar population[END_REF]

This value corresponds to the sum of legal hunting (32%) and poaching (13.6%) rates.

Natural mortality corresponds to mortality excluding hunting mortality.


R Vargas
email: r.vargas_m@ppgcem.ufscar.br

R B Canto

F Hild

On the Calibration of Cohesive Parameters for Refractories from Notch Opening Displacements in Wedge Splitting Tests

Keywords:

Digital Image correlation (DIC)

Finite Element Model Updating (FEMU) Notch Opening Displacement (NOD)

Refractory castable

Wedge Splitting Test (WST)

A B S T R A C T

Cohesive elements are commonly used to describe crack propagation in heterogeneous materials with toughening mechanisms. This work aims to provide a guideline on how these fracture parameters can be calibrated using notch opening displacements (NOD) measured via digital image correlation and force data from wedge splitting tests (WSTs). Weighted finite element model updating was applied to calibrate material and boundary condition parameters in the same framework. The influence of each parameter on force and NOD data are given together with uncertainties for the calibrated parameters. Numerical results showed very good agreement in terms of splitting force, NOD, displacement and gray level residual fields. It is shown that images obtained during WSTs focusing on the crack path (i.e., hiding the loading region) can be used to drive numerical simulations and obtain cohesive parameters.

Introduction

Refractories are known to maintain functional properties in very hazardous conditions such as high-temperature and corrosive environments [START_REF] Lee | Castable refractory concretes[END_REF][START_REF] Luz | Refractory Castable Engineering[END_REF]. Therefore, understanding their fracture mechanisms and modeling their development is of utmost importance to optimize processes and minimize risks [START_REF] Wachtman | Materials and Equipment -Whitewares -Refractory Ceramics -Basic Science[END_REF]. In such severe in-service conditions, crack initiation is unavoidable [START_REF] Lee | Evolution of in situ refractories in the 20th century[END_REF] and thus it is usual practice to limit propagation via toughening mechanisms [START_REF] Steinbrech | Toughening mechanisms for ceramic materials[END_REF][START_REF] Ritchie | The conflicts between strength and toughness[END_REF]. The latter ones lead to the formation of fracture process zones (FPZs) where cracks have some cohesion between the newly created surfaces. Linear elastic fracture mechanics (LEFM) was applied for structures with very small defects made of brittle materials, although severe deviations may arise in the presence of a developed FPZ, especially in small specimens [START_REF] Van Gils | Size effect predictions by fracture models for a refractory ceramic[END_REF]. Therefore, this work focuses on cohesive zone models (CZMs) suitable to be calibrated under usual laboratory characterization of materials, aiming to provide first estimates of parameters needed prior to simulating in-service applications.

The use of traction-separation laws to describe the FPZ [START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF][START_REF] Barenblatt | The mathematical theory of equilibrium of crack in brittle fracture[END_REF][START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF] has been successfully applied to refractory materials with different approaches. For instance, CZMs were used to study thermal shocks [START_REF] Cotterell | Thermal shock and size effects in castable refractories[END_REF] and to check size effects and deviations from LEFM [START_REF] Van Gils | Size effect predictions by fracture models for a refractory ceramic[END_REF]. Later on, they were utilized for analyzing damage during thermal cycles [START_REF] Andreev | FEM simulation of the thermo-mechanical behaviour and failure of refractories-a case study[END_REF] and fracture [START_REF] Jiménez-Piqué | Fictitious crack modeling of the fracture of a refractory ceramic: fitting through experimental measurements of the displacement field around the fracture process zone[END_REF] in various refractories. The same models may also be used to study interface cohesion in such heterogeneous materials [START_REF] Mathias | Homogenization of glass/alumina two-phase materials using a cohesive zone model[END_REF][START_REF] Özdemir | Modeling thermal shock damage in refractory materials via direct numerical simulation (DNS)[END_REF][START_REF] Hein | Influence of interfaces on crack propagation through a layered refractory loaded by thermal shock[END_REF]. More recently, several examples are found about its usage to simulate initiation and propagation of macrocracks [START_REF] Gruber | A laser irradiation disc test for fracture testing of refractory fine ceramics[END_REF][START_REF] Jin | Thermomechanical failure modeling and investigation into lining optimization for a Ruhrstahl Heraeus snorkel[END_REF][START_REF] Pan | Inverse simulation of fracture parameters for cement-bonded corundum refractories[END_REF][START_REF] Vargas | Analysis of a castable refractory using the wedge splitting test and cohesive zone model[END_REF][START_REF] Doitrand | Fracture and cohesive parameter identification of refractories by Digital Image Correlation up to 1200 °C[END_REF][START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF].

In few of the afore-cited references, the material parameters were estimated without experimental exploration [START_REF] Van Gils | Size effect predictions by fracture models for a refractory ceramic[END_REF][START_REF] Hein | Influence of interfaces on crack propagation through a layered refractory loaded by thermal shock[END_REF]. Some of them briefly comment on parameter calibration without giving many details [START_REF] Andreev | FEM simulation of the thermo-mechanical behaviour and failure of refractories-a case study[END_REF][START_REF] Jiménez-Piqué | Fictitious crack modeling of the fracture of a refractory ceramic: fitting through experimental measurements of the displacement field around the fracture process zone[END_REF][START_REF] Mathias | Homogenization of glass/alumina two-phase materials using a cohesive zone model[END_REF]. For the quantity to be optimized with respect to experimental information, the crack mouth opening displacement (CMOD) [START_REF] Cotterell | Thermal shock and size effects in castable refractories[END_REF][START_REF] Gruber | A laser irradiation disc test for fracture testing of refractory fine ceramics[END_REF] or elastic properties obtained from acoustic measurements [START_REF] Özdemir | Modeling thermal shock damage in refractory materials via direct numerical simulation (DNS)[END_REF][START_REF] Gruber | A laser irradiation disc test for fracture testing of refractory fine ceramics[END_REF] can be used. More recently, inverse algorithms were used to find the best fit of the loading curve [START_REF] Jin | Thermomechanical failure modeling and investigation into lining optimization for a Ruhrstahl Heraeus snorkel[END_REF][START_REF] Pan | Inverse simulation of fracture parameters for cement-bonded corundum refractories[END_REF]. Vargas et al. [START_REF] Vargas | Analysis of a castable refractory using the wedge splitting test and cohesive zone model[END_REF] used measured CMODs to prescribe boundary conditions and used the force measurements in the cost function of a Finite Element Model Updating (FEMU) procedure, while Doitrand et al. [START_REF] Doitrand | Fracture and cohesive parameter identification of refractories by Digital Image Correlation up to 1200 °C[END_REF] also used force measurement and crack lengths estimates to find the best fit through manual optimization.

Jiménez-Piqué et al. [START_REF] Jiménez-Piqué | Fictitious crack modeling of the fracture of a refractory ceramic: fitting through experimental measurements of the displacement field around the fracture process zone[END_REF] mentioned the use of digital image correlation (DIC) in the calibration procedure of a Hillerborg et al. [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF] model. DIC was applied to estimate the "transition from microcracking to bridging and the approximate contribution of both mechanisms." More recently, CMODs measured by DIC were applied as boundary conditions for the calibration of CZM parameters [START_REF] Vargas | Analysis of a castable refractory using the wedge splitting test and cohesive zone model[END_REF][START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF]. The authors also calculated displacement residuals between experimental and numerical results, as well as gray level residuals directly in the images using the kinematics from the Finite Element (FE) simulation. Doitrand et al. [START_REF] Doitrand | Fracture and cohesive parameter identification of refractories by Digital Image Correlation up to 1200 °C[END_REF] considered strain fields assessed via DIC to look for crack branches on the surface, to estimate the crack length using a user-defined threshold, and to define a tortuous crack path with cohesive elements. The authors also tested a set of different parameters to check which one would best fit the experimental force and crack lengths.

In a previous work, room temperature and 600°C Wedge Splitting Tests (WSTs) [START_REF] Tschegg | Prüfeinrichtung zur Ermittlung von bruchmechanishen Kennwerten sowie hiefür geeignete, Prüfkörper, Austrian Pat[END_REF][START_REF] Brühwiler | The wedge splitting test, a new method of performing stable fracture mechanics tests[END_REF] were performed to analyze how fracture parameters changed under these conditions for samples with different sintering temperatures [START_REF] Vargas | Effect of sintering temperature on fracture energy of Alumina-Mullite-Zirconia castable at 600°C[END_REF].

However, the window size of the furnace limited the field of view hiding the force application region. This difficulty led to the study of the relationship between notch opening displacements (NODs) and CMODs [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF]. In the present work, the NOD extrapolation to CMOD proposed in Ref. [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF] is used to drive numerical simulations. The corresponding parameters are calibrated together with material parameters in a specially tailored identification procedure.

First, the identification framework is introduced. Then, one benchmark experiment is tested to validate the proposed methodology before applying it to the room temperature experiment in which the CMOD was not accessible.

Calibration of CZM parameters

This work proposes to calibrate CZM parameters using NOD data to drive FE simulations. The so-called Finite Element Model Updating (or FEMU [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF][START_REF] Kavanagh | Extension of classical experimental techniques for characterizing composite-material behavior[END_REF][START_REF] Pagnacco | Parameter Identification from Mechanical Field Measurements using Finite Element Model Updating Strategies[END_REF]) framework is first introduced. Then, the geometry, boundary conditions, and all the necessary parameters for the methodology are described.

Identification strategy

The FEMU method is used hereafter. The identification technique, which is based on a Gauss-Newton scheme, consists in nonlinear least squares minimization of 2 with respect to parameters gathered in the column vector { }

2 ({ }) = 2 ({ }) + NOD 2 NOD ({ }) (1)
where 2 denotes the cost function for global equilibrium

2 ({ }) = 1 ∑ ( ) -FE ( , { }) 2 (2)
and 2 NOD that related to the Notch Opening Displacement (NOD)

2 NOD ({ }) = 1 ∑ NOD ( ) -NOD FE ( , { }) NOD 2 (3)
in which the subscript is related to the measured quantities (i.e., force and NOD from load cell and DIC, respectively)

and FE to computed quantities in FE analyses, the number of time steps, and NOD the standard measurement uncertainties, for force and NOD, respectively. Both weights ( and NOD ) in the total cost function are equal via

Bayesian statistics [30]

= 1 2 and NOD = 1 2 (4)
since the number of considered data is identical in each individual cost function. In such framework, and NOD are direct estimates of model quality since the closer they are to unity, the higher the probability that experimental noise be the source of error and not the chosen model.

The sensitivities for both FE and NOD FE with respect to the sought parameters { } are rectangular × matrices

[ ] and [ NOD ] = FE { } ( , { }) and NOD = NOD FE { } ( , { }) (5)
calculated by finite differences with a perturbation of 1% of each parameter with respect to the current estimates. After linearization of FE and NOD FE with respect to the parameter corrections { }, the minimization procedure consists in iteratively updating the current estimates { } of the parameter vector by solving linear systems

[ ] ⋅ { } = { } (6)
where [ ] is the dimensionless Hessian

[ ] = 1 2 F ⊤ + 1 2 NOD NOD ⊤ NOD (7)
and { } the right hand member

{ } = 1 2 F ⊤ ( ) -( , { }) + 1 2 NOD NOD ⊤ NOD ( ) -NOD FE ( , { }) (8)
Convergence is deemed successful when the root mean square (RMS) of the relative variation of the parameters is less than 10 -2 between two subsequent iterations.

The sensitivity matrices, and more importantly their condensed effect described by the Hessian matrix, give access to the covariance matrix [ ] of the calibrated parameters

[ ] = [ ] -1 (9)
in which the square root of each diagonal term expresses a fraction related to the standard deviation for each parameter when considered alone. The latter will be reported in the following results.

Parameters in the identification framework

The different regions considered herein are shown in Figure 1 in the reference images of the two analyzed tests.

The sample geometry that was used in the FEMU procedure corresponds to red contours, and the two nodes where displacement boundary conditions (BCs) were applied are highlighted in blue circles. The CMOD (extrapolated from NOD data) was used to drive FE simulations applying half of the opening for the horizontal component on each side.

Zero amplitudes were prescribed for the vertical component of these two points. The NODs were measured in the yellow squares and the cyan polygons depict the region where conventional DIC was performed. For the FE simulations, the PPR cohesive law [START_REF] Park | A unified potential-based cohesive model of mixed-mode fracture[END_REF][START_REF] Park | Computational implementation of the PPR potential-based cohesive model in ABAQUS: educational perspective[END_REF] was chosen. The shape parameter and the initial slope were equal to 7 and 0.005, as chosen for the same material investigated in Ref. [START_REF] Vargas | Analysis of a castable refractory using the wedge splitting test and cohesive zone model[END_REF]. It was assumed that the cohesive strength was identical for modes I and II, and the fracture energy for modes I and II was also the same since in this quasi pure mode I test no sensitivity was expected to uncouple the parameters for each mode in the identification procedure [START_REF] Affagard | Identification method for the mixed mode interlaminar behavior of a thermoset composite using displacement field measurements and load data[END_REF]. The traction-separation laws for the initial CZM parameters are shown in Figure 2. It is worth noting that only the cohesive strength and the fracture energy (integral of the traction-separation curve) were calibrated hereafter.

Figure 2: PPR traction-separation law using the initial parameters (see Table 1). The cohesive strength corresponds to the maximum traction (red circles), and the fracture energy is the integral of the traction-separation curve (dashed areas).

The calibration procedure was first tested in a benchmark case (Figure 1(a-b)) in which CMOD data were available as well as images from both sides of the WST specimen. The material for this experiment was fired at 540°C and was made using the DD40 grade made by IBAR-Brazil (i.e., an anti-erosive commercial refractory castable designed for petrochemical applications [START_REF] Cabrelon | Efeito do tempo de exposição a uma atmosfera coqueificante na microestrutura e nas propriedades de um concreto refratário usado na indústria petroquímica[END_REF][START_REF] Miyaji | The coke effect on the fracture energy of a refractory castable for the petrochemical industry[END_REF][START_REF] Pereira | A study about the contribution of the -phase transition of quartz to thermal cycle damage of a refractory used in fluidized catalytic cracking units[END_REF]). This refractory contains quartz, mullite, kyanite, -cristoballite and alumina [START_REF] Miyaji | The coke effect on the fracture energy of a refractory castable for the petrochemical industry[END_REF]. Then, the so-called WST-MZ1-S1450G experiment (Figure 1(c)) was studied where only NOD data were available. The nomenclature is after the designation MZ1-S1450G used in Refs. [START_REF] Vargas | Effect of sintering temperature on fracture energy of Alumina-Mullite-Zirconia castable at 600°C[END_REF][START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF], namely, the sample was made of alumina with mullite-zirconia aggregates (MZ), fired at 1450°C (S1450), and lateral grooves (G) were implemented to guide propagation crack. This material was also produced by IBAR-Brazil.

Although an earlier identification was performed using only force data for the benchmark case, the initial parameters were chosen from another experiment performed on the same material [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF] for comparison purposes, and to initialize the FEMU procedure farther away from a possibly optimal solution. For WST-MZ1-S1450G, was initially set from the value obtained in Ref. [START_REF] Vargas | Analysis of a castable refractory using the wedge splitting test and cohesive zone model[END_REF] (i.e., another material with similar geometry) multiplied by the ratio between the ultimate load in both experiments. The initial fracture energy was assumed as twice that measured from the loading curve of this experiment [START_REF] Vargas | Effect of sintering temperature on fracture energy of Alumina-Mullite-Zirconia castable at 600°C[END_REF].

Since the CMOD region (close to the blue circles in Figure 1) was not accessible in WST-MZ1-S1450G (Fig-

ure 1(c))
, the extrapolation proposed in Ref. [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF] was used hereafter. This proposal is based on the distinction of the NOD ( ) vs. CMOD (Δ) relationship into two regimes, namely, a quadratic interpolation for the pre-peak regime

( ) = Δ( max ) ( max ) -1 Δ Δ( max ) 2 + 2 - Δ( max ) ( max ) Δ Δ( max ) (10)
and an affine trend for the post-peak regime

-( ) ( ) = Δ( max ) ( max ) Δ -Δ( max ) Δ( max ) (11)
where is the slope of the latter Δ vs. interpolation. In the proposed framework, both portions are made C 1 (i.e., continuous first order derivative) at the peak load ( = max ). Therefore, two parameters are needed to calculate Δ from data, namely, Δ( max ) and , since ( max ) is experimentally measured. Both of these parameters are then added to the list of parameters so that Equations ( 10) and ( 11) provide the boundary conditions to the whole simulation. The NOD was measured by DIC and both parameters were initialized in the iterative procedure with those of the cyclic case reported in Ref. [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF]. For easier implementation, instead of identifying Δ( ), this parameter is substituted by the ratio = Δ( )∕ ( ).

Two more parameters were introduced in the identification scheme. The so-called Boundary Condition correction is defined by

Δ = ( -1) × Δ( 50% ) (12)
in which Δ( 50% ) is the CMOD at 50% of the ultimate load (pre-peak) and Δ consists in a correction for a nonzero force in the reference state and also corrects the kinematics of some possible wedge misalignment [START_REF] Vargas | Analysis of a castable refractory using the wedge splitting test and cohesive zone model[END_REF]. Δ is defined as a scalar and is interpreted as the percentage of the NOD from the time step where the force was 50% of the ultimate load. This parameter is taken as unity in the first iteration since it would be the case if no correction were needed.

Last, the Young's modulus is the sixth parameter, and its first estimate comes from an independent evaluation through impulse excitation technique. All the initial parameters are gathered in Table 1.

Benchmark case

The present methodology was first applied to one WST in which both NOD and CMOD were measured (Fig-

ure 1(a,b)). Moreover, the CZM parameters could be compared with the values from the very same monotonic test reported in Ref. [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF], in which the identification was performed using only force data since the simulations were driven by the measured CMOD. Although the geometry for the FE analysis is not entirely on the image frame, a large portion of the specimen that was imaged was available. One important source of independent information to check was displacement fields and gray level residuals as discussed hereafter. The mesh used in Ref. [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF] was also considered but CMODs were obtained by extrapolating NOD measurements using Equations ( 10) and ( 11) to obtain the boundary conditions to be applied on two nodes (see blue circles in Figure 1(a,b)). However, the parameters for such extrapolations will be considered unknown to simulate one case where the CMOD was not visible (Figure 1(c)). Last, the standard uncertainty = 5 N was obtained from the force fluctuations and NOD = 0.5 centi-px (≡ 0.24 µm) from NOD measurements (via DIC) for images in the pre-loaded step.

Before starting the identification procedure itself, a sensitivity analysis was performed [START_REF] Tarantola | Inverse Problems Theory. Methods for Data Fitting and Model Parameter Estimation[END_REF]. In the present case, it consisted in analyzing the temporal changes of each column of the sensitivity matrices and NOD , namely, for each parameter to be calibrated. Further, these sensitivities were first analyzed for the whole set of (360) images and then on a reduced set (one in ten images). Figure 3 shows the temporal changes of all sensitivies for the front face analysis in the converged state. They did not change qualitatively between each iteration and for each specimen side.

The parameter affects more the force at the start of the test, which was expected since it accounts for a correction for the unknown load-free configuration. It is followed by , , and . The cohesive strength influences the force in the time steps close to the peak load, while is dominant in the post-peak regime.

For the NOD, it is possible to see different trends, complementary to the force sensitivities, although considerably dominated by the parameter , whose sensitivity increases linearly with the same slope until the end while most of the other parameters exhibit post-peak plateaus. Moreover, after 1.7 mm, a fluctuation of the NOD sensitivities of and parameter occur. These fluctuations are probably related to numerical issues when very few cohesive elements are still carrying the load. Interestingly, using only one tenth of the available images was sufficient for faithfully describing the sensitivities, and possibly even less images could be used in this first analysis. Similarly, Figure 4 shows the temporal changes of each column of and NOD for the back face in the con-verged state. They are qualitatively very close to those shown for the front face in Figure 3. The quantitative differences are related to loading imperfections such as wedge misalignment throughout the experiment. The same oscillations are seen for the NOD sensitivities of and in the post-peak regime, which reinforces the hypothesis that this effect be correlated to the damage state of the cohesive elements. The Hessian [ ] and covariance [ ] matrices, as well as the eigen decomposition of [ ] are reported in Figure 5 for the front face. The eigen values reported in Figure 5 span over four orders of magnitude, which shows a reasonable conditioning of the system. If only the four most sensitive eigen parameters were considered, the conditioning becomes even better with only two orders of magnitude. The eigen vectors show small correlations between parameters. The main difference between the matrices calculated using the whole set of images or only few of them is one order of magnitude shift, but they remain qualitatively similar. This factor 10 between both covariance matrices allows us to anticipate that parameter uncertainties will decrease when using all images. This result may also be used as a way to estimate how many images would be required for a given accuracy, once the test is selected for the material of interest.

From the Hessian matrix [ ] alone, it is possible to infer that the highest sensitivity comes from the parameter , which is essentially the sixth eigen vector (i.e., related to the highest eigenvalue). The ordering of sensitivities for the other parameters is not straightforward by analyzing [ ]. However, it is easily obtained from the eigen vectors, and in decreasing order importance are the following parameters: , , , and . This analysis highlights that using NODs alone, the CMOD may be estimated using the proposed proposed framework, and it is very well conditioned to obtain the needed extrapolation. Moreover, the CZM parameters are also well conditioned, having the highest sensitivities after and . The parameter was expected to have a lower sensitivity since it implies a constant displacement, which is more important on the first time steps where displacement is very small and its influence vanishes as the test proceeds. One last point is related to the Young's modulus , which is the most difficult and uncertain parameter. If the cost function only contained the force measurements, the conditioning would have deteriorated to a point that it became meaningless to calibrate it. These same matrices were calculated for the back face and are reported in Figure 6. All the trends are very similar, as expected from the previously reported sensitivity analysis. For the back face however, the eigen decomposition leads to the third and fourth eigen vectors related to linear combinations of and , which reinforces that their sensitivity level is close. When using fewer images, there is also a correlation between the parameters and in the first and second eigenvector, which indicates that using more frames could be helpful to calibrate these lower sensitivity parameters. A similar factor 10 is obtained between both covariance matrices (using all or few images), which will also lead to lower parameter uncertainties. Experimental and simulated force and NOD curves are displayed for the front face analyzes in Figure 7. A very good agreement is observed (i.e., ≈ 40 N) with errors only about eight times the force uncertainty (also accounting for servohydraulic fluctuations), and six times the uncertainty related to NOD measurements (i.e., ≈ 1.5 µm). It is worth noting that = 2.9 was reported in Ref. [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF], but with a higher level for (i.e., based on 0.1% of the load cell capacity, or 29 N). After adding the NOD data in the identification procedure, it achieved an even better description of the horizontal force (i.e., it would have been equal to 16 with the present load uncertainty). The latter was very well captured for the pre-peak regime, apart from the very beginning where the noise to signal ratio was very high. In the post-peak regime, the simulation follows the trend of the experimental curve, although slightly underestimating it overall, and overestimating it only between 1.5 and 2.2 mm of actuator displacement. It is worth noting that the shape of the cohesive law was kept constant. Although out of the scope of the present work, further investigation on that parameter (for the same PPR cohesive law) may help to achieve even better agreement with experimental results. For the NOD, some very small differences are seen at the beginning of the affine trend but the overall trend is very well captured by the simulations. Moreover, the coarser temporal discretization using few images led to really close results, which also emphasizes how using only a fraction of the total data could be sufficient to achieve good results in the present case. Similar experimental and simulated forces, as well as NODs analyzed for the back face are shown in Figure 8. A slightly higher is observed especially for the case using all images (i.e., = 9.7), which is related to the higher oscillations at the beginning of the test. Imperfections of the sample geometry and wedge misalignment may cause one side to experience more load at the very beginning of the experiment before crack propagation. This effect induces a decrease of when fewer images are considered (i.e., only one image of the very beginning of the test was used).

For the NODs, NOD is even smaller for the back face ( = 3.2 in both cases) and no specific difference between both curves is observed. The parameters calibrated when analyzing the front and back faces are reported in Table 2. The set of reference parameters is also given from the identification performed with CMOD data utilized as boundary conditions, and for the experimental relationship between NOD and CMOD [START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF]. For the front face, the parameters obtained using few images are very close to those obtained when using all images. However, their standard uncertainties are about three times higher. This trend was expected from the covariance matrices shown in Figure 5(b,f). The difference in the parameter may be related to the low number of points in the first steps when few images are used. Comparing the cases where CMOD or NOD were used for driving boundary conditions, remains virtually the same while increased about 10% for the NOD case. This increase may also be related to different Young's moduli , which was not calibrated in the case using CMOD data. The parameters and , similar to , remain very close in both identifications with differences of the order of 1% to 3%. Although a value of 0.82 was necessary in the CMOD-driven case, when NOD data were used it remained closer to unity (i.e., a smaller correction was needed). The overall observations for the back face are similar, with now being slightly smaller, and closer to the reference level. Care should be taken to compare since the Young's Modulus is about twice the level reported for the front face. The Young's modulus was expected to be the least sensitive parameter (Figures 5 and6). Such difference between both faces highlights this point. Higher resolutions or more images in time steps where is most sensitive could improve this results. Moreover, 3D simulations using information from both opposite sides is another perspective to improve the identification of elastic parameters [START_REF] Carpiuc-Prisacari | Comparison between experimental and numerical results of mixedmode crack propagation in concrete: Influence of boundary conditions choice[END_REF]. The parameter is closer to the CMOD case as well, which is interesting since the higher the needed correction, the more CMOD and NOD are affected instead of only the CMOD, as observed for the front face.

Given the fact that only NOD data were used in addition to load recordings, it was possible to further probe the calibrated CZM with kinematic measurements. Standard T3-DIC was run over the cyan ROIs shown in Figure 1(a,b).

From these measurements instantaneous RMS displacement differences normalized by the corresponding measurement uncertainties (i.e., ( )) are displayed in Figure 9(a,b). There is a gradual increase of both residuals with time, which is due to the assumption of a straight crack path in a sample with no lateral grooves to guide crack propagation. Although out of the scope of the present work, this shortcoming could be improved by using the gray level residuals to position cohesive elements along the actual crack path [START_REF] Vargas | On the identification and validation of fracture mechanics models for curved crack in mortar[END_REF]. Even with such hypothesis, the RMS difference is on average 25 times the displacement uncertainty (i.e., ≈ 24 µm), which is deemed small given the present conditions. for the previous DIC analyses. There is a gradual increase that is related to the crack propagating through the sample.

These residuals are compared to a new DIC analysis using the displacement fields corresponding to the calibrated parameters augmented by unknown rigid body motions that were then determined in an integrated framework. The new residuals are rather close to the DIC analyses that did not account any mechanical model. Consequently, the kinematic predictions of the calibrated CZM are therefore in rather good agreement with experimental observations. This benchmark case provided insight into the consistency of NOD usage to guide FE simulations in order to calibrate CZM parameters. Very similar parameters were obtained in comparison to results using only CMOD as boundary conditions. Good conditioning was achieved by coupling force and NOD data in the FEMU cost function.

The proposed simulations capture very well the splitting force and NOD histories, with rather consistent kinematics confirmed by the displacement fields and gray level residual maps. Moreover, not only could the cohesive parameters be calibrated, but also the coefficients for extrapolating NOD data to CMODs, the Young's modulus, and a correction factor for boundary conditions as well. The presented analyses fully validate the procedure that is now applied to a case where the CMOD data were not accessible.

WST-MZ1-S1450G experiment

A WST in which the CMOD was not measurable (see Figure 1(c)) is now analyzed. It was a preliminary test at room temperature that was performed before higher temperature experiments [START_REF] Vargas | Effect of sintering temperature on fracture energy of Alumina-Mullite-Zirconia castable at 600°C[END_REF]. Therefore, the sample was put inside the (turned off) furnace and there was only one quartz window between the camera and the sample. This sample had lateral grooves to guide the crack, meaning that a straight crack path is a good assumption for the present case.

The load uncertainty was equal to 10 N and that of NODs NOD = 1.5 centi-px (or ≡ 0.3 µm). For this test, there were 584 images available during loading (apart from the reference). In order to obtain a first order estimate of the sought parameters, only one in ten images were used (i.e., a total of 59 equally spaced time steps). The identification procedure took about 15 minutes for each iteration and converged in seven iterations. Then, when all images are used with the improved initialization, each iteration lasted two hours and converged in three iterations. Last, a final analysis was run with even less images (i.e., one in 50 images), namely only keeping 12 in the identification framework. Each iteration took five minutes and converged in five iterations, when initialized with the same starting parameters as the case with 59 images (see Table 1).

The results for the sensitivity analyses for the converged state are presented in Figure 10. The overall trends are qualitatively identical to the benchmark case. This consistency is believed to be mostly related to the WST setup, since the materials were very different. Moreover, the benchmark case had no lateral grooves, contrary to WST-MZ1-S1450G, which further proves the robustness of the qualitative sensitivities. To exemplify the benefits of coupling force and NOD data in the cost function, it is worth noting that the fracture energy and slope show similar postpeak force sensitivities with an opposite trend. When their (very dissimilar) NOD sensitivities are also accounted for, they become decoupled and the conditioning is improved. The case using only 12 images shows that such a coarse discretization can already qualitatively capture all sensitivity features. The Hessian matrix [ ] and its diagonalization are reported in Figure 11 along with the covariance matrix [ ]. The conditioning for WST-MZ1-S1450G is better than the benchmark cases, given the fact that the eigen values span over three orders of magnitude. The covariance magnitude also changes by a factor 10 when 59 images are used instead of the whole set, and then fivefold when only 12 images. The same ordering of parameter importance is observed as in the benchmark case, with the and parameters coming first, followed by the CZM ones, and then the Young's modulus . Interestingly, when using only 12 images, the eigenvectors show smaller correlations between parameters (e.g., second and fifth eigen vectors). Both experimental and simulated splitting forces as well as NODs are shown in Figure 12 for the three identi-fications. Their goodness of fit measured by and ranges between 2.5 and 10, being better when only 12 images were used, deteriorating a bit using 59 images but being further improved when all the images were considered (Table 3). Apart from oscillations at the beginning of the test, the loading curve mainly shows small deviations below 50% of the ultimate load in the post-peak regime. Therefore, most of the crack initiation and propagation process is well captured with the selected CZM. Small deviations in the post-peak regime are observed, similar to the benchmark case. Although out of the scope of the present work, cohesive laws of slightly different shapes may be able to further improve such fit. For the NOD data, the agreement of the curves is excellent, with some very small deviations around 1 mm of actuator displacement (Figure 12(d)). The difference between experimental and simulated peak loads is less than 2% in all three cases, which shows that initiation is also consistently described. The calibrated parameters for the three identifications are shown in Table 3. These results show how robust the identification procedure was since all the parameters were very close even using very few images. This result is due to the good conditioning of the Hessian (Figure 11(c,g,k)). When 12 images were used, among all parameters the Young's modulus is the farther to that obtained with the full image set (with a 15% difference), followed by and with differences of 3% and less than 1%, respectively. When 59 images were selected, these differences were reduced to an order of 1% or below. These results can be used as guidelines for data acquisition during similar experiments and optimizing identification procedures, depending on the sought parameters, which was not straightforward before the present sensitivity analyses. The value > 1 means that a tensile correction was needed, which indicates that some compressive state was present in the reference image that is believed to be related to wedge misalignment, especially considering the setup inside the furnace [START_REF] Vargas | Effect of sintering temperature on fracture energy of Alumina-Mullite-Zirconia castable at 600°C[END_REF]. The standard uncertainties are also reported in Table 3. As expected, as more images are considered in the identification procedure, the uncertainties are lowered. Using only 12 images yields uncertainties one order of magnitude (i.e., ≈ √ 50) higher than the case using the full data set. The case using 59 images had standard uncertainties about three times (i.e., ≈ √ 10) higher than that with 584 images, thereby corroborating what was observed in the benchmark case. It is also worth noting that the parameter changed considerably when only 12 images were used. As seen in Figure 7(d,h,l), the influence of the parameter is the highest in for fifth eigen vector. The third row of the eigen vector matrices correspond to the influence of the parameter on each eigen vector. For the case with 59 and 584 images, its influence in the other eigen vectors is not negligible (e.g., in the second eigen vector coupled to ). When more images are added, the finer temporal sampling brings more uncorrelated information, which was not captured with 12 images. To exemplify this point, one may check the force sensitivity of and parameters in Figure 10.

When using 59 or 584 images (i.e., Figures 10(a,c)), the sensitivity goes downward from the start, while in-creases until ≈ 500 s. If this tendency is compared to the case with 12 images (Figure 10(e)), apart from the first image, both curves show very similar trends apart from one multiplicative factor between them (i.e., are essentially anti-correlated). Therefore, using more images led to more trustworthy parameters, since their full and detailed effects were better captured.

As for the benchmark case, the predictive quality of the model was further probed with additional DIC analyses.

Figure 13(a) shows the displacement residual between DIC and FEMU kinematics, normalized by the displacement uncertainty when compared in the cyan region of Figure 1. The grooved region (Figure 1(c)) was not accounted for in these calculations since the darker regions had very low contrast and the corresponding results would not be trustworthy.

On average, the RMS displacement difference (6 µm or 0.32 px) is 16 times the uncertainty level (0.02 px). The two displacement fields are probed independently thanks to the gray level residuals (Figure 13(b)). The difference between both gray level residuals is on average less than 1% of the dynamic range, which is even lower than what was observed in the benchmark case (Figure 9). As the crack was guided by the lateral grooves in WST-MZ1-S1450G, the crack path was better described by the assumption of a straight propagation, which explains the smaller residual levels and differences in the present case.

Conclusion

Cohesive zone model (CZM) parameters were calibrated using notch opening displacements (NODs) and force data from wedge splitting tests (WSTs). One benchmark case was first evaluated to validate the methodology, since the crack mouth opening displacements (CMODs) and images of both opposite faces were available. Then, a WST that had only the NOD visible (and only for one side) was analyzed. The proposed framework was especially interesting when imaging the crack propagation path (to have better access to data related to the fracture process zone) without losing information about the loading conditions for driving numerical simulations. Not only were CZM parameters calibrated, but also three parameters related to the boundary conditions together with the Young's modulus. Overall, the numerical results showed a good agreement with experimental data in terms of force, NODs, displacements fields, and gray level residuals. They showed that the selected CZM [START_REF] Park | A unified potential-based cohesive model of mixed-mode fracture[END_REF] was able to capture the main features of crack initiation and propagation in the two refractories studied herein.

The sensitivity analyses showed how each parameter influenced the splitting force and NODs, which were coupled in the cost function for identification purposes. By analyzing the Hessians and their diagonalized counterparts, the conditioning of the present framework could be assessed, as well as the correlation between parameters and also their influence. The diagonal of the covariance matrix also gave access to the uncertainties for the calibrated parameters.

The parameters in decreasing order of importance, for each analyzed case, were and (related to extrapolating NOD to CMOD), followed by and (CZM parameters), then (a correction for the boundary condition)

and (Young's modulus). If force or NOD alone had been used in the cost function, the conditioning would have deteriorated to a point that the calibration of some (if not all) the chosen parameters would not be trustworthy. Further, the effect on the calibration results of using only few images of the whole set was also investigated. Although very few images (down to 12) could be used (in particular to get initial estimates), the more time steps were used the more trustworthy the results and the higher the influence of each parameter.

On the benchmark case, it was shown that very similar results were obtained if NODs were used instead of CMODs to drive the boundary conditions in the simulation, thereby validating the proposed methodology. Analyzing both faces gave insight into the variability of calibrated parameters. Although the cohesive strength and the fracture energy were very similar, the Young's modulus had larger deviations, which are presumably related to 3D effects during loading and crack propagation [START_REF] Carpiuc-Prisacari | Comparison between experimental and numerical results of mixedmode crack propagation in concrete: Influence of boundary conditions choice[END_REF]. Consequently, it is advised to analyze both sides whenever possible, especially if the elastic parameters are sought. For the other WST, first estimates for the CZM parameters were obtained by simulating the full sample even if only part of it was imaged. In that case, there were lateral grooves guiding the crack, which further improved the kinematic residuals (i.e., in displacement and gray level) under the hypothesis of straight propagation.

Last, the absence of CMOD data in the last WST-MZ1-S1450G was due to the sample being inside a turned off furnace with limited size window. It was a preliminary test before higher temperature tests were performed [START_REF] Vargas | Effect of sintering temperature on fracture energy of Alumina-Mullite-Zirconia castable at 600°C[END_REF].

Therefore, using the same framework for these experiments would be a direct continuation of the present work to compare the parameters for the same material but at 600°C. 3D analyzes to better understand crack propagation and especially the influence on elastic parameters is another perspective, although information from both opposite faces would be needed.

Figure 1 :

 1 Figure 1: Sample geometry in the reference images for the benchmark case ((a) front and (b) back surfaces) and WST-MZ1-S1450G (c). The geometry used for the FE procedure is depicted in red (100 mm in height and width, and 72.5 mm (a,b) or 75 mm (c) in thickness), and the regions where Dirichlet boundary conditions were applied in blue circles. NODs were measured in the yellow squares, and the displacements were compared to DIC results in the cyan region.

Figure 3 :

 3 Figure 3: Parameter sensitivities of splitting force (a,c) and to the NOD NOD (b,d) for the front face using the whole image set (a,b) or one tenth of the latter (c,d). The vertical dashed line depicts the ultimate load instant.

Figure 4 :

 4 Figure 4: Parameter sensitivities of splitting force (a,c) and to the NOD NOD (b,d) for the back face using the whole image set (a,b) or one tenth of the latter (c,d). The vertical dashed line depicts the ultimate load instant.

Figure 5 :

 5 Figure 5: Dimensionless Hessian (a,e) and covariance (b,f) matrices. Diagonal (c,g) and eigen vector (d,f) matrices of the Hessian matrix. The matrices are shown for the whole image set (a-d) and a reduced set (e-h) for the front surface. The first three matrices are shown in decimal logarithms. The components that may have negative values are shown with the signs represented on it.

Figure 6 :

 6 Figure 6: Dimensionless Hessian (a,e) and covariance (b,f) matrices. Diagonal (c,g) and eigen vector (d,f) matrices of the Hessian matrix. The matrices are shown for the whole image set (a-d) and a reduced set (e-h) for the back surface. The first three matrices are shown in decimal logarithms. The components that may have negative values are shown with the signs represented on it.

Figure 7 :

 7 Figure 7: Experimental and simulated force (a,c) and NOD (b,d), using the full image set (a,b) or only a few images (c,d) for the front face.

Figure 8 :

 8 Figure 8: Experimental and simulated force (a,c) and NOD (b,d), using the full image set (a,b) or only a few images (c,d) for the back face.

Figure 9 :

 9 Figure 9: (a,b) Instantaneous RMS residuals between FEMU and DIC displacement fields. (c,d) Corresponding gray level residuals. The results are reported for the front (a,c) and back (b,d) faces.

Figure 9 (

 9 Figure 9(c,d) shows the change of gray level residuals normalized by the dynamic range of the reference image

Figure 10 :

 10 Figure 10: Parameter sensitivities of splitting force { } (a,c,e) and NOD { } (b,d,f) using the whole (584) image set (a,b), 59 (c,d) or 12 (e,f) images from the full set of WST-MZ1-S1450G. The vertical dashed line depicts the ultimate load instant.

Figure 11 :

 11 Figure 11: Dimensionless Hessian (a,e,i) and covariance (b,f,j) matrices of WST-MZ1-S1450G. Diagonal (c,g,k) and eigen vector (d,f,l) matrices of the Hessian matrix. The matrices are shown for the analysis using 584 (a-d), 59 (e-h), and 12 (i-l) images. The first three matrices are shown in decimal logarithms. The components that may have negative values are shown with the signs represented on it.

Figure 12 :

 12 Figure 12: Experimental and simulated force (a,c,e) and NOD (b,d,f) histories using the full (584) image set (a,b), 59 (c,d), or 12 images (e,f) of WST-MZ1-S1450G.

Figure 13 :

 13 Figure 13: (a) Instantaneous RMS residual between FEMU and DIC displacement fields. (b) Corresponding gray level residuals.

Table 1

 1 Initial parameters for both studied materials

	Parameter	Description	Benchmark WST-MZ1-S1450G
	[MPa]	Cohesive strength	1.7	3.8
	[J/m 2 ]	Fracture energy	108	286
	R	NOD vs. CMOD ratio at ultimate load	2.6	2.6
	c	Slope for the post-peak NOD vs. CMOD affine trend	1.54	1.54
	BCc	Boundary condition correction	1	1
	E [GPa]	Young's modulus	17	65

Table 2

 2 Converged parameters for both identification with 36 and 360 images for the front and back faces. Standard uncertainties (±) are also reported. Reference parameters after Ref.[START_REF] Vargas | Fracture energy evaluation of refractories in wedge splitting tests from notch opening displacements[END_REF].

			Front face			Back face
	Parameter Ref.	36 images	360 images	Ref.	36 images	360 images
	[MPa]	1.76	1.78 ± 0.005	1.794 ± 0.002	1.67	1.609 ± 0.004	1.610 ± 0.001
	[ J/m 2 ]	117	132.0 ± 0.2	132.3 ± 0.06	112	118.7 ± 0.2	118.9 ± 0.06
	R	2.31	2.36 ± 0.01	2.272 ± 0.004	2.37	2.31 ± 0.02	2.160 ± 0.005
	c	1.54 1.503 ± 0.0007 1.503 ± 0.0002 1.57 1.516 ± 0.0007 1.516 ± 0.0002
	BCc	0.82	0.970 ± 0.004	0.975 ± 0.001	0.73	0.73 ± 0.01	0.825 ± 0.002
	E [GPa]	-	20.5 ± 0.3	21.2 ± 0.1	-	43.7 ± 0.9	44.0 ± 0.3
		-	8.3	8.1	-	8.7	9.7
		-	6.2	6.1	-	3.2	3.2

Table 3

 3 Converged parameter sets for identifications using 12, 59 and 584 images with respective standard uncertainties (±).

	Parameter	12 images	59 images	584 images
	[MPa]	4.40 ± 0.02	4.25 ± 0.009	4.288 ± 0.003
	[J/m 2 ]	284 ± 0.8	281.6 ± 0.3	282.6 ± 0.1
	R	2.31 ± 0.01	2.16 ± 0.009	2.156 ± 0.003
	c	1.448 ± 0.001 1.474 ± 0.0007 1.4668 ± 0.0002
	BCc	1.57 ± 0.02	1.340 ± 0.005	1.263 ± 0.001
	E [GPa]	40 ± 0.8	47.0 ± 0.5	46.4 ± 0.2
		4.4	9.8	8.2
		2.5	5.0	4.2

Acknowledgments

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001 CAPES (Brazil) and #2018/15266-0, #2018/23081-0, and #2010/20920-9, São Paulo Research Foundation (FAPESP). The authors would like to thank Prof. J.A. Rodrigues for stimulating discussions about various aspects of WSTs and IBAR (Indústrias Brasileiras de Artigos Refratários, Poá, Brazil) for kindly providing the materials.

Credit authorship statement

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.