Elies Gherbi
email: elies.gherbi@irt-systemx.fr

Hamza Khemissa
email: hamza.khemissa@expleogroup.com

Mohammed Lamine Bouchouia
email: mohammed.bouchouia@irt-systemx.fr

Maxime Ayrault
email: maxime.ayrault@valeo.com

From Challenges to Opportunities: A Comprehensive Study of AI-based In-Vehicle Intrusion Detection Systems

Keywords: Machine learning, Intrusion detection, Forensics, in-vehicle network

While significant research has been conducted on ML-based In-Vehicle Intrusion Detection Systems (IV-IDS), the practical application of these systems needs further refinement. The safety-critical nature of IV-IDS calls for precise and objective evaluation and feasibility assessment metrics. This paper responds to this need by conducting a rigorous ML-based IV-IDS analysis. We offer a thorough review of recent automotive forensics studies spotlighting the constraints relevant to Invehicles networks and the associated security/safety requirements to reveal the current gaps in the existing literature. By addressing the limitations of AI in IV-IDS, this paper contributes to the existing research corpus and defines pertinent baseline metrics for in-vehicle networked systems. Essentially, we reconcile the requirements of real-world autonomous vehicles with those of the security domain, enabling an assessment of the viability of AI-based intrusion detection systems.

I. INTRODUCTION

Autonomous Vehicles (AVs) have been rapidly permeating everyday life. Over the past decades, continuous technological advancements have evolved these vehicles into sophisticated entities. AVs offer numerous advanced driver-assistance systems (ADAS) birthed from the progressive integration of Electronic Control Units (ECUs). The growing abundance of ECUs and networks within vehicles, while enhancing capabilities, makes them vulnerable to different possible cyber-attacks. As these AVs become more prevalent, such attacks pose everincreasing risks to the safety of passengers, pedestrians, and other vehicles.

In the face of these threats, deploying Intrusion Detection Systems (IDS) within vehicles has gained momentum. Such systems serve as vigilant sentinels, detecting unusual or suspect activities and alerting users or systems to potential breaches. By flagging these fraudulent attempts early on, IDS can deter them before they cause substantial damage.

We note the need for a precise definition of automotive forensics. However, a general understanding of principles and processes involved in investigations can be gained by following the guidelines set out in digital forensics, such as ISO/IEC 27043:2015 [START_REF]:2015 Information Technology -Security Techniques -Incident Investigation Principles and Processes[END_REF]. However, to enable resilience in these systems, a vehicle must defend itself against attacks and recover swiftly to its optimal operational mode. This is where digital forensics comes into play, to enhance the system's resilience throughout its lifecycle. Digital Forensics has evolved into a crucial part of cyber defense, traditionally understood as collecting, preserving, analyzing, and presenting data post-incident for evidentiary purposes [START_REF] Buquerin | A generalized approach to automotive forensics[END_REF]. While the conventional approach has been reactive, responding postincident, a proactive approach is being adopted increasingly. The proactive modality involves ongoing data collection, protection, detection of suspicious events, evidence acquisition, analysis, and building cases against potential threats [START_REF] Sharma | Cybersecurity and forensics in connected autonomous vehicles: A review of state-of-the-art[END_REF].

The realm of artificial intelligence (AI)-based IDS addresses new challenges. Given the complex and black-box nature of AI models, the forensic process differs significantly from rulebased systems. AI for IDS requires a nuanced understanding of the principles and processes involved in these investigations, following guidelines, recommendations, and standards in the digital forensics domain. As science evolves, a precise definition and standardization of automotive forensics must be established. Nevertheless, current AI-based IDS do not account for forensic requirements or specific internal automotive needs, which makes the use of the different in-vehicle IDS impractical.

In this article, we aim to build a framework for assessing the viability of IDS based on AI/machine learning (ML) algorithms specifically designed for in-vehicle use. Based on the analysis of current research, we have identified areas of improvement and suggested future research directions. Our main goal is to establish a set of basic evaluation metrics for AI/ML-based IDS that are tailored to the unique requirements of in-vehicle systems and ensure the models' practicality.

II. BACKGROUND AND RELATED WORK

In this section, we overview common attacks in AVs context, we revise the definition of resilience, and we outline recent research works on AI-based IDS for AVs.

A. Common Attacks

Over the past few years, Manufacturers have incorporated more driving assistance systems inside vehicles. These systems can control essential functions of the vehicle, which could compromise the driver's safety if misused. Additionally, the increased connectivity of these systems creates new opportunities for cyber-attacks, potentially compromising the privacy and integrity of the vehicle and its passengers. These attacks can occur in three ways: physically, close to the vehicle, or remotely.

• OBD-II: modifying the vehicle's configuration to unlock new functions , sending messages on the CAN bus. • USB port: installing malware or ransomware on the vehicle. • Physical attacks: cutting brakes or steering wires.

• Bluetooth: invading the user's privacy by tracking its movement.

• Wi-Fi: spreading worms to collect privacy data, install malware, or take remote control of the vehicle. • V2X: taking control of an entire fleet of vehicles.

B. Resilience definition

Resilience is a term that has been used extensively in the fields of psychology, economics, environment, and the medical field. In each of these domains a definition has appeared, giving meaning to this term, yet in the computer field a clear and precise definition has not been established. By merging different definitions that we encountered, we come up with the following definition:

• Resilience: being able to defend against an attack as long as possible, and once the defenses have fallen, being able to recover to the nominal operating state as quickly as possible. Resilience aims to improve the security of the vehicle but without any impact on the safety and limited impacts on the overall performance. To provide trustworthy systems, system architects are looking for a way to apply cyber-resilience concepts into architectures, designs, and operational systems [START_REF] Bodeau | Cyber resiliency metrics, measures of effectiveness, and scoring[END_REF]. According to NIST SP 800-160 Vol. 1 [START_REF] Ross | Nist sp 800-160[END_REF], cyber resilience metrics can be used as evidence in assurance cases.

C. AI-based IDS for AVs

ISO 26262 mandates the implementation of safety mechanisms in AVs components to maintain their functionality and mitigate known failures/malicious activities. However, AVs failures still occur due to compromised safety mechanisms which require additional countermeasures like IDS. AIpowered systems can analyze large amounts of data, identify patterns, and adapt detection methods in real-time. Many articles explored ML and deep learning (DL) methods to detect cyber threats in AVs.

• In a CNN-based IDS survey [START_REF] Mohammadpour | A survey of cnn-based network intrusion detection[END_REF], the authors compare the capabilities, features, and performance of ML approaches. The result varies depending on the used datasets, network architecture, and feature extraction techniques.

• In an overview of DL-based IDS in an automotive network [START_REF] Lampe | A survey of deeplearning-based intrusion detection in automotive application[END_REF], the authors categorized the different approaches on topology and techniques based on the accuracy, precision, recall, and F1 measures. • An improvement of in-vehicle security networks by using a hybrid blockchain framework to provide a secure and decentralized way to store and share data related to the in-vehicle network (IVN) has been discussed in [START_REF] Khatri | Security issues with invehicle networks, and enhanced countermeasures based on blockchain[END_REF]. • In a survey on AI/ML-based IDSs and misbehavior detection [START_REF] Ajibuwa | A survey on ai/ml-driven intrusion and misbehavior detection in networked autonomous systems: Techniques, challenges and opportunities[END_REF], the authors suggested that despite extensive research, few solutions have been implemented in practical applications. Partly due to the lack of real-world scenarios demonstration and the inadequacy of evaluation metrics used to measure IDS feasibility solutions. These articles share common points on methods, results, and applications. However, the evaluation is mainly done from a performance viewpoint. Feasibility is assessed based on the execution time and the computational cost. This brought to our attention a significant gap in the current approach to AVs. There is a need for metrics that address specific requirements in this domain. We also find that studies do not consider enough automotive cybersecurity and safety aspects, which are crucial in designing an effective IDS. To address this missing link in current methodologies and elucidate resultant system implications, we extract automotive and cybersecurity domainspecific requirements. Then, we outline the shortcomings of existing AI-based IDS solutions for AVs networks.

III. ARCHITECTURES CONSTRAINTS AND DOMAIN

REQUIREMENTS OF N-IDS IN AVS In this section, we discuss domain-specific requirements and constraints for building efficient IDS.

A. Physical constraints

Connected vehicles may offer a range of new functionalities, thanks to a large ECUs integration within their architecture, but they are nevertheless embedded systems with limited computing power. Most embedded ECUs within the vehicle are small 8-or 16-bit microcontrollers with only one simple task to perform, with no space or computing power to detect attacks or defend themselves. The ECU's available memory is more in the order of Kb than Gb, as on a conventional system making it complicated to implement defense processes.

So in order to defend such a system, an in-depth architecture has been implemented. The vehicle's various functionalities are grouped into domains, each with its own sub-network for communicating with each other. The sub-network is managed by a Domain Controller (DC), which handles traffic within the domain. Inter-domain communications are handled by a Secure Gateway with greater computing power, enabling the network to be monitored and various types of system defense to be implemented.

B. Resilience and forensics requirements

CAVs are critical embedded systems, which require efficient techniques to ensure safety. Mechanisms for threats detection and investigation methods are very important in the design of the vehicle architecture's defenses.

In order to strengthen the resilience level of a system, some cyber resiliency metrics established by MITRE can be used [START_REF] Bodeau | Cyber resiliency metrics catalog[END_REF]. From the selected metrics, we identified some approaches related to the metrics.

• Analytic monitoring: checking the system's integrity.

• Substantiated integrity: detecting adversary's attempts to deliver compromised data, software, and hardware. • Contextual awareness: situational awareness, revealing patterns or trends in adversary behavior. • Deception: misleading the adversary or hiding critical assets from the adversary. • Adaptive Response: optimizing the ability to react appropriately to adverse conditions. • Non-Persistence: limiting an adversary intrusion time.

• Redundancy: using hardware or software replication in a system.

Unfortunately, those approaches are tailored for forensic experts, and AI cannot take those directly into consideration. Thus, we discuss in the following section a new design to define AI applicable metrics.

IV. AI/ML-BASED IDS TECHNIQUES IN IVN

In this section, we introduce our taxonomy for AI/MLbased IDS techniques in IVN and an overview clarifying the decisions made throughout the various stages of constructing an AI model. This can assess the detection system efficacy and the suggested approaches practicality.

Our taxonomy presented in Figure 1 is divided into three parts: 1) data and monitoring, 2) modeling and learning tasks, and 3) deployment and detection strategies.

A. Data and monitoring

The necessity of well-curated datasets for effective anomaly detection in ML-based IDS is widely acknowledged. We aim to examine this relationship, structuring our investigation into two clear segments.

1) Data sources and modalities AVs are complex machines, comprised of sensors, ECU, and actuators. Those parts are interconnected through a variety of intra-vehicle networks that enable AVs monitoring and control. There are currently five primary intra-vehicle networks in use (LIN, CAN, FlexRay, Ethernet, and MOST), in addition to other V2X communication like WIFI and Bluetooth. There have been many proposed datasets for in vehicles communication [START_REF] Karopoulos | Demystifying in-vehicle intrusion detection systems: A survey of surveys and a meta-taxonomy[END_REF]- [START_REF] Vahidi | Systematic evaluation of automotive intrusion detection datasets[END_REF], which mainly focuses on capturing CAN traffic. Those datasets are usually recorded through the vehicle diagnostics port (OBD-II), or simulation software such as Simulink. In a study by [START_REF] Karopoulos | Demystifying in-vehicle intrusion detection systems: A survey of surveys and a meta-taxonomy[END_REF], the authors provide a comprehensive analysis of security-focused datasets, based on three primary factors: covered technology, attack types, and overall characteristics. In [START_REF] Verma | Addressing the lack of comparability & testing in can intrusion detection research: A comprehensive guide to can ids data & introduction of the road dataset[END_REF] the authors also explore several automotive IDS datasets and identify some issues, including inaccurate documentation or labeling and noisy attacks that can be easily detected through simple methods. Meanwhile, in [START_REF] Swessi | A comparative review of security threats datasets for vehicular networks[END_REF] the authors recently conducted a survey of automotive security datasets and evaluated each data-set based on 11 criteria. They recommend specific datasets for use in in-vehicle and inter-vehicle ML-based IDSs.

2) Information types and preprocessing

The extraction of relevant features from raw in-vehicle data is pivotal for effective attack detection using ML algorithms. IDS detects intrusions at two levels of inspections: flow-based and payload-based. Flow-based approaches group packets together based on their common properties during a specific period, considering the arbitration IDs' frequencies, timing, and sequencing. Payload-based approaches, on the other hand, focus on the information carried by the packets and reflect the behavior of the vehicle's ECUs. The message content can be represented either by the data frame (strings of bits) without explicitly recovering the signals that these bits represent or by the signal, which requires decoding the raw data field bits [START_REF] Verma | Addressing the lack of comparability & testing in can intrusion detection research: A comprehensive guide to can ids data & introduction of the road dataset[END_REF]. Other works use physical layer attributes such as voltage.

Some attacks, such as flooding attacks, can be identified by analyzing the communication flow as they significantly impact it. However, other types of attacks can only be detected by examining the payload. Therefore, detecting various types of attacks require analyzing both the flow and payload levels.

Besides the inspection level, preprocessing converts network traffic into a series of observations, represented as feature vectors. There are two modeling scenarios: point-wise and window-wise. Point-wise labels each packet as normal or malicious, while window-wise gives the network state over a specific time frame. However, outputting for each data point is impractical in real-time environments like IVN. Despite this, many in-vehicle ML-IDS methods use point-wise modeling [START_REF] Ma | A grubased lightweight system for can intrusion detection in real time[END_REF], [START_REF] Seo | GIDS: GAN based intrusion detection system for in-vehicle network[END_REF].

B. Modeling and learning tasks

In-vehicle IDS increasingly utilize DL and ML techniques. Many methods have been employed, from supervised to unsupervised anomaly detection. Various architectures have been implemented to showcase the effectiveness of DL in detecting attacks, compared to classical ML techniques [START_REF] Ajibuwa | A survey on ai/ml-driven intrusion and misbehavior detection in networked autonomous systems: Techniques, challenges and opportunities[END_REF]. These techniques include traditional neural networks and newer transformers, GPT [START_REF] Nam | Intrusion detection method using bidirectional gpt for in-vehicle controller area networks[END_REF], and diffusion models [START_REF] Tang | A diffusion model based on network intrusion detection method for industrial cyber-physical systems[END_REF], which have been used to improve the quality of intrusion detection. Although numerous techniques are available, there remains a considerable disparity between their application and practicality in real-world scenarios, especially in in-vehicle cybersecurity, forensics, and safety. This is because these methods must incorporate specific requirements as constraints in their modeling or evaluation processes and address the architectural limitations of in-vehicle systems, which have limited resources. Another critical factor is the need to reduce the "black box" aspect of deep neural networks to make forensic and investigation processes easier, which is the main goal of an IDS. Therefore, adapting and improving these methods is essential to meet the demands and challenges of the in-vehicle system and cybersecurity domains. To achieve this, the design of these approaches needs to consider the following aspects:

1) Explainability and interpretability Entrusting crucial decisions to a system without explanation presents apparent risks. The lack of transparency further exac- erbates the problem in the field of Cybersecurity. In [START_REF] Capuano | Explainable artificial intelligence in cybersecurity: A survey[END_REF], the authors provide an in-depth examination of the application of XAI in Cybersecurity and analyze the main Cybersecurity application fields concluding that machine-human integration is possible under one condition: the former must be explainable to the latter. Authors in [START_REF] Mahbooba | Explainable artificial intelligence (xai) to enhance trust management in intrusion detection systems using decision tree model[END_REF] explore the decision tree algorithm for malicious node identification by applying it to an openly available dataset. The resulting model generates rules that are understandable and help network security personnel enhance trust by taking a possible course of action in case of malicious traffic identification. In [START_REF] Neupane | Explainable intrusion detection systems (x-ids): A survey of current methods, challenges, and opportunities[END_REF], the authors consider human-inthe-loop, which can be used as a guideline when designing an X-IDS and introduce the use of SHAP (SHapley Additive exPlanations) in the field of IDS to generate explanations.

XAI in Cybersecurity can be a double-edged sword, as it can leave the system vulnerable to adversary attacks, meaning that, malicious actors that understand how the X-IDS works can exploit it to craft attacks challenging its robustness.

2) Robustness and generalization Recent years have witnessed a surge in research focusing on adversarial ML, a field that explores the vulnerabilities of ML models to adversarial attacks [START_REF] Bai | Recent advances in adversarial training for adversarial robustness[END_REF]. This is particularly relevant in the context of in-vehicle IDS, where the robustness of ML models is critical to ensure the safety and integrity of vehicular systems [START_REF] He | Adversarial machine learning for network intrusion detection systems: A comprehensive survey[END_REF]. Several methods have been proposed to enhance the robustness of ML models against adversarial attacks. However, these methods often involve a trade-off between adversarial robustness and detection accuracy, which may not be acceptable in certain domains where high detection accuracy is paramount. Given the current challenges, there is a requirement for strong and suitable methods to enhance the adversarial robustness of ML models while still retaining their detection accuracy. Additionally, it is essential to develop metrics that can evaluate the effectiveness of these methods and guide future research in this area, specifically for invehicle IDS.

To sum up, although adversarial ML has made notable advancements, its usage in the in-vehicle sector is still in its early stages [START_REF] Wickramasinghe | Rx-ads: Interpretable anomaly detection using adversarial ml for electric vehicle can data[END_REF], [START_REF] Sauka | Adversarial robust and explainable network intrusion detection systems based on deep learning[END_REF]. Further research should concentrate on defining the characteristics of adversarial attacks in this area, creating resilient approaches, and setting benchmarks for evaluating adversarial strength.

3) Architectures and topology

An important aspect that has been overlooked in current research is the unique architecture of future IVNs [START_REF] Mariño | The future roadmap of in-vehicle network processing: A hw-centric (r-)evolution[END_REF]. These networks are naturally distributed and hierarchical, with isolated probes for security purposes. This type of architecture requires a different approach when creating IDS models due to challenges related to communication overhead and information loss, mainly when compressing data. The impact of a distributed architecture on communication overhead is a crucial factor to consider. The difference between centralized and distributed models regarding communication cost can be substantial. Centralized models, while potentially more straightforward to implement, may suffer from high communication costs and increased latency due to the need to transmit data across the network. On the other hand, distributed models can reduce communication costs and latency by processing data locally. However, they may face challenges related to information loss, particularly when data compression is involved. Moreover, the fact that IVNs comprise multiple subnetworks, each with its modality and behaviors, introduces another layer of complexity. This scenario is reminiscent of the challenges faced in multi-modal ML, an advanced domain that seeks to harmonize different data modalities [START_REF] Baltrusaitis | Multimodal machine learning: A survey and taxonomy[END_REF]. IDS for IVN could draw inspiration from this field, aiming to harmonize the behaviors of different sub-networks within the system.

C. Detection and deployment strategies

When implementing AI-based IV-IDS, it is crucial to consider various factors. One key factor is the detection strategy. For instance, in a distributed setting, assessing the uncertainty of each probe's result is important. Cooperative aspects of the IDS output and different update strategies can help manage this uncertainty [START_REF] Abdar | A review of uncertainty quantification in deep learning: Techniques, applications and challenges[END_REF], [START_REF] Abusitta | A deep learning approach for proactive multi-cloud cooperative intrusion detection system[END_REF]. In [START_REF] Abusitta | A deep learning approach for proactive multi-cloud cooperative intrusion detection system[END_REF], the authors present a cooperative ML-IDS that leverages the output of the IDS by using a consensus algorithm that considers the uncertainty of the results from each probe. In [START_REF] Zhu | Mobile edge assisted literal multi-dimensional anomaly detection of in-vehicle network using lstm[END_REF], the authors proposed a distributed framework for IVN anomaly detection that utilizes multi-dimensional temporal and data properties. They suggest deploying it on a mobile edge for additional computational resources, which could cause delays and hinder real-time attack detection.

Another aspect to consider is the deployment of DL-based Embedded Intrusion Detection Systems (EIDS) in automotive applications. These systems need to be designed carefully considering the computational and memory costs. In the study [START_REF] Ma | A gru-based lightweight system for can intrusion detection in real time[END_REF], a lightweight DL model was proposed for automotive systems. The model was designed to minimize CPU and memory costs while maintaining high detection accuracy.

D. Discussion

Creating comprehensive automotive datasets is difficult due to various network technologies, leading to limited features for robust IDS models. More datasets are needed for cellular and Bluetooth communications. Incomplete datasets impact the evaluation of IDS approaches, making it hard to compare and benchmark ML-based in-vehicle IDS approaches. The absence of suitable data makes it challenging to reproduce research findings and develop dependable in-vehicle IDS.

Although there have been many studies on in-vehicle IDS, some fail to assess the effectiveness of their ML/DL solutions because the available datasets need to be aligned with the needs of AVs. Furthermore, more appropriate metrics must be used to evaluate and validate datasets and their preprocessing while considering the architecture and domain requirements of IVN IDS as mentioned in the preceding section III. A critical issue revolves around the semantic gap between the detection results produced by ML-based IDS and the actionable interpretations needed by network operators III-B. While these systems can identify potential security threats, translating these detections into practical, implementable defense strategies remains problematic. This gap stems primarily from the intricate nature of these systems, characterized by complex algorithmic processes and layers of abstraction that makes it difficult to interpret the outcomes in a human-understandable form. Also, Due to the opacity of their decision-making process, MLbased IDS can lead to false positives that may have severe implications for network operators, resulting in considerable hesitance to act based solely on their detection results. The root of these issues can be traced back to the lack of explainability. Explainability in AI refers to the degree to which a human can understand a system's reasoning. The need for explainability is pressing, particularly in IDSs where understanding the reasons behind an alert is crucial for effective response and mitigation. Current explanation methodologies developed for ML-based systems have been proven inadequate for In-vehicle IDS due to their inability to manage historical inputs and complex feature dependencies inherent in structured data. future research must explore ways to enhance the explainability of ML-based IDS. This would serve a dual purpose: firstly, to provide human operators with actionable insights for rapid forensic intervention, and secondly, to inform the development of automated defensive mechanisms. By addressing the fundamental root cause of the issues above, such efforts can significantly improve the effectiveness and reliability of ML-based IDS, bringing us closer to robust, AI-enabled network security.

While significant strides have been made in improving the accuracy of IDS through various machine-learning architectures, there is a pressing need to consider the unique characteristics of IVNs. Future research should address the challenges posed by these networks' distributed and hierarchical nature, the communication overhead, and the need for harmonization across different network modalities.

ML-based IDS deployment and detection strategies need to consider various aspects, including managing output uncertainty, cooperative aspects, update strategy, and computational and memory costs. Integrating DL methods can further enhance the performance and security of these systems.

V. METRICS FOR EVALUATING THE VIABILITY OF A PRACTICAL AI BASED IDS

When it comes to using AI, particularly ML and DL, for IVN IDS, many challenges and complexities must be addressed to make these systems more dependable and effective for AVs. In order to evaluate and tackle these challenges, it is essential to identify specific metrics. The following section outlines a set of metrics, each designed to address a particular issue mentioned in the previous section IV. These metrics serve as both a diagnostic tool to identify and quantify problems and a performance indicator to track improvements and interventions. Understanding the relevance of each metric and the issues they address is crucial in the context of IVN IDS.

• Data Adequacy: measures datasets available quantity and quality for IDS models training. Used to assess limited and incomplete datasets.

VI. CONCLUSION

In this paper, we examine the difficulties posed by AI/ML applications in IV-IDS. We take into account the constraints within the vehicle and the requirements of cybersecurity forensics. We identify several critical issues, including insufficient data, lack of transparency, and the requirement for protection against adversarial attacks. We propose a set of metrics to measure and tackle these challenges, which could guide improvements in system design and implementation. Future efforts should focus on creating appropriate benchmarks and enhancing the transparency of the system. This will enable effective detection and response strategies against intrusion. Overall, this study provides a solid foundation for researchers and practitioners, outlining essential considerations and potential directions for developing robust, AI-based IV-IDS.

Fig. 1 .

 1 Fig. 1. AI-based In-vehicle IDS requirements

•

  Data Compatibility: evaluates the compatibility of datasets with AVs' needs. Address the problem of misaligned datasets. • Harmonization: measures the system's ability to work harmoniously across different network modalities. Harmonization across different network modalities. • Evaluation Metric Suitability: measures evaluation metrics appropriateness used to validate datasets and preprocessing techniques. Communication Overhead: assesses the additional data being transmitted due to IDS implementation. Evaluate the challenges posed by the distributed and hierarchical nature of IVNs. • Output Uncertainty Management: evaluates the system's ability to handle and communicate the output uncertainty, given the need to manage output uncertainty in ML IDS. • Adversarial Robustness: assesses IDS resilience against adversarial attacks. Quantifies the system's ability to correctly identify threats even for inputs intentionally designed to deceive or mislead the system. • Cooperation Level: measures system collaboration with other systems and parts of the vehicle. • Update Efficiency: how effectively the system can update itself. • Computational and Memory Costs: quantifies computational resources required for running the IDS and the amount of used memory. • Detection Latency: measures the time taken to detect an intrusion.

• Explainability: quantifies the extent to which a human can understand a system's reasoning. Assess the lack of explainability in DL-NIDS.

•


Wilson F Espinoza

Jean-Michel Pereira

Timothy Kneafsey

Sheng Dai

Mechanical and creep properties of granitic minerals of albite, biotite, and quartz at elevated temperature Authors

Keywords: Granite, biotite, temperature, creep, indentation

Fundamental mechanical and creep behavior of minerals at elevated temperatures is relevant to many subsurface exploration and resource recovery processes, yet remains largely elusive. This study uses instrumented indentation to measure microscale deformation and nanoscale creep of biotite, albite, and quartz, which are the major mineral phases of granitic rocks, at temperatures up to 400 ⁰C. The results show that the elastic modulus of biotite is not only the lowest at room temperature but also decays the most at elevated temperature (i.e., by 25% reduction at 400 ⁰C) compared to that of albite and quartz. The creep deformation within 20 seconds doubles when the temperature increases from 20 ⁰C to 400 ⁰C for albite, quartz, and biotite, and biotite showed at least three times the overall creep deformation compared to quartz and albite under the same temperature. The indentation creep deformation can be well characterized using a logarithmic time model, showing both transient and constant-rate creep deformation. The secondary creep rate for biotite is about three times that of albite and quartz at the same temperature, and as the temperature increases from 20 ⁰C to 400 ⁰C, the rate of secondary creep becomes approximately 10 times faster for the three tested minerals. Both transient creep deformation and the rate of secondary creep can be empirically correlated to the elastic moduli, which allows quick estimates of the temperature-dependent creep behavior of granitic minerals using their elastic properties, especially when those creep constants are not readily available at elevated temperatures. These results enhance the understanding of temperature-dependent creep deformation at small scales and provide insight into mineral-level damage in granitic rocks due to temperature and long-term deformation.

Introduction

Understanding material behavior at constant stress is of interest in multiple disciplines and materials such as metals, polymers, ceramics, composites, concrete, and rocks [START_REF] Brinson | Effects of physical aging on long term creep of polymers and polymer matrix composites[END_REF][START_REF] Cannon | Creep of ceramics[END_REF][START_REF] Evans | Creep and time-dependent damage in argillaceous rocks[END_REF][START_REF] Griggs | Creep of rocks[END_REF][START_REF] Kassner | The effects of confining pressure and stress difference on static fatigue of granite[END_REF][START_REF] Sorelli | The nano-mechanical signature of ultra high performance concrete by statistical nanoindentation techniques[END_REF]. Most studies regarding the mechanical response of materials measured along multiple timescales aim to predict the serviceability of materials used in structural elements of machinery or infrastructures subjected to longterm mechanical or thermal loading. Engineering applications such as underground structures, mining sites, and rock slopes are subjected to stress levels and environmental effects that compromise their stability through creep deformation. The success of nuclear waste disposal sites, carbon sequestration facilities, and enhanced geothermal systems, which are important contributors to the reduction of current and future carbon footprint, relies on the ability of host rock formations to remain serviceable and structurally competent along the geological times. Hence, studying the time-dependent long-term deformation of rocks contributes to accelerating the deployment of these technologies that reduce the environmental risks of energy production and safely store hazardous materials in the subsurface.

Long-term constant stress tests have shown unstable crack propagation caused by slip mechanisms on clay particles rather than by the mechanical instability of quartzitic minerals within the structure of some sedimentary formations [START_REF] Evans | Creep and time-dependent damage in argillaceous rocks[END_REF]. Salt rocks have the most notorious creep deformation, attributed to dislocation at the grain scale in forms of glide, climb, and cross-slip [START_REF] Urai | Weakening of rock salt by water during long-term creep[END_REF]. Quite a few constitutive models to describe creep deformation in salt rock have been developed in the past decades [START_REF] Fei | A triaxial creep model for salt rocks based on variable-order fractional derivative[END_REF][START_REF] Firme | Enhanced double-mechanism creep laws for salt rocks[END_REF][START_REF] Firme | An assessment of the creep behaviour of Brazilian salt rocks using the multi-mechanism deformation model[END_REF][START_REF] Wu | A nonlinear creep damage model for salt rock[END_REF][START_REF] Zhou | A creep constitutive model for salt rock based on fractional derivatives[END_REF]. Creep deformation in crystalline rocks is influenced by confining stress [START_REF] Kassner | The effects of confining pressure and stress difference on static fatigue of granite[END_REF][START_REF] Kranz | Critical dilatant volume of rocks at the onset of Tertiary creep[END_REF],

water content [START_REF] Fujii | Circumferential strain behavior during creep tests of brittle rocks[END_REF][START_REF] Lockner | Room temperature creep in saturated granite[END_REF][START_REF] Masuda | EFFECTS OF WATER ON TIME-DEPENDENT BEHAVIOR OF GRANITE[END_REF], and stress and strain rate [START_REF] Fujii | Circumferential strain behavior during creep tests of brittle rocks[END_REF][START_REF] Lin | Time-dependent strength degradation of granite[END_REF][START_REF] Masuda | EXPERIMENTAL STUDY OF STRAIN-RATE DEPENDENCE AND PRESSURE DEPENDENCE OF FAILURE PROPERTIES OF GRANITE[END_REF]. The mechanisms to estimate damage in crystalline rocks consider stress corrosion and fracturing events measured and predicted as a function of strain measurements and acoustic emissions [START_REF] Fujii | Circumferential strain behavior during creep tests of brittle rocks[END_REF][START_REF] Lin | Time-dependent strength degradation of granite[END_REF][START_REF] Yanagidani | Localization of dilatancy in Ohshima granite under constant uniaxial stress[END_REF]. These experimental studies on crystalline rocks explore testing conditions to measure damage within mesoscale configurations, tracking fracture generation and propagation within granite structures. Experimental considerations on the possible effects of grain-scale mineral deformation, mineral interaction, and their contribution to creep deformation at elevated temperatures remain largely elusive. Some attempts to discretize rock behavior have used numerical approaches that predict the time to failure of rock specimens under triaxial conditions through grain boundary models [START_REF] Wang | A grain-based time-to-failure creep model for brittle rocks[END_REF].

Simplifications of grain distributions such as the Voronoi polygonal structure have been useful to simulate the intergranular interaction of crystalline rocks [START_REF] Ghazvinian | 3D random Voronoi grain-based models for simulation of brittle rock damage and fabric-guided micro-fracturing[END_REF][START_REF] Liu | A Voronoi element based-numerical manifold method (VE-NMM) for investigating micro/macro-mechanical properties of intact rocks[END_REF][START_REF] Wu | A zero-thickness cohesive element-based numerical manifold method for rock mechanical behavior with micro-Voronoi grains[END_REF].

Other studies model creep deformation based on the viscoelasticity or viscoplasticity of materials including rocks and metal composites [START_REF] Cristescu | Rock rheology[END_REF][START_REF] Cristescu | Time effects in rock mechanics[END_REF][START_REF] Perzyna | Fundamental Problems in Viscoplasticity[END_REF]. However, simplifications on the heterogeneity of inter-mineral structures may improperly estimate important characteristics of mechanical behavior, especially during long-term creep deformation. Recently, the indentation technique has been deployed to investigate the creep behavior in heterogeneous coal materials [START_REF] Sun | Creep characteristics of coal and rock investigated by nanoindentation[END_REF], hard minerals such as quartz considering mineral asperities [START_REF] Goldsby | Nanoindentation creep of quartz, with implications for rate-and state-variable friction laws relevant to earthquake mechanics[END_REF], and post-seismic transitory creep of quartz-rich rocks whose deformation mechanism is governed by dissolution processes [START_REF] Gratier | A pressure solution creep law for quartz from indentation experiments[END_REF]. Nanoscale indentation to understand the creep and mechanical characteristics of natural minerals for granitic rocks has not been fully explored.

This study uses instrumented indentation to measure microscale deformation and nanoscale creep of biotite, albite, and quartz, which are the major mineral phases of granitic rocks. The mechanical performance of these minerals is tested at temperatures up to 400 ⁰C and their time-dependent deformation is described by models that characterize both transient and steady-state creep. The results are useful to understand the temperature-dependent creep deformation at small scales and provide insight into mineral-level damage in granitic rocks due to temperature and long-term deformation.

Experimental design

Testing procedure

Samples tested in this study include albite, quartz, and biotite, which are typical mineral phases of granite composites in nature. All samples were cut using a 0.3 mm precision wafer blade installed on an Isomet-1000 cutting station. To ensure sample flatness during indentation testing, the minerals were subjected to two subsequent cuts that produced testing disks with 7mm of height. After cutting, all surfaces were polished using a sequence of sandpaper sheets with grits P800, P1200, and P2400, corresponding to averaged grit diameters of 25.8μm, 15.3μm, and 6.5μm, respectively.

After polishing, tested samples were fixed into a three-point chuck-grip holder installed into a furnace mounted on the base of the indenter frame. The sample holder prevented induced errors produced by improper sample seating. The mineral samples were tested perpendicular to the cleavage plane of biotite and parallel to the C-direction of both quartz and albite. The temperature was measured by Type K thermocouples in contact with the samples' surface and the heating elements of the furnace. All samples were heated to temperatures of 20 ⁰C, 50 ⁰C, 100 ⁰C, 200 ⁰C, 300 ⁰C, and 400 ⁰C, and maintained for 24 hours at each temperature to allow thermal stabilization between the indenter tip and the sample surfaces.

After thermal stabilization was reached, mechanical tests including loading, creep, and unloading stages were performed. The loading stage lets the indenter establish contact and penetrate the sample's surface until it reaches a load of 100mN (i.e., 0.1N) while measuring the indenter's penetration depth. The creep stage measures the indenter's penetration while the load of 0.1N is held constant for 20 seconds. And the unloading stage reduces the load back to 0N. Both loading and unloading used a loading rate of 200mN/min.

The same loading-creep-unloading testing conditions were repeated at least 50 times at each temperature for all tested samples.

Instrumented indentation

Indentation theory assumes that materials deform both elastically and plastically during loading, and any rebound during unloading is elastic [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF][START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]. Therefore, the material's response to a full indentation loading-unloading cycle can be used to calculate its mechanical and surface properties. In an indentation test, the measured secant elastic modulus must account for the contribution of both the indenter and the testing material to the elastic deformation at the contact. Thus, the indentation elastic modulus 𝐸 is given by [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF]:

𝐸 = (1 -𝜐 ! " ) ) # $ !"" - #%& # $ $ # * %# , (1)
where, 𝐸 ' and 𝜐 ' are the elastic modulus and the Poissonˈs ratio of the indenter, and 𝜐 ! is the Poissonˈs ratio of the tested material. The effective elastic modulus 𝐸 ()) is computed through:

𝐸 ()) = + * + % , #/" + - " ,, (2)
where 𝐴 . and 𝑆 are the contact area (between the indenter and the testing material) and the sample stiffness, respectively [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF]. Note also that the stiffness S can be determined using a power-law correlation of a segment from the unloading curve [ASTM-E2546, 2015. This segment starts at the onset of unloading and progresses to approximately 50% to 90% of load removal, and the S value can be calculated by:

𝑆 = 𝐵𝑚1𝑑 /01 -𝑑 ) 3 /%# , (3)
where 𝑑 /01 corresponds to the indenterˈs maximum penetration depth at maximum applied load; 𝑑 ) is the final penetration depth after load removal; the parameters 𝐵, 𝑚 and 𝑑 ) are the best-fit constants for the unloading force P versus displacement d curve using the power-law expression:

𝑃 = 𝐵1𝑑 -𝑑 ) 3 / . (4)
Because the stiffness S is simply the change of load with respect to deformation, Equation (3) results from the derivative of Equation ( 4) with respect to the penetration depth [ASTM-E2546-15].

For a Berkovich indenter, which is used in this study, the contact area Ac in Equation ( 2) correlates with the contact height dc as:

𝐴 . = 24.5 𝑑 . " .

(5)

Considering that the total height of penetration during load application does not represent full contact between the indenter and the testing material (some material surface around the indenter sinks in and separates from its faces as the indenter penetrates the sample), the contact height dc is calculated as [START_REF] Joslin | A new method for analyzing data from continuous depthsensing microindentation tests[END_REF]:

𝑑 . = 𝑑 /01 -0.75 2 &'( -, (6)
where Pmax is the maximum applied load. Finally, the hardness H of the tested material is defined by the ratio of the maximum applied load to the contact area:

𝐻 = 2 &'( + % . ( 7
)

Representative indentation data

Figure 1a shows the indentation curves for albite, quartz, and biotite loaded to 0.1N, creep at a constant load of 0.1N for 20 seconds, and unloaded to 0N. The maximum penetration depths for albite, quartz, and biotite are approximately 0.81 µm, 1.03 µm, and 1.86 µm (Figure 1a). Within 20 seconds of creep, biotite, quartz, and albite respectively show approximate maximum deformation depths of 76.9nm, 22.3nm, and 12.6nm (Figure 1b). The shaded areas under the creep curves of the tested minerals illustrate the transient creep with a deaccelerated deformation rate, while the straight lines indicate the steady-state creep where deformation rates become constant (Figure 1b). These indentation curves show that biotite undergoes more deformation than quartz and albite both during the loading and the creep stages of indentation.

Experimental results

Indentation signatures of all tested albite, quartz, and biotite samples at temperatures ranging from 20 ⁰C to 400 ⁰C are shown in Figure 2. The maximum penetration depth for biotite dmax ~ 3.5μm is greater than those of albite dmax ~ 1.7μm and quartz dmax ~ 1.6μm at room temperature conditions (Figure 2). Additionally, the upper bounds of dmax for quartz and albite approximately align with the lower bound of dmax for biotite. The three tested minerals all show a tendency of increasing dmax with increasing temperature (Figure 2). The creep deformation of all tested samples is summarized in Figure 3. At all tested temperatures, the creep deformation of biotite is the most pronounced. The maximum creep deformation within 20 seconds for albite, quartz, and biotite all increase with increasing temperature. Compared to that at room temperature, the maximum creep deformation within 20 seconds for albite, quartz, and biotite has increased by approximately two, five, and four fold at 400 ⁰C.

Analyses and discussion

Elastic modulus with temperature

The elastic moduli of the three minerals at all tested temperatures were calculated from the unloading portion of the indentation curves presented in Figure 2 through Equation (1). Measured elastic moduli of all tested minerals soften as the temperature increases. The peaks of the modulus distributions shift towards the left especially at 400 ⁰C, with standard deviations oscillating independent of temperature increase and with increasing coefficients of variation illustrating material softening with increasing temperature (Figure 4). Biotite has the lowest modulus compared to quartz and albite. The higher end of the moduli distribution for biotite approximately intersects the lower ends of the distributions for quartz and albite at all testing temperatures. At 400 ⁰C, the elastic modulus of biotite reduces by 25% compared with its room temperature condition, whereas those of quartz and albite reduce by approximately 10%. Therefore, biotite is not only the weakest of the three tested minerals but also the phase that might influence the possible weakening effects of granitic structures exposed to elevated temperatures.

The elastic modulus presented in Figure 4 suggests two competing effects of temperature in the mechanical response of tested minerals: strengthening due to thermal expansion caused by crack closure as well as softening due to thermal degradation resulting in softer minerals. The reduction of fracturing events during indentation (details in Section 3.3) as well as material dilation observed during testing suggest that, as temperature increases, fracture-closing mechanisms increase the material resistance to deformation under loading. This implies that the natural dislocations of pure minerals allow further compliance (deeper indenter penetration) at room temperature conditions than at high temperatures. The possible reduction of the internal spaces in fractures might have caused the transient increase in the elastic modulus of the tested minerals. This result is consistent with other studies that showed a rise in the elastic modulus of quartz and granites up to 300 °C, followed by a decrease at higher temperatures [START_REF] Trnik | Young's modulus of prefired quartz porcelain in a temperature range of 20-1200 °C[END_REF][START_REF] Yang | An experimental investigation on thermal damage and failure mechanical behavior of granite after exposure to different high temperature treatments[END_REF].

Hardness with temperature

Obtained hardness for biotite is lower than that of albite and quartz at all tested temperature conditions, which further indicates the mechanical contrast between strong and weak mineral phases as manifested in elastic modulus measurements. The temperature-dependent hardness also slightly increases followed by decreasing trends with increased temperature, similar to their temperature-dependent elastic modulus responses. The hardness peaks at 200 ⁰C for quartz and at 300 ⁰C for biotite and albite; strong minerals show sharp hardness increments, at 50 ⁰C, in addition to their peaks (Figure 5).

Fracture behavior with temperature

The fracture toughness of tested minerals is evaluated using the energy analysis method [START_REF] Cheng | Scaling relationships for indentation measurements[END_REF][START_REF] Liu | Applications of nano-indentation methods to estimate nanoscale mechanical properties of shale reservoir rocks[END_REF]. The fracture toughness 𝐾 . is calculated by:

𝐾 . = =𝐺 . 𝐸 , ( 8
)
where 𝐺 . is the critical energy release rate calculated by the ratio of fracture energy 𝑈 .30.4 to the maximum contact area 𝐴 5%678 . The former is calculated from the portion of fracturing energy comprised within the area in between the loading and unloading stages of indentation curves (see details in the Appendix), and the latter from Equation ( 5) using the maximum value of dc. The elastic modulus E is calculated using

Equation (1).

The results using this method show that fracture toughness slightly increases with increasing temperature for all tested minerals, with quartz showing the most pronounced fracture toughness increase upon heating (Figure 6a). Fracture toughness quantitatively describes a material's resistance to crack propagation, and materials with high fracture toughness tend to behave more ductile at failure. The ductility increase and decay in biotite and albite at elevated temperatures could respond to a combination of sequential crack healing and crack generation processes and material softening. The transition from increasing to decreasing fracture toughness has also been evidenced in other studies testing individual minerals [START_REF] Darot | Ductile-brittle transition investigated by micro-indentation: results for quartz and olivine[END_REF] and granites [START_REF] Feng | Fracture failure of granite after varied durations of thermal treatment: an experimental study[END_REF]Hu et al., 2018] [Meredith and[START_REF] Meredith | Fracture toughness and subcritical crack growth during high-temperature tensile deformation of Westerly granite and Black gabbro[END_REF] at similar temperatures.

Figure 6b shows an indentation curve that suffered a pop-in event, i.e., a sudden displacement burst typically associated with a fracture or crack initiation and propagation [START_REF] Bor | Nanoindentationbased study of the mechanical behavior of bulk supercrystalline ceramic-organic nanocomposites[END_REF]. This pop-in event occurred at a failure load Pf ~0.055N with a length of a ~0.25μm when the penetration depth is approximately at d ~0.7μm. Note that the hardness and elastic modulus data presented in previous sections are all based on indentation tests with no pop-in events. Except for quartz at 100 ⁰C and 400 ⁰C, and albite at 400 ⁰C, all other samples showed pop-in events (Figure 6c). Below 100⁰C, increased temperature enhances the modulus and hardness of tested minerals (as discussed in previous sections) and slightly reduces the occurrence of pop-in events during indentation (Figure 6c).

Overall, biotite evidently presents more fracture susceptibility than quartz and albite, suggesting a higher likelihood of brittle failure in the biotite phase in granitic composites, particularly at elevated temperature conditions. Hence, it can be inferred that fracture generation might be more prominent in granitic composites with high biotite contents than in granitic composites with high content of quartz and albite.

Thermal expansion contrast between biotite, quartz, and albite have been shown to induce stress concentration leading to fracturing mechanisms in granites [START_REF] Vázquez | Influence of mineralogy on granite decay induced by temperature increase: Experimental observations and stress simulation[END_REF]. Mineral orientation and rock texture also contribute to thermal fracturing [START_REF] Siegesmund | The bowing potential of granitic rocks: rock fabrics, thermal properties and residual strain[END_REF].

Maximum creep deformation with temperature

During the 20-second creep stage, the total creep deformation for biotite increased from an average of 58.9nm at 20 ⁰C to 125.7nm at 400 ⁰C (Figure 7). Below 300 ⁰C, the creep deformation increased slightly with increased temperature, and a more pronounced creep deformation with a higher variance was developed at 400⁰C. While temperature has less influence on the creep deformation of albite and quartz, their maximum creep deformation remained nearly unchanged up to 300 ⁰C, and then slightly increased at 400 ⁰C. The creep deformation for these minerals could respond to mechanisms such as dislocation creep [START_REF] Fukuda | Theoretical derivation of flow laws for quartz dislocation creep: Comparisons with experimental creep data and extrapolation to natural conditions using water fugacity corrections[END_REF], competing dislocations systems that transition from basal dislocations to prism dislocation glide [START_REF] Lister | The effect of the basal-prism mechanism switch on fabric development during plastic deformation of quartzite[END_REF] or crystal lattice rotation [START_REF] Takeshita | Plastic anisotropy and geometrical hardening in quartzites[END_REF], and can be exacerbated by high-temperature conditions. Under the same temperature, the creep deformation in biotite is about 3-4 times that in quartz and albite.

Such mismatch in creep deformation enlarges at higher temperature conditions. This attributes to the structural difference of tested minerals, as biotite can undergo more exacerbated deformation upon heating than the other two minerals, due to smaller elastic modulus and its characteristic one-dimensional cleavage, which undergoes ripplocations (layer ripples at the atomic level) and the formation of kink bands on their basal planes [START_REF] Anderson | KINK BANDSKink bands[END_REF][START_REF] Barsoum | Ripplocations: A Progress Report[END_REF][START_REF] Etheridge | Experiental deformation of single crystals of biotite[END_REF]. Additionally, biotite structures exposed to temperatures up to 400 ⁰C have shown axial elongation and separation of their interlayer bonds in the c-direction [START_REF] Chon | Crystal Structures of Biotite At High Temperatures and of Heat-treated Biotite Using Neutron Powder Diffraction[END_REF]. Such a mismatch in creep deformation among different granitic minerals can contribute to long-term granite damage due to temperature change.

Time-lapse creep analysis

Using the traditional unidimensional models to characterize the primary and the secondary creep [START_REF] Lomnitz | Creep measurements in igneous rocks[END_REF][START_REF] Vandamme | Nanogranular origin of concrete creep[END_REF], we here describe the indentation creep deformation, 𝑑 . (𝑡) [nm] following a logarithmic law with respect to time t [s] through: The magnitude of primary creep X1 increases exponentially with temperature for all tested samples (Figure 9a), with the magnitude for biotite about one order of magnitude larger than that for albite and quartz. The characteristic rate of primary creep X2 decreases with increasing temperature for albite and quartz (Figure 9b), suggesting longer primary creep duration, i.e., a more viscous and ductile response at elevated temperatures. In contrast, the X2 for biotite increases exponentially with increasing temperature (Figure 9b), presenting less viscous and more brittle deformation behaviors at a higher temperature. . ln(X 2 t + 1) + X 3 t

(b) X 1 1 1/X 2 Depth d c [nm] Time t [s] X 3 (a) X 3 = 1 nm/s X 2 = 1 /s Depth d c [nm] X 1 = 10 nm X 1 = 1 nm X 1 = 100 nm Depth d c [nm] X 2 = 1/s X 2 = 10/s X 2 = 100/s X 3 = 1 nm/s X 1 = 1 nm Depth d c [nm] X 3 = 10nm/s X 3 = 1nm/s Time t [s] X 3 =100nm/s X 2 = 1 /s X 1 = 1 nm
While elevated temperature causes biotite to accelerate its onset of steady-state creep, it causes the opposite effect on albite and quartz. Such divergent behavior might result from the interaction of competing deformation mechanisms typical of crystalline minerals including inter-grain space reduction through grain dilation, generation of intra-grain vacancies, grain boundary rotation, and grain sliding [START_REF] Soda | Crystallographic preferred orientations of plagioclase via grain boundary sliding in a lower-crustal anorthositic ultramylonite[END_REF].

Additionally, basal dislocation is the main deformation mechanism of biotite at high temperatures [START_REF] Etheridge | Experiental deformation of single crystals of biotite[END_REF]. Hence, the deformation of divergent nature among minerals might extend the As shown in Figure 9c, with the increase of temperature from 20 ⁰C to 400 ⁰C, all tested minerals show an exponentially increased rate of secondary creep, i.e., constant X3. Albite and quartz have almost identical secondary creep rates at all tested temperatures, which are about one-third to half of that of biotite at the same temperature. Hence, despite the increasing rate of deformation of all minerals during secondary creep, biotite shows the largest susceptibility to thermal damage and it is likely to aggravate the rate of deformation on biotite-rich granite composites.

Quartz

Biotite

Albite

Note that the large creep deformation in biotite at high temperatures (Figure 7) is mainly because both the magnitude of initial deformation (X1) and the rate of secondary deformation (X3) exponentially increase with temperature, and the values of X1 and X3 for biotite are about one order of magnitude larger than that for albite and quartz, as shown in Figure 9. On the other hand, phase transformation will not be a cause of the creep deformation in this study. At atmospheric pressure, the phase transformation for albite and quartz starts at temperatures of 1080°C [START_REF] Winter | A high-temperature structural study of high albite, monalbite, and the analbite--> monalbite phase transition[END_REF] and 573°C [START_REF] Ohno | Temperature variation of elastic constants of quartz across the α-β transition[END_REF] respectively, both of which are higher than the tested temperature in this study (up to 400°C). Tested temperatures up to 400°C cannot induce phase transformation for biotite either. However, biotite is of highly perfect basal cleavage and consists of flexible sheets or lamellae that can easily flake off. With increasing in temperature, there will be additional shrinkage in both the sheet thickness and the interlayer separation [START_REF] Chon | Crystal Structures of Biotite At High Temperatures and of Heat-treated Biotite Using Neutron Powder Diffraction[END_REF], which exacerbates the creep deformation.

In addition, although the same load of 100mN was applied during creep for all three tested minerals, the stress level may be different due to different penetration depths (or indentor-material contact areas). The stress level during creep can be calculated as a function of the maximum applied load 𝑃 /01 and the maximum penetration depth 𝑑 /01 through:

𝜎 = 2 &'( ">.@•B &'( $ . (10)
The result in Figure 10 shows that the creep deformation in biotite was measured under a stress level approximately five times smaller than the other two minerals, due to the larger contact area in biotite under identical indentation force. Therefore, at identical stress levels as albite and quartz, a larger creep deformation in biotite is expected. The rise and subsequent decline of stress during the creep phase also shows both the contact area reduction related to fracture-closure(strengthening) mechanisms and the material softening of all tested minerals at elevated temperatures. . This agreement implies that the transient creep of granitic minerals can be predicted using their elastic moduli for all temperatures. The observed material softening could be attributed to degradation mechanisms including the growth of intergranular and intragranular dislocations, preferential grain boundary sliding, and heterogeneous mineral expansion [START_REF] Ashby | A first report on deformation-mechanism maps[END_REF][START_REF] Hansen | Grain boundary sliding in San Carlos olivine: Flow law parameters and crystallographic-preferred orientation[END_REF][START_REF] Warren | Grain size sensitive deformation mechanisms in naturally deformed peridotite[END_REF]. Additionally, the rate of constant/secondary creep X3 is proportional to the magnitude of primary creep X1 logarithmically (Figure 11b). These relations provide quick estimates of the temperature-

Conclusions

Minerals of albite, biotite, and quartz, typically comprising granitic rocks, were tested at temperatures up to 400 °C using an instrumented indentation device in this study. Major conclusions include:

• Quartz and albite show a mechanical contrast with respect to biotite. The room temperature elastic moduli of albite (117.4GPa) and quartz (104.3GPa) are at least five times greater than that of biotite (18.5 GPa). At elevated temperatures, biotite undergoes greater mechanical degradation than quartz and albite. The elastic modulus of biotite reduces by 25%, whereas the elastic modulus of quartz and albite respectively reduce by 9.7% and 8.7%. X 3 = l . X 1 y Biotite

Albite

• Granitic rocks with high biotite content will compromise the most with increasing temperature, as biotite is not only the weakest mineral phase but also the one showing the most thermal degradation.

Elevated temperature tends to increase the fracture toughness of quartz and albite due to more ductile behavior evidenced by the reduced number of fracturing events during loading, while biotite presents a higher likelihood of brittle failure (i.e., pop-in events) during indentation loading, particularly at elevated temperature conditions.

• The temperature increase from 20 ⁰C to 400 ⁰C caused a two-fold increment in the creep deformation within 20 seconds for albite, quartz, and biotite. Biotite showed at least three times the overall creep deformation compared to quartz and albite under the same temperature.

• The indentation creep deformation in the tested minerals can be characterized using a logarithmic function of time. Such a logarithmic approach allows the estimation of the characteristic creep depth and characteristic time of transient creep.

• The transition from transient creep to steady-state creep for biotite is approximately three times faster than those of albite and quartz, possibly due to its sheet-like structure with an abundance of interfacial spacings that facilitate material deformation, especially at elevated temperatures.

• Transient creep accounts for the majority of the total creep deformation and steady-state creep plays a minor role within the 20 seconds creep measurement. However, over a longer time, the secondary creep deformation can continue to develop at a constant rate resulting in large deformation until failure. The rate of secondary creep for biotite is about three times that of albite and quartz at the same temperature. As temperature increases from 20 ⁰C to 400 ⁰C, the rate of secondary creep escalates about 10 times faster.

• The deformation of transient creep and the rate of secondary creep can be empirically correlated to the elastic moduli of albite, biotite, and quartz. These relations allow quick estimates of the temperature-dependent creep behavior of granitic minerals using their elastic properties, especially when those creep constants are not readily available at elevated temperatures.

Appendix. Energy method to estimate fracture toughness

The total energy Wtot used in the penetration process of an indenter into a testing sample corresponds to the area under the indentation loading curve. Such energy consists in an elastic portion We that corresponds to the sample's rebound under unloading conditions and the plastic portion Wp corresponding to an irrecoverable phase of materials caused by internal instabilities preventing further elastic deformation during unloading (Figure A-1). The plastic energy Wp can be calculated from the proportionality between the ratio of the plastic energy Wp to the total energy Wtot with the ratio of the final penetration depth df to the maximum penetration depth dmax through [START_REF] Cheng | Scaling relationships for indentation measurements[END_REF]:

𝑊 C 𝑊 DED = 1 - ⎩ ⎪ ⎨ ⎪ ⎧ 1 -3 P 𝑑 ) 𝑑 /01 Q " + 2 P 𝑑 ) 𝑑 /01 Q : 1 -P 𝑑 ) 𝑑 /01 Q " ⎭ ⎪ ⎬ ⎪ ⎫ (A.1)
The energy used for crack generation Ucrack is estimated from the irrecoverable energy Uirr that combines plastic mechanisms associated with material rearrangement and fracturing events using [START_REF] Liu | Applications of nano-indentation methods to estimate nanoscale mechanical properties of shale reservoir rocks[END_REF]:

𝑈 .30.4 = 𝑈 '33 -𝑊 C (A.2)
The critical energy release rate is measured as a ratio of the energy for crack generation Ucrack to the maximum contact area Ac -max by: 𝐺 . = 𝑈 .30.4 𝐴 .%/01 (A.

3)

The fracture toughness is calculated by [START_REF] Liu | Applications of nano-indentation methods to estimate nanoscale mechanical properties of shale reservoir rocks[END_REF]:

𝐾 . = =𝐺 . 𝐸 (A.4)

Figure 1 .

 1 Figure 1. Indentation results for albite, quartz, and biotite at room temperature. (a) Load vs. penetration depth during loading, creep, and unloading stages. (b) Time-lapse creep deformation depth.

Figure 2 .

 2 Figure 2. Indentation signatures of (a) albite, (b) quartz, and (c) biotite under various temperatures. Each temperature condition includes 50 indentations.

Figure 3 .

 3 Figure 3. Creep deformation of (a) albite, (b) quartz, and (c) biotite tested at temperatures of 20 ⁰C, 50 ⁰C, 100 ⁰C, 200 ⁰C, 300 ⁰C, and 400 ⁰C. Each temperature condition includes 50 indentations.

Figure 4 .

 4 Figure 4. (a) Histograms, (b) means, (c) standard deviations, and (d) coefficients of variation of indentation elastic modulus for albite, quartz, and biotite at various temperatures.

Figure 5 .

 5 Figure 5. Histograms, (b) means, (c) standard deviations, and (d) coefficients of variation of indentation hardness for albite, quartz, and biotite at various temperatures.

Figure 6 .

 6 Figure 6. Fracture responses. (a) Calculated fracture toughness using the energy analysis method. (b) Typical pop-in event during loading. (c) Percentage of fractured tests (i.e. comprising at least one pop-in event) during loading.

Figure 7 .

 7 Figure 7. Temperature-dependent maximum creep deformation (within 20 seconds) for biotite, quartz, and albite.

  𝑡) = 𝑋 # • ln(𝑋 " • 𝑡 + 1) + 𝑋 : • 𝑡 , (9) where X1 [nm] is a material constant describing the overall magnitude of the primary creep deformation; X2 [1/s] is the characteristic rate of primary creep, the reciprocal of which reflects the duration of primary creep; and X3 [nm/s] characterizes the creep rate of the secondary (i.e., constant) creep. The illustration and impacts of fitting constants X1, X2, and X3 on the creep deformation behavior are shown in Figure 8.

Figure 8 .

 8 Figure 8. Indentation creep deformation. (a) The time-dependent creep deformation is captured using a three-parameter creep model. (b) Creep deformation was calculated with unitary, tenfold, and hundredfold values of the constants X1 [nm], X2 [1/s], and X3 [nm/s] to illustrate their control of the creep behavior.

  transient creep time of quartz and albite but shorten that of biotite. Creep deformation triggered by thermally induced dislocations generated by the large coefficient of thermal expansion of biotite could also cause the increasing behavior of constant X2 (i.e., the reduction of transient creep time). At 400 ⁰C, the three tested minerals show almost identical transient creep duration.

Figure 9 .

 9 Figure 9. Creep model parameter values of (a) primary creep magnitude X1 [nm], (b) the reciprocal of primary creep duration X2 [1/s], (c) the rate of secondary creep X3 [nm/s] with temperature T. Data are fitted by 𝑿 𝒊 = 𝜶𝒆 𝜷𝑻 , where i = 1, 2, and 3; α and β are fitting parameters listed in the plots.

Figure 10 .

 10 Figure 10. The stress and strain rate of albite, quartz, and biotite during the creep deformation at all tested temperatures.

  of granitic minerals using their elastic properties at elevated temperatures, especially when those creep constants are not readily available.

Figure 11 .

 11 Figure 11. Creep constants and elastic modulus. (a) The primary creep depth X1 [nm] is inversely proportional to the elastic modulus E of biotite, albite, and quartz at all tested temperatures in a logarithmic scale. (b) The relationship between the primary creep depth X1[nm] and the secondary creep rate X3[nm/s] for albite, quartz, and biotite.

Figure A- 1

 1 Figure A-1 Indentation energy in a loading-unloading cycle. (a) Total energy distributed in elastic energy We during material unloading rebound and plastic energy Wp caused by plastic mechanisms excluding major cracking events. (b) Plastic energy Wp and irrecoverable energy Uirr including cracking events of approximate length a.

Acknowledgement

This material is based upon work supported by the National Science Foundation (EEC-1449501, CMMI-1943722, CMMI-2134311, and IRES-1854030). Any opinions, findings and conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the NSF. This study also benefited from the discussions with Dr. Chloe Arson (Georgia Institute of Technology) and Dr. Matthieu Vandamme (Ecole des Ponts, Paris Tech).