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DOE OSTI2024

A Thermo‐Flow‐Mechanics‐Fracture Model Coupling a Phase‐Field Interface Approach and Thermo‐Fluid‐Structure Interaction

This work proposes a novel approach for coupling non-isothermal fluid dynamics with fracture mechanics to capture thermal effects within fluid-filled fractures accurately. This method addresses critical aspects of calculating fracture width in enhanced geothermal systems, where the temperature effects of fractures are crucial. The proposed algorithm features an iterative coupling between an interface-capturing phase-field fracture method and interface-tracking thermo-fluid-structure interaction using arbitrary Lagrangian–Eulerian coordinates. We use a phase-field approach to represent fractures and reconstruct the geometry to frame a thermo-fluid-structure interaction problem, resulting in pressure and temperature fields that drive fracture propagation. We developed a novel phase-field interface model accounting for thermal effects, enabling the coupling of quantities specific to the fluid-filled fracture with the phase-field model through the interface between the fracture and the intact solid domain. We provide several numerical examples to demonstrate the capabilities of the proposed algorithm. In particular, we analyze mesh convergence of our phase-field interface model, investigate the effects of temperature on crack width and volume in a static regime, and highlight the method's potential for modeling slowly propagating fractures.

fracture↗

DOE OSTI2026

Predicting fracture behavior in single crystal nickel using a coupled crystal plasticity phase field damage approach

Understanding the fracture behavior of single crystal metals is critical for predicting material performance under mechanical loading. Here, in this study, we investigate the fracture characteristics of single crystal nickel tensile bars using a crystal plasticity coupled phase field damage (CP-PFD) model. Experimental tensile tests were conducted on 15 specimens spanning five crystallographic orientations and three thickness variants per orientation. The results revealed two distinct fracture modes: brittle fractures with 45-degree angled surfaces and ductile fractures characterized by significant necking. The CP-PFD model successfully replicated these fracture behaviors, demonstrating strong agreement with experimental observations. The model effectively predicted the strain at which necking and fracture occurred, as well as the orientation-dependent fracture mechanisms. By comparing experimental and simulated fracture surfaces, we establish the CP-PFD model as a robust tool for predicting single crystal behavior and damage evolution. This work provides insight into the microstructural dependence of fracture behavior and establishes a predictive framework for modeling orientation-dependent damage evolution in single-crystal nickel.

Crystal plasticity↗

DOE OSTI2025

Understanding Fracture Aperture and Permeability Evolution due to Carbonate Mineralization Utilizing 3D Printing

Caprock formations are a crucial part of subsurface-engineered systems. Composed largely of shale, caprocks act as natural barriers that prevent the upward migration of fluids, thereby ensuring the containment of stored substances in subsurface formations. Fractures in these formations are potential leakage pathways for stored fluids. Mineral precipitation reactions in these fractures, particularly calcite, can significantly restrict the fluid permeability, reducing leakage potential. However, predictive capabilities of mineral precipitation in fractures and associated permeability evolution are limited due to a lack of fundamental understanding of such reactions in natural samples, complicated by mineral heterogeneity and the complexity of the fracture structure. In this study, 3D-printed fracture samples are used to understand the impact of carbonate mineralization on fracture aperture and permeability evolution. Samples were printed using a digital light processing (DLP) 3D printer and commercial liquid resin. Calcite precipitation was first tested on printed 2D films before conducting plug flow column experiments aimed to understand fracture permeability changes due to mineral precipitation. Contact angle measurement and Fourier transform infrared (FTIR) spectroscopy on printed 2D films show evidence of a substantial amount of surface energy for calcite nucleation and precipitation. Surface topography analysis of printed fractured surfaces reveals comparable values, highlighting the high replicability of the printed samples. During the column experiments, the permeability reduces exponentially due to a decrease in fracture aperture. Reductions in fracture aperture estimated from effluent concentration and 3D X-ray computed tomography (CT) show comparable results. Moreover, 3D X-ray CT images suggest the impact of local flow velocities on precipitation. The insights gained from this research contribute to a deeper understanding of the permeability evolution due to carbonate mineralization in caprock formations.

3D printing↗

NASA NTRS2024

Risk of Bone Fracture due to Spaceflight-induced Changes to Bone

The Human Research Program (HRP) is taking a biomechanical approach to assessing subsequent fracture risk in active astronauts as a consequence of spaceflight exposure. Triennial testing of active and retired astronauts continues as Space Medicine monitors for a premature diagnosis of primary osteoporosis, which is associated with age-related bone loss and skeletal fragility. This updated 2023 Fracture Evidence Report has expanded the description of skeletal changes to capture the full effects of spaceflight on bone: • Recent analysis of fractures from health records of the full astronaut cohort, suggests an increased incidence rate of hip and spine fractures in astronauts following longer spaceflight duration flights compared to incident rates found in shorter duration flights. • Routine preflight-to-postflight surveillance by DXA (dual-energy x-ray absorptiometry) does not provide the full detection of loss and recovery in the long-duration (LD) astronaut nor the full recovery of hip trabecular bone. • The inclusion of hip quantitative computed tomography (QCT) in flight studies delineates effects of spaceflight, of countermeasures, and of post-flight recovery on cortical and trabecular bone parameters—some of which are verified predictors of hip fracture in the aged. • QCT detects and compares the distinct countermeasure effects of the pharmaceutical alendronate and of resistive exercise (on the Advanced Resistive Exercise Device, ARED) in specific cortical and trabecular bone sub-regions during spaceflight. • A published comparison of QCT-determined loss rates of hip trabecular bone in LD astronauts compared to terrestrial cohorts suggest that accelerated loss rates in trabecular volumetric BMD (vBMD) during spaceflight might be analogous to skeletal effects of accelerated loss rates in females due to menopause, potentially leading to disruptions in trabecular microarchitecture. • A dataset of finite element (FE) estimates of hip bone strength in aging terrestrial cohorts (spanning astronaut age-range) provides comparative context for changes in the FE of hip bone strength in LD astronauts, including comparison (force unit of newton) to percentiles (50th, 75th, and 100th) of sex-matched aged humans with hip fractures. • Risk of fracture, due to the mechanical overloading of bones, is being updated with IMPACT, the next-generation tool suite for probabilistic risk assessment (PRA) for exploration missions being created by the Exploration and Medical Capability Element (ExMC) at NASA. IMPACT is currently in development and will not be included in this 2024 update. The risk for fracture necessitates understanding the relationship between applied loads to bone and the biomechanical competence of bone. The Risk for Early Onset Osteoporosis focuses on the weakened condition of bone (including development of new technologies, measurements of novel skeletal attributes, translation of multiple measures to an index of bone fragility, and interpretations of data used to reflect a weakened bone), while the Risk for Fracture assesses factors that influence the probability an astronaut would encounter applied loads exceeding the biomechanical competence of bones, resulting in fracture. This Evidence Report combines the research gaps and tasks associated with both risks.

Fracture↗

DOE OSTI2024

Fracture Characterization Via AI‐Assisted Analysis of Temperature Logs

Abstract Fractures control fluid flow, mass transport, and heat transfer in a geothermal reservoir. This makes accurate characterization of fracture networks a prerequisite for optimal design and control of a reservoir's exploitation. We develop a deep‐learning procedure to identify fracture locations via interpretation of temporally and spatially continuous downhole temperature measurements. A long short‐term memory fully convolutional network (LSTM‐FCN) is used both to capture long‐term dependencies in sequential temperature data and to distill local features around fractures. A wellbore and fractured‐reservoir thermal model is established to generate temperature data for network training. The trained LSTM‐FCN exhibits a unique ability to detect multiple fractures intersecting a borehole. We use the LSTM‐FCN algorithm to evaluate the effectiveness of different‐stage wellbore temperature measurements on fracture detection in a complex fractured system. Our experiments reveal that the use of various‐stage temperature information as an input feature set improves the robustness of fracture detection to noise interference. This study indicates the practical feasibility of obtaining accurate fracture‐network reconstructions from temperature signals, at reasonable computational cost.

Yang, Xiaoyu↗

NASA NTRS1980

Fracture modes of high modulus graphite/epoxy angleplied laminates subjected to off-axis tensile loads

Angleplied laminates of high modulus graphite fiber/epoxy were examined in several ply configurations at various tensile loading angles to the zero ply direction to determine the effects of ply orientations on tensile properties, fracture modes, and fracture surface characteristics of the various plies. Experimental results consist of stress-strain data, selected plots, fracture stresses and strains, and scanning electron microscope (SEM) photographs of fracture surfaces. It was found that the stress-strain curves were linear to fracture, and that although fracture surface characteristics for a given fracture mode are similar to those for the same fracture mode in uniaxial specimens, no simple load angle range can be associated with a given fracture mode. It was also concluded that SEM results must be supplemented with ply stress calculations in order to identify ranges of fracture modes occurring as a function of ply orientation with respect to the load direction.

Sinclair, J. H.↗

NASA NTRS1979

Mechanics of fracture - Fundamentals and some recent developments

An overview is presented of the fundamental aspects of and recent developments in fracture mechanics. Reference is made to linear elastic fracture mechanics including the state of stresses and displacements in the vicinity of cracks, effects of crack geometry and orientation on stress intensity factors, energy balance of Griffith, Irwin's stress intensity concept, and linear elastic fracture mechanics testing for fracture toughness. Other aspects of this paper include the non-linear behavior of materials and their influence on fracture mechanics parameters, consideration of viscoelasticity and plasticity, non-linear fracture toughness parameters as C.O.D., R-curve and J-integral, and a non-linear energy method, proposed by Liebowitz. Finite element methods applied to fracture mechanics problems are indicated. Also, consideration has been given to slow crack growth, dynamic effects on K(IC), Sih's criterion for fracture, Lee and Liebowitz's criterion relating crack growth with plastic energy, and applications of fracture mechanics to aircraft design. Suggestions are offered for future research efforts to be undertaken in fracture mechanics.

Liebowitz, H.↗

NASA NTRS1990

Deformation and fracture of thin sheet aluminum-lithium alloys: The effect of cryogenic temperatures

The objective is to characterize the fracture behavior and to define the fracture mechanisms for new Al-Li-Cu alloys, with emphasis on the role of indium additions and cryogenic temperatures. Three alloys were investigated in rolled product form: 2090 baseline and 2090 + indium produced by Reynolds Metals, and commercial AA 2090-T81 produced by Alcoa. The experimental 2090 + In alloy exhibited increases in hardness and ultimate strength, but no change in tensile yield strength, compared to the baseline 2090 composition in the unstretched T6 condition. The reason for this behavior is not understood. Based on hardness and preliminary Kahn Tear fracture experiments, a nominally peak-aged condition was employed for detailed fracture studies. Crack initiation and growth fracture toughness were examined as a function of stress state and microstructure using J(delta a) methods applied to precracked compact tension specimens in the LT orientation. To date, J(delta a) experiments have been limited to 23 C. Alcoa 2090-T81 exhibited the highest toughness regardless of stress state. Fracture was accompanied by extensive delamination associated with high angle grain boundaries normal to the fatigue precrack surface and progressed microscopically by a transgranular shear mechanism. In contrast the two peak-aged Reynolds alloys had lower toughness and fracture was intersubgranular without substantial delamination. The influences of cryogenic temperature, microstructure, boundary precipitate structure, and deformation mode in governing the competing fracture mechanisms will be determined in future experiments. Results contribute to the development of predictive micromechanical models for fracture modes in Al-Li alloys, and to fracture resistant materials.

Wagner, John A.↗

NASA NTRS1992

Fracture mechanics; Proceedings of the 22nd National Symposium, Atlanta, GA, June 26-28, 1990. Vols. 1 & 2

Current research on fracture mechanics is reviewed, focusing on ductile fracture; high-temperature and time-dependent fracture; 3D problems; interface fracture; microstructural aspects of fatigue and fracture; and fracture predictions and applications. Particular attention is given to the determination and comparison of crack resistance curves from wide plates and fracture mechanics specimens; a relationship between R-curves in contained and uncontained yield; the creep crack growth behavior of titanium alloy Ti-6242; a crack growth response in three heat resistant materials at elevated temperature; a crack-surface-contact model for determining effective-stress-intensity factors; interfacial dislocations in anisotropic bimaterials; an effect of intergranular crack branching on fracture toughness evaluation; the fracture toughness behavior of exservice chromium-molybdenum steels; the application of fracture mechanics to assess the significance of proof loading; and a load ratio method for estimating crack extension.

Ernst, Hugo A.↗

NASA NTRS2004

Tensile Creep Fracture of Polycrystalline Near-Stoichiometric NiAl

Tensile creep fracture behavior of polycrystalline near-stoichiometric NiAl has been studied between 700 and 1200 K under initial applied stresses varying between 10 and 200 MPa. The stress exponent for fracture varied between 5.0 and 10.7 while the activation energy for fracture was 250 +/- 22 kJ/mol. The fracture life was inversely proportional to the secondary creep rate in accordance with the Monkman-Grant relation although there was extensive scatter in the data. This observation suggests that the fracture life for near-stoichiometric NiAl was influenced by creep under these stress and temperature conditions. Several different fracture morphologies were observed. Transgranular ductile cleavage fracture occurs at 700 K and at the higher stresses at 800 K. The fracture mode transitions to transgranular creep fracture at 900 and 1000 K and at lower stresses at 800 K, while plastic rupture and grain boundary cavitation occur at 1100 and 1200 K. An experimental fracture mechanism map is constructed for near-stoichiometric NiAl.

Raj, Sai V.↗

DOE OSTI2025

Utah FORGE: Well 16B(78)-32 Drill Core Fracture Analysis Images and Data

This dataset contains drilling core data from well 16B(78)-32, including PDF documents with flattened core images annotated by feature type and core interval, as well as spreadsheets detailing feature morphologies by depth, planar feature measurements, and planar feature orientations rotated to in situ conditions. Core was recovered from three intervals, one per stimulation stage, in the crystalline rocks affected by the stimulation of well 16A(78)-32. Seven core runs were conducted, yielding 135.8 feet of recovered core. Features in the core were categorized into planar fractures, semi-planar fractures, unbroken mineralized fractures, rough fractures, curviplanar fractures, concave-convex surfaces, and planar compositional features such as mylonite or dike-like structures. Planar features were measured while the core was positioned horizontally, with the core axis aligned to a downhole azimuth of 42 degrees. Planar core measurements from stimulations 2 and 3 that could be confidently correlated with FMI data were rotated to in situ orientations. This was done by rotating the planes along vertical and horizontal axes to match the azimuth and inclination data recorded in the directional survey of well 16B(78)-32, as well as applying an axial rotation to resemble the fracture orientations observed in the FMI log at corresponding depths. Coherent sets of planar fracture measurements were made by aligning the core within each 3-foot section of the dissected core barrel, and between adjacent 3-foot sections within a core run by matching rock fabrics, saw cuts and/or tool marks. Where coherent fracture measurements could not be made within a core run, data sets are denoted by a subscript (i.e. 2-Ta and 2-Tb both come from tangent core run number 2).

15 GEOTHERMAL ENERGY↗

DOE OSTI2025

Monitoring Fracture Hydromechanical Evolution in the Lab and Field Using Unsupervised Metric Learning

Fractures evolve in time through thermal‐hydraulic‐mechanical‐chemical (THMC) processes that alter their long‐range hydraulic transport properties and modify subsurface behavior and activities. The location of subsurface fractures makes it necessary to use remote sensing techniques such as passive or active seismic monitoring for fracture characterization. In this paper, we develop a machine learning approach to monitor the evolution of fracture properties using passive seismic sources in a laboratory setting and using active seismic monitoring from the Sanford Underground Research Facility in Lead, South Dakota, at a depth of 1.25 km in amphibolite rock during stimulation of natural fractures as well as during induced fracturing. The unsupervised metric learning technique applies tandem neural networks (twin (Siamese) or triplet) with contrastive loss and adaptive margins to track slowly varying systems for which class or similarity labels are not available. The approach adopts locality‐sensitive hashing to divide time‐ordered contiguous data into an arbitrary number of pseudo‐classes. Contrastive‐loss training with many hash bins generates an evolving latent‐space trajectory. This approach enables unsupervised metric learning for seismic data stacks under the condition of contiguous state sampling and slowly varying fracture properties. The displacement discontinuity theory provides a mechanistic foundation for the fracture‐dependent trajectories that are related to relaxation of fractures with time‐dependent specific stiffness responding to changes in stress or fluid saturation.

02 PETROLEUM↗

DOE OSTI2026

Extended Rice–Thomson analysis and atomistic simulations revealing grain boundary effects on fracture in refractory high-entropy alloys

Significance This work serves to extend the fundamental ductile vs. brittle fracture theory, specifically the Rice–Thomson criterion, by introducing a grain boundary ahead of an initiating crack which propagates at an oblique angle to impinge the boundary. Atomistic fracture simulations on two refractory complex concentrated alloys, the brittle NbMoTaW and the ductile Nb 45 Ta 25 Ti 15 Hf 15 , demonstrate qualitative correspondence with the extended Rice–Thomson criterion and experimental observations. Abstract Understanding how grain boundaries mediate fracture remains a critical challenge in designing ductile, high-performance refractory alloys. Here, we extend the Rice–Thomson criterion to account for the angle between cracks and the impinging grain boundaries (GBs), capturing the competition between intergranular fracture and dislocation-mediated plasticity. Using machine learning interatomic potentials, we performed molecular statics simulations to probe fracture mechanisms in nanocrystalline NbMoTaW and Nb 45 Ta 25 Ti 15 Hf 15 , each with two different grain sizes, revealing trends consistent with experimental observations and the extended Rice model. Comparison with averaged R-curves for bulk samples demonstrates that GBs enhance ductility in Nb 45 Ta 25 Ti 15 Hf 15 in both grain sizes investigated. In contrast, GBs only locally improve fracture resistance in NbMoTaW when cracks are temporarily pinned at GBs inclined at high angles from the crack, but generally promote brittle intergranular fracture. These contrasting behaviors are attributed to differences in GB cohesion, reflecting clear alloying trends that align with ab-initio calculations and trends observed experimentally. Our results bridge classical fracture theory, atomistic simulations, and experimental observations, providing a comprehensive understanding of the fracture mechanisms in nanocrystalline refractory complex concentrated alloys.

36 MATERIALS SCIENCE↗

NASA NTRSDate not supplied

Macro-scale Testing and Micromechanics Modeling of Fracture Behaviors for Boron Carbide Composites with Hierarchical Microstructures

With a unique combination of properties including high hardness, low density, wear and corrosion resistance, thermal stability, high neutron absorption, and semi-conductivity, boron carbide (B4C) is a candidate material for various engineering applications that involve extreme environment. The current applications of boron carbide, however, are limited by its intrinsic brittleness due to its strong covalent bonding. To toughen boron carbide, in this work hierarchical microstructure designs was used to provide multiple toughening mechanisms including crack deflection/bridging, micro-crack toughening, etc. Using field assisted sintering, B4C composites with hierarchical microstructure features including graphite platelets, micron and sub-micron sized TiB2 reinforcements were fabricated. The fracture toughness of fabricated B4C composites were previously measured at micro-scale using micro-indentation followed by post-testing microstructure inspection. However, questions including whether the fracture toughness enhancement measured at micro-scale can translate to macro-scale mechanical properties, and what are the fundamental mechanisms behind observed fracture toughness enhancement, remain to be answered. In this study, the fabricated B4C composites were tested using standardized four-point bending method to obtain fracture toughness at macro-scale. In addition, micromechanics modeling was conducted using MAC/GMC code and crack-band model to study the effect of residual stress and weak interphases on fracture behaviors of B4C composites reinforced with TiB2 particles. Through standardized four-point bending tests, fracture toughness enhancements up to 2.85, 3.32, and 3.65 MPa∙m(exp 1/2) (from 2.38 MPa∙m(exp 1/2)) were achieved for B4C composites with graphite platelets addition (micro/nano B4C), with TiB2 formation (micro B4C-TiB2), and with both graphite and TiB2 addition (micro/nano B4C-TiB2) respectively. Micromechanics modeling indicated that introduction of thermal residual stress and weak interphases caused enhanced micro-cracking behavior and resulted in the observed fracture toughness enhancement. These results furthered understanding about the mechanical behaviors at macro-scale and the mechanisms behind observed fracture toughness enhancement for B4C composites with hierarchical microstructures and can provide reference data for the future design of B4C composites with optimized microstructures for further fracture toughness enhancement.

Boron Carbide↗

DOE OSTI2025

A chain stretch-based gradient-enhanced model for damage and fracture in elastomers

Similar to quasi-brittle materials, it has been recently shown that elastomers can exhibit a macroscopically diffuse damage zone that accompanies the fracture process. In this study, we introduce a stretch-based gradient-enhanced damage (GED) model that allows the fracture to localize and also captures the development of a physically diffuse damage zone. This capability contrasts with the paradigm of the phase field method for fracture, where a sharp crack is numerically approximated in a diffuse manner. Capturing fracture localization and diffuse damage in our approach is achieved by considering nonlocal effects that encompass network topology, heterogeneity, and imperfections. These considerations motivate the use of a statistical damage function dependent upon the nonlocal deformation state. From this model, fracture toughness is realized as an output. While GED models have been classically utilized for damage modeling of structural engineering materials (e.g., concrete), they face challenges when trying to capture the cascade from damage to fracture, often leading to damage zone broadening (de Borst and Verhoosel, 2016). This deficiency contributed to the popularity of the phase-field method over the GED model for elastomers and other quasi-brittle materials. Other groups have proceeded with damage-based GED formulations that prove identical to the phase-field method (Lorentz et al., 2012), but these inherit the aforementioned limitations. To address this issue in a thermodynamically consistent framework, we implement two modeling features (a nonlocal driving force bound and a simple relaxation function) specifically designed to capture the evolution of a physically meaningful damage field and the simultaneous localization of fracture, thereby overcoming a longstanding obstacle in the development of these nonlocal strain- or stretch-based approaches. Here, we discuss several numerical examples to understand the features of the approach at the limit of incompressibility, and compare them to the phase-field method as a benchmark for the macroscopic response and fracture energy predictions.

Elastomers↗

DOE OSTI2024

Revealing complex subsurface dynamics with continuous seismic monitoring: Observations using distributed acoustic sensing and surface orbital vibrators during hydraulic fracturing

Understanding hydraulic fracturing is crucial to improving the stimulation of unconventional reservoirs and increasing fluid production. This study develops a novel seismic monitoring technology using distributed acoustic sensing (DAS) and surface orbital vibrators (SOV) to capture fracture seismic response and mechanical properties at high temporal intervals. We analyze continuous time-lapse vertical seismic profiling (VSP) data acquired every hour during the first nine days of treatment of an unconventional reservoir in the Austin Chalk/Eagle Field Laboratory. The VSP data contain clear seismic signals scattered from the activated fractures. The spatiotemporal changes of the fracture reflectivity revealed by the SOV/DAS data correlate well with the observations of fracture locations inferred from low-frequency DAS data. These results capture the fracture opening and closure processes, as well as highlight potential prestage activations of the fractures due to hydraulic connectivity with preexisting fracture systems. Therefore, analysis of the presented data set provides a unique opportunity to understand fracture initiation and subsequent evolution, not only in the context of unconventional resources but also in enhanced geothermal systems.

Correa, Julia↗

NASA NTRS1976

A fractographic study of the edge-sliding mode in fracture toughness testing

A fractographic study of Mode II fracture surfaces has been conducted for the purpose of identifying the microstructural mechanisms responsible for fracture in the edge-sliding mode. A compact shear (CS) specimen was employed to generate the fracture surfaces and also to establish Mode II fracture toughness values for Ti-6Al-4V, A533-B steel, and several aluminum alloys. In all tests, one of the two edge cracks sustained complete Mode II fracture while the other exhibited only a limited amount of subcritical crack growth. Mode II fracture surfaces, which were unique in appearance, have been examined by optical and scanning electron microscopy. It was determined that shear (or parabolic) microvoid coalescence was the dominant mechanism for Mode II fracture. It was also established that most of the surface abrasions were created by the relative sliding of the fracture surfaces during unstable fracture rather than the crack initiation process.

Jones, D. L.↗

NASA NTRS1976

Ductile fracture theories for pressurised pipes and containers

Two mechanisms of fracture are distinguished. Plane strain fractures occur in materials which do not undergo large-scale plastic deformations prior to and during a possible fracture deformation. Plane stress or high energy fractures are generally accompanied by large inelastic deformations. Theories for analyzing plane stress are based on the concepts of critical crack opening stretch, K(R) characterization, J-integral, and plastic instability. This last is considered in some detail. The ductile fracture process involves fracture initiation followed by a stable crack growth and the onset of unstable fracture propagation. The ductile fracture propagation process may be characterized by either a multiparameter (discrete) model, or some type of a resistance curve which may be considered as a continuous model expressed graphically. These models are studied and an alternative model is also proposed for ductile fractures which cannot be modeled as progressive crack growth phenomena.

Erdogan, F.↗