Fracture toughness evaluation across length scales in metallic materials and alloys: Methods,measurements,and insights

Abstract

Fracture in materials is a critical issue across various fields, including engineering, physics, and materials science. Fracture toughness, which quantifies a material’s resistance to crack propagation, can vary significantly with the scale of measurement. Traditional continuum mechanics often fail at small scales because the assumptions of homogeneity and continuity break down. At the nanoscale, fracture behavior differs due to the discrete nature of atoms and pronounced surface effects, leading to size-dependent fracture properties. Since experimental studies at this scale are limited, molecular dynamics (MD) simulations are frequently employed to investigate fracture mechanisms. The challenge of length-scale dependence remains a major obstacle in accurately determining fracture toughness. Metals and their alloys, particularly iron and aluminum are widely used structural materials, and their applications now extend from the continuum scale to micro and nanoscale systems. This review summarizes methods for determining fracture toughness of iron, aluminum, and their alloys across different scales, compares reported values, and discusses the implications of length-scale effects based on available results. The findings underscore the importance of deeper exploration into length-scale influences and fracture mechanisms to refine the determination of fracture toughness. Addressing these challenges will enable more reliable design of advanced materials for multi-scale applications. Future research should integrate experimental, computational, and theoretical approaches to bridge the gap between atomistic and continuum models.

Introduction and background

Metals are solid materials distinguished by their excellent electrical and thermal conductivity, as well as their opacity to light. They are typically malleable and ductile, allowing them to be easily shaped and formed into various structures. Known for their high tensile strength, metals are widely used in the construction of infrastructure, machinery, and tools. Their durability and resistance to corrosion make them ideal for applications in industries such as aerospace, automotive, and manufacturing.^1–3 However, structural integrity is very important to prevent fractures in critical infrastructure and machinery which helps avoid accidents and disasters. Fracture toughness determines a metal’s ability to withstand stress and prevent catastrophic failure in a preexisting crack condition. The presence of a crack acts as a stress riser and causes plastic deformation in the vicinity of crack-tip. In engineering applications, ensuring that metals can resist fracture is crucial for the safety and reliability of structures like bridges, buildings, aircraft, and many other structures.^4–6 Study of fracture mechanics is very important for various reasons as cracks and other flaws are generated in structure frequently and it can cause catastrophic failures in the structure. Also, strength of a material has inverse relations with fracture toughness. So, preference of a higher strength materials in structure may lead a dangerous path as fracture toughness decreases. In real materials, the very high elastic stresses in the vicinity of a crack tip exceed the yield strength of the materials, which creates a plastic zone. To determine the plastic zone at the crack tip, Irwin presented a simple model to solve for the theoretical distance from the crack tip at which the stresses are equal to the material’s yield strength. The stress field equation is,

$${\sigma }_{\mathrm{yy}}=\frac{K_{\mathrm{app}}}{\sqrt{2\pi r}}$$

(1)

Here, σ_yy is the stress applied perpendicular to the crack and yielding occurs when σ_yy = yield strength, r is the distance from the crack tip, K_app is the stress intensity factor at applied stress. This equation leads to find plastic zone size, which is out of scope to discussed here. Stress intensity factor predicts the stress state near the crack tip due to the remote applied stress and it also depends on crack size as well as part geometry. Failure occurs once the stress intensity factor exceeds the material’s fracture toughness also knows as critical stress concentration factor (K_IC). At this point the crack will grow in a rapid and unstable manner until fracture.^7–10

Linear Elastic Fracture Mechanics (LEFM) is primarily concerned with identifying the critical load at which a crack initiates propagation within a material. In cases involving fatigue or dynamic loading, its scope extends to predicting both the rate and direction of crack growth. The theoretical framework for fracture analysis is grounded in a qualitative interpretation of stress and deformation fields near crack tips in brittle, isotropic solids. Under the assumption of ideal linear elasticity, the asymptotic stress and displacement solutions derived from LEFM are regarded as accurate representations of the near-tip region. However, this asymptotic characterization becomes invalid at greater distances from the crack tip, where global geometry and boundary conditions exert significant influence. Moreover, the linear elastic solution is physically unrealistic in the immediate vicinity of the crack tip due to the singular (infinite) stresses it predicts. In practice, crack tips are not perfectly sharp, and localized nonlinear phenomena such as plastic deformation, microcracking, and other inelastic mechanisms govern the material response within this zone. Consequently, the linear elastic approximation must be supplemented or replaced by more comprehensive models to capture the actual behavior near the crack front.^11–15

Near the tip of a crack, there exists a process zone where the material experiences permanent damage. Just beyond this zone lies an area where the linear elastic stress field described by fracture mechanics accurately represents the behavior of the material. This is known as the region of K-dominance (Stress and strain fields are well-described by linear elasticity). Further away, the stress distribution is influenced by the overall shape of the structure and the applied boundary conditions. Although the actual failure of the material (like crack growth or fatigue) originates in the process zone, Linear Elastic Fracture Mechanics (LEFM) assumes that the detailed mechanics within this zone don’t need to be fully understood. Instead, it’s believed that the conditions in the process zone are primarily governed by the stress fields in the K-dominant region. These fields are characterized by three stress intensity factors, K_I (opening mode), K_II (sliding mode), and K_III (tearing mode), which are sufficient to describe the state of stress near the crack tip. K-dominance region decouples the complex, material-specific behavior in the process zone from the analysis.^16–19 Figure 1 represents different crack propagation modes and different zones under mode-I loading condition.

Figure 1.

Illustrations of (a) process zone and k-dominance zone in front of crack, (b) loading on cracked structure (i) mode I (K_I), (ii) mode II (K_II), and (iii) mode III (K_III) - crack- orange color.

Critical stress intensity factor (aka fracture toughness) depends on the applied load, crack size, geometry of the materials, etc. At all different length-scales, the calculated fracture toughness varies significantly. In addition to that, the plane stress and plane strain conditions also play an important role. In fracture mechanics, plane stress occurs in thin specimens where out-of-plane stress is negligible, resulting in higher apparent fracture toughness and ductile behavior. In contrast, thick specimens exhibit plane strain, where out-of-plane strain is constrained, leading to lower fracture toughness and brittle fracture. The plane strain fracture toughness is considered a true material property because it represents the minimum toughness and is independent of specimen geometry. There are many ways such as experimental, theoretical, computations, etc. were applied to evaluate the fracture toughness as different length scales.^20–24 Still, it is an ongoing debate to predict fracture toughness more accurately, especially at nanoscale because of the dynamics of materials evolution in the modern world.

This paper focuses on estimating fracture toughness across different length scales for the most widely used and technologically important structural materials. Among these materials, iron and aluminum, together with their alloys, remain the most common choices in construction and structural engineering due to their well-established mechanical performance.^25–27 Iron and its alloys generally crystallize in the body centered cubic (bcc) structure, whereas aluminum and its alloys adopt a face centered cubic (fcc) structure. It is worth noting that steel, which is primarily iron combined with controlled percentages of carbon and other alloying elements, is also discussed in this manuscript. Since the present study concentrates on these two dominant material systems, metallic systems possessing hexagonal close packed (hcp) crystal structures are not included in the analysis. Moreover, this review synthesizes current research efforts aimed at predicting fracture toughness for these metals and their alloys, highlighting variations in estimation methodologies and the resulting discrepancies in reported values. A comprehensive literature survey has been conducted to capture the prevailing approaches and understanding in this domain, which are presented herein. In essence, this manuscript consolidates relevant studies on fracture toughness, with particular emphasis on mode-I fracture, the most frequently encountered fracture mode. The review also aims to stimulate further investigations into newly developed materials across multiple length scales. The paper is organized as follows: Section 2 discusses various methodologies for measuring fracture toughness at different length scales, Section 3 presents reported results for aluminum, iron, and their alloys, and Section 4 concludes with key findings and future research directions.

Fracture toughness estimation

Since the occurrence of flaws is not completely avoidable in the processing, fabrication, or service of a material/component, the fracture toughness property is very important to estimate in all length scales. Many efforts are put to estimate fracture toughness property. Here, the endeavors to estimate in continuum and nanoscale are described.

Measurement of fracture toughness at the continuum scale

Evaluating fracture toughness at the continuum scale involves testing methods that focus on macroscopic behavior and crack propagation, typically relying on linear elastic fracture mechanics (LEFM) or elastic-plastic fracture mechanics (EPFM). The methods used at this scale provide important insights into how materials resist crack initiation and growth under applied stress.

Experimental methods

Several experimental approaches are taken to estimate critical stress intensity factor. Here, a few of the methods are briefly furnished.

Three-point and four-point bend test

This test is performed on a universal testing machine with a three-point or four-point bend fixture. This is a very popular method for easiness to prepare samples and widely accepted and standardized by professional organizations such as ASME E399. ²⁸ In 3-point bending test, the pick stress is observed at the mid-point where the load is applied. However, there is no single peak point for 4-point bending test it extends in a region between applied two loads. Schematic representation for 3-points and 4-points bending tests are shown in Figure 2.

Figure 2.

Schematic of flexure tests (three-point and four-point bending test). ²⁹

Young’s modulus and bending stress due to the loads are calculated by using flexure formula. The stress intensity factor is calculated by using the following equation.

$$K=Y\sigma \sqrt{\pi a}$$

(2)

Here, K is stress intensity factor, Y is geometry factor, σ is applied stress which is calculated from applied load, a is initial crack length. The are disadvantages of these tests as the 3-point tests generate stress concentration at the location of applied load and the set-up complexity is more prominent in 4-points bending tests. Hence, it is challenging to do these tests for thin materials and the results are more sensitive to geometry and loading conditions.^30–32 It is important to mention that another tests Similar to the three-point or four-point bending test but with a notch located at one edge which is known as Single-Edge Notched Beam (SENB). As the crack is a pre-requisite, hence all three points tests are not SENB.^33,34

Compact tension test (CT test)

Compact Tension (CT) specimens are the most widely used test specimens to measure fatigue crack-growth rates in metallic materials. A schematic of pre-cracked specimen for the compact tension (CT) test is represented in Figure 3. During the tensile fracture test, the CT specimen is loaded via two rigid pins passing through the holes either in monoclinic or cyclic manner. The failure of the specimens happens when the load reaches its maximum (i.e. at F = Fc) and the edge crack propagates abruptly, and the specimen breaks into two parts. Stress intensity factor can be calculated using the following equation.

$$K_{\mathrm{Ic}}=\frac{F_c}{B}\sqrt{\frac{\pi }{W}}f\left(\frac{a}{W}\right)$$

(3)

Figure 3.

Compact tension specimen for tensile fracture test in (a) 2D and (b) 3D.^35,36

This test follows standard procedures and endorsed by professional organizations. However, this test is good for ductile materials, especially metallic materials in plane strain condition.^35–37 In addition to these, biaxial loading tests for mixed mode conditions, center crack tensile test, etc. approaches are taken to estimate the critical stress intensity factor.

Computational methods

As computational power has increased drastically over the last couple of decades, computational methods have become popular due to their ability to reduce experimental costs and time. Moreover, they enable analysis at scales and under conditions that are difficult or impractical to achieve experimentally.

Finite element analysis (FEA)

At the advent of computational power, the application of the finite element methods (FEM) became popular because of handling complex geometries, mixed mode conditions, and different materials, etc. It also saves enormous time and provides approximation of fracture toughness calculations using linear elastic fracture mechanics (LEFM). FEA is used to simulate crack propagation under a given loading condition for different materials and approximate the stress and strain fields around the crack tip to eventually calculate the critical stress intensity factor.^38–40 Mesh refinement around the crack tip is necessary for accurate results and it becomes very crucial for the complex geometries. However, it requires advanced knowledge on FEA and the computational cost can be very high for complex geometries. The above-mentioned (in section 2.1.1) experimental method can be simulated by using finite element analysis. ⁴¹ It is important to note that while the FEM and FEA are closely related, FEM represents the mathematical and numerical foundation for solving complex engineering problems, whereas FEA refers to the practical implementation of this methodology through computational tools to model, analyze, and simulate real-world systems. Figure 4 represents the different mesh types in front of crack tip.

Figure 4.

Mesh pattern for (a) quadrilateral (b) triangular elements and typical strain and stress distributions near crack tip for corresponding (a) quadrilateral and (b) triangular elements. ⁴²

Another approach to find fracture toughness is cohesive zone modeling (CZM). It defines a cohesive zone in front of the crack tip where material damage, micro-cracking, and yielding occur. This method is effective for the materials with ductility and handles elastic-plastic fracture. However, it requires more detail in material modeling and sometimes uses experimental data.^43,44

Virtual crack closure technique (VCCT)

VCCT is a numerical technique for calculating the energy release rate at the crack tip during each step of crack extension. Critical stress intensity factor is calculated from energy release rate. The method assumes that the energy release when the crack is extended to a certain amount, Δa from its initial length, a is equivalent to the energy released when the crack face is closed. In finite element analysis, accurate meshing of crack faces is required. ⁴⁵ Critical energy release rate G_IC can be calculated using the following equation. ⁴⁶

$$G_I=\frac{1}{2\Delta a}[F_y^c(u_y^a-u_y^b)+F_y^g(u_y^p-u_y^q)]$$

(4)

where $F_y^c$ and $F_y^g$ are the vertical component of nodal force at nodes c and g. Also, u_y represents the vertical component of nodal displacement at corresponding points. Figure 5 shows the locations of nodes, which are schematically drawn.

Figure 5.

Virtual crack extension simulated by finite elements: (a) before crack extension and (b) after crack extension. ⁴⁷

Measurement of fracture toughness at the nanoscale

Conventional fracture mechanics assume materials are continuous and that stress becomes infinite at the crack tip. However, at the nanoscale, this assumption breaks down as materials exhibit discrete behavior, with atomic-level interactions playing a crucial role. Because of this nature, fracture toughness is hard to calculate. Additionally, precise measurements at such small scales are technically challenging. Several studies support that a singular stress field of only several nanometers still governs fracture as successfully as that at the macroscale. ⁴⁸ However, several experimental and computational methods have been developed to measure nanoscale fracture toughness.

Experimental methods

Nanoindentation

Nanoindentation is a popular method to find fracture toughness and other mechanical properties such as hardness. The nano-indentation test is used for non-destructive characterization and analysis of the mechanical properties of materials at the nano scale. In this method, an indenter with very small dimensions is used under a force of several millinewtons or less to prevent the influence of the substrate on the hardness value measured for the nanometer coating. Uses an instrumented nano indenter (e.g. Berkovich, cube-corner indenter, etc.) to create controlled cracks in thin films or bulk materials. The fracture toughness can be estimated by using the following equation,

$${\mathrm{K}}_{\mathrm{Ic}}=\alpha \sqrt{\frac{\mathrm{E}}{\mathrm{H}}}\frac{\mathrm{P}}{{\mathrm{C}}^{\frac{3}{2}}}$$

(5)

where P is the indentation load (the maximum load to generate crack), E is Young’s modulus, H is the hardness, c is the total length from the center of the indent to the end of crack and α is a constant related to the crack morphology and the shape of the indenter used.^49–51 However, crack geometry assumption can induce an error. In addition to that, this method is widely used primarily for brittle materials.

Fracture toughness using SEM and TEM

Transmission electron microscopy (TEM) and Scanning electron microscopy (SEM) are widely used to observe critical information such as real time crack propagation, indentation mark, crack tip dislocation emission, plastic zone, etc. Both TEM and SEM are used with the combination of the standard mechanical tests to get the complete picture. TEM provides more resolutions compared to SEM and is good for nano as well as sub-nano scale. On the other side, SEM is also good for nano scale as well as for micro scale. In both cases, the samples need to be made as ultra-thin (nano to micro scale). The cost and complexity are also very high. Several studies are reported using the TEM and SEM coupled with the experimental work to comprehend the fracture process.^52–54

Computational methods

Crack tip opening displacement (CTOD) methods

MD simulations are widely used to simulate crack propagation and estimate fracture toughness. One of the methods to find the fracture toughness is the Crack Tip Opening Displacement (CTOD). This method is widely used in fracture mechanics to characterize the fracture toughness of materials, especially in ductile metals. CTOD measures the displacement at the tip of a crack just before the material begins to fracture. It essentially quantifies how much the crack faces have opened due to plastic deformation at the crack tip. The relations between CTOD and stress intensity factor can be described with following two equations.^55,56

$$\mathrm{CTOD}=\frac{8K_I}{E}\sqrt{\frac{r_y}{2\pi }}$$

(6)

$$\mathrm{CTOD}=\frac{4}{\pi }\frac{K_I^2}{{\sigma }_{y}E}$$

(7)

Here, E is the modulus of elasticity, r_y is the plastic zone radius at crack tip, K_I is the critical stress intensity factor, σ_y is the yield strength. Figure 6 shows the measurement for CTOD. In MD simulations, the accuracy depends on the forcefield type and parameters.^57,58 Again, this method can also be used in continuum scale in experiments. However, it requires precise instrumentation and setup. It is also sensitive to specimen geometry and size.^7,59

Figure 6.

Crack tip opening displacement (CTOD) and plastic zone radius for a single-crystal Aluminum. ⁵⁵

Integrated MD-FEA (Irwin’s stress-intensity factor method)

Irwin’s approach to determining the stress intensity factor is a commonly adopted method for evaluating fracture toughness. This technique relies on knowing the local stress distribution near the crack tip. However, in MD simulations, defining local stress and linking it to continuum-level stress is inherently ambiguous. To address this issue, MD results are integrated with FEA, where material properties derived from MD are used as input for FEA. The stress-intensity factor, K_I, can then be computed using the following expression^60–62:

$$K_I=\underset{x\to 0}{\lim }\sqrt{2\pi x}{\sigma }_{\mathrm{yy}}$$

(8)

Where, x = distance measured from the crack tip; and ${\sigma }_{\mathrm{yy}}$  = opening stress normal to the initial crack surface as it is singular at the crack tip (x = 0), and K_I is determined by plotting a curve ${\sigma }_{\mathrm{yy}}\sqrt{2\pi x}$ versus x. K_I is the projected values of ${\sigma }_{\mathrm{yy}}\sqrt{2\pi x}$ at x = 0.

Figure 7 shows how the boundary conditions for finite element analysis (FEA) are obtained from the MD simulation. Failure displacement is calculated from the failure strain which is obtained from the stress strain curve.

Figure 7.

(a) Original MD model with crack of size 2a, (b) schematics showing how equivalent quarter model for finite-element analysis has been developed based on original MD model geometry shown in (a and c); strain-at-failure ε_yy data obtained from MD simulation models with different crack lengths, (d) sketch of quarter model shows the imposed boundary condition; BC 4 has been obtained from the strain-at-failure ε_yy data shown in (c). ⁴⁷

There are some other methods such as J-integral to find fracture toughness. ⁶³ J-integral is a way to calculate the strain energy release rate per unit surface area. It is a path independent integral and useful for analyzing nonlinear (elastic-plastic) fracture behavior. Figure 8 and equation (9) represent the J-integral. ⁶⁴

$$J={\int }_{\mathrm{\Gamma }}(\mathrm{w}{\mathrm{dx}}_2-T\frac{\mathrm{du}}{{\mathrm{dx}}_1}\mathrm{ds})$$

(9)

Figure 8.

Counter J-integral. ⁶⁴

Here, W (x₁, x₂) is the strain energy density, x₁, x₂ are the coordinate directions, t = n. σ is the traction vector, n is the normal to the curve Γ, σ is the Cauchy stress and u is the displacement. The J-integral is used in multiscale fracture analysis through both experimental and computational approaches, particularly in continuum-level methods like FEA; while its principles can be explored at the nanoscale using MD simulations, its direct applicability is limited due to continuum assumptions.^64–67

Fracture toughness can be evaluated at both continuum and atomic scales using distinct methodologies. At the continuum level, experimental techniques such as the compact tension (CT) and single-edge notch bending (SENB) tests are widely used to determine the critical stress intensity factor K_IC, which quantifies a material’s resistance to crack propagation under linear elastic conditions. These methods are standardized and applicable to a broad range of engineering materials. At the atomic level, MD simulations offer a powerful tool to study fracture behavior by modeling atomic interactions and observing crack initiation and growth under applied stress. MD simulations can reveal mechanisms such as dislocation emission, void formation, and bond breaking, providing insight into the intrinsic toughness of materials at the nanoscale.

In the following Table 1, the type of methods to find fracture toughness is summarized. It is worth mentioning that SEM and TEM are used along with many experimental methods to get the observations in real-time scenario.

Table 1.

Details of the different methods to find fracture toughness.

Method name	Type of method and scale	Dominant mechanism	Advantage	Drawback
Three-points bending test a	Experimental, Continuum.	Crack initiation and propagation from the bottom surface.	Easy to prepare specimens, widely accepted and standardized by professional organizations.	Generate stress concentration at the location of applied load, challenging to do for thin material.
Four-points bending test a	Experimental, Continuum.	Crack initiation and stable propagation between the inner loading points.	Similar advantage of three-point bending test but, no localized stress points as the stress is uniform between two load points.	Set-up is little more complex.
Single-Edge Notched Beam (SENB) a	Experimental, Continuum.	Crack initiation at the notch root followed by mode-I crack propagation under bending.	Pre-crack allows controlled crack growth, accurate estimation, and suitable for various types of materials.	Results depend on crack geometry, difficult to evaluate mode-II and III fracture toughness.
Compact tension (CT) Test a	Experimental, Continuum.	Mode-I crack initiation and propagation from the pre-crack location.	Ideal for ductile materials, easier lab set-up, stable crack growth propagation.	Not very suitable for brittle material, requires precise machining and set-up, the load holes may introduce stress concentration.
Cohesive Zone modeling (CZM).	Computational (using FEA), Continuum.	Progressive decohesion at the cohesive interface, representing gradual damage and crack propagation.	Can handle mix mode fracture, avoids singularity at crack tip, good for ductile materials.	Needs accurate parameters to implement traction-separation law, results are sensitive to crack-tip mesh density.
Virtual Crack Closure Technique (VCCT).	Computational (using FEA), Continuum.	Mode-I crack propagation evaluated through energy release rate computed from nodal forces and displacements at the crack tip.	Can work for all mode of fracture, work in standard mesh configuration, good for layered materials like composites.	Not suitable for materials with higher plasticity, challenging for 3D crack geometry and mixed mode condition.
Nano indentation b	Experimental, Continuum and nano-scale.	Crack formation around the indentation imprints due to localized high contact stress.	Ideal for thin films, minimal damage, applicable to various types of materials.	Require high resolution of image, fracture toughness is inferred, not suitable for ductile materials.
Crack tip opening displacement (CTOD).	Experimental and computational, Continuum and nano-scale.	Crack growth governed by plastic deformation and opening displacement at the crack tip.	Good for ductile material with large scale plasticity, directly measurable and physically intuitive.	Require precise measurement at crack tip, accurate measurement of CTOD is difficult for 3D simulation.
J-Integral.	Experimental and computational, Continuum and nano-scale.	Crack growth driven by the energy release rate associated with nonlinear (elastic–plastic) deformation around the crack tip.	It accounts for nonlinear material behavior, allowing flexibility in numerical implementation.	It does not describe irreversible plastic deformation; higher plastic deformation can affect the accuracy.
Irwin’s stress-intensity factor.	Experimental and computational, Continuum and nano-scale.	Crack propagation driven by the singular stress field at the crack tip.	Good for mixed mode analysis, can be implemented in FEA with low computational cost.	Meshing in front of crack tip needs special treatment, difficult to model for arbitrary crack growth.

a

These methods are also used in computational modeling using FEA.

b

This method is primary experimental, but FEA and MD simulations are also used.

Results on fracture toughness at the multiscale

The fracture toughness of metals exhibits considerable variation across different length scales, influenced by factors such as microstructural heterogeneity, plastic deformation, and size dependent effects. These variations arise from distinct mechanisms that govern crack initiation and crack propagation at each scale. At the nanoscale, which refers to dimensions within the nanometer range, toughness is primarily dictated by atomic level processes such as bond breaking, lattice trapping, and crack tip phenomena, as demonstrated through MD simulations.^68,69 As the length increases, the mesoscale becomes significant. This scale typically spans from tens of nanometers to several micrometers and serves as a transitional region where individual microstructural features begin to interact collectively. At this level, mechanisms such as dislocation pile ups, interactions among neighboring grains, phase boundaries, and the early formation of voids influence fracture resistance. The mesoscale is closely related to the microscale because their length ranges overlap, and both capture the effects of microstructural constituents on toughness. At the microscale, which corresponds to dimensions within the micrometer range, structural features such as grain boundaries, voids, second phase inclusions, and dislocation networks dominate the energy dissipation processes. These mechanisms play a particularly important role in heterogeneous materials such as bone, where microscopic components including osteons and cement lines contribute to fracture resistance. At the macroscale, which includes dimensions from millimeters to meters, mechanisms such as large-scale plasticity, crack tip blunting, and the development of a stable fracture process zone enhance energy absorption and generally increase the apparent toughness. However, larger specimens have a greater likelihood of containing critical flaws, which may reduce the apparent toughness under certain conditions.^23,70,71

A scale bridging fracture mechanics framework links each length scale to the measurement techniques best suited to the governing crack tip mechanisms. At the nanoscale, where crack advance is controlled by atomic separation, lattice level crack tip processes, and the initiation of dislocation emission or nucleation, molecular dynamics and coupled MD to FEM approaches are most appropriate, and small scale probing can also rely on nanoindentation based fracture methods and localized CTOD type metrics to characterize near tip separation processes.^72–74 At the lower mesoscale that overlaps the microscale, where collective dislocation activity, dislocation pile ups, grain boundary or phase boundary interactions, and early-stage void nucleation become dominant, small volume fracture measurements that isolate microstructural units, such as nanoindentation based microcantilever bending and related microscale tests, are well suited for quantifying microstructure sensitive resistance. ⁵¹ At the microscale and small structural scale, once plastic zones span multiple grains and elastic plastic crack tip fields are representative, standardized compact tension and single edge notch bend configurations are typically used to measure toughness via the J integral and CTOD under Mode-I loading, including initiation values and resistance curves.^75,76 At the macroscale, where large plastic zone development and specimen level geometry effects strongly influence apparent toughness, SENB and CT testing remain the primary methods for obtaining J integral and CTOD based resistance, with careful attention to specimen size and geometry to ensure transferability to component behavior.^76–78 Together, these techniques form a coherent scale bridging methodology connecting atomistic processes, microstructural mechanisms, and continuum fracture parameters.

The translation of atomistic fracture parameters obtained from MD simulations into continuum scale fracture toughness requires several simplifying assumptions. The most fundamental assumption is that the discrete atomic lattice can be treated as an equivalent homogeneous elastic continuum, allowing the extension of Griffith’s energy balance from macroscopic cracks to nanoscale fracture processes.^79,80 This approach further assumes that lattice specific effects and size dependent behavior, which are intrinsic to atomistic simulations, can be homogenized when relating atomistic fracture energy to continuum quantities. A major limitation is the extremely high strain rates inherent to MD simulations, often on the order of 10⁹–10¹¹ s⁻¹, which tend to suppress plastic mechanisms and produce artificially brittle responses. This rate dependence has been demonstrated clearly, for example, in MD studies of alpha alumina where the apparent fracture toughness decreases as loading rate increases. ⁸¹ In contrast, ASTM standard K_IC values are measured under quasi static conditions designed to ensure small scale yielding and valid continuum stress intensity fields. Additionally, experimental fracture toughness captures microstructural effects such as dislocation mediated plasticity, grain boundary interactions, and crack deflection, none of which are fully represented in idealized single crystal atomistic models. For these reasons, MD derived fracture parameters should be understood as intrinsic bond breaking energies, whereas ASTM K_IC represents an extrinsic macroscopic resistance that depends strongly on microstructure and loading conditions. ⁸²

Here the evaluated results from experimental and computational techniques at different length scales are furnished for aluminum and iron, and their alloys.

Fracture toughness of aluminum at different length scales

Aluminum and its alloys play a crucial role in modern engineering because of their low density, high strength-to-weight ratio, corrosion resistance, and excellent thermal and electrical conductivity. These properties make them indispensable in aerospace and automotive sectors, where lightweight and strength improve fuel efficiency and overall performance. Wrought aluminum alloys account for more than 70% of materials used in contemporary airframes, valued for their structural integrity and resistance to corrosion. Advances in Al–Li alloys, such as 2195-T8M4, have further enhanced specific strength and damage tolerance for aerospace applications.^83,84 Alloying combined with thermomechanical processing refines grain structure and optimizes precipitation, thereby improving mechanical and electrical characteristics. Microstructural control remains critical in wrought alloys, where precipitation behavior and texture evolution significantly influence performance.^83–85 Additionally, microalloying with rare earth elements like Sc and Er promotes the formation of stable nano-precipitates (e.g. Al₃Sc, Al₃Er), which boost strength, corrosion resistance, and thermal stability. Techniques such as interface segregation and core/shell architectures are being employed to develop heat-resistant aluminum alloys suitable for extreme environments.^86–88

Estimation of fracture toughness of pure aluminum

As aluminum and its alloys play an important role in modern world, their fracture properties and behavior are very important to comprehend. Most research of fracture toughness on aluminum alloys tends to focus on the higher-strength, more complex alloys (2000 series and higher) where fracture behavior is more of a concern. As there are many alloys, the initial focus is on Aluminum 1xxx series, which is nearly pure Aluminum. There are several reported works on fracture of aluminum 1000 series alloy. Vishnu et al. ⁸⁹ reported fracture energy for thin aluminum sheet using cohesive zone modeling in FEA. Patil et al. ⁹⁰ reported the fracture geometry and slip band near crack tip in combination of computational and experimental study where single edge notch specimen was considered. Ikumapayi et al. ⁹¹ reported fracture energy for aluminum 1050 alloy at the variation of temperature from −50°C to 125°C using Charpy test of v-notched specimen. Doan et al. ⁹² reported the fracture morphology of aluminum 1050 alloy at the variation of loading rate using the method of small punch test. These studies didn’t report the fracture toughness directly. Attar et al. ⁹³ reported the fracture toughness value at macroscale. The authors have used aluminum 1050 alloy and prepared the sample by warm accumulative roll bonding (WARB) in different cycles and ASTM-E561 standard was utilized to introduce standard compact tension (CT) specimens. The pre-cracked specimen dimensions are 22.5 mm × 21.6 mm × 2 mm. Figure 9 shows the specimen’s dimension, experimental set-up, and the fracture toughness curve.

Figure 9.

(a) Dimensions of CT sample according to ASTM E561 standard (all dimensions in mm), (b) experimental setup of the CT test in Zwick/Roell machine, and (c) R-curves for AA1050 processed by WARB at five cycles. ⁹³

The fracture toughness is calculated by using the following equation which is also reported in other studies.^94,95

$$K_{\mathrm{ri}}=\frac{P_i}{b\sqrt{w}}{f}_i\left(\frac{a}{w}\right)$$

(10)

The crack length, a shows relative to the applied force (Pi), and b and w are the thickness and width of the CT sample, respectively. In Figure 9(c), Kr is plotted for three fixed forces as a function of the crack length (dashed line). The intersection of the R-curve with the constant lines indicates the value of Kc. This means that when the stress intensity factor is equal to the fracture resistance, the crack will unsteadily start to grow. So, one step increment after the intersection of the two curves leads to a situation in which the stress intensity factor supersedes the resistance of the substance indicating the unsteady growth of the crack. At least three specimens were tested, and the reported average results of the fracture toughness is 22.85 MPa $\sqrt{m}$ . A similar study on aluminum 1050 alloys reported where the specimens were prepared by using accumulative roll bonding process. ⁹⁴ Evaluated fracture toughness is fracture toughness is 25.40 MPa $\sqrt{m}$ . Taktak et al. reported ⁹⁶ the effect of cold rolling on fracture toughness aluminum alloy 1050. The thickness of one cold rolled specimen reaches 0.85 mm, which can be considered as micro scale. The fracture toughness tests were performed on a center cracked specimen by using universal tensile machine. The fracture energy, J is calculated ⁹⁷ in both rolling direction (RD and transversal direction (TD). The crack growth Δa was calculated through images obtained by the high-resolution video camera. The obtained fracture energy is 2.15 kJ/m² in RD and 1.05 kJ/m² in TD. The elongated grains, micro-cracks, and dislocations in metal produced during cold rolling ⁹⁸ make the transition from ductile to brittle fracture which negatively impacts the fracture toughness. Fracture toughness value can be calculated from equation (11) in plane strain condition, where E is modulus of elasticity and ν is Poisson’s ratio. ⁹⁹

$$K_I=\sqrt{J.\frac{E}{1-{\vartheta }^2}}$$

(11)

The E for aluminum is considered as 69 GPa and ν as 0.33.^100,101 So, the fracture toughness at this scale is 12.90 MPa $\sqrt{m}$ in RD and 9.02 MPa $\sqrt{m}$ in TD.

Velilla-Diaz et al. reported ¹⁰² the fracture toughness at nanoscale of single crystal and bi-crystal with a misorientation GB tilt angle of 30° of aluminum by using MD simulation. In the MD simulations, EAM potential is used which are proposed by Mendelev et al. ¹⁰³ The authors evaluated the fracture toughness by using J-integral, energy release rate, and CTOD methods. The dimensions of the both single and bi-crystal are 24.3 nm × 16.2 nm × 8.1 nm where bi-crystal system has equal dimensions of each grain. The crack lengths varied from 2.025 to 8.1 nm. The fractured single crystal exhibited brittle behavior, showing no signs of dislocation emission. The grain boundary acted as a barrier, halting the unstable propagation of the initial crack. This led to crack blunting and a transition to a more ductile-like fracture behavior. Figure 10 shows the fracture behavior of single and bi-crystal system. In all three methods, the evaluated fracture energy numbers are very close, and J-integral provides the fracture energy for single crystal 2.36 J/m² and in bicrystal system as 13.61 J/m².

Figure 10.

Centro symmetry parameter for (a) crack propagation in a single crystal at ε _zz  = 6.8%, (b) first tearing at ε _zz  = 6.5%, and (c) final fracture showing void coalescence in the second grain at ε _zz  = 30%.

The E for aluminum is considered as 60.58 GPa and ν as 0.36 according to the authors and also reported by Haque for 30–50 nm thin films. ¹⁰⁴ By using equation (11), the fracture toughness value for single crystal is 0.41 MPa $\sqrt{m}$ and for bicrystal is 0.97 MPa $\sqrt{m}$ . In another study, the fracture toughness of single crystal is reported similar values, the methods were compared with the results of other metals. ⁵⁵ The results in molecular dynamics simulations can be interatomic potential dependent.

Measurement of fracture toughness of aluminum alloys

The alloy designation system classifies the wrought aluminum alloy based on its major alloying elements and the fracture toughness in aluminum alloys varies significantly across different series due to differences in alloying elements, microstructure, and processing routes. As previously mentioned, Alloys in the 1xxx series are composed of 99.00% or more pure aluminum and are known for excellent corrosion resistance and high thermal and electrical conductivity.^105–107 For instance, 2xxx series alloys, which are copper-rich, typically exhibit high strength but lower fracture toughness due to coarse intermetallic particles and grain boundary segregation.^108,109 In contrast, 3xxx series alloys, primarily alloyed with manganese, are non-heat-treatable and offer moderate toughness suitable for non-structural applications. 6xxx series alloys, containing magnesium and silicon, strike a balance between strength and toughness, making them ideal for automotive and structural use. ¹¹⁰ 7xxx series alloys, rich in zinc and often combined with magnesium and copper, achieve very high strength but are prone to intergranular fracture and reduced toughness due to fine precipitate formation and stress corrosion susceptibility. ¹¹¹ Lastly, the 8xxx series include miscellaneous elements such as lithium or iron, tailored for specialized applications like packaging and electrical cables. Also, Metallurgical factors and microstructure affect the fracture toughness.^112,113 Processing methods such as forging, extrusion, and heat treatment further influence grain size, precipitate distribution, and residual stress, all of which affect fracture behavior.^114,115

In different aluminum alloys, the trend of decreasing value for fracture toughness across the scales is similar to pure aluminum. Fracture toughness in aluminum alloy 7075 varies depending on specimen orientation in rolling, long transverse, and short transverse directions, as reported by Iric et al., ¹¹⁶ with values ranging from 21.81 to 41.92 MPa $\sqrt{m}$ . Gariboldi et al. ¹⁰⁹ reported the fracture toughness of aluminum alloy 2xxx series. The authors manufactured the compact tension (CT) specimen with thickness 20 mm and used crack opening displacement to estimate the fracture toughness, which ranges from 19 to 26 MPa $\sqrt{m}$ . Kumar et al. ¹¹⁷ reported the fracture toughness of aluminum alloy 6061 with a diameter of 12 mm. The fracture toughness is obtained from bending test around 35.8 MPa $\sqrt{m}$ . Barker ¹¹⁸ reported fracture toughness for alloy 7075 is 23.5 MPa $\sqrt{m}$ and alloy 2014 is 26.8 MPa $\sqrt{m}$ , respectively. Kumar et al. ¹¹⁹ also reported fracture toughness for 7075 alloy as 22.28 MPa $\sqrt{m}$ by following ASTM E399 standard and using a CT specimen with thickness of 12.5 mm. ASM handbook ¹²⁰ provides the fracture toughness of aluminum alloys except 1xxx series by estimating in the traditional experimental methods.

The reported work is very limited in microscale for aluminum alloys. Ibrahim reported ¹²¹ fracture toughness of Al 6061 alloy with addition of different percentage of graphite nanoparticles. The specimen size is 10 mm in diameter and 3 mm thickness, which is very close to the micro scale. The fracture toughness is obtained by indentation method using Vickers’ indenter in a pyramid shape. The reported fracture toughness of Al 6061 with 1% graphite is 14.48 MPa $\sqrt{m}$ and with 2% graphite is 16.62 MPa $\sqrt{m}$ . Hence, we can argue that the fracture toughness of Aluminum alloy is around 14 MPa $\sqrt{m}$ in microscale. Gucci et al. ¹²² reported fracture energy at nano scale for aluminum alloy with 5% magnesium, which represents the 5xxx and 6xxx series aluminum alloy. The dimension of molecular model is 19.85 × 19.85 × 1.22 nm, where EAM potential is used. J-integral is used to calculate energy. The reported energy is 3.55 J/m² which converts to 0.57 MPa $\sqrt{m}$ by using equation (11). This value changes in small percentage with the variation of crack length. It is worth mentioning that few studies reported very high fracture toughness (>100 MPa $\sqrt{m}$ ) for aluminum alloys. ¹²³ Figure 11 illustrates the variation in fracture toughness across different scales for the reviewed manuscripts. “Max” denotes the maximum values, “Min” the minimum values, and “Avg” the average values. Since the variation at the nanoscale is small, these metrics (Max, Min, Avg) are not shown for this scale.

Figure 11.

Fracture toughness of pure aluminum and aluminum alloys.

Fracture toughness of iron at different length scales

Iron and its alloys are foundational to engineering applications due to their diverse mechanical properties and adaptability. Armco iron, a nearly pure form of iron with minimal impurities, is prized for its excellent magnetic characteristics, corrosion resistance, and weldability, making it suitable for electromagnetic and high-purity applications.^124,125 In contrast, iron alloys are tailored for structural and functional uses. Ductile cast iron, also known as nodular iron, contains spheroidal graphite that imparts high strength and toughness, making it ideal for automotive and pressure systems. White cast iron features carbon in the form of cementite, resulting in a hard, brittle structure with excellent wear resistance, commonly used in abrasion-prone environments. Malleable cast iron is produced by heat-treating white iron to form temper carbon, enhancing ductility and machinability for use in brackets and fittings. Gray cast iron, the most widely used cast iron, contains flake graphite in a ferritic or pearlitic matrix. Its unique microstructure provides excellent damping capacity, good machinability, and wear resistance, though it is relatively brittle under tensile loads. It is extensively used in machine bases, engine blocks, and structural components where vibration damping is critical.^126–129 Steel and steel alloys are included in iron alloys. In addition to being cheaper materials with which to manufacture, cast irons have some physical properties that give them a competitive advantage when compared with cast steel.

Estimation of fracture toughness of pure iron

Hohenwarter et al ¹³⁰ reported the fracture toughness value for armco iron which is nearly a pure iron contains 99.97% of iron (Fe). The final CT specimens were produced with a dimension of 5 mm width and thickness 2.5 mm with a grain size of 300 nm in high pressure torsion (HPT) approach. The fracture analysis is conducted according to ASTM standard E399. The reported fracture toughness is 48.6 MPa $\sqrt{m}$ . This research group extended their study ¹³¹ for a temperature range of −200°C to 200°C with a grain size of 300 nm, 1 µm, 5 µm in HPT approach to observe the effect of ductile to brittle transition in fracture behavior. Also, severe plastic deformation (SPD) techniques were also used for grain refinement and compared the results. Their new results also aligned with a previous result and shows that the larger grain size represents the lower fracture toughness. Figure 12 shows the specimen and the fracture toughness results.

Figure 12.

(a) Fracture specimens with their different crack orientations machined from the HPT disk, (b) summary of the fracture toughness measurements for the tangential orientation.

Srinivas et al. ¹³² also reported fracture toughness value of armco iron for different grain size with a range of 38–1050 µm. The trend is similar as the fracture toughness decreases with increasing the grain size.

Snartland et al. reported ¹³³ the fracture toughness of 99.99% pure iron in micro scale. In their test, micro-sized cantilevered beam was used to estimate the fracture toughness at a range from 37.5 K to room temperature. The orientation of crack system is (100; 0 $\overline{1}$ 1). Two experiments were conducted at two different institutions, and the dimensions of the cantilevers were 8 µm × 1 µm × 2 µm and 8 µm × 2 µm × 2 µm with a pentagonal cross-section. The estimated elastic-plastic fracture toughness is around 4.02 MPa $\sqrt{m}$ at room temperature and little higher at 37.5 K. The relative size of the sample compared to the plastic zone size ahead of the crack tip is the cause of increasing fracture toughness at lower temperature. As the plastic zone size is larger than the fracture sample itself, the cross-section of the cantilever will be plasticized, moving the behavior of the crack from fracture toward plastic collapse. The same research group ¹³⁴ conducted similar experiment with the crack system of (100; 0 $\overline{1}$ 1) and (10 $\overline{1}$ ). ⁹⁹ The reported values were in the range of 4.5–8.7 MPa $\sqrt{m}$ for different dimensions and methods where no clear full brittle fracture was observed, however, fast crack growth was evident. Scanning electron microscope was used to measure the initial and final crack lengths after loading. Micrographs with crack mouth opening displacement were obtained before and after each unloading segment. Dislocation density is reduced when the cantilevers are partially unloaded, indicating plastic recovery. Hence, a sharpening of the notch tip can take place. A sharper tip means a higher stress concentration at the tip which could enable crack propagation compared to the continuously loaded cantilevers. ¹³⁵

At nanoscale, the fracture behavior of α-iron (bcc iron) is reported by several researchers. Ma ¹³⁶ shows the fracture behavior by using molecular dynamics simulation. Vatne et al. ¹³⁷ reported the fracture toughness value of iron at nanoscale at different crystallographic orientation. Figure 13 shows the crack opening for different crystallographic orientation. EAM potential is used to model with a sizing of 150 nm × 150 nm. The crack tip boundary conditions such as isotropic or anisotropic and mechanisms such as cleavage crack propagation, twinning, and dislocation emission are considered in this analysis. The calculated fracture toughness is in the range of 0.87–1.23 MPa $\sqrt{m}$ .

Figure 13.

Crack opening at different crystallographic orientations: (a) orientation (010) ⁹⁹ , (b) orientation ( $\overline{1}10$ ; 001), (c) 01 $\overline{1}$ ; 011, and (d) 010; 001.

Thaulow reported the fracture toughness value as 0.86 MPa $\sqrt{m}$ of bcc iron by applying CTOD approach in molecular dynamics simulations. ¹³⁸ Zhao et al ¹³⁹ reported the atomic scale fracture behavior of iron under the impact load. The generated a polycrystal model is 60 × 60 × 30 nm with an average of 15 nm grain diameter. The carried out the simulations under different loading rate and obtained Griffith energy release rate 3.58 J/m², which is equivalent to around 0.9 MPa $\sqrt{m}$ .

Measurement of fracture toughness of iron alloys

Iron and steel alloys are foundational materials in modern engineering due to their exceptional mechanical properties and versatility. Iron alloys which contain much higher carbon content compare to steel alloys, including cast iron and wrought iron, are widely used in construction, automotive, and machinery applications. Steel alloys, which are iron mixed with elements like carbon, chromium, or nickel, offer enhanced strength, corrosion resistance, and durability, making them indispensable in industries ranging from aerospace and infrastructure to medical devices and consumer goods. Their broad applicability stems from the ability to tailor their properties to meet specific performance requirements.^140,141

Bradley and Srinivasan et al. ¹⁴² reported the fracture properties of different cast iron alloys where the effect of microscopic factors is portrayed. Gray iron, malleable iron, ductile iron, compacted Iron are well-knows alloys microstructure that consists of a graphite phase (usually 8–14 vol.%) in a matrix which may be ferritic, pearlitic, bainitic, or tempered martensitic. At room temperature, the fracture toughness of cast irons varies widely with graphite morphology and matrix structure. Gray cast iron exhibits the lowest toughness (19–40 MPa $\sqrt{m}$ ), while ductile iron shows the highest around 90 MPa $\sqrt{m}$ due to its nodular graphite and ferritic–pearlitic matrix. Malleable and compacted graphite irons display intermediate toughness levels about 25–67 MPa $\sqrt{m}$ . However, white cast iron remains the most brittle (14–31 MPa $\sqrt{m}$ ), because of its different microstructure which is rich in cementite and pearlite and makes it better candidate for abrasive resistance but limits widely structural applications. These values were obtained mainly from Charpy, and fatigue-pre-cracked compact-tension tests. The variations reflect differences in composition—principally carbon (2–4 wt%) and silicon (1–3 wt%) with magnesium additions for nodularity and determine their use from wear-resistant parts (white iron) to structural components (ductile iron). The fracture toughness of white cast iron is also reported in the similar range. ¹⁴³ The effect of microstructural modifications, contents of different elements such as C, Si, and Mn on fracture toughness property of ductile cast iron were investigated and the reported results are also in the range of above-mentioned values.^20,144,145 In micro and nano scale, to the best of authors’ knowledge, the proper manuscripts were not found for iron alloys for fracture toughness prediction.

Measurement of fracture toughness of steel alloys

Steel alloys are engineered materials composed primarily of iron and carbon, with additional alloying elements such as chromium, nickel, molybdenum, vanadium, and manganese to enhance specific mechanical properties. These elements are added in varying proportions to tailor the steel for particular applications. Chromium improves corrosion resistance and hardness, forming the basis of stainless steels. Nickel enhances toughness and resistance to impact and fatigue, especially at low temperatures. Molybdenum contributes to high-temperature strength and creep resistance. Vanadium refines grain structure and increases wear resistance, while manganese aids in hardenability and deoxidation during steelmaking. Steel alloys are classified into low, medium, and high alloy steels based on the total percentage of alloying elements. Low alloy steels (less than 5%) are widely used in structural applications due to their balance of strength and cost. Medium and high alloy steels are used in aerospace, automotive, and energy sectors where enhanced performance is critical. Applications include aircraft landing gear (e.g. 4340 steel), chemical reactors (316 stainless), and construction components like bridges and buildings.^146–148

Timofeev et al. ¹⁴⁹ reported the fracture toughness of different types of low carbon steel in temperature variation which contains less than 0.25% carbon and mostly used in power engineering related industry. Three point bending test is done according to ASTM E399 standard which produced the fracture toughness value in the range of 75–90 MPa $\sqrt{m}$ . Sakamoto et al. ¹⁵⁰ reported the fracture toughness of medium to high carbon steel for railroad wheel where the specimens contain carbon in the range of 0.41%–0.68%. It is reported that the fracture toughness can vary from ∼100 to ∼50 MPa $\sqrt{m}$ with the increasing of carbon content from 0.41% to 0.68%. Shimada et al ¹⁵¹ studied the bending test at micro scale for the low alloy steel with a carbon percentage of 0.1% where the specimen size is approximately 1250 × 250 × 250 µm. The crack tip opening displacement (CTOD) was measured using a video to estimate the fracture toughness value. The estimated value of fracture toughness is 19.8 MPa $\sqrt{m}$ . As the specimen shape doesn’t comply with ASTM E1304, the fracture toughness is expressed as K_Q, rather K_IC. Chabok et al. ¹⁵² also executed microcantilever bending tests of high strength steel welds. J-integral and CTOD methods are used to calculate fracture toughness. The dimensions are 15 × 5.5 × 4.5 µm. The experiments were conducted for different welded zones and the results were in the range of approximately 5–22 MPa $\sqrt{m}$ . Figure 14 illustrates the variation in fracture toughness across different scales for the reviewed manuscripts. Labeling is done similar to Figure 11.

Figure 14.

Fracture toughness of pure iron, iron alloys, and steel alloys.

Steel alloys exhibit wide range of fracture toughness value as it depends on grain size, the added elements, temperature variation, heat treatment, manufacturing process, etc. Here sone additional manuscripts are reviewed to comprehend the wide range of fracture toughness variation. Sakamoto et al. ¹⁵⁰ also reported that increasing the content of manganese (Mn) will increase the fracture toughness value until the ratio of Mn/C reaches 2.2. Tunnicliffe ¹⁵³ studied the fracture toughness low carbon steel with high and low Nitrogen content and the effect of the grain size and temperature. CT specimens were prepared and the fracture toughness was evaluated in CTOD method. The reported fracture toughness is 110 MPa $\sqrt{m}$ for low nitrogen content and 70 MPa $\sqrt{m}$ for high nitrogen content steel. Nickel (Ni) content also affect on fracture toughness property as it increases with the increase of Ni. ¹⁵⁴ Meshref ¹⁵⁵ reported the change of fracture toughness of AISI 4140 low alloy due to the different heat treatment such as stress relief annealing, full annealing, hardening and tempering followed by quenching. The fracture toughness diminished to 34 MPa $\sqrt{m}$ in quenching heat treatment where annealing exhibits 127 MPa $\sqrt{m}$ . Moreover, Upadhyaya et al. ¹⁵⁶ reported the fracture toughness around 45 MPa $\sqrt{m}$ for AISI 4140 steel without any heat treatment for CT specimen by following ASTM E399-90 standard. Fracture toughness is also enhanced by tailoring the microstructure and the fracture toughness also depends on the loading directions. Zhang et al. ¹⁵⁷ manufactured a strong yet tough steel alloy by adding fibrous ferrite into martensite matrix where Mn and Al are added. The authors reported fracture toughness about 105 MPa $\sqrt{m}$ without sacrificing the high strength around 1.5 GPa where the loading was in the parallel to fibrous ferrite. The fracture toughness reduces nearly 50% if the loading direction is perpendicular to fibrous ferrite. Hence, the fracture toughness is not reported as a single value rather reported in a range as it varied widely.

Conclusion

In this study, current scenario and insights regarding fracture toughness measurement methods across different length scales and evaluated values for aluminum and iron and their alloys (including steel alloys) are discussed. The fracture toughness of pure metal is also emphasized because the alloys of both metals are so versatile that fracture toughness variation is in a wide range. In addition to that, different testing standards for finding fracture toughness can provide different values which can put the engineers into a dilemma. Li et al. ¹⁵⁸ reported fracture toughness for steel alloy by using ASTM E1820 and ISO 12135 standards. Between these two standards, ISO provides the lower value around 40% in lower temperature and the difference reduces as the temperature goes up more than 200°C. Hence, the results depend on many parameters which are discussed throughout the manuscript. Beyond these issues at the bulk scale, significant challenges exist in standardizing fracture toughness measurements at the micro and nano scales. At these small dimensions, existing ASTM and ISO standards are not directly applicable, and no universally accepted protocols exist for micro or nanoscale specimens. This has led to large scatter in reported toughness values because results depend strongly on specimen geometry, preparation method, and testing technique, as shown in this manuscript. Moreover, size effects, intrinsic material length scales, and focused ion beam fabrication artifacts further complicate reproducibility and hinder the establishment of unified standards. Recent reviews emphasize that emerging micro and nano fracture methods, including micro cantilever approaches, are actively being refined to improve consistency and may eventually serve as a foundation for developing future standardized testing protocols. Hence, while the present manuscript highlights challenges, methods, and estimation of fracture toughness, it also underscores that substantial research and methodological development are still required before micro and nanoscale fracture toughness testing can achieve the level of standardization currently available at the macroscale.

When comparing fracture toughness across different length scales for aluminum and iron and their respective alloys, clear trends emerge due to variations in crystal structure, alloying strategies, and microstructural design. At the atomic or nanoscale, both metals exhibit low intrinsic toughness (∼0.5–2 MPa·m^1/2), because dislocation motion is restricted and plasticity is minimal. The overall reduction in toughness from macro to nano levels is primarily linked to suppressed plastic deformation, diminished crack-tip blunting, and the prevalence of brittle fracture modes.

At the microstructural scale, alloying and phase morphology become critical. Ductile irons, particularly those with spheroidal graphite, exhibit high toughness (∼75–106 MPa·m^1/2) through crack deflection and energy absorption mechanisms. In contrast, aluminum alloys, even those strengthened by precipitation hardening (e.g. 2xxx and 7xxx series), remain lower (∼15–40 MPa·m^1/2), as precipitates often serve as crack initiation sites, limiting toughness. At the mesoscale, engineered microstructures further enhance toughness: cast irons with optimized graphite morphology reach ∼60–120 MPa·m^1/2, while aluminum alloys, despite grain refinement and heat treatments, typically stay within ∼15–40 MPa·m^1/2 due to their inherently lower strain-hardening capacity.

At the macroscopic or structural scale, iron and its alloys display the widest toughness range (∼50–150 MPa·m^1/2), benefiting from complex phase interactions and transformation toughening. Aluminum alloys, although optimized for aerospace and automotive applications, remain the least tough (∼22–40 MPa·m^1/2), prioritizing low density and corrosion resistance over fracture resistance. Overall, iron-based alloys dominate structural applications because of their superior energy absorption and crack propagation resistance, whereas aluminum alloys emphasize weight reduction and manufacturability, often at the expense of toughness.

Challenges and promising prospects for estimating fracture toughness across scales for metallic materials are discussed in the manuscript. In future, the integration of advanced computational models with high-resolution experimental techniques and recently developed data driven methodologies can improve fracture toughness predictions across multiple length scales.