Indentation Load Research Articles

Study of damage behavior and repair effectiveness of patch repaired carbon fiber laminate under quasi-static indentation loading

Damage caused due to low-velocity impacts in composites leads to substantial deterioration in their residual strength and eventually provokes structural failure. This work presents an experimental investigation on the effects of different patch and parent laminate stacking sequences on the enhancement of impact strength of Carbon Fiber Reinforced Polymers (CFRP) composites by utilising the adhesively bonded external patch repair technique. Damage evolution study is also performed with the aid of Acoustic Emission (AE). Two different quasi-isotropic configurations were selected for the parent laminate, viz., [45°/45°/0°/0°]s and [45°/0°/45°/0°]s. Quasi Static Indentation (QSI) test was performed on both the pristine laminates, and damage areas were detected by using the C-scan inspection technique. Damaged laminates were repaired by using a single-sided patch of two different configurations, viz., [45°/45°/45°/45°] and [45°/0°/0°/45°], and employing a circular plug to fill the damaged hole. Four different combinations of repaired laminates with two configurations of each parent and patch laminate were produced, which were further subjected to the QSI test. The results reveal the effectiveness of the repair method, as all the repaired laminates show higher impact resistance compared to the respective pristine laminates. Patches of [45°/0°/0°/45°] configuration when repaired by taking [45°/45°/0°/0°]s and [45°/0°/45°/0°]s as parents exhibited 68% and 73% higher peak loads, respectively, than the respective pristine laminates. Furthermore, parent and patch of configuration [45°/0°/45°/0°]s and [45°/0°/0°/45°], respectively, attain the highest peak load, whereas [45°/45°/0°/0°]s and [45°/45°/45°/45°] combinations possess the most gradual decrease in the load.

Effect of crystal orientation on the nanoindentation deformation behavior of TiN coating based on molecular dynamics

In order to investigate the mechanism of nanoindentation deformation of TiN and the protective properties of TiN coating in [001] and [111] crystal orientation, molecular dynamics simulations were established for single-crystal TiN and bilayer TiN/FeCrNi models in this study. The hardness and reduced elastic modulus were obtained by analyzing the indenter load displacement curves obtained from the simulations. Combined with the atomic-scale analysis, it was found that the trigonal defect structure generated in the [111] crystal orientation of the single-crystal TiN enhanced the hardness of the material. The analysis of the simulation results of bilayer TiN/FeCrNi found that the coating withstood high stresses to effectively protect the substrate from damage, and the lower substrate hardness also improved the plastic deformation of the coating to avoid fragmentation, which showed basically the same hardness in both crystal orientations. The results of the study can provide theoretical support for the preparation and application of TiN coatings and provide useful insights to defect-formation, defect/defect interactions and plausible strengthening mechanisms upon loading.

Wear and Subsurface Stress Evolution in a Half-Space under Cyclic Flat-Punch Indentation

Wear is a tremendously important phenomenon, which takes place on the surfaces of two solids in contact under cyclic loads and constitutes one of the most-significant ways of failure for mechanical elements. However, it is not the only source of failure in contacting solids. The subsurface stresses should also be considered, due to the fatigue and crack initiation problems. Nevertheless, these stresses (i.e., their maximum values and distributions) evolve with the solids’ surface wear (i.e., with the load cycles) and also depend on the friction intensity. Therefore, their evolution should be properly computed to predict failures in mechanical elements under wear conditions. This work focused on the study of the evolution of the surface wear and the subsurface stress distributions generated—in an elastic half-space—by a cylindrical flat-ended punch, under cyclic indentation loading (i.e., radial fretting wear conditions). Based on a numerical scheme recently presented by the authors, this is the first time that, for this contact problem, the surface wear and subsurface stress distribution (i.e., maximum value and its location)—and its evolution—were simultaneously analyzed when orthotropic friction and fretting wear conditions were considered. The studies presented in this work were developed for purely elastic contact assumptions.

Glass hardness: Predicting composition and load effects via symbolic reasoning-informed machine learning

Glass hardness varies in a non-linear fashion with the chemical composition and applied load, a phenomenon known as the indentation size effect (ISE), which is challenging to predict quantitatively. Here, using a curated dataset of over 3,000 inorganic glasses from the literature comprising the composition, indentation load, and hardness, we develop machine learning (ML) models to predict the composition and load dependence of Vickers hardness. Interestingly, when tested on new glass compositions unseen during the training, the standard data-driven ML model failed to capture the ISE. To address this gap, we combined an empirical expression (Bernhardt's equation) to describe the ISE with ML to develop a framework that incorporates the symbolic equation representing the domain reasoning in ML, namely Symbolic Reasoning-Informed ML Procedure (SRIMP). We show that the resulting SRIMP outperforms the data-driven ML model in predicting the ISE. Finally, we interpret the SRIMP model to understand the contribution of the glass network formers and modifiers toward composition and load-dependent (ISE) and load-independent hardness. The deconvolution of the hardness into load-dependent and load-independent terms paves the way toward a holistic understanding of the composition effect and ISE in glasses, enabling efficient and accelerated discovery of new glass compositions with targeted hardness.

Load-induced local phase transformation and modulus of shape memory alloys under spherical indentation by finite element method

Shape memory alloys are a unique class of materials that are capable of large reversible deformations under external stimuli such as stress or temperature. The present study examines the phase transformations and mechanical responses of NiTi and NiTiHf shape memory alloys under the loading of a spherical indenter by using a finite element model. It is found that the indentation unloading curves exhibit distinct changes in slopes due to the reversible phase transformations in the SMAs. The normalized contact stiffness (F/S2) of the SMAs varies with the indentation load (depth) as opposed to being constant for conventional single-phase materials. The load-induced phase transformation that occurred under the spherical indenter was simulated numerically. It is observed that the phase transformation phenomenon in the SMA induced by an indentation load is distinctly different from that induced by a uniaxial load. A pointed indenter produces a localized deformation, resulting in a stress (load) gradient in the specimen. As a result, the transformation of phases in SMAs induced by an indenter can only be partially completed. The overall modulus of the SMAs varies continuously with the indentation load (depth) as the average volumetric fraction of the martensite phase varies. For NiTi (Ea > Em), the modulus decreases with the depth, while for NiTiHf (Ea < Em), the modulus increases with the depth. The predicted young modules during indentation modeling agree well with experimental results. Finally, the phase transformation of the SMAs under the indenter is not affected by the post-yield behavior of the materials.

Analysis of transverse elastic modulus and compressive deformation mechanism of cellulose crystals based on molecular dynamics

Cellulose nanocrystals (CNCs) have been widely used as reinforcing agents for composite materials due to their excellent mechanical properties. CNC composites are inevitably deformed by external forces during use. In order to reduce the damage of CNC composites in application and improve their service life, their mechanical properties and deformation mechanisms need to be studied in depth. In this study, cellulose Iβ crystals were selected for molecular dynamics simulation of nanoindentation to predict the transverse elastic modulus of cellulose crystals. The simulation results were compared with existing nanoindentation tests to verify the simulation accuracy. The impact of the loading depth, loading speed, and radius of the spherical indenter on elastic modulus and surface deformation of a system was explored. The results show that the loading depth and loading speed of the indenter have weak influence on the elastic modulus, and the small radius of the indenter (1 nm) will lead to the distortion of the elastic modulus. The compression deformation mechanism of cellulose crystal at the molecular level was analyzed by analyzing changes in cellulose crystal indentation morphology, molecular potential energy and hydrogen bond number during compression simulation. As the indenter radius and loading depth increase, the deformation degree of cellulose crystal also increases, whereas the loading speed within the 50–150 m/s range does not significantly influence on the deformation degree of cellulose crystal. These results may guide the production of nanocomposites with cellulose crystal reinforcements, which can be useful for preventing material damage and enhancing material efficiency.

Nanomechanical characterization of the hydrogen effect on high strength steels

The effect of hydrogen charging on nanohardness (HIT), elastic work (Welast), and plastic work (Wplastic) was investigated using nanoindentation. To accomplish this, two high-strength low-alloy steels (AISI 4130 M and AISI 4137 M) were tested using electrochemical nanoindentation with different loading rates (5 mN/min and 15 mN/min). Additionally, a continuous multicycle (CMC) indentation loading profile was proposed to assess the hydrogen embrittlement (HE) effect on AISI 4130 M steel. Under a current density of 0.5 mA/cm2 and 5 mN/min loading rate, AISI 4137 M steel exhibited a 30.6% reduction in Welast and a 9.8% increase in HIT value. In contrast, under the same conditions, Welast increased by 5.0% for AISI 4130 M steel. For the CMC indentation tests conducted on AISI 4130 M steel, both HIT and Welast values were affected, as evidenced by a reduction between 7% and 10% in HIT and between 4% and 22% in Welast. The lower deformation rates resulting from long-term CMC-type indentations enable hydrogen to follow the dislocation movement. In the experimental setup under study, the hydrogen-induced softening caused by the shielding of the elastic fields was observed as a reduction in nanohardness.

Mechanical properties of ZrB2 ceramics determined by two laboratories

AbstractThe mechanical properties for zirconium diboride (ZrB2) were measured at two laboratories and compared. Two billets of ZrB2 were prepared by hot‐pressing commercial powder. The relative densities of the billets were >99% and with an average grain size of 5.9 ± 4.5 µm. Both laboratories prepared American Society for Testing and Materials (ASTM) C1161 B‐bars for strength and ASTM C1421 bars with notch configuration A for fracture toughness. Specimens were machined by diamond grinding at the Army Research Laboratory (ARL) and electrical discharge machining (EDM) at Missouri S&T. Strength bars tested at Missouri S&T were polished to a .25 μm finish while the bars were tested as‐ground at ARL. Strengths were 473 ± 79 MPa for the Missouri S&T bars and 438 ± 68 for the ARL bars while the fracture toughness values were 3.9 ± .7 MPa•m1/2 for the Missouri S&T bars and 4.4 ± .6 MPa•m1/2 for the ARL bars. Vickers hardness was measured by both laboratories over a range of indentation loads. The resulting hardness values were on the low end of previously reported values and were quite different from each other especially at indentation loads ≤20N. The study demonstrated that the properties of materials tested to ASTM standards at different laboratories can be compared directly. In addition, strength and fracture toughness were nearly identical for bars prepared by conventional diamond grinding or EDM.

Achieving Diamond-Like Wear in Ta-Rich Metallic Glasses.

Most metals and alloys suffer from high friction and wear due to their low hardness and lack of self-lubrication. Although plenty of strategies have been proposed, it is still a long-standing challenge to achieve diamond-like wear in metals. Metallic glasses (MGs) are supposed to possess low coefficient of friction (COF) because of their high hardness and fast surface mobility. However, their wear rate is larger than that of diamond-like materials. Here, this work reports the discovery of Ta-rich MGs that exhibit diamond-like wear. This work develops an indentation approach for high-throughput characterization of crack resistance. By employing deep indentation loading, this work is able to efficiently identify the alloys that exhibit better plasticity and crack resistance according to the differences of indent morphology. With high temperature stability, high hardness, improved plasticity, and crack resistance, the discovered Ta-based MGs exhibit diamond-like tribological properties, featured by COF as low as ≈0.05 for diamond ball test and ≈0.15 for steel ball test, and specific wear rate of only ≈10-7 mm3 N- 1 m-1 . The discovery approach and the discovered MGs exemplifie the promise to substantially reduce friction and wear of metals and may unleash the potential of MGs in tribological applications.

Field strength effect on elastoplastic behavior of aluminoborosilicate glass: I. Elastic moduli and indentation size effect

AbstractThe modifier field strength (FS) is believed to play an important role in determining the elastic–plastic responses of aluminoborosilicate (ABS) glasses, but its effect is not well understood. Three novel alkali and three alkaline earth (AE) ABS compositions were created for this study which is the first part of two studies that explored the elastoplastic responses of these glasses. Six compositions were designed using different network modifiers (NWMs) to cover a range of cation FS. The glasses were also designed such that the concentrations of NWM and Al2O3 were similar, which maximized the three‐coordinated boron fraction in the network. It is well known that modifier FS can affect the coordination number (CN) of Al and B in an ABS glass structure, for example, a higher FS modifier can promote B3 → B4 and higher [Al5,6], but the degree of this depends on network former (NWF) ratios. Previous work used solid‐state NMR spectroscopic analysis on the current glasses to find that there was variation between [B4] and [Al4] between the two glass series (alkali vs. AE) but that was attributed to synthesis factors and no trend with FS was associated with the varying NWF CN. Further, 29Si ssNMR showed no evidence of NBOs which made sense based on composition. The conclusion, therefore, was that there was a far greater correlation with modifier FS for the increased mechanical and physical properties rather than the CN of Al and B. Part I of the current work focused on the elastic moduli, Poisson's ratio, the indentation size effect (ISE), and the bow‐in parameter. This part laid out the foundation to investigate the intersection of these elastoplastic properties with hardness and crack resistance as a function of NWM FS. Results showed that: (i) the Young's, bulk, and shear moduli increased with modifier FS, whereas Poisson's ratio did not trend with FS; (ii) the alkali glasses had a significantly higher magnitudes of ISE compared to the AE glasses; and (iii) the bow‐in parameter was load dependent and decreased with modifier FS at the highest indentation load.

Micromechanical characterization of microwave dielectric ceramic BaO–Sm 2O 3–5TiO 2 by indentation and scratch methods

Mechanical characterization of dielectric ceramics, which have drawn extensive attention in wireless communication, remains challenging. Micromechanical properties with microstructures of dielectric ceramic BaO-Sm₂O₃-5TiO₂ (BST) were assessed by nanoindentation, microhardness, and microscratch tests under different indenters, along with X-ray diffraction, scanning electron microscopy, and Raman spectroscopy. Accurate determination of elastic modulus (i.e., 260 GPa) and indentation hardness (i.e., 16.2 GPa) of brittle BST ceramic by instrumented indentation technique requires low loads with little indentation-induced damage. Elastic modulus and indentation hardness were analyzed by different methodologies such as energy-based and displacement-based approaches, and elastic recovery of Knoop imprint. Consistent values (about 3.1 MPa·m^1/2) of fracture toughness of BST ceramic were obtained by different methods such as Vickers indenter-induced cracking method, energy-based nanoindentation approaches, and linear elastic fracture mechanics-based scratch approach with a spherical indenter, demonstrating successful applications of indentation and scratch methods in characterizing fracture properties of brittle solids. The deterioration of elastic modulus or indentation hardness with the increase in indentation load is caused by indentation-induced damage, and can be used to determine fracture toughness of material by energy-based nanoindentation approaches, and the critical void volume fraction is 0.27 (or 0.18) if elastic modulus (or indentation hardness) of the brittle BST ceramic is used. The fracture work at the critical load corresponding to the initial decrease in elastic modulus or indentation hardness can also be used to assess fracture toughness of brittle solids, opening new venues of application of nanoindentation test as a means to characterize fracture toughness of brittle ceramics.

Damage characterisation of amine-functionalized MWCNT reinforced carbon/epoxy composites under indentation loading

Damage resistance of carbon fibre reinforced composites is crucial parameter to be considered at both primary selection and in-service maintenance stages. High stiffness of carbon-epoxy system and stacked configuration make it susceptible to impact induced brittle damages. Delamination is one such life-limiting damage modes that can severely inhibit load carrying capacity of laminates. Hence to improve the damage resistance of composites demands tailoring tough microstructure and altering brittle damage modes with ductile ones. This work attempted to design damage resistant carbon composites through modification of epoxy matrix with amine functionalized multi walled carbon nanotubes (NH2-MWCNT). Laminates with different nanotube concentrations (0.3, 0.6, 0.9 and 1.2 wt. %) were fabricated to investigate its influence on load bearing capacity, toughness and damage resistance of fibre reinforced nanocomposites. Generally, peak force, displacement and toughness until maximum force improved at all nanoparticle concentrations. Matrix cracking was suppressed owing to optimum cross-linking of nanoparticle with matrix. External damage area increased with nanoparticle concentration although delaminated area, mapped through c-scan, was suppressed up to 27.73%. A combination of higher degree of interfacial interactions, induced by nanoparticles, and energy absorbing microstructure was concluded to be behind improvement in the damage resistance of composites.

Damage evolution of the coating/substrate system TiN/X2CrNi18-9 under high indentation loads: experimental and numerical study

Hard coatings, in particular TiN, are widely used as coatings for cutting tools and in the agri-food industry. In the literature, however, few characterizations of hard coatings can be found which define the minimum applied load when the coating starts to fail. In the present study TiN coating was deposited on stainless steel X2CrNi18-9. Vickers and Brinell indentation tests with a wide load range were performed. The main results revealed that the increase of the applied load in Vickers and Brinell indentation influenced the coating and coating/substrate damage evolution. SEM investigation of the Vickers indentation area shows five modes of damage: inclined cracks, radial cracks, lateral cracks, edge cracks, and shear steps. Each damage mode occurs at a specific load range. Parallel cracks already appeared at the edges of the indents at the lowest load of 2 N. For Brinell indentation, cracks start in the coating only at loads higher than F = 307 N. The SEM examinations present damage modes such as circumferential cracks in the border and additional circular cracks in the center of the indent, creating a crack network. Numerical simulations of Brinell indentation were carried out in order to determine the stress distribution in the indent. The comparison of the numerical simulation results with the experimental findings revealed that the coating started to fail at a stress range above 5735 MPa which corresponds to a normal load range of higher than 307 N in Brinell indentation tests. At a load of 613 N cracks were observed.

Study on parameter measurement of elastic thin films with substrate effect based on contact mechanics

The development of contact theories for films is valuable for high-precision machining and measurement. Accordingly, accurate contact mechanics theories for films under common indenter loading are established based on the constitutive equations and potential theory method, Here, the film structures are “film with two free boundaries” and “film/rigid substrate.” The displacement and stress at arbitrary positions can be obtained using exact solutions represented by elementary functions. The application of the foregoing theories to resolve the contact problem encountered in films is convenient for engineers. The influences of substrate and film parameters on contact radius and indentation depth were analyzed. Indentation curves are slightly influenced by Poisson’s ratio, but primarily affected by the elastic modulus. An effective method for measuring of film material parameters is presented. Using numerical software platforms, the method’s analytic algorithm can be transformed into numerical codes and be integrated into special software.

Development of a Detailed 3D Finite Element Model for a Lithium-Ion Battery Subject to Abuse Loading

<div class="section abstract"><div class="htmlview paragraph">Lithium-ion batteries (LIBs) have been used as the main power source for Electric vehicles (EVs) in recent years. The mechanical behavior of LIBs subject to crush loading is crucial in assessing and improving the impact safety of battery systems and EVs. In this work, a detailed 3D finite element model for a commercial vehicle battery was built, in order to better understand battery failure behavior under various loading conditions. The model included the major components of a prismatic battery jellyroll, i.e., cathodes, anodes, and separators. The models for these components were validated against the corresponding material coupon tests (e.g., tension and compression). Then the components were integrated into the cell level model for simulation of jellyroll loading and damage behavior under three types of compressive indenter loading: (1) Flat-end punch, (2) Hemispherical punch and (3) Round-edge wedge. The comparisons showed reasonable agreement between modeling and experiments. With the validated numerical model, parametric studies were further performed to analyze the effect of separator anisotropy, to highlight its important role in the overall structural response of LIBs.</div></div>

Nanoindentation test of a DLC coated high-speed steel substrate using a two-dimensional axisymmetric finite element method

Diamond-Like Carbon is a thin coating that has high hardness, chemical inertia and low coefficient of friction characteristics that can improve the performance of cutting tools and mechanical components. To identify the characteristics of this film it is important to carry out mechanical tests, such as nanoindentation, which can provide important information in understanding and improving coating techniques and, when combined with the finite element method, leads to more detailed results of material behavior during testing. Considering this, the present work aimed to study and apply a nanoindentation test simulation of a Diamond-Like Carbon coated high-speed steel specimen. The results of the indenter penetration versus load were compared with experimental data seeking correlation of the model. A great proximity was observed between the experimental results and the results obtained in the simulation. When comparing the final indentation force, a proximity of 93.9 % and 95.5 % was found for the model with only one indentation and with partial indentations, respectively, thus validating the created model. The impact of modifying the indenter tip radius for the partial indentation method was also evaluated. The tip radius variation did not have a significant impact on the results of final indentation force and material hardness. The material hardness was also obtained in the simulation, which was equal to the experimental: 9.25 GPa.

An analytical model of Brinell hardness for power-law hardening materials

In the Brinell hardness test a hard spherical indenter is pressed into a metallic material. Due to the inhomogeneous distribution of stresses and strains the hardness measure is not a unique function of the material properties alone, but also depends on the setup geometry notably the diameter of the indenter. To this date there exists no analytic description of these interrelations. Therefore, it seems desirable to derive an analytical model which directly relates the material properties to the hardness value. Based on extensive numerical simulations analytic formulas are derived describing the dependence of Brinell hardness on the applied indentation load. The formulas are capable to represent the entire plastic range relevant for hardness measurements. Based on these results the indentation depths proposed in ISO 6506 can be directly related to a range of force-diameter indices, which serve as a means for selecting the appropriate Brinell test.

Ocular Deformations in Spaceflight-Associated Neuro-Ocular Syndrome and Idiopathic Intracranial Hypertension.

Spaceflight-associated neuro-ocular syndrome (SANS) shares several clinical features with idiopathic intracranial-hypertension (IIH), namely disc edema, globe-flattening, hyperopia, and choroidal folds. Globe-flattening is caused by increased intracranial pressure (ICP) in IIH, but the cause in SANS is uncertain. If increased ICP alone causes SANS, then the ocular deformations should be similar to IIH; if not, alternative mechanisms would be implicated. Using optical coherence tomography (OCT) axial images of the optic nerve head, we compared "pre to post" ocular deformations in 22 patients with IIH to 25 crewmembers with SANS. We used two metrics to assess ocular deformations: displacements of Bruch's membrane opening (BMO-displacements) and Geometric Morphometrics to analyze peripapillary shape changes of Bruch's membrane layer (BML-shape). We found a large disparity in the mean retinal nerve-fiber layer thickness between SANS (108um; 95% confidence interval [CI]=105-111um) and IIH (300um; 95% CI=251-350.1um). The pattern of BML-shape and BMO-displacements in SANS were significantly different from IIH (P < 0.0001). Deformations in IIH were large and preponderantly anterior, whereas the deformations in SANS were small and bidirectional. The degree of disc edema did not explain the differences in ocular deformations. This study showed substantial differences in the degree of disc edema and the pattern of ocular deformations between IIH and SANS. The precise cause for these differences is unknown but suggests that there may be fundamental differences in the underlying biomechanics of each consistent with the prevailing hypothesis that SANS is consequent to multiple factors beyond ICP alone. We propose a hypothetical model to explain the differences between IIH and SANS based on the pattern of indentation loads.

Effect of poling direction on microhardness and indentation fracture toughness in PMN-0.28PT and PMN-0.32PT single crystals

The relaxor ferroelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT) has been considered to be a potential choice for high performance actuators and sensors owing to their ultrahigh electromechanical properties. Their devices are often subjected to contact loading which may nucleate surface cracks leading to catastrophic failure. Therefore, in order to understand the deformation and fracture behavior, Vickers micro-indentation experiments are performed on [001] oriented poled PMN-0.28PT and PMN-0.32PT single crystals along parallel and perpendicular to the poling direction. Results show that the hardness, H is primarily governed by the orientation of domains with respect to the loading direction. The finite element simulations demonstrated that H increases with rise in piezoelectric coupling coefficient, d33. Further, the length of crack parallel to poling direction is found to be larger than that of perpendicular to poling direction when indentation load is perpendicular to the poling direction. This behavior is in contrast to the trend reported for Pb(ZrxTi1−x)O3 (PZT), BaTiO3, and lead-free piezoelectric materials in the past. The Switching spectroscopy piezoresponse force microscopy (SS-PFM) performed along the cracks shows different patterns of d33* for crack parallel and perpendicular to the poling direction. Motivated from the observed trend in d33*, the mechanism of crack propagation is presented and the reasons for anisotropy in KIC is also explained.

Buckling of an imperfect spherical shell subjected to external pressure

This paper presents numerical results focusing on the buckling behaviour of externally pressurised imperfect spherical shells. The numerical model followed the recommendations of the European Standard EN1993-1-6. A good agreement between the model FE and the experimental results with an average difference of less than 7% was found. The perfect steel spherical shell is then subjected to several imperfection approaches, such as (a) eigenmode imperfection and (b) single load indentation (SLI) imperfection. The eigenmode shape imperfection proves to be the worst imperfection for the externally pressurised spherical shells, yet the SLI is an attractive technique to simulate the realistic imperfection (i.e., dent/dimple). Based on the recommendation of the European Standard EN1993-1-6 for imperfection tolerance, the SLI imperfection was used for the realistic-case imperfection approach for different spherical shell configurations. A lower-bound closed-form empirical formula for a spherical shell under external pressure was proposed for the spherical shell with the single load indentation (SLI). The results show that for a shell shape parameter λ ≥ 10, the EBC provides a much higher knockdown factor compared to the guidelines of NASA SP -8032, Wagner, and lower-bound closed-form empirical formula (the differences are in the range of 48%–56%).