A reinforced mixed-mode bending apparatus (RMMBA) is provided. The RMMBA is a new fixture design for testing fracture and characterizing delamination in layered materials under combined mode I (pulling/tension) and mode II (in-plane shear) loads. The purpose of this design is to provide a much less compliant fixture for conducting the Mixed Mode Bending (MMB) test. Embodiments described herein improve the accuracy of the MMB test and reduce the complexity of post-processing the collected data.

TECHNICAL FIELD

The present disclosure relates generally to fixtures for testing fracture and characterizing delamination in layered materials and more specifically to a reinforced mixed-mode bending apparatus.

BACKGROUND

Delamination is a primary failure mode for layered materials, such as composites and nanocomposites. In practice, delamination cracks initiate and propagate under a mixture of mode I (pulling/tension) and mode II (in-plane shear) load conditions, leading to separation of the layers of the material. In the past 40 years, several mixed-mode characterization techniques have been developed to determine a material's characteristics, such as the single-leg four-point bend (SLFPB), prestressed end-notched flexure (PENF), cracked lap shear (CLS), edge delamination tension (EDT), Arcan, asymmetric double cantilever beam, mixed-mode flexure, variable mixed-mode, and mixed-mode bending (MMB) test. The fixture complexity, the inconsistency of the mixed-mode ratio vs. crack length, and the complexity of the post-processing data are the main drawbacks of some of these techniques.

MMB is a pulling-mode load added to a mid-span end notched flexure (ENF) specimen. The MMB test has been recognized as a standard mixed-mode characterization technique due to clear advantages: (i) the use of simple beam theory equations of double cantilever beam (DCB) and ENF tests for analyzing data; (ii) the stable delamination growth; (iii) the applicability of the technique to a wide range of mode I/II ratios; and (iv) the consistency of the mixed-mode ratio during crack growth.

MMB was initially designed to characterize unidirectional composites' static toughness; however, researchers have used it to characterize fatigue fracture and adhesive toughness. MMB has been used to characterize glass/epoxy composite, carbon fiber/PEEK, and stitch-bonded composites. The lever rotation in MMB's original design caused a significant error in toughness calculation and changed the mixed-mode ratio.

FIG.1is a schematic diagram of the MMB fixture according to the current standard (ASTM 6671M-19). The MMB fixture contains eight main components: yoke (Y), saddle (S), lever (L), top roller (TR), a top hinge (TH), bottom roller (BR), a bottom hinge (BH), and base (B). The top and bottom rollers consist of three subcomponents: roller holder, roller, and bearing. The top and bottom hinges have two sub-components: hinge clamp and hinge. The MMB fixture also has a top connecting rod (TCR) to attach the yoke to the testing machine, and it is supported underneath the thick steel base by a base slider (BS) in a slot of the base rather than being clamped to a flat surface.

The bearing-mounted roller applies the midspan load and the left support reaction to the split-beam specimen to minimize the frictional forces. Aluminum hinges were bonded at the right end of the split-beam specimen arms to transfer the pulling forces. The mode I to mode II load ratio is changed by changing the distance between the saddle and the top-roller (length c).

In the MMB test, load and load-point displacements are recorded. The load-point displacement is usually determined from the crosshead position, including the compliance of the loading system and testing fixture. Scientists have emphasized the importance of the compliance of the fixture's internal components for different testing protocols.

The displacement measurements should be corrected for the compliance of the load frame and the MMB fixture. One approach to measuring the loading system's compliance (Csys) is by measuring the stiffness of a bar of known stiffness (e.g., steel) and the slope of the load vs. load-point displacement curve (mbar) as in Equation 1.

where Ebarand bbar, are the elastic modulus and width of the bar, t is total thickness, L is the length, and c is the horizontal distance between the yoke and the top roller. The system compliance depends on c, and the compliance should be calculated for each mixed-mode ratio. The data needs to be corrected to account for the compliance, making post-processing of the data relatively complicated.

SUMMARY

A reinforced mixed-mode bending apparatus (RMMBA) is provided. Embodiments of the RMMBA provide new fixture designs for testing fracture and characterizing delamination in layered materials under combined mode I (pulling/tension) and mode II (in-plane shear) loads. The purpose of these designs is to provide a much less compliant fixture for conducting the Mixed Mode Bending (MMB) test. Embodiments described herein improve the accuracy of the MMB test and reduce the complexity of post-processing the collected data.

An exemplary embodiment provides a system for mixed mode characterization of a specimen. The system includes a lever; a saddle at least partially surrounding and laterally fixed to the lever; a yoke over the saddle and configured to carry an applied load to a specimen through axial force; and a top roller holder coupled below the lever and holding a top roller which transfers the applied load to a top surface of the specimen.

Another exemplary embodiment provides an RMMBA. The RMMBA includes a lever; a saddle at least partially surrounding the lever; a yoke over the saddle and configured to carry an applied load to a specimen, wherein the yoke comprises a top member connected with two angled yoke members; and a top roller holder coupled below the lever and holding a top roller which transfers the applied load to a top surface of the specimen.

In some embodiments, a reinforced mixed-mode bending apparatus (RMMBA) for testing fracture and characterizing delamination in layered materials includes a lever, a saddle disposed above a top surface of the lever and adjacent to two lateral sides of the lever (the saddle laterally fixed to the lever), a top roller holder disposed below and coupled to the lever (the top roller holder holding a top roller), and a yoke disposed over the saddle and configured to transfer an applied load to the top roller via the saddle, the lever, and the top roller holder. In some embodiments, the top roller is configured to transfer the applied load to a specimen being tested with the RMMBA.

In some embodiments, the saddle includes saddle legs. In some embodiments, each saddle leg is disposed adjacent to one of the two lateral sides of the lever. In some embodiments, the RMMBA also includes a plurality of fasteners. In some embodiments, each fastener extends from one of the saddle legs to the lever to laterally fix the saddle to the lever. In some embodiments, each fastener includes a bolt extending through one of the saddle legs and threaded into the lever.

In some embodiments, the top roller holder includes a support pillar in contact with and reinforcing the top roller. In some embodiments, the support pillar reduces deformation of the top roller under the applied load. In some embodiments, the yoke includes a top member disposed between angled yoke arms.

In some embodiments, the RMMBA also includes a bottom roller holder coupled to a base and holding a bottom roller that is configured to support a bottom surface of the specimen, a top hinge disposed below and coupled to an end of the lever, and a bottom hinge coupled to the base. In some embodiments, the top hinge and the bottom hinge accommodate the specimen therebetween.

In some embodiments, a reinforced mixed-mode bending apparatus (RMMBA) for testing fracture and characterizing delamination in layered materials includes a lever, a saddle disposed above a top surface of the lever and adjacent to two lateral sides of the lever, a top roller holder disposed below and coupled to the lever, and a yoke disposed over the saddle and configured to transfer an applied load to the top roller via the saddle, the lever, and the top roller holder. In some embodiments, the top roller holder includes a middle section and two roller arms extending from the middle section. In some embodiments, the top roller holder holds a top roller. In some embodiments, the top roller holder includes a support pillar disposed between the two roller arms. In some embodiments, the support pillar is in contact with and reinforces the top roller. In some embodiments, the top roller is configured to transfer the applied load to a specimen being tested with the RMMBA.

In some embodiments, the support pillar reduces deformation of the top roller under the applied load. In some embodiments, the middle section and the two roller arms of the top roller holder form a U shape. In some embodiments, the support pillar extends from the middle section to contact the top roller. In some embodiments, an end of the support pillar which contacts the top roller is rounded to accommodate the top roller. In some embodiments, the saddle is laterally fixed to the lever. In some embodiments, the yoke includes a top member disposed between angled yoke arms.

In some embodiments, a reinforced mixed-mode bending apparatus (RMMBA) for testing fracture and characterizing delamination in layered material includes a lever, a saddle disposed above a top surface of the lever and adjacent to two lateral sides of the lever, a top roller holder disposed below and coupled to the lever (the top roller holder holding a top roller), and a yoke disposed over the saddle. In some embodiments, the yoke includes a top member disposed between angled yoke arms. In some embodiments, the angled yoke arms are angled to reinforce the yoke. In some embodiments, the yoke is configured to transfer an applied load to a top roller via the saddle, the lever, and the top roller holder. In some embodiments, the top roller is configured to transfer the applied load to a specimen being tested with the RMMBA.

In some embodiments, each of the angled yoke arms include a low portion perpendicular to the top member and an angled upper portion. In some embodiments, the top roller holder includes a support pillar in contact with and reinforcing the top roller. In some embodiments, the saddle is laterally fixed to the lever. In some embodiments, the lever is an I-beam lever having a bottom flange which is thicker than a top flange.

DETAILED DESCRIPTION

In the following detailed description, reference is made to the accompanying drawings which form a part hereof, and which are shown by way of illustration embodiments that may be practiced. The embodiments set forth below represent the necessary information to enable those skilled in the art to practice the embodiments and illustrate the best mode of practicing the embodiments. Upon reading the following description in light of the accompanying drawing figures, those skilled in the art will understand the concepts of the disclosure and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure and the accompanying claims.

A reinforced mixed-mode bending apparatus (RMMBA) is provided. The RMMBA is a new fixture design for testing fracture and characterizing delamination in layered materials under combined mode I (pulling/tension) and mode II (in-plane shear) loads. The purpose of this design is to provide a much less compliant fixture for conducting the Mixed Mode Bending (MMB) test. Embodiments described herein improve the accuracy of the MMB test and reduce the complexity of post-processing the collected data.

The MMB test is commonly used to investigate the fracture toughness of layered materials such as carbon fiber/epoxy composites, glass/epoxy composites, and stitch-bonded composites. An analytical model was developed herein to be used along with a numerical based optimization technique to improve the traditional MMB fixture, resulting in the RMMBA. The analytical model provides an understanding of the complex interactions between the components of the fixture. The change in the geometry, load transfer mechanism, and boundary conditions in RMMBA compared to the initial fixture reduces the fixture's compliance and enhances the accuracy of the mixed-mode tests significantly. The accurate fracture characterization of materials improves safety and security and saves money for industries such as space technology, aerospace, defense, energy, transportation, and public health.

A. Material Properties and Modeling

The traditional MMB fixture ofFIG.1was modeled using SolidWorks2019according to the dimensions per ASTM standard 6671M-19. All holes were left unthreaded, and the bearings were modeled as simple cylinders to simplify the model. These changes have a negligible effect on the stiffness calculation of the fixture. The saddle, lever, and roller holders are made from aluminum6061with a yield strength of 276 megapascals (MPa) and elastic modulus of 69 gigapascals (GPa).

The split-beam specimen is modeled as a rigid body. All other components were modeled as low carbon steel1018having a yield strength of 370 MPa and an elastic modulus of 200 GPa. The split-beam specimen contains a typical pre-crack that helps load transfer to the MMB's lower assembly in the FEA model. The linear elastic 3D models were built in ANSYS version 2019R3 simulation software. A quadratic hexahedral-dominated 3D solid element was chosen for higher accuracy. Mesh convergence studies were performed for each component, and the most efficient element size of 1 millimeter (mm) was used for all subsequent simulations. Bonded contacts and weak springs were implemented for all models to avoid under-constrained boundary conditions.

B. Stiffness of MMB Components

The maximum applied force to the traditional MMB fixture is limited to the maximum load-carrying capacity of the fixture's weakest component to avoid excessive local stress within the fixture. The components' free-body diagrams are illustrated inFIGS.2A-2Jbased on the fixture's load transfer mechanism.

FIG.2Ais a free body diagram of the top connecting rod of the traditional MMB fixture ofFIG.1.FIG.2Bis a free body diagram of the yoke of the traditional MMB fixture ofFIG.1.FIG.2Cis a free body diagram of the saddle of the traditional MMB fixture ofFIG.1.FIG.2Dis a free body diagram of the top hinge of the traditional MMB fixture ofFIG.1.FIG.2Eis a free body diagram of the bottom hinge of the traditional MMB fixture ofFIG.1.FIG.2Fis a free body diagram of the top roller of the traditional MMB fixture ofFIG.1.FIG.2Gis a free body diagram of the bottom roller of the traditional MMB fixture ofFIG.1.FIG.2His a free body diagram of the base slider of the traditional MMB fixture ofFIG.1.FIG.2Iis a free body diagram of the lever of the traditional MMB fixture ofFIG.1.FIG.2Jis a free body diagram of the base of the traditional MMB fixture ofFIG.1.FIG.2Kis a legend for the symbols used inFIGS.2A-2J.

The Distortion Energy (DE) theory was considered to calculate the maximum load-carrying capacity of every component. For the saddle, the critical point is at the uppermost fillets, where a stress concentration factor of 2 is presumed based on the fillet radius ratio to the height of the cross-section. The maximum effective stress at the critical point determines the saddle's load-carrying capacity as 4689 newtons (N).

For the lever, the load, shear, and moment diagrams vary with the parameter c. The expression for the maximum moment remains unchanged for c<65 mm and c>65 mm, (Pc(c+16.7))/(c+50). The values of c for the extremum of the moment expression are not within the permissible range between 0 and 108.9 mm. The load-carrying capacity for the maximum moment of 86.08 pascals (P) at c=108.9 mm is 16474 N.

Table 1 shows the load-carrying capacity for all the components. A loading force of 1000 N was chosen based on Table 1 to maintain a minimum design factor of —2for all of the MMB fixture. The components were modeled separately, considering the free body diagrams shown inFIGS.2A-2J. The largest deformation area was investigated, and the equivalent stiffness for each component was calculated, as shown in Table 2. The vertical deformation was recorded in the applied force line of action for the lever and base, allowing them to be modeled as linear springs with the other components.

C. Stiffness of MMB Fixture

FIG.3is a schematic diagram of an analytical spring model for the traditional MMB fixture ofFIG.1. The analytical model was developed to determine the relationship between the components' stiffnesses and the MMB fixture's stiffness. The model assumes each component as a linear spring with a spring constant corresponding to the values shown in Table 2. The springs are combined in series except for the two hinges and two rollers that are not in series with the other components.

FIG.4Ais a schematic diagram of the analytical spring model of top and bottom rollers and hinges before deformation.FIG.4Bis a schematic diagram of the analytical spring model of the top and bottom rollers and hinges after deformation.FIGS.4A and4Billustrate how the springs of these four components are situated in the model. a and b represent the undeformed distances between the rigid specimen, base, and the lever. The equivalent stiffness of the top and bottom rollers and hinges is equal to the ratio of the applied load, P, to the displacement, δeq, in the force's line of action.

Similarly, the displacement of each member is related to its internal force and stiffness:

where K and δ are stiffness and deformation, and the subscripts (TR, TH), (BR, BH) represents top roller and hinge, and bottom roller and hinge, respectively.

The model's total displacement inFIG.4Bis related to the individual components' displacements following the deformation compatibility.

where h is the average of two distances.

By combining Equations 2 and 3, one can find an expression for the equivalent stiffness of the subsystem (Keq) inFIG.4B, which depends on the parameter c.

The equivalent spring of the subsystem is combined in series with the other components giving the following expression for MMB stiffness:

The compliance of the fixture depends on the parameter c. As c is increased for higher mode I applications, Keqdecreases, reducing the fixture's stiffness. Using Equation 6 and Table 2 shows that the MMB fixture's stiffness is almost 3.95 KN/mm for a c value of 50 mm.

A finite element model of the MMB fixture was used to verify the analytical prediction of fixture stiffness. The MMB was simulated with a c value of 50 mm using the same loading force (1000 N) and the mesh quality as the components. The measured stiffness was 4.46 KN/mm and 12.9% higher than the analytical model. This discrepancy could be due to the bonded contacts in the fixture model and extra fixity due to the 3D stiffness properties in the finite element model, creating more fixity than those present in the individual component simulations. The analytical model, Equations 5 and 6, and Table 2 show that the MMB stiffness will be particularly sensitive to changes in the most compliant components: saddle and yoke.

FIG.5is a graphical representation of the effects of top roller, top hinge, bottom roller, and bottom hinge on their combined equivalent stiffness (Keq). Equation 5 indicates that the top roller has a disproportionate contribution to the fixture's compliance because it bears a higher load than all other subsystem components, especially at higher c values, as shown inFIG.5.

FIG.6is a graphical representation of the effects of change in components' stiffness on the fixture stiffness. Percentage changes in each component's stiffness on the normalized MMB stiffness are presented for c=50 mm using the analytical model. Based on this analysis, the saddle, yoke, and top roller were chosen to be redesigned.

II. Design Improvements

Embodiments of the RMMBA were invented for energy measurement of fracture specimens. An embodiment of the RMMBA consists of several components, as depicted inFIGS.7A-20B. Compliance in a test fixture introduces error into the fracture energy measurement, and the current complex data post-processing correction technique requires multiple assessments of the fixture's compliance. The RMMBA avoids all of the corrections and results in a more accurate characterization. The RMMBA described herein was designed using the analytical model and finite-element based optimization described above. The loading mechanism of the RMMBA was designed as a roller and support pillar with an optimized stiffness to weight ratio to add enough rigidity and reduce or eliminate the errors in the fracture energy measurement.

Three sources of improvements were identified to design embodiments of the RMMBA. One is the saddle, in which deformation is mostly due to inward rotation of its legs due to the moment created by the applied force. Secondly, the top roller's maximum displacement occurs in the center of the steel roller, where the deflection under the applied load is largest. Third, it is the yoke's right angle that causes higher rotation due to relatively larger moments.

In embodiments of the RMMBA, the inward rotation of the saddle was countered by designing a reinforcing mechanism. Dimensions of the saddle and top toller were optimized to increase the fixture's stiffness. The saddle was further designed to accommodate a lateral mechanism that increases the vertical rigidity of the apparatus. The yoke was designed to transfer the load to the saddle by combining axial and flexural components rather than all flexural to reduce deformation and enhance rigidity.

Embodiments of the RMMBA leverage all the benefits of traditional MMB techniques while increasing the accuracy significantly and improving the fracture experiment. An exemplary embodiment of the RMMBA has shown an 87% increase in stiffness over the current techniques. This enhanced stiffness improves the experiment's quality and reduces the complexity of post-processing calculations, leading to a safer crack-resistant design. The increase in the accuracy also allows for materials with a more extensive range of elastic modulus to be tested. Stiffer laminated composites that were previously beyond the current testing techniques can now be characterized with reliability.

FIG.7Ais an isometric view of an RMMBA10according to embodiments described herein.FIG.7Bis a front view of the RMMBA10ofFIG.7A. Similar to the traditional MMB fixture, the RMMBA10includes a top connecting rod12, a yoke14, a saddle16, a lever18, a top roller holder20, a top roller22, a bottom roller holder24, a bottom roller26, a top hinge28, a bottom hinge30, a base32, and a base slider34. A test specimen36is placed between the top roller22and the bottom roller26, as well as between the top hinge28and the bottom hinge30. Each of the bottom hinge30and the top hinge28includes a hinge clamp38, a hinge pin40, and a hinge tab42which contacts the test specimen36.

In an exemplary aspect, the bottom roller holder24and bottom hinge30(at the hinge clamp38) are attached to the base32. The span length2L can be set to a desired value, where span length is the lateral distance between the center of the bottom roller26and the center of the hinge pin40of the bottom hinge30. Generally, an axis of the bottom roller26is aligned parallel to an axis of the hinge pin40of the bottom hinge30.

The top roller holder20and top hinge28(at the hinge clamp38) are attached to the lever18such that the lateral distance between the center of the hinge pin40of the top hinge28and the center of the top roller22is half the span length, or L. This attachment is generally made such that an axis of the top roller22is parallel to an axis of the hinge pin40of the top hinge28and that both are perpendicular to a longitudinal axis of the lever18. In addition, the hinge pin40of the top hinge28is generally vertically aligned with the hinge pin40of the bottom hinge30.

The saddle16includes saddle legs44which extend past the lever18and a saddle bearing46attached to each saddle leg44. In some embodiments, each saddle leg44is disposed adjacent to a lateral side of lever18. The saddle16is attached to the lever18such that the saddle16surrounds the lever18on three sides. For example, as shown inFIGS.7A and7B, saddle16is disposed above a top surface of lever18and adjacent to two lateral sides of lever18. Saddle16may be attached to lever18at a location that results in a length along a line of the lever18between the top roller22and a center line of each saddle bearing46that equals the desired lever18length c. The center line of the saddle bearings46and the center line of the top roller22should also be parallel. This can be accomplished by making sure that they are both perpendicular to the length of the lever18.

The test specimen36is attached to the base32by holding the test specimen36flush against the bottom roller26while tightening the hinge tab42in the bottom hinge30. The hinge tab42should be inserted into the bottom hinge30far enough so that the longitudinal axis of the test specimen36is parallel to the top plane of the base32.

Next, the lever18is attached by holding the top roller22flush to the test specimen36while tightening the hinge tab42in the top hinge28. The hinge tab42should be inserted far enough into the top hinge28so that the lower plane of the lever18is parallel with the longitudinal axis of the test specimen36. The loading yoke14is placed over the saddle16until it contacts the saddle bearings46. The RMMBA10is placed in a load frame (not shown) clamping the base32firmly to a bottom plate of the machine such that the axis of the saddle bearings46is parallel to the axis of the loading yoke14.

As described above, the RMMBA10is used to load split laminate test specimens36to determine the delamination fracture toughness at various ratios of Mode I to Mode II loading. The composite test specimen36can consist of a rectangular, uniform thickness, unidirectional laminated composite specimen, containing a nonadhesive insert at the midplane which serves as a delamination initiator. Loading forces are applied to the test specimen36via the hinge tabs42of the top hinge28and bottom hinge30that are applied near the ends of the delaminated section of the test specimen36and through the rollers22,26that bear against the test specimen36in the nondelaminated region.

The base32of the RMMBA10holds the test specimen36stationary while the lever18loads the test specimen36. The base32attaches to the bottom hinge30and also bears on the test specimen36near the far end with the bottom roller26. The lever18attaches to the top hinge28and bears down on the test specimen36halfway between the bottom roller26and the hinge tabs42. The top roller22acts as a fulcrum so by pushing down on the lever18opposite the top hinge28, the top hinge28is pulled up.

The length of the lever18, c, can be changed to vary the ratio of the load pulling on the top hinge28to the load bearing through the top roller20, thus changing the mode mixture of the test. The load is generally applied to the lever18such that the load remains vertical during the loading process. To reduce geometric nonlinear effects as a result of lever18rotation, the lever18is generally loaded such that the height of loading is slightly above the pivot point where the lever18attaches to the test specimen36.

A record of the applied load versus opening displacement can be recorded on an x-y recorder, or equivalent real-time plotting device or stored digitally. The interlaminar fracture toughness and mode mixture are calculated from critical loads read from the load displacement curve.

FIG.8Ais an isometric view of the saddle16and lever18of the RMMBA10ofFIG.7A.FIG.8Bis a front view of the saddle16and lever18ofFIG.8A.

With reference toFIGS.7A,7B,8A, and8B, embodiments of the RMMBA10include one or more of the following improvements. First, in some embodiments the saddle16is laterally fixed to the lever18to reduce bending of the saddle legs44. For example, bolts48or other fasteners may be inserted through the saddle legs44and come in contact with or thread into the lever18. Second, in some embodiments the top roller holder20includes a support pillar50to reinforce the top roller22(e.g., reduce its deflection). Third, in some embodiments the yoke14includes a top member52(e.g., where the top connecting rod12attaches) between angled yoke arms54, where the angled yoke arms54are angled to reinforce the yoke14.

These improvements are further described below at Sections II.A and II.B with reference toFIGS.7A-14B. The components of the RMMBA10are further illustrated, along with example dimensions (in mm), inFIGS.9A-14B. It should be understood that the example dimensions shown are for an optimized embodiment of the RMMBA10, and the dimensions of other embodiments may vary according to design and/or manufacturing differences. A standard machining tolerance of ±0.13 mm is acceptable for all of the components.

A. Optimized Dimensions of the Saddle and Top Roller

The analytical model described above proved the upper part of the traditional MMB fixture, above the specimen, has a significant role in the fixture's compliance. The top roller22is one of the crucial components for rigidity, makes contact with the top of the test specimen36, and it is centered between the bottom hinge30and bottom roller26. The diameter of the top roller22should be larger than the bottom roller26since it is exposed to a larger load. The top roller holder20includes the support pillar50to reduce its deflection. The saddle legs44that extend downwards on either side of the lever18rotate and deform significantly.

In a first aspect, the stiffness of the RMMBA10is increased by altering critical dimensions of the saddle16and top roller22using an adaptive multi-objective optimization algorithm. The deformation and weight were minimized while ensuring the Von Mises stress never exceeds 120% of its original value, and it is less than the yield strength. The algorithm varies dimensional parameters of the finite-element models to find an optimal solution.

The width and thickness of the saddle legs44were selected as the saddle16optimization parameters to resist the inward bending moment due to the loading force of the yoke14. The revised dimension of the saddle16adds additional stiffness and reduces the deformation. The optimization algorithm confirms that a slight increase in the primary diameter of the top roller22reduces deflection in the steel roller. The converged solution for the two components resulted in a ˜34% increase in the stiffness of the saddle16and a 54% increase in the stiffness of the top roller22; however, to maintain clearance between the yoke14and the new saddle16, the span of the yoke14needed to be widened. This increased span length creates a larger moment arm on the yoke14leading to a reduction in stiffness. An additional optimization was utilized to mitigate this effect by angling the angled yoke arms54to carry more axial stress and experience less rotation from bending. According to the analytical model, these three new components' combined effect should result in a 27% increase in fixture stiffness. A simulation based on these optimizations resulted in a higher stiffness (5.32 KN/mm) but only differed from the predicted value (5 KN/mm) by 6.4%.

The detailed dimensions of these components are shown in the figures: the yoke14(FIGS.10A and10B), the top roller holder20(FIGS.13A-13D), and the top roller22(FIGS.14A and14B).

B. Lateral Reinforcement of the Saddle and Further Improvements

In a second aspect, a reinforcing mechanism was designed to pass through the saddle legs44and screw into either side of the lever18to provide extra rigidity to the saddle16, as shown inFIGS.8A and8B. Maintaining the system's symmetry during the fracture experiment was pondered by securing equal tightening with equal torque. The yoke14fits over the top of the saddle16and rests on two saddle bearings46. The geometry of the yoke14was designed to carry a significant part of the applied load through axial force rather than bending moment to reduce the deformation and enhance the rigidity of the upper part of RMMBA10.

The base slider34and top connecting rod12are attached to two ends of a testing machine, which applies a known vertical load and measures load-point displacement. The base32and base slider34have high rigidity, and their weight has no detrimental effect on the accurate measurement of fracture properties. When a vertical load is applied, the top roller22pushes down on the middle of the test specimen36and acts as a fulcrum for the lever18while the top hinge28pulls upward on the cracked end of the test specimen36. By recording the force and the load-point displacement, the fracture toughness of the test specimen36under mixed-mode loading conditions is determined.

To further reduce bending of the saddle legs44, in some embodiments, two steel M5 bolts48or other fasteners48are used as a lateral reinforcement. In some embodiments, the bolts48screw into threaded holes on both sides of the saddle16(e.g., at a saddle leg44) and are tightened until they contact the lever18. This contact creates a moment to counter the moment generated by the applied load, thus reducing the legs' rotation and the maximum deformation. Thus, in some embodiments, each fastener48extends from one of saddle legs44to lever18to laterally fix saddle16to lever18. In some embodiments, the bolts are cut to the required length (e.g., 23 mm) and have a low-profile head height (e.g., <2 mm) to ensure clearance between the yoke14and saddle16. The careful alignment of the saddle16, lever18, and bolts48is necessary to measure fracture energy correctly. A small difference between the rotational angles of bolts48with the lever18could lead to asymmetric crack propagation.

In some embodiments, the lever18also includes threaded holes, and the bolts48screw into both the lever18and the saddle16. This avoids asymmetry due to inconsistent contacts between the lever18and saddle16. Similar to the above embodiments, the bolts48are cut to the required length (e.g., 27 mm) and have a low-profile head height (e.g., <2 mm) to ensure clearance between the yoke14and saddle16.

Furthermore, to create more available surface area for the contact between the bolts48and lever18, the bottom flange height of the lever18was increased by 2 mm while keeping the total lever18height the same. The lever's original design is an aluminum I-beam, which is much stiffer than the split-beam specimen. The revised lever18is still an I-beam with symmetry to the vertical axis. The stiffness of the revised lever18reduces; however, due to its low participation in the overall stiffness (seeFIG.6), its impact on the stiffness is negligible.

In some embodiments, the top roller22was further redesigned by adding a support pillar50underneath the steel roller, which significantly reduces deformation. The support pillar50is part of the aluminum top roller holder20and is designed to make lubricated contact with the steel roller to prevent frictional effects.

Further details of the components are illustrated in the figures, including the design of the saddle16(FIGS.11A-11C), the lever18(FIGS.12A-12C), and the top roller holder20(FIGS.13A-13D). The RMMBA10deforms evenly across its width. The apparatus should fulfill symmetrical and alignment criteria for a uniform crack growth of the split-beam test specimen36.

With these further modifications, the measured stiffness of the finite-element model of the RMMBA10was 8.36 KN/mm, within an 8.9% difference of the analytical model (9.11 KN/mm), validating the developed spring model. Unlike the original MMB, the finite-element model showed a lower stiffness than the analytical model. This is likely because in this design, the bolts couple the lever18and saddle16together. They can no longer be considered separate springs in series, which causes a slight difference between the analytical model and the validation finite-element model. Additionally, the bolts' lateral compression force on the lever18reduces the stiffness of the lever18in the vertical direction, leading to a lower overall stiffness than the prediction from the spring model. The finite-element model showed an ˜87% improvement in stiffness over the finite-element model of the original MMB, validating the results of the analytical model and the improvements of the RMMBA10described herein.

One potential drawback of the RMMBA10design could be the upper assembly's additional weight, contributing to the system's preloading. The effects of the additional weight on the center of gravity and the mixed-mode fracture toughness are critical for accurate fracture characterization. The saddle16is moved to create different mixed-mode ratios. Moving the saddle16changes the center of gravity of the subsystem, including saddle16and lever18. The extra weight of the top roller22handle acts along the centerline of the fulcrum and contributes to the pure mode II. The roller assembly mass was increased from ˜105 grams (g) in the original MMB to ˜128 g in RMMBA10. The mass increase has a negligible effect on preloading the split-beam test specimen36in mode II. The saddle16mechanism moves when the lever18load point is changed; thus, the center of gravity of the lever18and saddle16cgalso changes. The expression of cgfor MMB and RMMBA10is (0.338c+21.62) mm and (0.393c+19.84) mm, respectively.

The largest error in G1happens at large c values, while the largest error in G11occurs at small c values. Assuming c=97.5 mm and c=16.8 mm, the estimates of error in SERR calculation for mode I and II (eICand eIIC) are presented in Equations 7 and 8

where L is the half span length, a is the crack length, b and I are width and moment of inertia of the split-beam test specimen36, E11is the flexural longitudinal elastic modulus of the composite material. GIc, and GIIcare the critical SERR for modes I and II. Pgis the weight of the lever18and saddle16, which is 6.07N for MMB and 6.61N for RMMBA10.

Data of stitch-bonded biaxial polymer matrix composites were used from a mixed-mode fracture study (L=50 mm, b=20 mm, a=36 mm, I=853.3 mm4,GIc=0.3 Nmm/mm2, G11=3.2 Nmm/mm2). The estimated errors of SERR for mode I and II are 1.1% and 0.23% using the traditional MMB fixture. In the case of RMMBA10, the corresponding estimated errors are 1.3% and 0.24%, confirming the negligible effect of the added weight on SERR.

III. Example RMMBA with Optimized Dimensions

FIGS.9A-14Billustrate the components of the RMMBA10with exemplary optimized dimensions (in mm).FIG.9Ais a top schematic diagram of the top connecting rod12of the RMMBA10ofFIG.7A. The top connecting rod12includes a standoff nut56with opposing flat sides and rounded ends. The flat sides are 12.70 mm long and the rounded ends are 22.00 mm wide with a 12.70 mm radius to center. The standoff nut56includes a 9.90 mm diameter rod58extending from its top.

FIG.9Bis a front schematic diagram of the top connecting rod12ofFIG.9A. The standoff nut56is 25.40 mm high and a threaded stud with a 7.00 mm diameter extends from its bottom.

FIG.10Ais a front schematic diagram of the yoke14of the RMMBA10ofFIG.7A. The yoke14includes the top member52, which includes a tapped 1/4-28UNC-2B opening which is centered and may extend through the top member52(e.g., to accept the threaded stud of the top connecting rod12). The top member52has a top length of 75.67 mm and a bottom length of 61.20 mm. The top member52is between two angled yoke arms54with angled upper portions60and bottom portions62which are perpendicular to the top member52. Angled upper portions60are positioned at a non-perpendicular and non-parallel angle with respect to top member52. The angled yoke arms54each have a thickness of 11.18 mm. The angled upper portions60have an outside height of 33.55 mm and an inside height of 24.77 mm. The bottom portions62are spaced 83.42 mm apart.

FIG.10Bis a side schematic diagram of the yoke14ofFIG.10A. The yoke14has an overall width of 25.40 mm and an overall height of 100.08 mm.

FIG.11Ais a top schematic diagram of the saddle16of the RMMBA10ofFIG.7A. The saddle16includes two saddle legs44extending as one piece outward and downward from a middle section64. The middle section64meets each saddle leg44at a rounded corner with a radius of 3.30 mm. The middle section64is rectangular, with a length of 63.50 mm (extending 22.28 mm from an edge of each saddle leg44) and two oblong mounting holes66aligned along the length of the middle section64. The mounting holes66are rectangles with rounded ends, where the ends have a radius of 2.65 mm and are spaced 14.22 mm on center, with one end aligned with an edge of the saddle legs44and the other end 8.15 mm from one of the long ends of the middle section64.

FIG.11Bis a front schematic diagram of the saddle16ofFIG.11A. An overall width of the saddle16is 77.94 mm. Each saddle leg44includes an upper section67which accommodates the lever18which thins to a lower section68ending in a saddle bearing holder70. The upper sections67are spaced 40.50 mm apart and include fastener openings (e.g., threaded openings) through which the bolts48or other fasteners pass. The lower sections68meet the upper sections67with rounded corners having a radius of 8.00 mm. Each lower section68is 10.62 mm wide. Each saddle bearing holder70extends outward (e.g., away from the middle section64) in an L shape from the corresponding lower section68. A width of the combined lower section68and saddle bearing holder70is 25.22 mm. A width from the outside of the lower section68and upper section67to the inside of the saddle bearing holder70is 9.80 mm. A height of the saddle bearing holder70is 22.30 mm, and an outside width of its L shape is 4.80 mm.

FIG.11Cis a side schematic diagram of the saddle16ofFIG.11A. A width of each saddle leg44is 18.94 mm. A height from a bottom edge of the saddle16to an axis of the saddle bearing46is 17.70 mm, and the saddle bearing holder70(and the lower section68) includes a 6.60 mm diameter opening to accommodate the saddle bearing46. A height from the bottom edge of the saddle16to a center line of the fastener opening is 55.24 mm. In an exemplary aspect, the fastener opening accommodates an M5×0.8 or M4×0.7 bolt. A height from the bottom edge of the saddle16to a bottom of the middle section64is 78.74 mm. A height from the bottom edge of the saddle16to a top of the saddle16is 91.44 mm.

FIG.12Ais a top schematic diagram of the lever18of the RMMBA10ofFIG.7A. The lever18has a length of 215.90 mm and a width of 31.86 mm. A top surface of the lever18includes a series of eleven 10-32 UNC holes with a depth of 7.87 mm aligned on center with the width of the lever18for mounting to the saddle16. The first hole is 12.70 mm from center to an edge of the lever18away from the top hinge28, and the holes are spaced 12.70 mm on center. A bottom surface of the lever18includes four 10-32 UNC holes with a depth of 12.71 mm for mounting to the top hinge28. Each hole is 7.98 mm away from a wide edge of the lever18, with a first pair being 14.22 mm from an end of the lever18and a second pair being 36.45 mm from the end.

FIG.12Bis a side schematic diagram of the lever18ofFIG.12A. The bottom edge further includes five 10-32 UNC holes with a depth of 7.87 mm aligned on center with the width of the lever18. The first hole is 50.67 mm from center to the edge of the lever18where the top hinge28attaches, and the holes are spaced 12.70 mm on center (e.g., such that three of these align with holes in the top surface). Each side of the lever18includes twenty-eight M5×0.8 or M4×0.7 threaded holes with a depth of 5 mm to accommodate the bolts48or other fasteners. The first hole is 4.05 mm from center to the edge of the lever18away from the top hinge28, the last hole is 75.35 mm from center to the edge attached to the top hinge28, and the holes are spaced 10.50 mm on center. The holes are 8.36 mm from center to the bottom surface of the lever18.

FIG.12Cis a front schematic diagram of the lever18ofFIG.12A. The lever18is 31.86 mm high, with beveled sides. The bevels begin at 12.71 mm and end at 25.15 mm from the bottom surface. The bevels extend 12.74 mm into the lever18, and have rounded inner corners with a radius of 4.83 mm.

FIG.13Ais a bottom schematic diagram of the top roller holder20of the RMMBA10ofFIG.7A. The top roller holder20includes two roller arms72extending as one piece outward and downward from a middle section74. The middle section74meets each roller arm72at a rounded corner with a radius of 4.83 mm. The middle section74is rectangular, with a length of 66.55 mm and a width of 19.30 mm. The middle section74includes two oblong mounting holes76aligned along the length of the middle section74. The mounting holes76are rectangles with rounded ends, where the ends have a radius of 2.63 mm and are spaced 15.75 mm on center, with one end 3.98 mm from an edge of the roller arms72and the other end 3.98 mm from one of the long ends of the middle section74.

FIG.13Bis a front schematic diagram of the top roller holder20ofFIG.13A(with the top roller holder20flipped vertically). An overall width of the top roller holder20is 50.80 mm. Each roller arm72extends downward (shown upward inFIG.13B) from the middle section74with an inner radius of 11.18 mm and outer radius of 9.65 mm at the joint such that the top roller holder20is U-shaped with the middle section74being the bottom of the U. The support pillar50may be disposed between the two roller arms72. The support pillar50extends from the middle section74toward the top roller22(not shown inFIG.13B), and provides support and resilience against deformation of the top roller22as a load is applied to the RMMBA10. A width of the support pillar50is 22.94 mm.

FIG.13Cis a cross-sectional diagram taken along line A-A of the top roller holder20ofFIG.13B. An end of the support pillar50which comes in contact with the top roller22is rounded, with a radius of 5.74 mm. At its highest, the support pillar50extends 30.21 mm from a bottom surface (shown inFIG.13Cas a top surface) of the middle section74.

FIG.13Dis a side schematic diagram of the top roller holder20ofFIG.13A. The roller arms72accommodate the top roller22at rounded ends having a radius of 10.67 mm, at a height of 34.0 mm from the bottom surface (shown inFIG.13Das a top surface) of the middle section74. The middle section74has a height of 6.35 mm.

FIG.14Ais a front schematic diagram of the top roller22of the RMMBA10ofFIG.7A. The top roller22is cylindrical in shape, with a thicker inner section78which contacts the test specimen36and thinner outer sections80which are inserted into the top roller holder20. The inner section78is 36.58 mm wide, and each of the outer sections80is 7.11 mm wide.

FIG.14Bis a side schematic diagram of the top roller22ofFIG.14A. A diameter of the outer sections80is 6.35 mm and a diameter of the inner section78is 11.40 mm.

FIG.15Ais a top schematic diagram of the bottom roller holder24of the RMMBA10ofFIG.7A. The bottom roller holder24includes two roller arms82extending as one piece outward and upward from a middle section84. The middle section84meets each roller arm82at a rounded corner with a radius of 4.83 mm. The middle section84is rectangular, with a length of 66.55 mm and a width of 19.30 mm. The middle section84includes two oblong mounting holes86aligned along the length of the middle section84. The mounting holes86are rectangles with rounded ends, where the ends have a radius of 2.63 mm and are spaced 15.75 mm on center, with one end 3.98 mm from an edge of the roller arms82and the other end 3.98 mm from one of the long ends of the middle section84.

FIG.15Bis a front schematic diagram of the bottom roller holder24ofFIG.15A. An overall width of the bottom roller holder24is 50.80 mm. Each roller arm82extends upward from the middle section84with an inner radius of 11.18 mm and outer radius of 9.65 mm at the joint such that the bottom roller holder24is U-shaped with the middle section84being the bottom of the U.

FIG.15Cis a side schematic diagram of the bottom roller holder24ofFIG.15A. The roller arms82accommodate the bottom roller26at rounded ends having a radius of 10.67 mm, at a height of 34.93 mm from a top surface of the middle section84. The middle section84has a height of 6.35 mm.

FIG.16Ais a front schematic diagram of the bottom roller26of the RMMBA10ofFIG.7A. The bottom roller26is cylindrical in shape, with a thicker inner section88which contacts the test specimen36and thinner outer sections90which are inserted into the bottom roller holder24. The inner section88is 36.58 mm wide, and each of the outer sections90is 7.11 mm wide.

FIG.16Bis a side schematic diagram of the bottom roller26ofFIG.16A. A diameter of the outer sections90is 6.35 mm and a diameter of the inner section88is 9.53 mm.

FIG.17Ais a top schematic diagram of the hinge clamp38of the top hinge28of the RMMBA10ofFIG.7A. Although the hinge is labeled as top hinge28, the same principles apply to the bottom hinge30, and the hinge inFIGS.17A-17Crepresents bottom hinge30as well. The hinge clamp38has a base with four mounting holes in pairs spaced symmetrically on center 22.23 mm apart, with each hole having a diameter of 4.83 mm and being spaced 15.90 mm from its pair.

FIG.17Bis a front schematic diagram of the hinge clamp38ofFIG.17A. The hinge clamp38is U-shaped, with the base being 34.93 mm wide and 7.37 mm high. An inner width between arms of the hinge clamp38is 25.40 mm.

FIG.17Cis a side schematic diagram of the hinge clamp38ofFIG.17A. The base of the hinge clamp38is 31.75 mm long. An overall height of the hinge clamp38is 46.04 mm. The hinge pin40has a diameter of 3.18 mm and is spaced 43.66 mm from the base of the hinge clamp38and 15.88 mm from edges of the arms (e.g., centered on the arms).

FIG.18Ais a top schematic diagram of the hinge tab42of the top hinge28of the RMMBA10ofFIG.7A. Although the hinge is labeled as top hinge28, the same principles apply to the bottom hinge30, and the hinge inFIGS.18A-18Brepresents bottom hinge30as well. The hinge tab42has a width of 24.77 mm.

FIG.18Bis a front schematic diagram of the hinge tab42ofFIG.18A. The hinge tab42further has a length of 31.75 mm, with a thickness of 5.08 mm and the hinge pin40positioned 15.88 mm from edges of the hinge tab42. With reference toFIGS.17C,18A, and18B, the hinge pin40is inserted through the hinge tab42and at least partially through the hinge clamp38to hold the hinge tab42in place.

FIG.19Ais a top schematic diagram of the base32of the RMMBA10ofFIG.7A. The base32is 63.50 mm wide and 254.00 mm long. A top surface of the base32includes seven holes for mounting the bottom roller holder24, each of which is aligned along a centerline of its width. The seven holes are positioned at 89.03 mm, 101.73 mm, 114.43 mm, 127.13 mm, 139.83 mm, 152.53 mm, and 165.23 mm on center from one end of the base32. Two pairs of mounting holes for the bottom hinge30are spaced 23.80 mm and 39.70 mm from one side of the base32. The first pair of holes is positioned at 217.55 mm on center from the one end of the base32, and the second pair of holes is positioned at 239.78 mm. Each of the seven holes and two pairs of holes has a 4.83 mm diameter and 12.7 mm depth.

FIG.19Bis a front schematic diagram of the base32ofFIG.19A. A bottom surface of the base32(shown on the right inFIG.19B) defines a channel with an undercut groove. The opening of the channel at the bottom surface extends from 23.81 mm to 39.69 mm from the one side of the base32. The groove of the channel extends from 19.56 mm to 43.94 mm from the one side of the base32. The groove of the channel further extends from 14.29 mm to 23.81 mm from the top surface of the base32. The base32has a height of 31.75 mm.

FIG.20Ais a bottom schematic diagram of the base slider34of the RMMBA10ofFIG.7A. The base slider34includes a body and tongues extending therefrom for fitting into the groove of the base32. The base slider34has an overall width of 22.00 mm and a length of 28.50 mm. The body of the base slider34includes an opening which is 11.01 mm in diameter.

FIG.20Bis a front schematic diagram of the base slider34ofFIG.20A. The body of the base slider34has a width of 14.00 mm and an overall height of 16.00 mm. Each tongue extends 4.00 mm from the body with a height of 8.50 mm.

IV. Example RMMBA with Parametric Dimensions

FIGS.21A-25Billustrate the components of the RMMBA10with exemplary variable primary and dependent parameters. The primary parameters are indicated in Table 3, along with a range of distances. Dependent parameters vary according to a relationship with the primary parameters, as indicated in Table 4.

FIG.21Ais a front schematic diagram of the yoke14, similar toFIG.10A, with variable parameters indicated.FIG.21Bis a side schematic diagram of the yoke14, similar toFIG.10B, with variable parameters indicated. The variable parameters of the yoke14are as indicated: the bottom length C, the overall width E, the thickness Y of the angled yoke arms54, and the space Z between the bottom portions62.

FIG.22Ais a top schematic diagram of the saddle16, similar toFIG.11A, with variable parameters indicated.FIG.22Bis a front schematic diagram of the saddle16, similar toFIG.11B, with variable parameters indicated.FIG.22Cis a side schematic diagram of the saddle16, similar toFIG.11C, with variable parameters indicated. The variable parameters of the saddle16are as indicated: a width A of each lower section68and a width B of each saddle leg44.

FIG.23Ais a top schematic diagram of the lever18, similar toFIG.12A, with variable parameters indicated.FIG.23Bis a side schematic diagram of the lever18, similar toFIG.12B, with variable parameters indicated.FIG.23Cis a front schematic diagram of the lever18, similar toFIG.12C, with variable parameters indicated. The variable parameters of the lever18are as indicated: the bevels of the lever18begin at F from the bottom surface.

FIG.24Ais a bottom schematic diagram of the top roller holder20, similar toFIG.13A, with variable parameters indicated.FIG.24Bis a front schematic diagram of the top roller holder20, similar toFIG.13B, with variable parameters indicated.FIG.24Cis a cross-sectional diagram taken along line A-A of the top roller holder20, similar toFIG.13C, with variable parameters indicated.FIG.24Dis a side schematic diagram of the top roller holder20, similar toFIG.13D, with variable parameters indicated. The variable parameters of the top roller holder20are as indicated: a width G of the support pillar50, a height W of the top roller22from the bottom surface of the middle section74and a radius X of an end of the support pillar50which comes in contact with the top roller22.

FIG.25Ais a front schematic diagram of the top roller22, similar toFIG.14A, with variable parameters indicated.FIG.25Bis a side schematic diagram of the top roller22, similar toFIG.14B, with variable parameters indicated. The variable parameters of the top roller22are as indicated: a diameter D of the inner section78.