Strength distributions of laminated FeNi-based metal amorphous nanocomposite ribbons

Abstract

Metal Amorphous Nanocomposite (MANC) materials offer low losses at high magnetic switching frequency, enabling high power density motors with increased rotational speed. While MANCs have high strength, they are brittle. The use of motor components such as a rotor consisting of brittle material presents a reliability concern. Here, a promising MANC alloy is subjected to tensile tests and failure is observed with high-speed photography. A method is developed to prepare tensile specimens of laminated MANC and epoxy layers, simulating the stacking of an epoxy-impregnated tape-wound core. Tensile tests are conducted for single layer ribbon and for five- and ten-layer stacks of laminated material with thin layers of thermosetting epoxy. Failure distributions are shown to have increasing Weibull modulus with increasing layer count. The composite MANC material system is modeled using chain-of-bundles models. Using a k-failure model, we show that single ribbon strength distribution data can be used to predict well the failure distribution of laminated stacks. The agreement occurs when the assumed ineffective length, over which load is recovered in a failed layer, is comparable to the observed interlaminar separation length.

Keywords

metal amorphous nanocomposite soft magnetic materials high-speed electric motors power magnetic devices laminated composite failure chain-of-bundles models

Introduction

An increase in power density of electric motors is beneficial in many applications.^1,2 This can be achieved by increasing the rotational speed, but higher operating frequency induces higher eddy current loss in the soft magnetic portion of the motor. Conventional motors are typically made using stamped Fe-Si laminates as the soft magnetic material. The laminates limit the magnetic frequency because they are relatively thick at ∼500 μm and eddy current losses cause significant temperature rise. A solution to the high-frequency loss problem in motors is introduced by Metal Amorphous Nanocomposite (MANC) soft magnetic materials,^3–6 which exhibit high saturation magnetization while having much smaller thickness (∼20 μm) and ∼5 times higher electrical resistivity than Fe-Si competitors.⁷ Eddy current losses decrease with increased resistivity and with the thickness squared. Hence these losses are ∼1000 times lower in MANCs than in Fe-Si.

With lower magnetic switching losses, mechanical limits have an increased significance in motor design, as stress increases with the square of rotational frequency.⁸ MANCs have been successfully employed in power magnetic devices such as inductors and transformers, which are relatively insensitive to stress. However, commercial MANCs like FINEMET are excessively brittle,⁹ making them less attractive for stress-sensitive applications like high-speed motors (HSMs). New high saturation induction MANCs are designed to exhibit even lower energy loss at high magnetic switching frequency,^7,10 and qualitative bend tests have indicated higher bending strain to failure. Given the limited documentation on the mechanical properties of these new alloys, a deeper understanding of MANC mechanical performance and failure behavior is required to better predict their limitations as soft magnets in motor applications. If the limitations arising from their brittleness can be managed, MANCs present significant opportunities in increasing motor power density.

MANCs are produced by annealing a precursor amorphous magnetic ribbon (AMR) between a primary ( $T_{x 1}$ ) and secondary ( $T_{x 2}$ ) crystallization temperature to yield high-induction nanocrystals embedded in a residual amorphous matrix.¹¹

Crystallization kinetics are quantified by Johnson Mehl Avrami Kolmogorov (JMAK) kinetics^12–17 in conjunction with the Kissenger equation and differential scanning calorimetry experiments.^10,18 Successful casting and processing of these alloys requires careful stoichiometry selection to provide adequate glass-forming ability and amorphous phase stability.¹⁹ The nanocrystallization step improves magnetic properties by minimizing coercivity and maximizing saturation magnetization,¹² but eliminates mechanical ductility that is characteristic of some AMRs.²⁰ AMRs become brittle at room temperature after a critical volume fraction, ranging from 20% to 50%,^21–23 has been crystallized, which is associated with a material’s Poisson’s ratio (ν) being less than 0.31–0.32.²⁴

The failure strength of brittle materials is typically dictated by flaw size. Weibull analysis is a commonly used method for measuring the failure distribution of a brittle material sample population^25–27 and has been used to measure the strength of brittle amorphous and nanocrystalline ribbons.^28–30 Two-parameter Weibull analysis reports a characteristic strength ( $σ_{o}$ ), at which 63.2% of the population has failed, and a Weibull modulus ( $m$ ).²⁵ Larger $m$ indicates a smaller failure stress range. Typically, the literature concerning MANCs shows little plastic deformation and a low $m$ . Reported $m$ for AMRs and MANCs are typically 3 to 10.^28–30 However, MANCs exhibiting some fcc phase such as that used in this study tend to exhibit more favorable mechanical properties than bcc-based MANCs.^20,31

Measuring the statistical likelihood of ribbon failure is important to understand the strength limitations of MANCs for HSMs. However, this metric alone does not fully inform the tensile behavior of materials to be used in these applications. MANCs are typically used in epoxy impregnated tape wound cores (TWCs), where the ribbon is wound to manufacture a toroid with hundreds of ribbon layers. The epoxy impregnation process used in this work is described in the Methods section. Given the composite nature of an impregnated TWC, it is expected that the tensile behavior will be different from that of a MANC ribbon alone. This composite structure can be expected to perform mechanically similar to other composites, where progressive irreversible damage occurs during tensile tests prior to failure. In this paradigm, load sharing enables specimen survival after some localized “damage” such as fiber or layer failure or layer debonding.^32,33 While laminated composites are known to mechanically survive some local cracking, it must be noted that cracked magnetic material is associated with localized gaps, which will alter magnetic properties such as effective coercivity and permeability.^34,35

While MANCs in composite form have not been studied in the literature, the tensile response of AMR in composite form has been investigated by Leng and Courtney^36,37 and Li et al.³⁸ Both studies found an increase in ductility and strength by plating an AMR in another ductile metal. However, these benefits were achieved at the expense of volume fraction of AMR with a substantial increase in the amount of high strength ductile material. In these studies, the AMR volume fraction was 20% and 23% respectively, with the balance being ductile brass and Ni, respectively. For magnetic device applications, high packing factors are required to achieve power density requirements. TWCs are routinely wound with at least 80% AMR or MANC packing factor, with the remaining being filled by the electrically insulating but mechanically soft epoxy. This structure is substantially different from those created by Leng^36,37 and Li et al.³⁸ The study here is conducted to characterize the tensile response of the TWC/epoxy composite with ∼80% fill factor, and to evaluate methods to model their failure distributions. Weibull distributions of single ribbons and stacks of five and ten layers are measured. Chain of bundles models of increasing complexity^39–48 are used to understand the results in the context of relating single ribbon failure to multi-ribbon failure.

Methods

AMR alloys of composition (Fe₇₀Ni₃₀)₈₀B₁₄Nb₄Si₂ were fabricated using a METGLAS batch mode planar flow casting process.⁴⁹ The ribbon tested exhibited a calculated average thickness of 19.8 μm. Figure 1(a) shows an optical microscope view of ribbon edge. Because the surface roughness (or defect size) of brittle materials limits their strength, it is important to test a material with similar surface characteristics to those present in a motor component. Experiments were conducted using full-width ribbons rather than “dog bone” specimens because cutting the ribbon sides into these shapes alters the edge roughness from that developed by the planar flow casting process. Ribbons were cut across their width with shears to a length of 150 mm and annealed in an air atmosphere at 440°C, between $T_{x 1}$ and $T_{x 2}$ , for 15 min, as determined to be optimal for the soft magnetic properties of this alloy in prior work.^10,50 Initial work⁵⁰ used differential scanning calorimetry to find the range of $T_{x 1}$ and $T_{x 2}$ and study strain annealing.⁵¹ This information was used by Byerly et al.¹⁰ to determine optimal strain annealing and re-annealing temperatures. Annealing in this temperature range oxidizes the surface to a thickness of ∼10 nm, as explored by Egbu et al.⁵² Thus, the surfaces of the ribbons tested here are coated with this oxide. The ends of the annealed ribbons were attached to sheet metal grip tabs using cyanoacrylate adhesive, (Figure 1(b)). The grip tab assembly prevents tensile machine grip serrations from fracturing the ribbon when tightening prior to the test. Resulting tensile specimens were 100 mm long between grip tabs and 25 mm wide.

Figure 1.

(a) Edge of ribbon in used in tensile tests. (b) Full tensile test specimen attached to grip tabs.

Similar processes of annealing and grip tab adhesion were followed in preparing tensile specimens from laminated ribbon stacks of five or ten layers. To prepare laminated stacks, multiple annealed MANC ribbons were placed between Teflon blocks and the stack was bound with wire (Figure 2(a) and (b)). These ribbon stacks consisted of five or ten annealed ribbons, followed by an as-cast ribbon protruding from the end of the stack. This stacking method allowed for the preparation of many stacks in a single vacuum pressure impregnation (VPI) cycle, equivalent to that used to impregnate TWCs with epoxy, as shown in Figure 3. The TWC is placed in a vacuum to remove air from between the layers (Figure 3(a)). While remaining under vacuum, the TWC is submerged into a liquid epoxy bath (Figure 3(b)). Next, positive pressure is applied (Figure 3(c)), forcing epoxy between ribbon layers. After reducing to atmospheric pressure (Figure 3(d)), the TWC is removed from the bath (Figure 3(e)). The epoxy impregnated TWC is then cured in air at 160°C for 4 h. After curing the laminated stacks in this study, the stack was separated by peeling into smaller stacks, with the number of layers determined by the interspersed as-cast ribbons. The as-cast ribbons shown in Figure 2(a) facilitate peeling, as the epoxy does not bond as well to their surfaces.

Figure 2.

(a) Schematic and (b) image of stack of ribbons prepared for VPI process.

Figure 3.

Vacuum pressure impregnation (VPI) process schematic.

A random sample of laminated ribbon stacks was measured with a micrometer to determine overall stack thickness in ten locations across the stack. The total ribbon thickness in the stack was subtracted from this measurement to find the total epoxy thickness. From these measurements, the average epoxy layer thickness ranged from 3.7 to 6.5 μm, similar to that achieved in an 80% fill factor TWC. Since the epoxy cross-sectional area is ∼20% of the total cross-section and the epoxy has a Young’s Modulus ∼0.8% of the ribbon, the epoxy stiffness contribution to the stack is negligible. Thus, so long as the epoxy adequately wets and bonds the ribbon layers, a change in epoxy thickness should not change the elastic behavior significantly. Accordingly, the area considered for stress calculation from tensile tests is only the cross-sectional area of the ribbon layers. Though this is not a conventional method of documenting composites, it is more appropriate to describe stress in terms of ribbon cross-section because this provides a direct comparison with the single-layer stress calculation.

Annealed ribbons rapidly shatter into multiple fragments and specimens had no reduced test section to ensure fracture away from tensile grips. A high-speed camera was used to determine if the initial ribbon fracture location was sufficiently far from the stress concentrator associated with the grips. Tensile tests were conducted using an Instron model 4469 with a deformation rate of 3.0 mm/min (strain rate of 0.0005/s) while the ribbons were monitored with a Phantom Vision v1211 high-speed camera at 66,000 frames/s (15.2 μs between frames, 7 μs exposure time). If a specimen failed away from the grips (where stress concentrators are located), that specimen was used in the tensile strength ( $σ$ ) population to determine Weibull statistics. To produce Weibull plots, equation (1) was first used to estimate the probability of failure for a given failure ranked test. Accordingly,

P_{f} = (R_{f} - 0.5) / n_{s}

(1)

where

P_{f}

is the probability of ribbon failure,

n_{s}

is the number of specimens, and

R_{f}

is the failure stress rank of a given test. The two-parameter Weibull equation,

P_{f} (σ) = 1 - \exp (- {(σ / σ_{o})}^{m})

(2)

is used to fit such data.

Results and discussion

Experimental results

For single ribbons, most high-speed camera observations indicate that cracks initiate at the edges far from the grips. In only six of 27 tests did a surface flaw away from the ribbon edge constitute the critical flaw, leading us to conclude that edge flaws generally limit the ribbon strength. Different single ribbon specimens failed with different crack propagation patterns. Figure 4 shows the high-speed camera setup (Figure 4(a)) and images of what we call failure Types I, II, and III. In each, the first image shows the incipient propagation, and the second is 75 μs later. Observations of crack branching help to verify the origin of the critical flaws. In failure Type I (Figure 4(b)), the crack branches into many cracks which form a fanlike pattern. In Type II (Figure 4(c)), the primary crack is more dominant, with fewer, smaller branches. In Type III (Figure 4(d)), the crack originates in the middle of the strip, away from an edge, implying that the critical flaw is a surface rather than edge defect. These types of propagation events have been widely studied,⁵³ and are further discussed below only briefly.

Figure 4.

(a) High-speed camera setup. Ribbon failure Types (b) I, (c) II, and (d) III near initiation and after 75 μs.

Young’s Modulus was determined using Instron software from single-layer tensile tests, with a 27-specimen average of 144 GPa and 8.5 GPa standard deviation. Stress–strain curves from two selected samples (Figure 5(a)) are shifted on the strain axis for clarity. These exhibit elastic behavior until brittle failure, with no plastic deformation. A stress–strain curve of the epoxy is shown in Figure 5(b). The epoxy has much lower failure stress than the ribbon, but its failure strain is more than 15 times larger.

Figure 5.

(a) MANC ribbon tensile test stress versus strain results from two selected tests. Curves are shifted by 0.001 strain for clarity. (b) Epoxy tensile test.

Weibull results from single-, five- and ten-layer tests are shown in Figure 6 and are summarized in Table 1. The number of specimens used for the single-, five- and ten-layer tests are 27, 24, and 16, respectively. The $m$ value of 3.8 for the single layer ribbon (grey) is similar to existing literature on other AMR alloys.^28–30 All single ribbons exhibiting failure Type I failed at lower $σ$ values. The higher $σ$ failures exhibit either Type II or III behavior. Although single layer data points result from different dynamic propagation types, their Weibull moduli are not significantly different. That is, the Weibull modulus of the single ribbon line does not vary over the range of stress. This suggests there is no substantial difference in initial failure mechanism. Upon crack extension, dynamics of elastic energy dissipation may lead to the different failure types observed. As Ravi-Chandar^53,54 and others^55,56 have shown, crack instability and branching occurs when some critical crack velocity is exceeded. Katzav et al.⁵⁵ notes that this critical velocity does not depend on applied traction or specimen geometry, indicating that the difference in Types I, II, and III is more likely due to flaw profile than stress magnitude. While the distinct appearance of the different propagation patterns is notable, the focus of the present work concerns the failure strength distributions. As seen in Figure 6 and Table 1, as more layers are added to the laminate $σ_{o}$ increases modestly from 1000 to 1100 MPa, but $m$ increases significantly from 3.8 to 11.2.

Figure 6.

Weibull distribution single layer ribbons (gray), five-layer laminated stack (black), and ten-layer laminated stack (blue).

Table 1.

Summary of results from Weibull distributions in single-layer and laminated stack tensile tests, including fitting quality, $r^{2}$ .

	Characteristic strength, σ₀ (MPa)	Weibull Modulus, m	$r^{2}$
Single ribbon	1004	3.81	0.923
5 stack	1058	6.85	0.969
10 stack	1099	11.16	0.975

Table 2.

Comparison of Figure 10 k-failure model with data for δ values which lead to $σ_{o}^{*}$ matching $σ_{o}$ .

Layers	Method	δ (mm)	$m^{*}$	m _data	$m^{*}$ /m _data
5	1	6.8	7.4	6.2	1.19
5	2	11.0	7.6	6.2	1.23
10	1	4.1	11.0	11.2	0.98
10	2	9.1	11.4	11.2	1.02

Experiment discussion

The tighter failure distribution with increasing layers is qualitatively understood from observations of stacks during testing. Figure 7(a) shows a five-layer laminated ribbon stack strained to failure, as captured with 33.3 μs time resolution over a 6 s span, at illustrative times of $t =$ −5.6 s, −3.6 s, −1.8 s, −33 μs, 0 μs, +33 μs and +466 μs. Tensile stresses over this time increase from 679 to 894 MPa. Black dots in Figure 7(a) are used for strain measurements and can be ignored in this discussion. In the leftmost frame of Figure 7(a) at $t =$ −5.63 s, the outermost ribbon layer has already fractured across the full width, but the stack remains intact. Viewing Figure 7(a) from left to right, as the stress increases with time, the ribbon gradually peels away from the adjacent layer at the location of the first fracture. A second full width fracture on the outermost ribbon has developed by $t =$ −3.6 s, and the ribbon also peels away from that fracture over time, yet the stack does not fail. That failure is ∼ 20 mm away from the first fracture, while the epoxy thickness is ∼ 3 μm. Hence the fracture-to-fracture separation distance to epoxy thickness ratio here is ∼ 7,000, and in general it varies at random. A new and fatal fracture develops near $t =$ 0 μs. Soon thereafter, the adjacent material recoils, as shown in Figure 7(a), $t =$ +33 and +466 μs. The fatal fracture may have originated inside the stack. The neighboring ribbons can no longer support the load, they fail, and the stack fails.

Figure 7.

(a) High speed camera front view images of laminated ribbon stacks near failure point after sustaining multiple individual ribbon fractures, shown progressively from left to right with increasing $σ$ . Black dots were used for optical strain measurement and may be ignored. Side view schematics of ribbon stack sustaining failure of single layer (b) immediately after layer fracture, (c) after further loading progression, and (d) near probable failure when regions A and B encompasses another dominant flaw. Length of region A indicated in (b)–(d) by arrows.

This sequence is represented schematically in Figure 7(b), (c) and (d), where a dominant flaw causes the outside ribbon layer to fracture in Figure 7(b). The total tensile load is then borne by only four layers in Region A. Because the epoxy bond sustains shear stress, a portion of the tensile load is transferred into the fractured ribbon layer at a sufficient distance away from Region A. Region B is a transition region, where the tensile stress in the failed ribbon gradually increases with distance from the flaw. The total length, $δ$ , of both Regions B is estimated using shear-lag models by,⁵⁷

δ \approx {2 F}_{\max} / w τ_{i}

(3)

where

F_{\max}

is the maximum force transmitted between layers,

w

is the ribbon width, and

τ_{i}

is the shear strength of the interface. Assuming

w =

25 mm,

F_{\max} =

100 N (derived from 1 GPa for the five-layer stack), and τ_i = 35 MPa (an upper bound based on epoxy tensile tests),

δ \approx

229 μm. This value is shown to scale in Region B of Figure 7(b). Thus,

δ \approx

11 times the thickness of the ribbon layer, in reasonable agreement with an analogous rule of thumb often used in the fiber composites literature of five times the fiber diameter.⁴⁷ Incipient edge flaws of various sizes are distributed along the length of each ribbon layer. Because the length of Region B is ∼ 1,000x shorter than the total specimen length, the likelihood of another critical flaw being present in another ribbon in Region B is low. Therefore, the tensile test continues with the remaining ribbons carrying the load. Beyond Region B, the tensile stress becomes evenly distributed among all layers in Region C.

After a ribbon layer fractures, it begins to peel away from the adjacent ribbon layer, shown pictorially in Figure 7(a), $t =$ −3.6 s to −33 μs, and schematically in Figure 7(c) and (d). Because the peel length in Figure 7(a) is much longer than Region B, the peel length is scaled down in Figure 7(c) and (d) to illustrate proper detail of the layers. As peeling extends due to the limited epoxy to ribbon bond strength, Region A grows, increasing the likelihood that another critical incipient flaw will be contained in Region A or B, increasing the probability of failure. While the load is increasing and Region A is extending, other ribbon fractures create more zones like region A, further increasing the probability of failure. Two such regions are shown in Figure 7(a), beginning at t = −3.6 s.

Region A will eventually grow to encompass another incipient edge flaw of sufficient size to induce an additional layer fracture (Figure 7(d)), though this flaw is necessarily smaller than that which caused the first layer to fail. Upon failure of a second layer due to the local load concentration from the first failure, the probability of system failure increases again. System failure is finally shown in Figure 7(a), t = 0, after sufficient peel extension. Importantly, the first fracture of any given layer would be the point of failure in a single-layer test. This means that, due to good layer adhesion, the laminated stack can prevent specimen failure at the low-stress end of the distribution, as shown by the five- and ten-layer distributions in Figure 6. We note that Figure 7(b)–(d) assume peeling occurs across the full ribbon width, and as such do not account for peel length variation along ribbon width, which we assume to have little impact on overall behavior. We observe a similar sequence of events for internal ribbon failures in other specimens.

Modeling MANC-epoxy composite failure

Statistical models describing phenomena similar to the sequence described in the experiment discussion section have been developed in the context of impregnated fiber bundle failure. In these models, the matrix has negligible contribution to strength, in contrast with fiber-reinforced composites that have the matrix constituting a significant volume fraction. We apply fiber bundle models in the following treatment, and consider whether single layer ribbon failure data can be used to model five- and ten-layer stack strength data. The first model adapted to the present dataset is the chain-of-bundles model using Global Load Sharing (GLS). Early work by Weibull²⁵ in failure statistics and Daniels³⁹ in weakest link theory contributed to the chain-of-bundles model. It was later applied to carbon fiber composites by Zweben,^40,58 Rosen,⁴¹ and Harlow and Phoenix.^42,43

The GLS model assumes that there is an ineffective length, $δ$ , over which the load previously borne by a layer that has failed is absorbed equally by all surviving layers. Thus, $δ$ is the sum of the length of Regions A and B in Figure 7 and the single-layer or multi-layered specimens consist of $n = L / δ$ links or bundles, respectively, in series. In fiber failure models, a bundle represents a discrete length of a group of parallel load-carrying fibers. Here, the bundle takes the form of a short segment of several laminated ribbons. An estimation for $δ$ in the absence of layer peeling is given in the experiment discussion section. However, the peeling of ribbons (Figure 7(a)) will dominate this quantity. Thus, we consider which value of $δ$ leads to the best correlation between failure data and model prediction. We then assess whether these $δ$ values are realistic. From Figure 7(a), it is apparent that a realistic average value of $δ$ is in the range of a few mm to ∼12 mm. An average $δ$ across the specimen width is used.

The GLS model analysis begins with a Weibull distribution for ribbon specimen failure, $P_{f}$ , given by equation (2). This function is called a fiber cumulative distribution function (CDF) and gives the probability of failure for a given specimen length $L$ at stress $σ$ . Because the ribbon specimen is modeled as a chain of $n$ links in series, a separate CDF can then be calculated from $P_{f}$ for the failure distribution of individual links. This individual link CDF is denoted $F$ and is given by,

F (σ) = 1 - {(1 - P_{f} (σ))}^{1 / n}

(4)

which is a form of the weakest link theorem first described by Peirce.⁴⁴

While the details of load sharing between bonded layers can be explained using solid mechanics arguments, these effects are simplified in chain-of-bundles models. Load sharing is typically considered using a stress concentration factor, $K$ , applied to surviving layers in a bundle. $K$ varies with the number of failures in a bundle, $r$ , and is calculated for GLS assumptions by,

K_{r} = N / (N - r)

(5)

Where

N

is the number of layers.

F

can be used to determine the probability of bundle failure,

G

, by,

G (σ) = B_{N} F (σ) Π_{r = 1}^{N - 1} F (K_{r} σ)

(6)

where

B_{N} = 2^{N - 1}

is the number of possible failure sequences for arriving at

N

failed layers, or a fully failed bundle.⁴⁵

K_{r}

denotes the stress amplification in surviving layers after

r

layer failures. Finally, the probability of system failure,

P_{f},

is calculated in using the weakest link theorem by,

P_{f} = 1 - {[1 - G (σ)]}^{n}

(7)

Under these assumptions, $δ$ is the only free variable in the model and increasing the value of δ increases F for a given $σ$ . Figure 8 shows the failure distribution predicted by equation (7) with GLS for the 5-layer data for various $δ$ values. The Weibull distributions predicted by the model exhibit $m$ values that are unrealistically high for each $δ$ value. We also see that increasing δ results in decreasing the $σ_{o}$ value. In the following, the estimated parameters for the laminated stacks are indicated by “*”. The model predicts $m^{*} ⋍$ 17.4 for each δ value, with the best $σ_{o}$ agreement with the data fit occurring when δ = 5.2 mm, a realistic value. With a model prediction of $m * ⋍$ 17.4, the GLS model does not adequately describe the data because $m *$ values do not agree well with the experimental result of $m =$ 6.2.

Figure 8.

GLS model prediction (dashed blue lines) for five-layer composite failure, compared with experimental data (solid red line). Here $m^{*} ⋍$ 17.4.

The next model we attempt is the chain-of-bundles model with Local Load Sharing (LLS), which requires the following steps to calculate a predicted failure distribution. First, ribbon data is used to write $P_{f}$ as a Weibull distribution as in equation (1), and this is used to calculate the individual link CDF, $F$ , using equation (4). Next, an algorithm generates all possible states of failed and surviving fibers, of which there are $2^{N}$ , where $N$ is the number of layers. The stress concentration factors, $K$ , are then calculated for each layer in each state, assuming the stress previously carried by a failed layer is absorbed in the immediately adjacent surviving layers. There may be 1 or 2 neighboring surviving layers, depending on whether the failed layers are adjacent to the edge of the stack. $K_{i, j}$ denotes $K$ for the $i^{t h}$ layer failure of the $j^{t h}$ sequence. Next, all possible failure sequences are generated, where each stage in a sequence adds one new layer failure to the previously failed layers. There are $N!$ possible failure sequences. The probability of a bundle failure by the $j^{t h}$ failure sequence is,

F_{j} = {F (σ) Π}_{i = 1}^{N} [F (K_{i, j} σ_{f})]

(8)

The bundle failure probability is given as the sum of the probability of each possible sequence,

G (σ) = Σ_{j = 1}^{N!} (F_{j})

(9)

Finally,

P_{f}

is calculated using equation (7). This method is analogous to that used in the two-part work of Harlow and Phoenix,^42,43 but with linear rather than circular bundles, which leads to different load sharing.

The failure distribution predicted by equation (7) with LLS for the five-layer data is shown in Figure 9 for various δ values, chosen to predict a failure distribution in a range close to $σ_{o}$ from the data. The model predicts $m^{*} ⋍$ 15.9 for each $δ$ value shown, with the best $σ_{o}$ agreement with the data fit being when $δ =$ 1.5 mm. Thus, the LLS model also fails to accurately model the laminated MANC failure distributions of Figure 6, as the $m^{*}$ values for five-layer ribbon stacks are again estimated to be too high. This model also requires extensive computational resources at large number of layers, as the number of possible failure sequences increases dramatically.

Figure 9.

LLS model prediction (dashed blue lines) for five-layer composite failure, compared with experimental data (solid red line). Here $m^{*} ⋍$ 15.9.

As a remedy to this limitation, some researchers^45–48 have used LLS principles with simplified failure conditions in a model for bundle failure called k-failure. This model assumes bundle failure to occur when a critical number of immediately adjacent fibers, $k^{*}$ , fail in a bundle. Properly predicting $k^{*}$ is important for the accuracy of this method. While there has been much discussion in the literature concerning proper estimation of $k^{*}$ , Harlow and Phoenix use k* = 2 in a relevant statistical study,⁴⁶ and Smith offers a few methods of calculating it.⁴⁸ The method from Smith used to calculate k* in the following analysis uses the expression,

γ (r) = r l n (K_{r}) - \sum_{i = 1}^{r - 1} \ln (K_{i})

(10)

to calculate the inequality,

γ (r - 1) < \ln (N n) / m < γ (r)

(11)

where r gives the number of adjacent failures and

K_{i}

gives the local stress amplification for

i

adjacent failures. The value of

r

which satisfies the inequality of equation (11) is designated as

k^{*}

. For the laminated stacks of five and ten ribbons considered in this work, these equations yield

k^{*}

values of two and three, respectively.

There are various forms of the k-failure model, but the two methods used in the present treatment are based on work by Smith and will be referred to as Method 1⁴⁵ and Method 2.^47,48 Method 1 is an explicit calculation of P_f, similar to the LLS model but only considering adjacent failures up to ${r = k}^{*}$ . Method 2 is mathematical augmentation of the Weibull parameters, given by,

m^{*} = m k^{*}

(12)

σ_{o}^{*} = N^{- 1 / m^{*}} [\prod_{i = 1}^{k^{*} - 1} (q_{i}^{1 / m} K_{i})] δ^{1 / m^{*} - 1 / m} σ_{o}

(13)

where the estimated parameters for the laminated stacks are indicated by “*”. In both methods, we adjust

K

and

q

to accommodate for edge effects, where the number of adjacent surviving layers is one rather than two. This is done by considering the total number of configurations (

N_{t o t} = N - k^{*} + 1

) of failed and surviving fibers and the portion of those configurations which include a failed edge fiber (

R_{e} = 2 / N_{t o t}

). Then,

K_{r}

for

r

adjacent failures is calculated by,

K_{r} = R_{e} K_{e} + ({1 - R}_{e}) K_{c}

(14)

where

K_{c}

for center and

K_{e}

for edge failures, are defined,

K_{c} (r) = 1 + r / 2, K_{e} (r) = 1 + r

(15)

and

q

, the number of ribbons subjected to stress amplification

K_{r}

, is given by,

q = 2 R_{c} + R_{e}

(16)

To evaluate whether Methods 1 and 2 match the data, $δ$ was again adjusted to find the value for which $σ_{o}^{*}$ matches $σ_{o}$ . Figure 10(a) shows the k-failure model results for the five-layer data for the δ value for which $σ_{o}^{*}$ matches $σ_{o}$ for Method 2. Method 1 is also included in Figure 10(a) for the same δ value. By adjusting $δ$ , either Method 1 or 2 can be made to best represent the data.

Figure 10.

Results from k-failure model Method Method 1 (dashed cyan) and Method 2 (dashed blue) for (a) 5 and (b) 10 layers, compared with data fit line (solid red). The associated $m^{*}$ values are given in Table 2.

Given that the k-failure models describe the failure data well for the five-layer case, the models were then applied to the ten-layer case, with $σ_{o}^{*}$ matching $σ_{o}$ for Method 2, as shown in Figure 10(b). Again, by adjusting $δ$ , Method 1 could be made to represent the data as well as Method 2. This is demonstrated in Table 2, where the $δ$ inputs yielding $σ_{o}^{*}$ matching $σ_{o}$ are given along with the resultant $m^{*}$ . While better data agreement of ∼2% error is shown in the ten-layer case, both models have $m^{*}$ predictions within 23% of data for the five-layer case as well. This is far better agreement than the previous GLS and LLS models produced. These results are achieved with a higher δ value for Method 2 in both cases, but all $δ$ values are in reasonable agreement with Figure 7(a).

Composite failure modeling discussion

The existing literature concerning fiber composite failure data comparison with chain-of-bundles failure models contains mixed results.⁴⁷ Multiple researchers have found GLS assumptions to perform poorly in modeling fiber systems, especially when compared with Local Load Sharing.^46,59,60 LLS assumptions align better with theory of load distribution, but direct calculations using this method become computationally expensive at large bundle sizes,²⁷ making this approach unreasonable for modeling real systems. Watson and Smith⁴⁷ obtain good agreement between data and k-failure theory for impregnated fiber bundles. Other researchers have used Monte Carlo methods using LLS or k-failure models to accurately model failure distributions from experiments.^59–61 Despite being developed ∼40 years ago, these models continue to be cited recently^62–64 as the main modeling approaches for fiber bundles, or are used as the basis for models of different material systems.⁶⁵ Ongoing efforts to understand related phenomena such as the size effect⁶⁶ utilize the same weakest link and load sharing principles referenced in this work. In fiber reinforced composite systems, fiber packing is more random than simplified bundle geometries, meaning the models being used to predict the performance of such materials need to account for these load-sharing irregularities. Because laminated MANCs have a simpler arrangement of layers stacked linearly, the present material system is perhaps better suited to these simple models than the fiber systems for which they were developed.

As explored in the modeling MANC-epoxy composite failure section, k-failure models can be adapted to MANC laminations to have good agreement with experiment. Two such methods produce similar results while assuming different values of $δ$ . While $δ$ was estimated in absence of layer peeling to be ∼ 0.23 mm, the peel length at failure will dominate this quantity. Using Figure 7(a), $t =$ −33 μs, the maximum peel length is approximately 19 mm, while other areas have negligible peeling. Although the average internal and external layer peel length at failure cannot be measured, an estimate is made that this quantity is likely in the range of 2 to 12 mm for the system studied. Therefore, all four $δ$ values listed in Table 2 are considered reasonable, and Methods 1 and 2 produce good estimations of failure distributions. It is worth noting that improvements in interlaminar bond strength are expected to improve the strength distributions further, as this will reduce peel length.

Motors constructed from MANC epoxy-impregnated TWCs will have hundreds of layers, and it is of interest to consider what the k-failure model predicts as $N$ increases. As given by equation (12), under Method 2, m scales with $k^{*}$ . Using the $k^{*}$ calculation used in this work with δ = 10mm, $k^{*} =$ 3 from N = 9 to 61. At $N =$ 62 and 884, $k^{*}$ becomes 4 and 5, respectively. Thus, $k^{*}$ , and therefore $m^{*}$ , increases slowly with $N$ – a rotor constructed from 1000 layers would have $m^{*} =$ 19. As given in equation (13), $σ_{o}^{*} \propto N^{- 1 / m^{*}} δ^{1 / m^{*} - 1 / m}$ , representing a slight decrease in $σ_{o}$ with $N$ for a fixed $δ$ . However, the experimental results show a slight increase in $σ_{o}$ with $N$ , which is accounted for by a modest decrease in $δ$ (Table 2). It is unclear whether the reduction in peel length is physically significant or if it represents a misalignment between data and model.

The predicted increase in $m$ from 11 to 19 would mean that motors are much less likely to fail at lower stresses than the ten-layer stacks in this work. For example, the experiments indicate 0.1% of specimens to fail at 160, 390, and 590 MPa for single-, five-, and ten-layered specimens. The scaling of these improvements with $N$ is uncertain, however, due to uncertainty in $δ$ . With $N =$ 1000 and $δ =$ 9.1 mm, constant above $N =$ 10, $σ_{o}^{*}$ decreases to 870 MPa and 0.1% of specimens fail at 600 MPa. However, if $δ$ continues decreasing to $δ =$ 3 mm, $σ_{o}^{*} \approx$ 1100 MPa and 0.1% of specimens fail at 760 MPa. More tensile experiments of larger laminated stacks are needed to verify the trends in $σ_{o}$ and $δ$ with $N$ increasing beyond ten.

Conclusions

The results presented here show that TWCs laminated with relatively thin layers of thermosetting epoxy show a substantial increase in Weibull Modulus, $m$ , from 4 to 11 as the characteristic strength, $σ_{o}$ , increases 10% from 1000 to 1100 MPa with layer count increasing from one to ten. This occurs because of a sufficiently strong interlaminar bond which prevents a single layer fracture from inducing immediate specimen failure. Our results suggest that stress-sensitive components such as electric motors made as a bonded stack of MANC ribbons can safely operate at higher stresses, and thus higher speeds, than would be inferred from single ribbon data. This is because the largest flaws in ribbons lead to local fracture without component failure. The effect of composite strengthening was studied by employing various chain-of-bundles models to predict the composite failure distributions. Of the models considered, the k-failure models had the best alignment with data. This work demonstrates the suitability of these models for predicting multilayer strength distributions from single-layer failure data.

Although localized cracking may not lead to catastrophic mechanical failure, it is important to note that this cracking is expected to alter the magnetic performance of the component. While more work is needed to study the effects of the types of cracks observed on magnetic performance, existing literature suggests that these cracks would increase coercivity and decrease permeability.^34,35 Thus, a TWC subjected to a small stress which develops few distributed cracks may exhibit a minimal change in magnetic performance, but many cracks may significantly influence magnetic properties.

Other future work should investigate the impact of different cutting methods on ribbon edge roughness and the subsequent impact on the failure strength of laminated MANC ribbon stacks. Further work is also needed to determine the optimal processing steps to improve bond strength between layers to minimize interlaminar separation, as this should improve mechanical strength of the stack. Work should also examine the change in failure distribution of ribbon produced using a continuous casting process, as this ribbon is known to be of different quality than batch casting processes. More testing is required to quantify the $m$ and $σ_{o}$ improvement as layer count is increased beyond ten layers, and whether the k-failure models employed successfully to the data in this study appropriately model failure data from larger laminated ribbon stacks.

Footnotes

Acknowledgements

The authors graciously acknowledge the contributions of Eric Theisen, James Egbu, and William Pingitore to the experiments conducted in this work.

Declaration of conflicting interests

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Michael McHenry reports a relationship with CorePower Magnetics that includes equity. Michael McHenry has patent #US2019/0368013 A1 issued to Carnegie Mellon University.”

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Department of Energy (DOE) Advanced Manufacturing Office (AMO) [DE-EE0007867] and the DOE Vehicle Technology Office (VTO) [DE-EE0008870].

Data availability statement

The raw data required to reproduce these findings are available to download from [].

CRediT statement

Kyle Schneider: Conceptualization, Methodology, Formal analysis, Investigation, Data Curation, Writing – Original Draft, Visualization Michael McHenry: Conceptualization, Supervision, Writing - Review and Editing, Funding Acquisition Maarten de Boer: Conceptualization, Supervision, Writing – Review and Editing.

ORCID iD

Maarten P de Boer

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