Experimental Investigation of Friction at Buckling-Restrained Brace Debonding Interfaces

The bearing pressure (${\sigma }_N$) was calculated by averaging the Abaqus field output CPRESS for the nodes in contact at each wavecrest, and the mean is shown in Fig. 3 at three strain levels. The bearing pressure was primarily determined by the debonding gap ratio ($s_w/t_c$) and was not meaningfully correlated with the other dimensions, location along the yield length, nor friction model. The low yield point (LY100, LY225) and mild steels (SN400B, SN490B) stabilized to similar bearing pressures at large strains ($\overline{\epsilon }>2\%$), while the high-strength steels (SA440B, SA700) with yield strengths exceeding $f_y>400\mathrm{MPa}$ produced higher bearing pressures. This was due to the mechanics of higher-mode buckling and material properties, which are explained as follows.

Normal force is generated at the higher-mode buckling wavecrests during compression half-cycles due to the core axial force acting at an incline determined by the sinusoidal wavelength and wave height (Fig. 3). Both are smallest at the core ends and gradually increase along the yield length as friction sheds force from core to the restrainer, which reduces the compressive stress (${\sigma }_c$) and strain (${\epsilon }_c$). The wave height equals the total as-built debonding gap ($2s_w$) less Poisson expansion of the core ($-{\nu }_p|{\epsilon }_c|t_c$, where ${\nu }_p=0.5$ is the plastic Poisson ratio), while the sinusoidal wavelength ($l_{p,w}$) is a function of the material-specific axial stress (${\sigma }_c$) and tangent stiffness ($E_t$) and may be estimated as $l_{p,w}/t_c\approx 2\pi \surd (E_t/3{\sigma }_c)$ (Sitler and Takeuchi 2021). The lower tangent stiffnesses of the high-strength (SA440B, SA700) and low-yield-point (LY100, LY225) steels result in shorter wavelengths than SN400B and SN490B, while the axial stress increases with the steel grade. This suggests that the bearing pressure will be substantially higher for SA400B and SA700, but almost identical for LY100, LY225, SN400B, and SN490B, which is consistent with the numerical results.

Based on this understanding, an analytical estimate of the wavecrest bearing pressure (${\sigma }_N$) is given by Eq. (1), where the compressive stresses (${\sigma }_c$), wave heights ($2s_w-{\nu }_p|{\epsilon }_c|t_c$), and sinusoidal wavelengths ($l_{p,w}$) may be obtained by cycling the engineering stress-strain curve to ${\epsilon }_c=-\overline{\epsilon }$. The contact length ($l_c$) was complex because highly concentrated initial contact tended to be alleviated as the wavecrests flattened, until new waves eventually formed via snap-through buckling. Nevertheless, the total contact length was about $l_c\approx 0.9t_c$ for fully developed wavecrests, but about half this length at initial contact (Fig. 3). Both the average and peak bearing pressure may be obtained as

Given that the debonding gap ratio ($s_w/t_c$) was the dominant parameter at large strains for most steel grades, Eq. (1) may be further simplified to Eq. (2), where the pressure concentration factor (${\alpha }_P$) may be taken as ${\alpha }_P=1$ for fully developed wavecrests with $l_c=0.9t_c$ and ${\alpha }_P=2$ at initial contact. Eq. (2) may be used by engineers to estimate the wavecrest bearing pressure

As the most common debonding gap ratios ($0.01<s_w/t_c<0.05$) produced bearing pressures anywhere from ${\sigma }_N=5–50\mathrm{MPa}$ (with ${\alpha }_P=1–2$), values up to the mortar’s compressive strength were considered for the friction test. Note that the bearing pressure is not a function of friction.

Next, the slip velocity was calculated from the slip rate (FSLIPR), but was nearly equal to the instantaneous rate of relative axial deformation between the core and restrainer at the contact node in question. The relative motion is directly determined by the externally applied stroke displacement (${\delta }_{\mathrm{stroke}}$) at the core ends and gradually decreases to zero at the fixity point, which is typically a midspan shear key. Assuming that the axial restrainer and connection deformations are negligible and structural drift is predominately shear, ${\delta }_{\mathrm{stroke}}$ is given by Eq. (3), where $\mathrm{\Delta }$ is the interstory drift angle, $H_{wp}$ the workpoint height, and ${\theta }_{\mathrm{BRB}}$ the horizontal inclined angle (Fig. 4). With a competent shear key dividing the axial displacement evenly between each end and ${\theta }_{\mathrm{BRB}}=30°–50°$, the displacement stroke at each end is just 30% to 45% of the horizontal story displacement:

The maximum slip velocity ($V_{\mathrm{slip}}$) at the core ends is then equal to the stroke velocity ($V_{\mathrm{stroke}}$), which is given as follows by Eq. (4) for a structure vibrating in a sinusoidal motion with period $T$:

The slip velocity distribution along the core is depicted in Fig. 4 for the final cycle of an example model with SN400B, $t_c=25\mathrm{mm}$, $L_p=2\mathrm{m}$, $s_w/t_c=0.05$, and $\mu =0.3$. The longitudinal distribution was sublinear because the axial deformation rate decreased along the core as friction shed force to the restrainer. Thus, most wavecrests experience much lower slip velocities, even during dynamic earthquake events (Fig. 4). Because wavecrests along the entire core contribute to the cumulative friction force, the test considered quasi-static to peak stroke velocities, which are independent of friction.

Unlike the slip velocity, the slip distances included an essentially random component and so must be assessed probabilistically. Specifically, the maximum slip distance (${\delta }_{\mathrm{slip}}$) over a compressive half-cycle from $+{\delta }_{\mathrm{stroke}}$ to $-{\delta }_{\mathrm{stroke}}$ was always less than the relative displacement between the core and restrainer ($2{\delta }_{\mathrm{stroke}}$) since points were ever only occasionally in contact. This was due to the small deformation required for force reversal and initiation of buckling, and to the constant change in the higher-mode buckling shape. Following the arbitrarily selected point in Fig. 5, the higher-mode buckling waves occasionally rolled along the core toward midspan (G and H), while the wavecrest flattening alleviated contact at the center of each wavecrest and eventually permitted new waves to form via snap-through buckling (D and F). Therefore, no point was ever in contact for a full cycle, and the points in contact constantly changed from cycle to cycle. Nevertheless, the critical point experiencing the maximum slip was usually, but not always, located at the endmost wavecrest.

The maximum single-cycle (${\delta }_{\mathrm{slip}}$) and cumulative ($\sum {\delta }_{\mathrm{slip}}$) slip displacements were obtained from FSLIPEQ and are depicted in Fig. 6, normalized by $2{\delta }_{\mathrm{stroke}}$ or $\sum {\delta }_{\mathrm{stroke}}$. These were always significantly less than the applied stroke but increased as the higher-mode buckling shape stabilized during the large-amplitude cycles later in the loading protocol. Furthermore, no correlation was observed with the steel grade or core dimensions. The variation across all models followed a normal distribution, and the final-cycle mean slip ratios and COVs for the preliminary 576 models are given by Eqs. (5) and (6)

The nonlinear friction model slightly increased the duration of contact at some points and resulted in slip ratios of ${\delta }_{\mathrm{slip}}=0.62·2{\delta }_{\mathrm{stroke}}$ ($\mathrm{COV}=0.09$) and $\sum {\delta }_{\mathrm{slip}}=0.22\sum {\delta }_{\mathrm{stroke}}$ ($\mathrm{COV}=0.14$), or one standard deviation above the original results. This information was not available a priori but is offset by conservatism in the original results (i.e., 5%–20% of FSLIPEQ occurs at negligible bearing pressures, and the original test specification rounded up the target displacements), and so is not considered further. Nevertheless, future studies may consider these higher slip ratios.

A probabilistic estimate of the maximum slip displacement at the core ends may then be estimated from the stroke displacement imposed by a BRB qualification protocol. AISC 341-16 prescribes a BRB qualification protocol of $2\times 0.5{\mathrm{\Delta }}_{bm}$, $2\times 1.0{\mathrm{\Delta }}_{bm}$, $2\times 1.5{\mathrm{\Delta }}_{bm}$, $2\times 2.0{\mathrm{\Delta }}_{bm}$, $2\times 1.5{\mathrm{\Delta }}_{bm}$, where ${\mathrm{\Delta }}_{bm}$ is the design drift, which is permitted to vary from 1% to 2.5% for normal buildings (AISC 2016). Nevertheless, well-designed BRBs exhibit sufficient fatigue capacity to withstand multiple design-level events and so even larger cumulative slip displacements are of interest.

The cumulative slip displacement at fracture may be obtained from the empirical low cycle fatigue (LCF) curves developed for BRBs, which are reduced from the uniaxial base material capacities due to higher-mode buckling, friction-induced strain ratcheting, triaxiality and strain concentration. In the plastic regime, the number of cycles to fracture ($N_f$) is given by Eq. (7) as a function of the Coffin-Manson exponent ($m$) and index strain range ($\mathrm{\Delta }{\epsilon }_0$). This study adopted $m=0.49$ and $\mathrm{\Delta }{\epsilon }_0=0.2048$ (Yoshikawa et al. 2010), while the average axial strain range of $2\overline{\epsilon }=2{\delta }_{\mathrm{stroke}}/L_p$ was computed at the design drift ($\mathrm{\Delta }={\mathrm{\Delta }}_{bm}$) using Eq. (3), assuming that $L_p<0.8L_{wp}$ and $L_{wp}=H_{wp}/\sin {\theta }_{\mathrm{BRB}}$. The cumulative slip displacement at fracture was then obtained from Eq. (6), but with the cumulative stroke displacement [Eq. (8)] estimated by cycling to fracture at the average axial design strain

The maximum slip velocities and slip displacements at the core end were calculated from Eqs. (3)–(8) for several archetype structures in Table 1. The maximum slip velocity was typically on the order of $V_{\mathrm{slip}}=50–200\mathrm{mm}/\mathrm{s}$ but increased up to $500\mathrm{mm}/\mathrm{s}$ for short period structures with long BRBs (e.g., warehouses). Similarly, the maximum slip distance varied from ${\delta }_{\mathrm{slip}}=30–80\mathrm{mm}$, while the cumulative slip during the compression half-cycles was less than $\sum {\delta }_{\mathrm{slip},\mathrm{AISC}}<0.5\mathrm{m}$ for the AISC 341-16 protocol. The cumulative slip at fracture increased to $\sum {\delta }_{\mathrm{slip},\mathrm{LCF}}<1–2\mathrm{m}$ for most archetypes but was sensitive to the yield length and may exceed 4 m for long BRBs subjected to dozens or hundreds of small-amplitude low cycle fatigue cycles.

Considering that the friction force accumulates from all wavecrests along the entire yield length, slip values from zero up to the calculated maximums are of interest, while bearing pressures were previously shown to vary from ${\sigma }_N=3–50\mathrm{MPa}$, depending on the debonding gap ratio. Therefore, the characteristic demands for low- and midrise structures include slip velocities from $V_{\max }=0–200\mathrm{mm}/\mathrm{s}$, single-cycle slip distances from ${\delta }_{\mathrm{slip}}=0–80\mathrm{mm}$ and one-way cumulative slip distances from $\sum {\delta }_{\mathrm{slip}}=0–2.0\mathrm{m}$. Note that the velocities applied in the experiment were slightly lower, but the friction velocity-dependency saturated well below $200\mathrm{mm}/\mathrm{s}$. Additionally, although cyclic heating has a small beneficial effect on the friction coefficient (Kumar et al. 2015), it is difficult to monitor the debonding material temperature and the sides were left open to permit visual monitoring, limiting the potential temperature gain. Consequently, temperature dependency was outside the scope of this study, other than to report the friction coefficient at the third cycle.

A unique test rig (Fig. 7) was designed to accommodate dynamic horizontal reciprocating motion while applying a static, load-controlled vertical force (i.e., compression half-cycles) or displacement-controlled lifted position for the return motion (i.e., tension half-cycles). This was achieved using a lower horizontal actuator that drove the slide table, and an upper vertical actuator and jointed horizontal load cell stabilized by removable ball bearings and cylindrical guide plates. This test setup was capable of $\pm 150\mathrm{mm}/\mathrm{s}$, $\pm 300\mathrm{mm}$, and $\pm 500\mathrm{kN}$ in the horizontal direction, and $+25\mathrm{mm}/-5\mathrm{mm}$ and $+500\mathrm{kN}$ in the vertical direction, with positive values referring to compressive bearing pressures, and down/rightwards movement of the upper surface. Testing was conducted by the authors from October 2019 to January 2020 at the Tokyo Institute of Technology.

Eight specimens were prepared, each consisting of a curved core slider bolted to the upper stiff-arm assembly and a mortar-filled restrainer bed bolted to the lower sliding table (Fig. 8). The debonding materials were applied after curing and leveling the mortar to a $\pm 1\mathrm{mm}$ finish, or after curing the mortar for 6 h, leveling and rewetting to form a bond with the textile, where applicable. Stiffened cover plates were fixed with $8\times \mathrm{M}6$ Grade 4.6 bolts to introduce a 0.5 MPa hold-down pressure representing the hydrostatic pressure imposed by wet mortar, but they were removed the morning of the test to mark out a white $20\times 20\mathrm{mm}$ grid.

The core slider geometry was designed to capture friction scale effects and may reasonably represent any wavecrest of a rectilinear core plate. Each core slider featured a $50\times 100\mathrm{mm}$ apparent contact area surrounded by surfaces inclined at $1:5$ and finished to a 100 mm radius in the direction of sliding, and $1:4$ inclined surfaces finished to a 25 mm radius on the other two sides. The 50 mm contact length helped to promote uniform bearing pressures in the presence of the test rig eccentricities. To ensure a consistent finish, only four core sliders were fabricated, and each was cleaned with oil and sanded using a coarse 120-grit sandpaper prior to each test to replicate an unpainted core plate with mill scale removed. Nevertheless, plowing wear dominated, minimizing the influence of corrosive wear if oxides were present (Stachowiak and Batchelor 2014).

Two types of restrainer beds were employed with a 736 or 838 mm long mortar surface, which permitted multiple test positions per specimen. The restrainer beds of three specimens (N-1, A1-1, and A1-2 in Table 2) were elevated on steel angles, cast with 25 MPa concrete (A1-1 and A1-2) or 50 MPa mortar (N-1), and later reused for bulging tests, which are not reported in this paper. The remaining five specimens employed nonelevated restrainer beds with 50 MPa mortar (54% water/cement, 5 mm crushed sand). The mortar was cast on September 13, 2019, and the concrete on October 23, producing the cylinder strengths reported in Table 2.

The debonding interfaces are denoted in Table 2 by the material (N, A, or B), nominal thickness (1 or 2 mm), specimen number (1–4) and position (A–C). As a control, one specimen (N-1) was tested with no debonding material (Interface type IV), while the debonded specimens were of Interface type I. Three of these featured a soft polymer with textile backing facing the mortar (A1-1, A1-2) or core slider (A1-3) and a combined 1 mm thickness, while another four employed 1 mm (B1-1, B1-2, B1-3) or 2 mm (B2-4) of a soft polymer. These were selected to be representative of the unique debonding materials used in a variety of BRBs from international practice and academia, and were provided by a major BRB supplier.

A standard protocol was developed consisting of 15 sets of 4 cycles (Table 3), with a minimum 5 min rest period between each set. Each test position targeted a fixed bearing pressure (${\sigma }_N=3$, 5, 10, 20, 25, 30, 40, or 45 MPa) and horizontal displacement amplitude (${\delta }_H=\pm 20$ or $\pm 40\mathrm{mm}$), although smaller amplitudes (${\delta }_H/10$, ${\delta }_H/4$ and ${\delta }_H/2$) were applied in the first three breaks in sets. The target velocity was varied in subsequent sets ($V_{\max }=0.8–125\mathrm{mm}/\mathrm{s}$), but a reference velocity of $V_{\max ,0}=30\mathrm{mm}/\mathrm{s}$ was applied every third set to track the effect of the cumulative slip distance. Additional sets were applied at $V_{\max ,0}$ with lower bearing pressures for B1-3A, B1-3B, and B2-4A, while A1-3B was subjected to an extended protocol of over 650 cycles at $V_{\max }=80\mathrm{mm}/\mathrm{s}$, and the earlier (N, A1) specimens omitted the low-velocity sets below $10\mathrm{mm}/\mathrm{s}$.

Two different sliding procedures were used, denoted as two-way sliding and one-way sliding. For two-way sliding, the vertical load was slowly applied once and then held constant throughout the entire test. Sequential sets of reciprocating sinusoidal motion were applied, with the first half-cycles also applied sinusoidally to avoid an impulse. One-way sliding also imposed dynamic horizontal reciprocating motion, but the vertical force was reapplied at the start of each cycle and only imposed during the motion in one direction. This represents the fact that BRB higher-mode buckling wavecrests only form during compression half-cycles, and contact at the wavecrests is removed as the core straightens and contracts during the tension half-cycles. Each one-way cycle started at $-{\delta }_H$; contact was initiated using displacement control and the full vertical force applied using load control. The core slider was then displaced horizontally in a sinusoidal motion from $-{\delta }_H$ to $+{\delta }_H$ over a duration $T/2$, lifted and returned from $+{\delta }_H$ to $-{\delta }_H$ before immediately starting the next cycle. The distance lifted for the return motion was to the point of zero load (i.e., as-built condition), plus an additional $+0.5\mathrm{mm}$ to represent tensile Poisson contraction of the core. While only one-way sliding occurs in real BRBs, two-way sliding retains substantial parallels given the similar horizontal motion and so was used to establish most of the friction coefficient dependencies. Two-way sliding was also used for validation purposes and corresponds to the stabilized dynamic friction coefficients commonly cited in the literature.

The vertical (i.e., normal) ($P_V$) and horizontal (i.e., shear) ($P_H$) forces were measured using load cells in the stationary upper test rig to exclude the test rig friction. No postprocessing was required for the normal force, while a 25 Hz lowpass filter was applied to the shear force data. The horizontal pins had a total slack of 1.5 mm, which introduced small impact forces at each load reversal, but these accounted for less than 20% of the amplification in the first cycle and rapidly decayed. Unless stated otherwise, the friction coefficient ($\mu =P_H/P_V$) is always reported at the zero-displacement intercept (i.e., peak velocity) of the third cycle, with the positive and negative intercepts averaged for the two-way sliding tests. The average bearing pressure (${\sigma }_N$) was taken as $P_V$ divided by the apparent contact area ($50\times 100\mathrm{mm}$), although in-plane rotational slack of the stiff-arm assembly and wear resulted in a slightly uneven contact surface. Finally, the relative vertical (${\delta }_V$) and horizontal (${\delta }_H$) displacements at the sliding interface were directly measured using lasers (one horizontal and four vertical), and the maximum slip velocity ($V_{\max }$) confirmed from the slip displacement rate obtained using a fifth-order Savitzky-Golay polynomial with a $T/8$ window.

A bare steel-mortar specimen (N-1) of Interface type IV was tested as a control. This specimen exhibited a constant friction coefficient of $\mu \approx 0.75$ at zero displacement in all cycles, but neither pressure nor velocity dependency (Fig. 9). The friction hysteresis was rectangular during the first few sets, which is characteristic of Coulomb friction. However, the abraded surface began to spall during the 6th set (22nd cycle), even with a bearing pressure of just ${\sigma }_N=20\mathrm{MPa}$ (i.e., 40% of the mortar compressive strength). The spalled mortar grains were then plowed into a slope, inclining the tangent plane and increasing the shear force, which is visible as small bumps in Fig. 9.

Spalling increased the vertical displacement by about 0.5 mm per four cycles, which would significantly increase the higher-mode buckling amplitude. Although stress concentrations may have occurred at the leading edges, none of the corresponding debonded specimens crushed at these low pressures, nor did any spall even at much higher pressures. It is more likely that damage accumulated from the large longitudinal shear stresses in the unreinforced mortar because the large, spalled flakes were inconsistent with a crushing failure. Regardless of the specific cause, the high friction coefficient and spalling are significant performance limitations because they increase the compressive overstrength force and risk of restrainer bulging, and so an antifriction debonding material (Interface types I, II, and III) is strongly recommended for mortar-filled steel tube BRBs.

The friction coefficients achieved for the debonded Interface type I specimens are shown in Fig. 10 for a range of bearing pressures, slip velocities and cumulative slip distances. Although only a subset of the B1 (1 mm polymer) specimen data is shown, the A1 (1 mm polymer with textile) and B2 (2 mm polymer) specimens exhibited similar performance. Comprehensive third-cycle friction coefficients are illustrated later in Fig. 15 and complete results in Sitler (2021). The friction coefficient increased at low bearing pressures, doubled from ${\sigma }_N=20$ to 3 MPa, and then slightly decreased at higher pressures. Friction coefficients were recorded between $\mu =0.03$ and 0.05 during quasi-static loading ($V_{\max }<1.5\mathrm{mm}/\mathrm{s}$), increased to $\mu =0.06–0.10$ at $V_{\max ,0}=30\mathrm{mm}/\mathrm{s}$, and then slightly decreased at higher velocities with high bearing pressures, likely due to increased cyclic heating. Furthermore, the friction coefficient was significantly amplified during the first cycle of the high-velocity tests, but there was negligible amplification at the first cycle of the quasi-static tests. This is discussed further in the context of the one-way sliding results.

An extended dynamic protocol was applied to the A1-3B specimen with 5, 10, and 50 cycle sets at $V_{\max }=80\mathrm{mm}/\mathrm{s}$ to saturate the potential cyclic heating and give an indication of the temperature dependency. Extrapolating to exclude the first-cycle dynamic amplification, the friction coefficient experienced an average 30% reduction from the 1st to 10th cycles, but half of this total reduction occurred in the first 3 cycles and there was negligible change after the 10th cycle. Also, the debonding material formed a hard, crystalline residue by the end of the test and may have melted.

A unique wear effect was observed due to the mortar countersurface, which is weaker than the steel countersurfaces more commonly used in antifriction applications. As depicted in Fig. 11, the transfer material fully coated both countersurfaces and exhibited significant plowing wear. This was accompanied by an increased friction coefficient (Fig. 10), which was tracked at the reference velocity $V_{\max ,0}$ every third set and increased linearly with cumulative slip distance, higher bearing pressure, and thinner debonding material. Although wear does not usually immediately increase the friction coefficient, this appeared to be a system effect, as small patches of exposed mortar were observed during the visual inspections. The bare steel and mortar exhibited a constant friction coefficient of $\mu =0.75$, and so these patches increased the smeared friction coefficient. However, the previously displaced debonding material eventually resurfaced the exposed mortar, creating a self-healing effect that limited the increase to a wear coefficient of ${\mu }_W<0.15$.

This upper bound wear coefficient was also tested in Specimen A1-3B by applying an extended protocol with over 28 m of positive sliding, which is about 10 times the expected cumulative slip distance at fracture for a high-performance BRB and so would never be experienced in practice. Nevertheless, while the friction coefficient continued to gradually increase set to set, the wear contribution always remained below ${\mu }_W<0.15$ and the total friction coefficient below $\mu <0.25$, which is one third of the friction coefficient of the bare steel-mortar control specimen. High bearing pressures (${\sigma }_N=45\mathrm{MPa}$) applied to the A1-3A specimen caused localized crushing. The dislodged, interspersed mortar grains temporarily increased the friction coefficient by 0.15 to $\mu =0.22$, but this subsequently recovered, suggesting that the ${\mu }_W<0.15$ limit also applies to crushed mortar.

The wear effect was also influenced by the debonding material composition and thickness. Specifically, mortar patches were exposed earlier for thinner polymers, which had less transfer material available to resurface the exposed mortar patches. This may be observed by comparing the otherwise identical 1 mm (B1-3A) and 2 mm (B2-4A) specimens in Fig. 12. These exhibited identical pressure and velocity dependencies, but the friction coefficient increased 50% faster for the thinner B1-3A specimen. The bearing pressure at which wear began to increase the friction coefficient was a function of thickness, while the rate of increase was directly proportional to how much the applied pressure exceeded this wear-initiating bearing pressure.

Conversely, the textile backing of the A1-1 and A1-2 specimens tore just outside the contact area. While this did not affect the friction coefficient in this test, the large, exposed patches may significantly degrade the performance should the core slide over this area. Conversely, the A1-3 specimen had the textile backing placed upside down facing the core slider. The textile tore near the center and was subsequently ejected from the contact area, causing a spike in the friction coefficient for several cycles (Fig. 10), until the transfer material on the mortar countersurface was able to resurface the exposed patch. Therefore, the effect of textile backing is neutral at best and potentially detrimental, as a large area of high-friction bare mortar may be exposed if the backing material tears, making it more difficult to resurface than when the transferred debonding material directly coats the mortar. This leads to the counterintuitive conclusion that adding a weak supplemental bond breaker layer (such as a textile or wax paper backing) may be detrimental because it blocks the protective transfer material from forming.

The one-way sliding tests were somewhat erratic owing to small deviations in the slip path and should be treated with a measure of caution. Nevertheless, the wear behavior and pressure dependency were similar, and the velocity dependency greater, than the two-way sliding tests. This was attributed to a recurring amplification at the start of each cycle, which was of similar magnitude as the first cycle of the two-way tests (Fig. 13). Neither the test rig inertial nor impact forces explain the amplification since these were also present during the two-way sliding reversals but found to be minimal. Instead, this appears to be a dynamic breakaway effect associated with transient viscoelastic hardening, which is also observed during shear tests (Kasai and Nishizawa 2010). The friction coefficient was $+40\%$ greater at the zero-displacement intercept for the ${\delta }_H=\pm 20\mathrm{mm}$ tests, on average, but attenuated to the stabilized dynamic value with increasing slip distance. No amplification was observed at the zero-displacement intercept for the ${\delta }_H=\pm 40\mathrm{mm}$ tests and only minimal amplification for the quasi-static tests.

The vertical load application at the start of each test series provided useful information as to the compressibility of the debonding materials, or effective debonding gap, which is important to accommodate Poisson expansion and controls the higher-mode buckling amplitude, thereby also contributing to the wavecrest friction force. It is important to note that the full debonding material thickness may not necessarily be easily compressed nor displaced, and the available thickness may be further reduced by permanent set induced by the hydrostatic pressure of the wet mortar.

The vertical force-displacement plots are shown in Fig. 14 with the 30 MPa tests emphasized. The four vertical laser displacements were averaged and normalized by the polymer thickness ($t_{\mathrm{poly}}$) measured prior to fabrication. Note that the vertical force starts from zero, because the 0.5 MPa precompression was removed to install the specimen. The debonding material was compressed by about 10% of the polymer thickness at ${\sigma }_N=0.5\mathrm{MPa}$, then rapidly stiffened while undergoing plastic flow at higher pressures, with the displaced material forming permanent bulges on all four sides of the contact area, as well as some air pockets. Note that the open-sided test setup minimally confined the debonding material, such that it provides an upper bound of the debonding gap. Nevertheless, the vertical displacement nearly reached the polymer thickness by ${\sigma }_N=25\mathrm{MPa}$, after which the mortar stiffness controlled. This produced a permanent clearance, with subsequent one-way sliding cycles displacing about 50% of the polymer thickness before registering a significant pressure. In design situations where a larger debonding gap is conservative, the polymer thickness may be directly adopted ($s_w=t_{\mathrm{poly}}$), while textile backing should be excluded.

The nonlinear friction model presented in Eqs. (9)–(13) was implemented as Abaqus/Explicit 2017 user subroutines (Sitler 2021). Given the complex dependencies and variable slip along the core yield length, simplified friction models indirectly calibrated against the BRB force-displacement hysteresis may be inaccurate when the core geometry or loading protocol significantly differs from the test specimen. Specifically, a larger friction coefficient is expected for thicker cores with smaller debonding gap ratios ($s_w/t_c$), as these produce lower bearing pressures, and for dynamic loading tests. Similarly, the greater cumulative slip displacement of longer cores increases the wear effect, which may be artificially suppressed in the small-scale specimens often used in academic studies.

Nevertheless, in some cases it may be reasonable to simplify the nonlinear friction model, specifically when the cumulative friction effects are small or to validate a complex model. In these cases, the small amplitude ($k_1=1.8$), high pressure (${\sigma }_{N,0}=30\mathrm{MPa}$, $k_P=0.9$), slow speed ($V_{\max }<5\mathrm{mm}/\mathrm{s}$, $k_V=0.7$), and final cycle ($\sum {\delta }_{\mathrm{slip}}=0.3\mathrm{m}$, $t_{\mathrm{poly}}=1\mathrm{mm}$, ${\mu }_W=0.02$) friction coefficient of $\mu =0.09$ is nearly equal to the $\mu =0.1$ used by many researchers and may be a reasonable starting point. However, in general, experimentally validated friction models that include the unique dynamic breakaway and wear effects are recommended, such as the proposed nonlinear friction model.

Dynamic testing is not currently required by the AISC 341-16 specification owing to cost and test facility limitations (§Comm.K3.4, AISC 2016), while dynamic device-level tests have produced mixed results (Hasegawa et al. 1999; Lanning et al. 2016; Qu et al. 2020). However, the argument for incorporating loading rate effects fundamentally differs for the steel core, which is discussed in AISC 341-16, and the debonding material, which is not. Dynamic loading increases both the BRB forces and yield strengths of the force-controlled members, minimizing the change in the demand-to-capacity ratio. However, velocity-dependent friction exclusively affects demand, specifically by increasing the friction component (${\beta }_F$) of the compressive adjustment factor ($\beta$).

This is an important distinction because the 2016 edition of AISC 341 §K3.8.d increased the acceptance criteria to $\beta <1.5$, such that friction may now constitute up to approximately 40% of the quasi-static BRB compressive force, significantly increasing the potential influence of the friction coefficient on the overall behavior. Because dynamic testing introduces its own challenges, a practical approach may be to (1) obtain $\beta$ from full-length quasi-static tests, (2) estimate the friction component as ${\beta }_F\approx \beta /(1+2\overline{\epsilon })$, (3) scale the quasi-static ${\beta }_F$ to an average dynamic value (${\beta }_{F,dyn}$) using Eq. (14), and (4) recombine with the Poisson component to produce a design value of ${\beta }_{dyn}={\beta }_{F,dyn}\times (1+2\overline{\epsilon })$. This effectively reimposes the previous acceptance criteria of $\beta <1.3$ for quasi-static tests, assuming $C_{\mathrm{slow}}=0.6/k_1\approx 0.45$, while increasing the compression design forces by up to 20%. The unadjusted $\beta$ test value may still be used for full-length dynamic testing, velocity-insensitive debonding materials (e.g., Interface type IV) or if friction is negligible (e.g., $\beta <1.1$), otherwise, the following correction applies:

Finally, dynamic tests with peak stroke velocities exceeding about $100\mathrm{mm}/\mathrm{s}$ should produce similar friction velocity dependency due to the low saturation velocity of $k_V$. Cyclic heating may partially mitigate this adverse effect, but there are currently no comparative temperature data for full-scale BRBs, a prerequisite to take advantage of beneficial temperature dependency.